README

# MGL Manual

###### \[in package MGL\]
## MGL ASDF System

- Version: 0.1.0
- Description: MGL is a machine learning library for backpropagation
  neural networks, boltzmann machines, gaussian processes and more.
- Licence: MIT, see COPYING.
- Author: Gábor Melis <mega@retes.hu>
- Mailto: [mega@retes.hu](mailto:mega@retes.hu)
- Homepage: [http://melisgl.github.io/mgl](http://melisgl.github.io/mgl)
- Bug tracker: [https://github.com/melisgl/mgl/issues](https://github.com/melisgl/mgl/issues)
- Source control: [GIT](https://github.com/melisgl/mgl.git)

## Introduction

### Overview

MGL is a Common Lisp machine learning library by [Gábor
Melis](http://quotenil.com) with some parts originally contributed
by Ravenpack International. It mainly concentrates on various forms
of neural networks (boltzmann machines, feed-forward and recurrent
backprop nets). Most of MGL is built on top of MGL-MAT so it has
BLAS and CUDA support.

In general, the focus is on power and performance not on ease of
use. Perhaps one day there will be a cookie cutter interface with
restricted functionality if a reasonable compromise is found between
power and utility.

### Links

Here is the [official repository](https://github.com/melisgl/mgl)
and the [HTML
documentation](http://melisgl.github.io/mgl-pax-world/mgl-manual.html)
for the latest version.

### Dependencies

MGL used to rely on [LLA](https://github.com/tpapp/lla) to
interface to BLAS and LAPACK. That's mostly history by now, but
configuration of foreign libraries is still done via LLA. See the
README in LLA on how to set things up. Note that these days OpenBLAS
is easier to set up and just as fast as ATLAS.

[CL-CUDA](https://github.com/takagi/cl-cuda) and
[MGL-MAT](https://github.com/melisgl/mgl) are the two main
dependencies and also the ones not yet in quicklisp, so just drop
them into `quicklisp/local-projects/`. If there is no suitable GPU
on the system or the CUDA SDK is not installed, MGL will simply
fall back on using BLAS and Lisp code. Wrapping code in
MGL-MAT:WITH-CUDA\* is basically all that's needed to run on the GPU,
and with MGL-MAT:CUDA-AVAILABLE-P one can check whether the GPU is
really being used.

### Code Organization

MGL consists of several packages dedicated to different tasks.
For example, package `MGL-RESAMPLE` is about
MGL-RESAMPLE::@MGL-RESAMPLE and `MGL-GD` is about MGL-GD::@MGL-GD
and so on. On one hand, having many packages makes it easier to
cleanly separate API and implementation and also to explore into a
specific task. At other times, they can be a hassle, so the MGL
package itself reexports every external symbol found in all the
other packages that make up MGL and MGL-MAT (see
MGL-MAT::@MAT-MANUAL) on which it heavily relies.

One exception to this rule is the bundled, but independent
MGL-GNUPLOT library.

The built in tests can be run with:

    (ASDF:OOS 'ASDF:TEST-OP '#:MGL)

Note, that most of the tests are rather stochastic and can fail once
in a while.

### Glossary

Ultimately machine learning is about creating **models** of some
domain. The observations in the modelled domain are called
**instances** (also known as examples or samples). Sets of instances
are called **datasets**. Datasets are used when fitting a model or
when making **predictions**. Sometimes the word predictions is too
specific, and the results obtained from applying a model to some
instances are simply called **results**.

## Datasets

###### \[in package MGL-DATASET\]
An instance can often be any kind of object of the user's choice.
It is typically represented by a set of numbers which is called a
feature vector or by a structure holding the feature vector, the
label, etc. A dataset is a SEQUENCE of such instances or a
@MGL-SAMPLER object that produces instances.

- [function] MAP-DATASET FN DATASET

    Call FN with each instance in DATASET. This is basically equivalent
    to iterating over the elements of a sequence or a sampler (see
    @MGL-SAMPLER).

- [function] MAP-DATASETS FN DATASETS &KEY (IMPUTE NIL IMPUTEP)

    Call FN with a list of instances, one from each dataset in
    DATASETS. Return nothing. If IMPUTE is specified then iterate until
    the largest dataset is consumed imputing IMPUTE for missing values.
    If IMPUTE is not specified then iterate until the smallest dataset
    runs out.
    
    ```common-lisp
    (map-datasets #'prin1 '((0 1 2) (:a :b)))
    .. (0 :A)(1 :B)
    
    (map-datasets #'prin1 '((0 1 2) (:a :b)) :impute nil)
    .. (0 :A)(1 :B)(2 NIL)
    ```
    
    It is of course allowed to mix sequences with samplers:
    
    ```common-lisp
    (map-datasets #'prin1
                  (list '(0 1 2)
                        (make-sequence-sampler '(:a :b) :max-n-samples 2)))
    .. (0 :A)(1 :B)
    ```


### Samplers

Some algorithms do not need random access to the entire dataset and
can work with a stream observations. Samplers are simple generators
providing two functions: SAMPLE and FINISHEDP.

- [generic-function] SAMPLE SAMPLER

    If SAMPLER has not run out of data (see FINISHEDP)
    SAMPLE returns an object that represents a sample from the world to
    be experienced or, in other words, simply something the can be used
    as input for training or prediction. It is not allowed to call
    SAMPLE if SAMPLER is FINISHEDP.

- [generic-function] FINISHEDP SAMPLER

    See if SAMPLER has run out of examples.

- [function] LIST-SAMPLES SAMPLER MAX-SIZE

    Return a list of samples of length at most MAX-SIZE or less if
    SAMPLER runs out.

- [function] MAKE-SEQUENCE-SAMPLER SEQ &KEY MAX-N-SAMPLES

    Create a sampler that returns elements of SEQ in their original
    order. If MAX-N-SAMPLES is non-nil, then at most MAX-N-SAMPLES are
    sampled.

- [function] MAKE-RANDOM-SAMPLER SEQ &KEY MAX-N-SAMPLES (REORDER #'MGL-RESAMPLE:SHUFFLE)

    Create a sampler that returns elements of SEQ in random order. If
    MAX-N-SAMPLES is non-nil, then at most MAX-N-SAMPLES are sampled.
    The first pass over a shuffled copy of SEQ, and this copy is
    reshuffled whenever the sampler reaches the end of it. Shuffling is
    performed by calling the REORDER function.

- [variable] *INFINITELY-EMPTY-DATASET* #\<FUNCTION-SAMPLER "infinitely empty" \>

    This is the default dataset for MGL-OPT:MINIMIZE. It's an infinite
    stream of NILs.

#### Function Sampler

- [class] FUNCTION-SAMPLER

    A sampler with a function in its GENERATOR that
    produces a stream of samples which may or may not be finite
    depending on MAX-N-SAMPLES. FINISHEDP returns T iff MAX-N-SAMPLES is
    non-nil, and it's not greater than the number of samples
    generated (N-SAMPLES).
    
        (list-samples (make-instance 'function-sampler
                                     :generator (lambda ()
                                                  (random 10))
                                     :max-n-samples 5)
                      10)
        => (3 5 2 3 3)


- [reader] GENERATOR FUNCTION-SAMPLER (:GENERATOR)

    A generator function of no arguments that returns
    the next sample.

- [accessor] MAX-N-SAMPLES FUNCTION-SAMPLER (:MAX-N-SAMPLES = NIL)

- [reader] NAME FUNCTION-SAMPLER (:NAME = NIL)

    An arbitrary object naming the sampler. Only used
    for printing the sampler object.

- [reader] N-SAMPLES FUNCTION-SAMPLER (:N-SAMPLES = 0)

## Resampling

###### \[in package MGL-RESAMPLE\]
The focus of this package is on resampling methods such as
cross-validation and bagging which can be used for model evaluation,
model selection, and also as a simple form of ensembling. Data
partitioning and sampling functions are also provided because they
tend to be used together with resampling.

### Partitions

The following functions partition a dataset (currently only
SEQUENCEs are supported) into a number of partitions. For each
element in the original dataset there is exactly one partition that
contains it.

- [function] FRACTURE FRACTIONS SEQ &KEY WEIGHT

    Partition SEQ into a number of subsequences. FRACTIONS is either a
    positive integer or a list of non-negative real numbers. WEIGHT is
    NIL or a function that returns a non-negative real number when
    called with an element from SEQ. If FRACTIONS is a positive integer
    then return a list of that many subsequences with equal sum of
    weights bar rounding errors, else partition SEQ into subsequences,
    where the sum of weights of subsequence I is proportional to element
    I of FRACTIONS. If WEIGHT is NIL, then it's element is assumed to
    have the same weight.
    
    To split into 5 sequences:
    
    ```common-lisp
    (fracture 5 '(0 1 2 3 4 5 6 7 8 9))
    => ((0 1) (2 3) (4 5) (6 7) (8 9))
    ```
    
    To split into two sequences whose lengths are proportional to 2 and
    3:
    
    ```common-lisp
    (fracture '(2 3) '(0 1 2 3 4 5 6 7 8 9))
    => ((0 1 2 3) (4 5 6 7 8 9))
    ```


- [function] STRATIFY SEQ &KEY (KEY #'IDENTITY) (TEST #'EQL)

    Return the list of strata of SEQ. SEQ is a sequence of elements for
    which the function KEY returns the class they belong to. Such
    classes are opaque objects compared for equality with TEST. A
    stratum is a sequence of elements with the same (under TEST) KEY.
    
    ```common-lisp
    (stratify '(0 1 2 3 4 5 6 7 8 9) :key #'evenp)
    => ((0 2 4 6 8) (1 3 5 7 9))
    ```


- [function] FRACTURE-STRATIFIED FRACTIONS SEQ &KEY (KEY #'IDENTITY) (TEST #'EQL) WEIGHT

    Similar to FRACTURE, but also makes sure that keys are evenly
    distributed among the partitions (see STRATIFY). It can be useful
    for classification tasks to partition the data set while keeping the
    distribution of classes the same.
    
    Note that the sets returned are not in random order. In fact, they
    are sorted internally by KEY.
    
    For example, to make two splits with approximately the same number
    of even and odd numbers:
    
    ```common-lisp
    (fracture-stratified 2 '(0 1 2 3 4 5 6 7 8 9) :key #'evenp)
    => ((0 2 1 3) (4 6 8 5 7 9))
    ```


### Cross-validation

- [function] CROSS-VALIDATE DATA FN &KEY (N-FOLDS 5) (FOLDS (ALEXANDRIA:IOTA N-FOLDS)) (SPLIT-FN #'SPLIT-FOLD/MOD) PASS-FOLD

    Map FN over the FOLDS of DATA split with SPLIT-FN and collect the
    results in a list. The simplest demonstration is:
    
    ```common-lisp
    (cross-validate '(0 1 2 3 4)
                    (lambda (test training)
                     (list test training))
                    :n-folds 5)
    => (((0) (1 2 3 4))
        ((1) (0 2 3 4))
        ((2) (0 1 3 4))
        ((3) (0 1 2 4))
        ((4) (0 1 2 3)))
    ```
    
    Of course, in practice one would typically train a model and return
    the trained model and/or its score on TEST. Also, sometimes one may
    want to do only some of the folds and remember which ones they were:
    
    ```common-lisp
    (cross-validate '(0 1 2 3 4)
                    (lambda (fold test training)
                     (list :fold fold test training))
                    :folds '(2 3)
                    :pass-fold t)
    => ((:fold 2 (2) (0 1 3 4))
        (:fold 3 (3) (0 1 2 4)))
    ```
    
    Finally, the way the data is split can be customized. By default
    SPLIT-FOLD/MOD is called with the arguments DATA, the fold (from
    among FOLDS) and N-FOLDS. SPLIT-FOLD/MOD returns two values which
    are then passed on to FN. One can use SPLIT-FOLD/CONT or
    SPLIT-STRATIFIED or any other function that works with these
    arguments. The only real constraint is that FN has to take as many
    arguments (plus the fold argument if PASS-FOLD) as SPLIT-FN
    returns.

- [function] SPLIT-FOLD/MOD SEQ FOLD N-FOLDS

    Partition SEQ into two sequences: one with elements of SEQ with
    indices whose remainder is FOLD when divided with N-FOLDS, and a
    second one with the rest. The second one is the larger set. The
    order of elements remains stable. This function is suitable as the
    SPLIT-FN argument of CROSS-VALIDATE.

- [function] SPLIT-FOLD/CONT SEQ FOLD N-FOLDS

    Imagine dividing SEQ into N-FOLDS subsequences of the same
    size (bar rounding). Return the subsequence of index FOLD as the
    first value and the all the other subsequences concatenated into one
    as the second value. The order of elements remains stable. This
    function is suitable as the SPLIT-FN argument of CROSS-VALIDATE.

- [function] SPLIT-STRATIFIED SEQ FOLD N-FOLDS &KEY (KEY #'IDENTITY) (TEST #'EQL) WEIGHT

    Split SEQ into N-FOLDS partitions (as in FRACTURE-STRATIFIED).
    Return the partition of index FOLD as the first value, and the
    concatenation of the rest as the second value. This function is
    suitable as the SPLIT-FN argument of CROSS-VALIDATE (mostly likely
    as a closure with KEY, TEST, WEIGHT bound).

### Bagging

- [function] BAG SEQ FN &KEY (RATIO 1) N WEIGHT (REPLACEMENT T) KEY (TEST #'EQL) (RANDOM-STATE \*RANDOM-STATE\*)

    Sample from SEQ with SAMPLE-FROM (passing RATIO, WEIGHT,
    REPLACEMENT), or SAMPLE-STRATIFIED if KEY is not NIL. Call FN with
    the sample. If N is NIL then keep repeating this until FN performs a
    non-local exit. Else N must be a non-negative integer, N iterations
    will be performed, the primary values returned by FN collected into
    a list and returned. See SAMPLE-FROM and SAMPLE-STRATIFIED for
    examples.

- [function] SAMPLE-FROM RATIO SEQ &KEY WEIGHT REPLACEMENT (RANDOM-STATE \*RANDOM-STATE\*)

    Return a sequence constructed by sampling with or without
    REPLACEMENT from SEQ. The sum of weights in the result sequence will
    approximately be the sum of weights of SEQ times RATIO. If WEIGHT is
    NIL then elements are assumed to have equal weights, else WEIGHT
    should return a non-negative real number when called with an element
    of SEQ.
    
    To randomly select half of the elements:
    
    ```common-lisp
    (sample-from 1/2 '(0 1 2 3 4 5))
    => (5 3 2)
    ```
    
    To randomly select some elements such that the sum of their weights
    constitute about half of the sum of weights across the whole
    sequence:
    
    ```common-lisp
    (sample-from 1/2 '(0 1 2 3 4 5 6 7 8 9) :weight #'identity)
    => ;; sums to 28 that's near 45/2
       (9 4 1 6 8)
    ```
    
    To sample with replacement (that is, allowing the element to be
    sampled multiple times):
    
    ```common-lisp
    (sample-from 1 '(0 1 2 3 4 5) :replacement t)
    => (1 1 5 1 4 4)
    ```


- [function] SAMPLE-STRATIFIED RATIO SEQ &KEY WEIGHT REPLACEMENT (KEY #'IDENTITY) (TEST #'EQL) (RANDOM-STATE \*RANDOM-STATE\*)

    Like SAMPLE-FROM but makes sure that the weighted proportion of
    classes in the result is approximately the same as the proportion in
    SEQ. See STRATIFY for the description of KEY and TEST.

### CV Bagging

- [function] BAG-CV DATA FN &KEY N (N-FOLDS 5) (FOLDS (ALEXANDRIA:IOTA N-FOLDS)) (SPLIT-FN #'SPLIT-FOLD/MOD) PASS-FOLD (RANDOM-STATE \*RANDOM-STATE\*)

    Perform cross-validation on different shuffles of DATA N times and
    collect the results. Since CROSS-VALIDATE collects the return values
    of FN, the return value of this function is a list of lists of FN
    results. If N is NIL, don't collect anything just keep doing
    repeated CVs until FN performs a non-local exit.
    
    The following example simply collects the test and training sets for
    2-fold CV repeated 3 times with shuffled data:
    
    ```commonlisp
    ;;; This is non-deterministic.
    (bag-cv '(0 1 2 3 4) #'list :n 3 :n-folds 2)
    => ((((2 3 4) (1 0))
         ((1 0) (2 3 4)))
        (((2 1 0) (4 3))
         ((4 3) (2 1 0)))
        (((1 0 3) (2 4))
         ((2 4) (1 0 3))))
    ```
    
    CV bagging is useful when a single CV is not producing stable
    results. As an ensemble method, CV bagging has the advantage over
    bagging that each example will occur the same number of times and
    after the first CV is complete there is a complete but less reliable
    estimate for each example which gets refined by further CVs.

### Miscellaneous Operations

- [function] SPREAD-STRATA SEQ &KEY (KEY #'IDENTITY) (TEST #'EQL)

    Return a sequence that's a reordering of SEQ such that elements
    belonging to different strata (under KEY and TEST, see STRATIFY) are
    distributed evenly. The order of elements belonging to the same
    stratum is unchanged.
    
    For example, to make sure that even and odd numbers are distributed
    evenly:
    
    ```common-lisp
    (spread-strata '(0 2 4 6 8 1 3 5 7 9) :key #'evenp)
    => (0 1 2 3 4 5 6 7 8 9)
    ```
    
    Same thing with unbalanced classes:
    
    ```common-lisp
    (spread-strata (vector 0 2 3 5 6 1 4)
                   :key (lambda (x)
                          (if (member x '(1 4))
                              t
                              nil)))
    => #(0 1 2 3 4 5 6)
    ```


- [function] ZIP-EVENLY SEQS &KEY RESULT-TYPE

    Make a single sequence out of the sequences in SEQS so that in the
    returned sequence indices of elements belonging to the same source
    sequence are spread evenly across the whole range. The result is a
    list is RESULT-TYPE is LIST, it's a vector if RESULT-TYPE is VECTOR.
    If RESULT-TYPE is NIL, then it's determined by the type of the first
    sequence in SEQS.
    
    ```common-lisp
    (zip-evenly '((0 2 4) (1 3)))
    => (0 1 2 3 4)
    ```


## Core

###### \[in package MGL-CORE\]
### Persistence

- [function] LOAD-STATE FILENAME OBJECT

    Load weights of OBJECT from FILENAME. Return OBJECT.

- [function] SAVE-STATE FILENAME OBJECT &KEY (IF-EXISTS :ERROR) (ENSURE T)

    Save weights of OBJECT to FILENAME. If ENSURE, then
    ENSURE-DIRECTORIES-EXIST is called on FILENAME. IF-EXISTS is passed
    on to OPEN. Return OBJECT.

- [function] READ-STATE OBJECT STREAM

    Read the weights of OBJECT from the bivalent STREAM where weights
    mean the learnt parameters. There is currently no sanity checking of
    data which will most certainly change in the future together with
    the serialization format. Return OBJECT.

- [function] WRITE-STATE OBJECT STREAM

    Write weight of OBJECT to the bivalent STREAM. Return OBJECT.

- [generic-function] READ-STATE* OBJECT STREAM CONTEXT

    This is the extension point for READ-STATE. It is
    guaranteed that primary READ-STATE\* methods will be called only once
    for each OBJECT (under EQ). CONTEXT is an opaque object and must be
    passed on to any recursive READ-STATE\* calls.

- [generic-function] WRITE-STATE* OBJECT STREAM CONTEXT

    This is the extension point for WRITE-STATE. It is
    guaranteed that primary WRITE-STATE\* methods will be called only
    once for each OBJECT (under EQ). CONTEXT is an opaque object and must
    be passed on to any recursive WRITE-STATE\* calls.

### Batch Processing

Processing instances one by one during training or prediction can
be slow. The models that support batch processing for greater
efficiency are said to be *striped*.

Typically, during or after creating a model, one sets MAX-N-STRIPES
on it a positive integer. When a batch of instances is to be fed to
the model it is first broken into subbatches of length that's at
most MAX-N-STRIPES. For each subbatch, SET-INPUT (FIXDOC) is called
and a before method takes care of setting N-STRIPES to the actual
number of instances in the subbatch. When MAX-N-STRIPES is set
internal data structures may be resized which is an expensive
operation. Setting N-STRIPES is a comparatively cheap operation,
often implemented as matrix reshaping.

Note that for models made of different parts (for example,
MGL-BP:BPN consists of MGL-BP:LUMPs) , setting these
values affects the constituent parts, but one should never change
the number stripes of the parts directly because that would lead to
an internal inconsistency in the model.

- [generic-function] MAX-N-STRIPES OBJECT

    The number of stripes with which the OBJECT is
    capable of dealing simultaneously. 

- [generic-function] SET-MAX-N-STRIPES MAX-N-STRIPES OBJECT

    Allocate the necessary stuff to allow for
    MAX-N-STRIPES number of stripes to be worked with simultaneously in
    OBJECT. This is called when MAX-N-STRIPES is SETF'ed.

- [generic-function] N-STRIPES OBJECT

    The number of stripes currently present in OBJECT.
    This is at most MAX-N-STRIPES.

- [generic-function] SET-N-STRIPES N-STRIPES OBJECT

    Set the number of stripes (out of MAX-N-STRIPES)
    that are in use in OBJECT. This is called when N-STRIPES is
    SETF'ed.

- [macro] WITH-STRIPES SPECS &BODY BODY

    Bind start and optionally end indices belonging to stripes in
    striped objects.
    
        (WITH-STRIPES ((STRIPE1 OBJECT1 START1 END1)
                       (STRIPE2 OBJECT2 START2)
                       ...)
         ...)
    
    This is how one's supposed to find the index range corresponding to
    the Nth input in an input lump of a bpn:
    
         (with-stripes ((n input-lump start end))
           (loop for i upfrom start below end
                 do (setf (mref (nodes input-lump) i) 0d0)))
    
    Note how the input lump is striped, but the matrix into which we are
    indexing (NODES) is not known to WITH-STRIPES. In fact, for lumps
    the same stripe indices work with NODES and MGL-BP:DERIVATIVES.

- [generic-function] STRIPE-START STRIPE OBJECT

    Return the start index of STRIPE in some array or
    matrix of OBJECT.

- [generic-function] STRIPE-END STRIPE OBJECT

    Return the end index (exclusive) of STRIPE in some
    array or matrix of OBJECT.

- [generic-function] SET-INPUT INSTANCES MODEL

    Set INSTANCES as inputs in MODEL. INSTANCES is
    always a SEQUENCE of instances even for models not capable of batch
    operation. It sets N-STRIPES to (LENGTH INSTANCES) in a :BEFORE
    method.

- [function] MAP-BATCHES-FOR-MODEL FN DATASET MODEL

    Call FN with batches of instances from DATASET suitable for MODEL.
    The number of instances in a batch is MAX-N-STRIPES of MODEL or less
    if there are no more instances left.

- [macro] DO-BATCHES-FOR-MODEL (BATCH (DATASET MODEL)) &BODY BODY

    Convenience macro over MAP-BATCHES-FOR-MODEL.

### Executors

- [generic-function] MAP-OVER-EXECUTORS FN INSTANCES PROTOTYPE-EXECUTOR

    Divide INSTANCES between executors that perform the
    same function as PROTOTYPE-EXECUTOR and call FN with the instances
    and the executor for which the instances are.
    
    Some objects conflate function and call: the forward pass of a
    MGL-BP:BPN computes output from inputs so it is like a
    function but it also doubles as a function call in the sense that
    the bpn (function) object changes state during the computation of
    the output. Hence not even the forward pass of a bpn is thread safe.
    There is also the restriction that all inputs must be of the same
    size.
    
    For example, if we have a function that builds bpn a for an input of
    a certain size, then we can create a factory that creates bpns for a
    particular call. The factory probably wants to keep the weights the
    same though. In @MGL-PARAMETERIZED-EXECUTOR-CACHE,
    MAKE-EXECUTOR-WITH-PARAMETERS is this factory.
    
    Parallelization of execution is another possibility
    MAP-OVER-EXECUTORS allows, but there is no prebuilt solution for it,
    yet.
    
    The default implementation simply calls FN with INSTANCES and
    PROTOTYPE-EXECUTOR.

- [macro] DO-EXECUTORS (INSTANCES OBJECT) &BODY BODY

    Convenience macro on top of MAP-OVER-EXECUTORS.

#### Parameterized Executor Cache

- [class] PARAMETERIZED-EXECUTOR-CACHE-MIXIN

    Mix this into a model, implement
    INSTANCE-TO-EXECUTOR-PARAMETERS and MAKE-EXECUTOR-WITH-PARAMETERS
    and DO-EXECUTORS will be to able build executors suitable for
    different instances. The canonical example is using a BPN to compute
    the means and convariances of a gaussian process. Since each
    instance is made of a variable number of observations, the size of
    the input is not constant, thus we have a bpn (an executor) for each
    input dimension (the parameters).

- [generic-function] MAKE-EXECUTOR-WITH-PARAMETERS PARAMETERS CACHE

    Create a new executor for PARAMETERS. CACHE is a
    PARAMETERIZED-EXECUTOR-CACHE-MIXIN. In the BPN gaussian process
    example, PARAMETERS would be a list of input dimensions.

- [generic-function] INSTANCE-TO-EXECUTOR-PARAMETERS INSTANCE CACHE

    Return the parameters for an executor able to
    handle INSTANCE. Called by MAP-OVER-EXECUTORS on CACHE (that's a
    PARAMETERIZED-EXECUTOR-CACHE-MIXIN). The returned parameters are
    keys in an EQUAL parameters->executor hash table.

## Monitoring

###### \[in package MGL-CORE\]
When training or applying a model, one often wants to track various
statistics. For example, in the case of training a neural network
with cross-entropy loss, these statistics could be the average
cross-entropy loss itself, classification accuracy, or even the
entire confusion matrix and sparsity levels in hidden layers. Also,
there is the question of what to do with the measured values (log
and forget, add to some counter or a list).

So there may be several phases of operation when we want to keep an
eye on. Let's call these **events**. There can also be many fairly
independent things to do in response to an event. Let's call these
**monitors**. Some monitors are a composition of two operations: one
that extracts some measurements and another that aggregates those
measurements. Let's call these two **measurers** and **counters**,
respectively.

For example, consider training a backpropagation neural network. We
want to look at the state of of network just after the backward
pass. MGL-BP:BP-LEARNER has a MONITORS event hook corresponding to the moment after
backpropagating the gradients. Suppose we are interested in how the
training cost evolves:

    (push (make-instance 'monitor
                         :measurer (lambda (instances bpn)
                                     (declare (ignore instances))
                                     (mgl-bp:cost bpn))
                         :counter (make-instance 'basic-counter))
          (monitors learner))

During training, this monitor will track the cost of training
examples behind the scenes. If we want to print and reset this
monitor periodically we can put another monitor on
MGL-OPT:ITERATIVE-OPTIMIZER's MGL-OPT:ON-N-INSTANCES-CHANGED
accessor:

    (push (lambda (optimizer gradient-source n-instances)
            (declare (ignore optimizer))
            (when (zerop (mod n-instances 1000))
              (format t "n-instances: ~S~%" n-instances)
              (dolist (monitor (monitors gradient-source))
                (when (counter monitor)
                  (format t "~A~%" (counter monitor))
                  (reset-counter (counter monitor)))))
          (mgl-opt:on-n-instances-changed optimizer))

Note that the monitor we push can be anything as long as
APPLY-MONITOR is implemented on it with the appropriate signature.
Also note that the ZEROP + MOD logic is fragile, so you will likely
want to use MGL-OPT:MONITOR-OPTIMIZATION-PERIODICALLY instead of
doing the above.

So that's the general idea. Concrete events are documented where
they are signalled. Often there are task specific utilities that
create a reasonable set of default monitors (see
@MGL-CLASSIFICATION-MONITOR).

- [function] APPLY-MONITORS MONITORS &REST ARGUMENTS

    Call APPLY-MONITOR on each monitor in MONITORS and ARGUMENTS. This
    is how an event is fired.

- [generic-function] APPLY-MONITOR MONITOR &REST ARGUMENTS

    Apply MONITOR to ARGUMENTS. This sound fairly
    generic, because it is. MONITOR can be anything, even a simple
    function or symbol, in which case this is just CL:APPLY. See
    @MGL-MONITOR for more.

- [generic-function] COUNTER MONITOR

    Return an object representing the state of MONITOR
    or NIL, if it doesn't have any (say because it's a simple logging
    function). Most monitors have counters into which they accumulate
    results until they are printed and reset. See @MGL-COUNTER for
    more.

- [function] MONITOR-MODEL-RESULTS FN DATASET MODEL MONITORS

    Call FN with batches of instances from DATASET until it runs
    out (as in DO-BATCHES-FOR-MODEL). FN is supposed to apply MODEL to
    the batch and return some kind of result (for neural networks, the
    result is the model state itself). Apply MONITORS to each batch and
    the result returned by FN for that batch. Finally, return the list
    of counters of MONITORS.
    
    The purpose of this function is to collect various results and
    statistics (such as error measures) efficiently by applying the
    model only once, leaving extraction of quantities of interest from
    the model's results to MONITORS.
    
    See the model specific versions of this functions such as
    MGL-BP:MONITOR-BPN-RESULTS.

- [generic-function] MONITORS OBJECT

    Return monitors associated with OBJECT. See various
    methods such as MONITORS for more
    documentation.

### Monitors

- [class] MONITOR

    A monitor that has another monitor called MEASURER
    embedded in it. When this monitor is applied, it applies the
    measurer and passes the returned values to ADD-TO-COUNTER called on
    its COUNTER slot. One may further specialize APPLY-MONITOR to change
    that.
    
    This class is useful when the same event monitor is applied
    repeatedly over a period and its results must be aggregated such as
    when training statistics are being tracked or when predictions are
    begin made. Note that the monitor must be compatible with the event
    it handles. That is, the embedded MEASURER must be prepared to take
    the arguments that are documented to come with the event.

- [reader] MEASURER MONITOR (:MEASURER)

    This must be a monitor itself which only means
    that APPLY-MONITOR is defined on it (but see @MGL-MONITORING). The
    returned values are aggregated by COUNTER. See
    @MGL-MEASURER for a library of measurers.

- [reader] COUNTER MONITOR (:COUNTER)

    The COUNTER of a monitor carries out the
    aggregation of results returned by MEASURER. The See @MGL-COUNTER
    for a library of counters.

### Measurers

MEASURER is a part of MONITOR objects, an embedded monitor that
computes a specific quantity (e.g. classification accuracy) from the
arguments of event it is applied to (e.g. the model results).
Measurers are often implemented by combining some kind of model
specific extractor with a generic measurer function.

All generic measurer functions return their results as multiple
values matching the arguments of ADD-TO-COUNTER for a counter of a
certain type (see @MGL-COUNTER) so as to make them easily used in a
MONITOR:

    (multiple-value-call #'add-to-counter <some-counter>
                         <call-to-some-measurer>)

The counter class compatible with the measurer this way is noted for
each function.

For a list of measurer functions see @MGL-CLASSIFICATION-MEASURER.

### Counters

- [generic-function] ADD-TO-COUNTER COUNTER &REST ARGS

    Add ARGS to COUNTER in some way. See specialized
    methods for type specific documentation. The kind of arguments to be
    supported is the what the measurer functions (see @MGL-MEASURER)
    intended to be paired with the counter return as multiple values.

- [generic-function] COUNTER-VALUES COUNTER

    Return any number of values representing the state
    of COUNTER. See specialized methods for type specific
    documentation.

- [generic-function] COUNTER-RAW-VALUES COUNTER

    Return any number of values representing the state
    of COUNTER in such a way that passing the returned values as
    arguments ADD-TO-COUNTER on a fresh instance of the same type
    recreates the original state.

- [generic-function] RESET-COUNTER COUNTER

    Restore state of COUNTER to what it was just after
    creation.

#### Attributes

- [class] ATTRIBUTED

    This is a utility class that all counters subclass.
    The ATTRIBUTES plist can hold basically anything. Currently the
    attributes are only used when printing and they can be specified by
    the user. The monitor maker functions such as those in
    @MGL-CLASSIFICATION-MONITOR also add attributes of their own to the
    counters they create.
    
    With the :PREPEND-ATTRIBUTES initarg when can easily add new
    attributes without clobbering the those in the :INITFORM, (:TYPE
    "rmse") in this case.
    
        (princ (make-instance 'rmse-counter
                              :prepend-attributes '(:event "pred."
                                                    :dataset "test")))
        ;; pred. test rmse: 0.000e+0 (0)
        => #<RMSE-COUNTER pred. test rmse: 0.000e+0 (0)>


- [accessor] ATTRIBUTES ATTRIBUTED (:ATTRIBUTES = NIL)

    A plist of attribute keys and values.

- [method] NAME (ATTRIBUTED ATTRIBUTED)

    Return a string assembled from the values of the ATTRIBUTES of
    ATTRIBUTED. If there are multiple entries with the same key, then
    they are printed near together.
    
    Values may be padded according to an enclosing
    WITH-PADDED-ATTRIBUTE-PRINTING.

- [macro] WITH-PADDED-ATTRIBUTE-PRINTING (ATTRIBUTEDS) &BODY BODY

    Note the width of values for each attribute key which is the number
    of characters in the value's PRINC-TO-STRING'ed representation. In
    BODY, if attributes with they same key are printed they are forced
    to be at least this wide. This allows for nice, table-like output:
    
        (let ((attributeds
                (list (make-instance 'basic-counter
                                     :attributes '(:a 1 :b 23 :c 456))
                      (make-instance 'basic-counter
                                     :attributes '(:a 123 :b 45 :c 6)))))
          (with-padded-attribute-printing (attributeds)
            (map nil (lambda (attributed)
                       (format t "~A~%" attributed))
                 attributeds)))
        ;; 1   23 456: 0.000e+0 (0)
        ;; 123 45 6  : 0.000e+0 (0)


- [function] LOG-PADDED ATTRIBUTEDS

    Log (see LOG-MSG) ATTRIBUTEDS non-escaped (as in PRINC or ~A) with
    the output being as table-like as possible.

#### Counter classes

In addition to the really basic ones here, also see
@MGL-CLASSIFICATION-COUNTER.

- [class] BASIC-COUNTER ATTRIBUTED

    A simple counter whose ADD-TO-COUNTER takes two
    additional parameters: an increment to the internal sums of called
    the NUMERATOR and DENOMINATOR. COUNTER-VALUES returns two
    values:
    
    - NUMERATOR divided by DENOMINATOR (or 0 if DENOMINATOR is 0) and
    
    - DENOMINATOR
    
    Here is an example the compute the mean of 5 things received in two
    batches:
    
         (let ((counter (make-instance 'basic-counter)))
           (add-to-counter counter 6.5 3)
           (add-to-counter counter 3.5 2)
           counter)
         => #<BASIC-COUNTER 2.00000e+0 (5)>


- [class] RMSE-COUNTER BASIC-COUNTER

    A BASIC-COUNTER with whose nominator accumulates
    the square of some statistics. It has the attribute :TYPE "rmse".
    COUNTER-VALUES returns the square root of what BASIC-COUNTER's
    COUNTER-VALUES would return.
    
        (let ((counter (make-instance 'rmse-counter)))
          (add-to-counter counter (+ (* 3 3) (* 4 4)) 2)
          counter)
        => #<RMSE-COUNTER rmse: 3.53553e+0 (2)>


- [class] CONCAT-COUNTER ATTRIBUTED

    A counter that simply concatenates
    sequences.
    
    \`\`\`cl-transcript
    (let ((counter (make-instance 'concat-counter)))
      (add-to-counter counter '(1 2 3) #(4 5))
      (add-to-counter counter '(6 7))
      (counter-values counter))
    => (1 2 3 4 5 6 7)
    \`\`\`\`

- [reader] CONCATENATION-TYPE CONCAT-COUNTER (:CONCATENATION-TYPE = 'LIST)

    A type designator suitable as the RESULT-TYPE
    argument to CONCATENATE.

## Classification

###### \[in package MGL-CORE\]
To be able to measure classification related quantities, we need to
define what the label of an instance is. Customization is possible
by implementing a method for a specific type of instance, but these
functions only ever appear as defaults that can be overridden.

- [generic-function] LABEL-INDEX INSTANCE

    Return the label of INSTANCE as a non-negative
    integer.

- [generic-function] LABEL-INDEX-DISTRIBUTION INSTANCE

    Return a one dimensional array of probabilities
    representing the distribution of labels. The probability of the
    label with LABEL-INDEX `I` is element at index `I` of the returned
    arrray.

The following two functions are basically the same as the previous
two, but in batch mode: they return a sequence of label indices or
distributions. These are called on results produced by models.
Implement these for a model and the monitor maker functions below
will automatically work. See FIXDOC: for bpn and boltzmann.

- [generic-function] LABEL-INDICES RESULTS

    Return a sequence of label indices for RESULTS
    produced by some model for a batch of instances. This is akin to
    LABEL-INDEX.

- [generic-function] LABEL-INDEX-DISTRIBUTIONS RESULT

    Return a sequence of label index distributions for
    RESULTS produced by some model for a batch of instances. This is
    akin to LABEL-INDEX-DISTRIBUTION.

### Classification Monitors

The following functions return a list monitors. The monitors are
for events of signature (INSTANCES MODEL) such as those produced by
MONITOR-MODEL-RESULTS and its various model specific variations.
They are model-agnostic functions, extensible to new classifier
types. 

- [function] MAKE-CLASSIFICATION-ACCURACY-MONITORS MODEL &KEY OPERATION-MODE ATTRIBUTES (LABEL-INDEX-FN #'LABEL-INDEX)

    Return a list of MONITOR objects associated with
    CLASSIFICATION-ACCURACY-COUNTERs. LABEL-INDEX-FN is a function
    like LABEL-INDEX. See that function for more.
    
    Implemented in terms of MAKE-CLASSIFICATION-ACCURACY-MONITORS\*.

- [function] MAKE-CROSS-ENTROPY-MONITORS MODEL &KEY OPERATION-MODE ATTRIBUTES (LABEL-INDEX-DISTRIBUTION-FN #'LABEL-INDEX-DISTRIBUTION)

    Return a list of MONITOR objects associated with
    CROSS-ENTROPY-COUNTERs. LABEL-INDEX-DISTRIBUTION-FN is a
    function like LABEL-INDEX-DISTRIBUTION. See that function for more.
    
    Implemented in terms of MAKE-CROSS-ENTROPY-MONITORS\*.

- [function] MAKE-LABEL-MONITORS MODEL &KEY OPERATION-MODE ATTRIBUTES (LABEL-INDEX-FN #'LABEL-INDEX) (LABEL-INDEX-DISTRIBUTION-FN #'LABEL-INDEX-DISTRIBUTION)

    Return classification accuracy and cross-entropy monitors. See
    MAKE-CLASSIFICATION-ACCURACY-MONITORS and
    MAKE-CROSS-ENTROPY-MONITORS for a description of paramters.

The monitor makers above can be extended to support new classifier
types via the following generic functions.

- [generic-function] MAKE-CLASSIFICATION-ACCURACY-MONITORS* MODEL OPERATION-MODE LABEL-INDEX-FN ATTRIBUTES

    Identical to MAKE-CLASSIFICATION-ACCURACY-MONITORS
    bar the keywords arguments. Specialize this to add to support for
    new model types. The default implementation also allows for some
    extensibility: if LABEL-INDICES is defined on MODEL, then it will be
    used to extract label indices from model results.

- [generic-function] MAKE-CROSS-ENTROPY-MONITORS* MODEL OPERATION-MODE LABEL-INDEX-DISTRIBUTION-FN ATTRIBUTES

    Identical to MAKE-CROSS-ENTROPY-MONITORS bar the
    keywords arguments. Specialize this to add to support for new model
    types. The default implementation also allows for some
    extensibility: if LABEL-INDEX-DISTRIBUTIONS is defined on MODEL,
    then it will be used to extract label distributions from model
    results.

### Classification Measurers

The functions here compare some known good solution (also known as
*ground truth* or *target*) to a prediction or approximation and
return some measure of their \[dis\]similarity. They are model
independent, hence one has to extract the ground truths and
predictions first. Rarely used directly, they are mostly hidden
behind @MGL-CLASSIFICATION-MONITOR.

- [function] MEASURE-CLASSIFICATION-ACCURACY TRUTHS PREDICTIONS &KEY (TEST #'EQL) TRUTH-KEY PREDICTION-KEY WEIGHT

    Return the number of correct classifications and as the second
    value the number of instances (equal to length of TRUTHS in the
    non-weighted case). TRUTHS (keyed by TRUTH-KEY) is a sequence of
    opaque class labels compared with TEST to another sequence of
    classes labels in PREDICTIONS (keyed by PREDICTION-KEY). If WEIGHT
    is non-nil, then it is a function that returns the weight of an
    element of TRUTHS. Weighted cases add their weight to both
    counts (returned as the first and second values) instead of 1 as in
    the non-weighted case.
    
    Note how the returned values are suitable for MULTIPLE-VALUE-CALL
    with #'ADD-TO-COUNTER and a CLASSIFICATION-ACCURACY-COUNTER.

- [function] MEASURE-CROSS-ENTROPY TRUTHS PREDICTIONS &KEY TRUTH-KEY PREDICTION-KEY (MIN-PREDICTION-PR 1.0d-15)

    Return the sum of the cross-entropy between pairs of elements with
    the same index of TRUTHS and PREDICTIONS. TRUTH-KEY is a function
    that's when applied to an element of TRUTHS returns a sequence
    representing some kind of discrete target distribution (P in the
    definition below). TRUTH-KEY may be NIL which is equivalent to the
    IDENTITY function. PREDICTION-KEY is the same kind of key for
    PREDICTIONS, but the sequence it returns represents a distribution
    that approximates (Q below) the true one.
    
    Cross-entropy of the true and approximating distributions is defined
    as:
    
        cross-entropy(p,q) = - sum_i p(i) * log(q(i))
    
    of which this function returns the sum over the pairs of elements of
    TRUTHS and PREDICTIONS keyed by TRUTH-KEY and PREDICTION-KEY.
    
    Due to the logarithm, if q(i) is close to zero, we run into
    numerical problems. To prevent this, all q(i) that are less than
    MIN-PREDICTION-PR are treated as if they were MIN-PREDICTION-PR.
    
    The second value returned is the sum of p(i) over all TRUTHS and all
    `I`. This is normally equal to `(LENGTH TRUTHS)`, since elements of
    TRUTHS represent a probability distribution, but this is not
    enforced which allows relative importance of elements to be
    controlled.
    
    The third value returned is a plist that maps each index occurring
    in the distribution sequences to a list of two elements:
    
         sum_j p_j(i) * log(q_j(i))
    
    and
    
        sum_j p_j(i)
    
    where `J` indexes into TRUTHS and PREDICTIONS.
    
        (measure-cross-entropy '((0 1 0)) '((0.1 0.7 0.2)))
        => 0.35667497
           1
           (2 (0.0 0)
            1 (0.35667497 1)
            0 (0.0 0))
    
    Note how the returned values are suitable for MULTIPLE-VALUE-CALL
    with #'ADD-TO-COUNTER and a CROSS-ENTROPY-COUNTER.

- [function] MEASURE-ROC-AUC PREDICTIONS PRED &KEY (KEY #'IDENTITY) WEIGHT

    Return the area under the ROC curve for PREDICTIONS representing
    predictions for a binary classification problem. PRED is a predicate
    function for deciding whether a prediction belongs to the so called
    positive class. KEY returns a number for each element which is the
    predictor's idea of how much that element is likely to belong to the
    class, although it's not necessarily a probability.
    
    If WEIGHT is NIL, then all elements of PREDICTIONS count as 1
    towards the unnormalized sum within AUC. Else WEIGHT must be a
    function like KEY, but it should return the importance (a positive
    real number) of elements. If the weight of an prediction is 2 then
    it's as if there were another identical copy of that prediction in
    PREDICTIONS.
    
    The algorithm is based on algorithm 2 in the paper 'An introduction
    to ROC analysis' by Tom Fawcett.
    
    ROC AUC is equal to the probability of a randomly chosen positive
    having higher KEY (score) than a randomly chosen negative element.
    With equal scores in mind, a more precise version is: AUC is the
    expectation of the above probability over all possible sequences
    sorted by scores.

- [function] MEASURE-CONFUSION TRUTHS PREDICTIONS &KEY (TEST #'EQL) TRUTH-KEY PREDICTION-KEY WEIGHT

    Create a CONFUSION-MATRIX from TRUTHS and PREDICTIONS.
    TRUTHS (keyed by TRUTH-KEY) is a sequence of class labels compared
    with TEST to another sequence of class labels in PREDICTIONS (keyed
    by PREDICTION-KEY). If WEIGHT is non-nil, then it is a function that
    returns the weight of an element of TRUTHS. Weighted cases add their
    weight to both counts (returned as the first and second values).
    
    Note how the returned confusion matrix can be added to another with
    ADD-TO-COUNTER.

### Classification Counters

- [class] CLASSIFICATION-ACCURACY-COUNTER BASIC-COUNTER

    A BASIC-COUNTER with "acc." as its :TYPE
    attribute and a PRINT-OBJECT method that prints percentages.

- [class] CROSS-ENTROPY-COUNTER BASIC-COUNTER

    A BASIC-COUNTER with "xent" as its :TYPE
    attribute.

#### Confusion Matrices

- [class] CONFUSION-MATRIX

    A confusion matrix keeps count of classification
    results. The correct class is called `target' and the output of the
    classifier is called`prediction'.

- [function] MAKE-CONFUSION-MATRIX &KEY (TEST #'EQL)

    Classes are compared with TEST.

- [generic-function] SORT-CONFUSION-CLASSES MATRIX CLASSES

    Return a list of CLASSES sorted for presentation
    purposes.

- [generic-function] CONFUSION-CLASS-NAME MATRIX CLASS

    Name of CLASS for presentation purposes.

- [generic-function] CONFUSION-COUNT MATRIX TARGET PREDICTION

- [generic-function] MAP-CONFUSION-MATRIX FN MATRIX

    Call FN with `TARGET`, PREDICTION,
    COUNT paramaters for each cell in the confusion matrix. Cells with a
    zero count may be ommitted.

- [generic-function] CONFUSION-MATRIX-CLASSES MATRIX

    A list of all classes. The default is to collect
    classes from the counts. This can be overridden if, for instance,
    some classes are not present in the results.

- [function] CONFUSION-MATRIX-ACCURACY MATRIX &KEY FILTER

    Return the overall accuracy of the results in MATRIX. It's computed
    as the number of correctly classified cases (hits) divided by the
    name of cases. Return the number of hits and the number of cases as
    the second and third value. If FILTER function is given, then call
    it with the target and the prediction of the cell. Disregard cell
    for which FILTER returns NIL.
    
    Precision and recall can be easily computed by giving the right
    filter, although those are provided in separate convenience
    functions.

- [function] CONFUSION-MATRIX-PRECISION MATRIX PREDICTION

    Return the accuracy over the cases when the classifier said
    PREDICTION.

- [function] CONFUSION-MATRIX-RECALL MATRIX TARGET

    Return the accuracy over the cases when the correct class is
    TARGET.

- [function] ADD-CONFUSION-MATRIX MATRIX RESULT-MATRIX

    Add MATRIX into RESULT-MATRIX.

## Features

###### \[in package MGL-CORE\]
### Feature Selection

The following *scoring functions* all return an EQUAL hash table
that maps features to scores.

- [function] COUNT-FEATURES DOCUMENTS MAPPER &KEY (KEY #'IDENTITY)

    Return scored features as an EQUAL hash table whose keys are
    features of DOCUMENTS and values are counts of occurrences of
    features. MAPPER takes a function and a document and calls function
    with features of the document.
    
    ```common-lisp
    (sort (alexandria:hash-table-alist
           (count-features '(("hello" "world")
                             ("this" "is" "our" "world"))
                           (lambda (fn document)
                             (map nil fn document))))
          #'string< :key #'car)
    => (("hello" . 1) ("is" . 1) ("our" . 1) ("this" . 1) ("world" . 2))
    ```


- [function] FEATURE-LLRS DOCUMENTS MAPPER CLASS-FN &KEY (CLASSES (ALL-DOCUMENT-CLASSES DOCUMENTS CLASS-FN))

    Return scored features as an EQUAL hash table whose keys are
    features of DOCUMENTS and values are their log likelihood ratios.
    MAPPER takes a function and a document and calls function with
    features of the document.
    
    ```common-lisp
    (sort (alexandria:hash-table-alist
           (feature-llrs '((:a "hello" "world")
                           (:b "this" "is" "our" "world"))
                         (lambda (fn document)
                           (map nil fn (rest document)))
                         #'first))
          #'string< :key #'car)
    => (("hello" . 2.6032386) ("is" . 2.6032386) ("our" . 2.6032386)
        ("this" . 2.6032386) ("world" . 4.8428774e-8))
    ```


- [function] FEATURE-DISAMBIGUITIES DOCUMENTS MAPPER CLASS-FN &KEY (CLASSES (ALL-DOCUMENT-CLASSES DOCUMENTS CLASS-FN))

    Return scored features as an EQUAL hash table whose keys are
    features of DOCUMENTS and values are their *disambiguities*. MAPPER
    takes a function and a document and calls function with features of
    the document.
    
    From the paper 'Using Ambiguity Measure Feature Selection Algorithm
    for Support Vector Machine Classifier'.

### Feature Encoding

Features can rarely be fed directly to algorithms as is, they need
to be transformed in some way. Suppose we have a simple language
model that takes a single word as input and predicts the next word.
However, both input and output is to be encoded as float vectors of
length 1000. What we do is find the top 1000 words by some
measure (see @MGL-FEATURE-SELECTION) and associate these words with
the integers in \[0..999\] (this is `ENCODE`ing). By using for
example [one-hot](http://en.wikipedia.org/wiki/One-hot) encoding, we
translate a word into a float vector when passing in the input. When
the model outputs the probability distribution of the next word, we
find the index of the max and find the word associated with it (this
is `DECODE`ing)

- [generic-function] ENCODE ENCODER DECODED

    Encode DECODED with ENCODER. This interface is
    generic enough to be almost meaningless. See ENCODER/DECODER for a
    simple, MGL-NLP:BAG-OF-WORDS-ENCODER for a slightly more involved
    example.
    
    If ENCODER is a function designator, then it's simply `FUNCALL`ed
    with DECODED.

- [generic-function] DECODE DECODER ENCODED

    Decode ENCODED with ENCODER. For an DECODER /
    ENCODER pair, `(DECODE DECODER (ENCODE ENCODER OBJECT))` must be
    equal in some sense to `OBJECT`.
    
    If DECODER is a function designator, then it's simply `FUNCALL`ed
    with ENCODED.

- [class] ENCODER/DECODER

    Implements O(1) ENCODE and DECODE by having an
    internal decoded-to-encoded and an encoded-to-decoded EQUAL hash
    table. ENCODER/DECODER objects can be saved and loaded (see
    @MGL-PERSISTENCE) as long as the elements in the hash tables have
    read/write consitency.
    
    ```common-lisp
    (let ((indexer
            (make-indexer
             (alexandria:alist-hash-table '(("I" . 3) ("me" . 2) ("mine" . 1)))
             2)))
      (values (encode indexer "I")
              (encode indexer "me")
              (encode indexer "mine")
              (decode indexer 0)
              (decode indexer 1)
              (decode indexer 2)))
    => 0
    => 1
    => NIL
    => "I"
    => "me"
    => NIL
    ```


- [function] MAKE-INDEXER SCORED-FEATURES N &KEY (START 0) (CLASS 'ENCODER/DECODER)

    Take the top N features from SCORED-FEATURES (see
    @MGL-FEATURE-SELECTION), assign indices to them starting from START.
    Return an ENCODER/DECODER (or another CLASS) that converts between
    objects and indices.

Also see MGL-NLP::@MGL-NLP-BAG-OF-WORDS.

## Gradient Based Optimization

###### \[in package MGL-OPT\]
We have a real valued, differentiable function F and the task is to
find the parameters that minimize its value. Optimization starts
from a single point in the parameter space of F, and this single
point is updated iteratively based on the gradient and value of F at
or around the current point.

Note that while the stated problem is that of global optimization,
for non-convex functions, most algorithms will tend to converge to a
local optimum.

Currently, there are two optimization algorithms:
MGL-GD::@MGL-GD (with several variants) and MGL-CG::@MGL-CG both of
which are first order methods (they do not need second order
gradients) but more can be added with the @MGL-OPT-EXTENSION-API.

- [function] MINIMIZE OPTIMIZER GRADIENT-SOURCE &KEY (WEIGHTS (LIST-SEGMENTS GRADIENT-SOURCE)) (DATASET \*INFINITELY-EMPTY-DATASET\*)

    Minimize the value of the real valued function represented by
    GRADIENT-SOURCE by updating some of its parameters in WEIGHTS (a MAT
    or a sequence of MATs). Return WEIGHTS. DATASET (see
    MGL-DATASET::@MGL-DATASET) is a set of unoptimized parameters of the
    same function. For example, WEIGHTS may be the weights of a neural
    network while DATASET is the training set consisting of inputs
    suitable for SET-INPUT. The default
    DATASET, (*INFINITELY-EMPTY-DATASET*) is suitable for when all
    parameters are optimized, so there is nothing left to come from the
    environment.
    
    Optimization terminates if DATASET is a sampler and it runs out or
    when some other condition met (see TERMINATION, for example). If
    DATASET is a SEQUENCE, then it is reused over and over again.
    
    Examples for various optimizers are provided in MGL-GD::@MGL-GD and
    MGL-CG::@MGL-CG.

### Iterative Optimizer

- [class] ITERATIVE-OPTIMIZER

    An abstract base class of MGL-GD::@MGL-GD and
    MGL-CG::@MGL-CG based optimizers that iterate over instances until a
    termination condition is met.

- [reader] N-INSTANCES ITERATIVE-OPTIMIZER (:N-INSTANCES = 0)

    The number of instances this optimizer has seen so
    far. Incremented automatically during optimization.

- [accessor] TERMINATION ITERATIVE-OPTIMIZER (:TERMINATION = NIL)

    If a number, it's the number of instances to train
    on in the sense of N-INSTANCES. If N-INSTANCES is equal or greater
    than this value optimization stops. If TERMINATION is NIL, then
    optimization will continue. If it is T, then optimization will
    stop. If it is a function of no arguments, then its return value
    is processed as if it was returned by TERMINATION.

- [accessor] ON-OPTIMIZATION-STARTED ITERATIVE-OPTIMIZER (:ON-OPTIMIZATION-STARTED = NIL)

    An event hook with parameters `(OPTIMIZER
    GRADIENT-SOURCE N-INSTANCES)`. Called after initializations are
    performed (INITIALIZE-OPTIMIZER*, INITIALIZE-GRADIENT-SOURCE*) but
    before optimization is started.

- [accessor] ON-OPTIMIZATION-FINISHED ITERATIVE-OPTIMIZER (:ON-OPTIMIZATION-FINISHED = NIL)

    An event hook with parameters `(OPTIMIZER
    GRADIENT-SOURCE N-INSTANCES)`. Called when optimization has
    finished.

- [accessor] ON-N-INSTANCES-CHANGED ITERATIVE-OPTIMIZER (:ON-N-INSTANCES-CHANGED = NIL)

    An event hook with parameters `(OPTIMIZER
    GRADIENT-SOURCE N-INSTANCES)`. Called when optimization of a batch
    of instances is done and N-INSTANCES is incremented.

Now let's discuss a few handy utilities.

- [function] MONITOR-OPTIMIZATION-PERIODICALLY OPTIMIZER PERIODIC-FNS

    For each periodic function in the list of PERIODIC-FNS, add a
    monitor to OPTIMIZER's ON-OPTIMIZATION-STARTED,
    ON-OPTIMIZATION-FINISHED and ON-N-INSTANCES-CHANGED hooks. The
    monitors are simple functions that just call each periodic function
    with the event parameters (OPTIMIZER GRADIENT-SOURCE N-INSTANCES).
    Return OPTIMIZER.
    
    To log and reset the monitors of the gradient source after every
    1000 instances seen by OPTIMIZER:
    
        (monitor-optimization-periodically optimizer
                                           '((:fn log-my-test-error
                                              :period 2000)
                                             (:fn reset-optimization-monitors
                                              :period 1000
                                              :last-eval 0)))
    
    Note how we don't pass it's allowed to just pass the initargs for a
    PERIODIC-FN instead of PERIODIC-FN itself. The :LAST-EVAL 0 bit
    prevents RESET-OPTIMIZATION-MONITORS from being called at the start
    of the optimization when the monitors are empty anyway.

- [generic-function] RESET-OPTIMIZATION-MONITORS OPTIMIZER GRADIENT-SOURCE

    Report the state of MONITORS of
    OPTIMIZER and GRADIENT-SOURCE and reset their counters. See
    MONITOR-OPTIMIZATION-PERIODICALLY for an example of how this is
    used.

- [method] RESET-OPTIMIZATION-MONITORS (OPTIMIZER ITERATIVE-OPTIMIZER) GRADIENT-SOURCE

    Log the counters of the monitors of OPTIMIZER and GRADIENT-SOURCE
    and reset them.

- [generic-function] REPORT-OPTIMIZATION-PARAMETERS OPTIMIZER GRADIENT-SOURCE

    A utility that's often called at the start of
    optimization (from ON-OPTIMIZATION-STARTED). The default
    implementation logs the description of GRADIENT-SOURCE (as in
    DESCRIBE) and OPTIMIZER and calls LOG-MAT-ROOM.

### Cost Function

The function being minimized is often called the *cost* or the
*loss* function.

- [generic-function] COST MODEL

    Return the value of the cost function being
    minimized. Calling this only makes sense in the context of an
    ongoing optimization (see MINIMIZE). The cost is that of a batch of
    instances.

- [function] MAKE-COST-MONITORS MODEL &KEY OPERATION-MODE ATTRIBUTES

    Return a list of MONITOR objects, each associated with one
    BASIC-COUNTER with attribute :TYPE "cost". Implemented in terms of
    MAKE-COST-MONITORS\*.

- [generic-function] MAKE-COST-MONITORS* MODEL OPERATION-MODE ATTRIBUTES

    Identical to MAKE-COST-MONITORS bar the keywords
    arguments. Specialize this to add to support for new model types.

### Gradient Descent

###### \[in package MGL-GD\]
Gradient descent is a first-order optimization algorithm. Relying
completely on first derivatives, it does not even evaluate the
function to be minimized. Let's see how to minimize a numerical lisp
function with respect to some of its parameters.

```commonlisp
(cl:defpackage :mgl-example-sgd
  (:use #:common-lisp #:mgl))

(in-package :mgl-example-sgd)

;;; Create an object representing the sine function.
(defparameter *diff-fn-1*
  (make-instance 'mgl-diffun:diffun
                 :fn #'sin
                 ;; We are going to optimize its only parameter.
                 :weight-indices '(0)))

;;; Minimize SIN. Note that there is no dataset involved because all
;;; parameters are being optimized.
(minimize (make-instance 'sgd-optimizer :termination 1000)
          *diff-fn-1*
          :weights (make-mat 1))
;;; => A MAT with a single value of about -pi/2.

;;; Create a differentiable function for f(x,y)=(x-y)^2. X is a
;;; parameter whose values come from the DATASET argument passed to
;;; MINIMIZE. Y is a parameter to be optimized (a 'weight').
(defparameter *diff-fn-2*
  (make-instance 'mgl-diffun:diffun
                 :fn (lambda (x y)
                       (expt (- x y) 2))
                 :parameter-indices '(0)
                 :weight-indices '(1)))

;;; Find the Y that minimizes the distance from the instances
;;; generated by the sampler.
(minimize (make-instance 'sgd-optimizer :batch-size 10)
          *diff-fn-2*
          :weights (make-mat 1)
          :dataset (make-instance 'function-sampler
                                  :generator (lambda ()
                                               (list (+ 10
                                                        (gaussian-random-1))))
                                  :max-n-samples 1000))
;;; => A MAT with a single value of about 10, the expected value of
;;; the instances in the dataset.

;;; The dataset can be a SEQUENCE in which case we'd better set
;;; TERMINATION else optimization would never finish.
(minimize (make-instance 'sgd-optimizer :termination 1000)
          *diff-fn-2*
          :weights (make-mat 1)
          :dataset '((0) (1) (2) (3) (4) (5)))
;;; => A MAT with a single value of about 2.5.
```

We are going to see a number of accessors for optimizer paramaters.
In general, it's allowed to SETF real slot accessors (as opposed to
readers and writers) at any time during optimization and so is
defining a method on an optimizer subclass that computes the value
in any way. For example, to decay the learning rate on a per
mini-batch basis:

```commonlisp
(defmethod learning-rate ((optimizer my-sgd-optimizer))
  (* (slot-value optimizer 'learning-rate)
     (expt 0.998
           (/ (n-instances optimizer) 60000))))
```


#### Batch Based Optimizers

First let's see everything common to all batch based optimizers,
then discuss @MGL-GD-SGD-OPTIMIZER, @MGL-GD-ADAM-OPTIMIZER and
@MGL-GD-NORMALIZED-BATCH-GD-OPTIMIZER. All batch based optimizers
are `ITERATIVE-OPTIMIZER`s, so see
MGL-OPT::@MGL-OPT-ITERATIVE-OPTIMIZER too.

- [class] BATCH-GD-OPTIMIZER

    Another abstract base class for gradient based
    optimizers tath updates all weights simultaneously after chewing
    through BATCH-SIZE inputs. See subclasses SGD-OPTIMIZER,
    ADAM-OPTIMIZER and NORMALIZED-BATCH-GD-OPTIMIZER.
    
    PER-WEIGHT-BATCH-GD-OPTIMIZER may be a better choice when some
    weights can go unused for instance due to missing input values.

- [accessor] BATCH-SIZE GD-OPTIMIZER (:BATCH-SIZE = 1)

    After having gone through BATCH-SIZE number of
    inputs, weights are updated. With BATCH-SIZE 1, one gets
    Stochastics Gradient Descent. With BATCH-SIZE equal to the number
    of instances in the dataset, one gets standard, 'batch' gradient
    descent. With BATCH-SIZE between these two extremes, one gets the
    most practical 'mini-batch' compromise.

- [accessor] LEARNING-RATE GD-OPTIMIZER (:LEARNING-RATE = 0.1)

    This is the step size along the gradient. Decrease
    it if optimization diverges, increase it if it doesn't make
    progress.

- [accessor] MOMENTUM GD-OPTIMIZER (:MOMENTUM = 0)

    A value in the \[0, 1) interval. MOMENTUM times the
    previous weight change is added to the gradient. 0 means no
    momentum.

- [reader] MOMENTUM-TYPE GD-OPTIMIZER (:MOMENTUM-TYPE = :NORMAL)

    One of :NORMAL, :NESTEROV or :NONE. For pure
    optimization Nesterov's momentum may be better, but it may also
    increases chances of overfitting. Using :NONE is equivalent to 0
    momentum, but it also uses less memory. Note that with :NONE,
    MOMENTUM is ignored even it it is non-zero.

- [accessor] WEIGHT-DECAY GD-OPTIMIZER (:WEIGHT-DECAY = 0)

    An L2 penalty. It discourages large weights, much
    like a zero mean gaussian prior. WEIGHT-DECAY \* WEIGHT is added to
    the gradient to penalize large weights. It's as if the function
    whose minimum is sought had WEIGHT-DECAY\*sum\_i\{0.5 \* WEIGHT\_i^2\}
    added to it.

- [accessor] WEIGHT-PENALTY GD-OPTIMIZER (:WEIGHT-PENALTY = 0)

    An L1 penalty. It encourages sparsity.
    SIGN(WEIGHT) \* WEIGHT-PENALTY is added to the gradient pushing the
    weight towards negative infinity. It's as if the function whose
    minima is sought had WEIGHT-PENALTY\*sum\_i\{abs(WEIGHT\_i)\} added to
    it. Putting it on feature biases consitutes a sparsity constraint
    on the features.

- [reader] USE-SEGMENT-DERIVATIVES-P GD-OPTIMIZER (:USE-SEGMENT-DERIVATIVES-P = NIL)

    Save memory if both the gradient source (the model
    being optimized) and the optimizer support this feature. It works
    like this: the accumulator into which the gradient source is asked
    to place the derivatives of a segment will be SEGMENT-DERIVATIVES
    of the segment. This allows the optimizer not to allocate an
    accumulator matrix into which the derivatives are summed.

- [accessor] AFTER-UPDATE-HOOK GD-OPTIMIZER (:AFTER-UPDATE-HOOK = NIL)

    A list of functions with no arguments called after
    each weight update.

- [accessor] BEFORE-UPDATE-HOOK BATCH-GD-OPTIMIZER (:BEFORE-UPDATE-HOOK = NIL)

    A list of functions of no parameters. Each
    function is called just before a weight update takes place (after
    accumulated gradients have been divided the length of the batch).
    Convenient to hang some additional gradient accumulating code
    on.

##### SGD Optimizer

- [class] SGD-OPTIMIZER BATCH-GD-OPTIMIZER

    With BATCH-SIZE 1 this is Stochastic Gradient
    Descent. With higher batch sizes, one gets mini-batch and Batch
    Gradient Descent.
    
    Assuming that ACCUMULATOR has the sum of gradients for a mini-batch,
    the weight update looks like this:
    
    $$
    \Delta\_w^\{t+1\} = momentum \* \Delta\_w^t
      + \frac\{accumulator\}\{batchsize\}
      + l\_2 w + l\_1 sign(w)
    $$
    
    $$
    w^\{t+1\} = w^\{t\} - learningrate \* \Delta\_w,
    $$
    
    which is the same as the more traditional formulation:
    
    $$
    \Delta\_w^\{t+1\} = momentum \* \Delta\_w^\{t\}
      + learningrate \* \left(\frac\{\frac\{df\}\{dw\}\}\{batchsize\}
                           + l\_2 w + l\_1 sign(w)\right)
    $$
    
    $$
    w^\{t+1\} = w^\{t\} - \Delta\_w,
    $$
    
    but the former works better when batch size, momentum or learning
    rate change during the course of optimization. The above is with
    normal momentum, Nesterov's momentum (see MOMENTUM-TYPE) momentum is
    also available.
    
    See @MGL-GD-BATCH-GD-OPTIMIZER for the description of the various
    options common to all batch based optimizers.

##### Adam Optimizer

- [class] ADAM-OPTIMIZER BATCH-GD-OPTIMIZER

    Adam is a first-order stochasistic gradient descent
    optimizer. It maintains an internal estimation for the mean and raw
    variance of each derivative as exponential moving averages. The step
    it takes is basically `M/(sqrt(V)+E)` where `M` is the estimated
    mean, `V` is the estimated variance, and `E` is a small adjustment
    factor to prevent the gradient from blowing up. See version 5 of the
    [paper](http://arxiv.org/abs/1412.6980) for more.
    
    Note that using momentum is not supported with Adam. In fact, an
    error is signalled if it's not :NONE.
    
    See @MGL-GD-BATCH-GD-OPTIMIZER for the description of the various
    options common to all batch based optimizers.

- [accessor] LEARNING-RATE ADAM-OPTIMIZER (= 2.0e-4)

    Same thing as LEARNING-RATE but with the default suggested by the Adam paper.

- [accessor] MEAN-DECAY ADAM-OPTIMIZER (:MEAN-DECAY = 0.9)

    A number between 0 and 1 that determines how fast
    the estimated mean of derivatives is updated. 0 basically gives
    you RMSPROP (if VARIANCE-DECAY is not too large) or AdaGrad (if
    VARIANCE-DECAY is close to 1 and the learning rate is annealed.
    This is $\beta\_1$ in the paper.

- [accessor] MEAN-DECAY-DECAY ADAM-OPTIMIZER (:MEAN-DECAY-DECAY = (- 1 1.0d-7))

    A value that should be close to 1. MEAN-DECAY is
    multiplied by this value after each update. This is $\lambda$ in
    the paper.

- [accessor] VARIANCE-DECAY ADAM-OPTIMIZER (:VARIANCE-DECAY = 0.999)

    A number between 0 and 1 that determines how fast
    the estimated variance of derivatives is updated. This is
    $\beta\_2$ in the paper.

- [accessor] VARIANCE-ADJUSTMENT ADAM-OPTIMIZER (:VARIANCE-ADJUSTMENT = 1.0d-7)

    Within the bowels of adam, the estimated mean is
    divided by the square root of the estimated variance (per weight)
    which can lead to numerical problems if the denominator is near
    zero. To avoid this, VARIANCE-ADJUSTMENT, which should be a small
    positive number, is added to the denominator. This is `epsilon` in
    the paper.

##### Normalized Batch Optimizer

- [class] NORMALIZED-BATCH-GD-OPTIMIZER BATCH-GD-OPTIMIZER

    Like BATCH-GD-OPTIMIZER but keeps count of how many
    times each weight was used in the batch and divides the accumulated
    gradient by this count instead of dividing by N-INSTANCES-IN-BATCH.
    This only makes a difference if there are missing values in the
    learner that's being trained. The main feature that distuinguishes
    this class from PER-WEIGHT-BATCH-GD-OPTIMIZER is that batches end at
    same time for all weights.

- [accessor] N-WEIGHT-USES-IN-BATCH NORMALIZED-BATCH-GD-OPTIMIZER

    Number of uses of the weight in its current batch.

#### Segmented GD Optimizer

- [class] SEGMENTED-GD-OPTIMIZER

    An optimizer that delegates training of segments to
    other optimizers. Useful to delegate training of different segments
    to different optimizers (capable of working with segmentables) or
    simply to not train all segments.

- [reader] SEGMENTER SEGMENTED-GD-OPTIMIZER (:SEGMENTER)

    When this optimizer is initialized it loops over
    the segment of the learner with MAP-SEGMENTS. SEGMENTER is a
    function that is called with each segment and returns an optimizer
    or NIL. Several segments may be mapped to the same optimizer.
    After the segment->optimizer mappings are collected, each
    optimizer is initialized by INITIALIZE-OPTIMIZER with the list of
    segments mapped to it.

- [reader] SEGMENTS SEGMENTED-GD-OPTIMIZER

SEGMENTED-GD-OPTIMIZER inherits from `ITERATIVE-OPTIMIZER`, so see
MGL-OPT::@MGL-OPT-ITERATIVE-OPTIMIZER too.

#### Per-weight Optimization

- [class] PER-WEIGHT-BATCH-GD-OPTIMIZER

    This is much like @MGL-GD-BATCH-GD-OPTIMIZER but it
    is more clever about when to update weights. Basically every weight
    has its own batch independent from the batches of others. This has
    desirable properties. One can for example put two neural networks
    together without adding any connections between them and the
    learning will produce results equivalent to the separated case.
    Also, adding inputs with only missing values does not change
    anything.
    
    Due to its very non-batch nature, there is no CUDA implementation of
    this optimizer.

- [accessor] N-WEIGHT-USES-IN-BATCH PER-WEIGHT-BATCH-GD-OPTIMIZER

    Number of uses of the weight in its current batch.

#### Utilities

- [function] CLIP-L2-NORM MATS L2-UPPER-BOUND &KEY CALLBACK

    Scale MATS so that their $L\_2$ norm does not exceed L2-UPPER-BOUND.
    
    Compute the norm of of MATS as if they were a single vector. If the
    norm is greater than L2-UPPER-BOUND, then scale each matrix
    destructively by the norm divided by L2-UPPER-BOUND and if non-NIL
    call the function CALLBACK with the scaling factor.

- [function] ARRANGE-FOR-CLIPPING-GRADIENTS BATCH-GD-OPTIMIZER L2-UPPER-BOUND &KEY CALLBACK

    Make it so that the norm of the batch normalized gradients
    accumulated by BATCH-GD-OPTIMIZER is clipped to L2-UPPER-BOUND
    before every update. See CLIP-L2-NORM.

### Conjugate Gradient

###### \[in package MGL-CG\]
Conjugate gradient is a first-order optimization algorithm. It's
more advanced than gradient descent as it does line searches which
unfortunately also makes it unsuitable for non-deterministic
functions. Let's see how to minimize a numerical lisp function with
respect to some of its parameters.

```
;;; Create an object representing the sine function.
(defparameter *diff-fn-1*
  (make-instance 'mgl-diffun:diffun
                 :fn #'sin
                 ;; We are going to optimize its only parameter.
                 :weight-indices '(0)))

;;; Minimize SIN. Note that there is no dataset involved because all
;;; parameters are being optimized.
(minimize (make-instance 'cg-optimizer
                         :batch-size 1
                         :termination 1)
          *diff-fn-1*
          :weights (make-mat 1))
;;; => A MAT with a single value of about -pi/2.

;;; Create a differentiable function for f(x,y)=(x-y)^2. X is a
;;; parameter whose values come from the DATASET argument passed to
;;; MINIMIZE. Y is a parameter to be optimized (a 'weight').
(defparameter *diff-fn-2*
  (make-instance 'mgl-diffun:diffun
                 :fn (lambda (x y)
                       (expt (- x y) 2))
                 :parameter-indices '(0)
                 :weight-indices '(1)))

;;; Find the Y that minimizes the distance from the instances
;;; generated by the sampler.
(minimize (make-instance 'cg-optimizer :batch-size 10)
          *diff-fn-2*
          :weights (make-mat 1)
          :dataset (make-instance 'function-sampler
                                  :generator (lambda ()
                                               (list (+ 10
                                                        (gaussian-random-1))))
                                  :max-n-samples 1000))
;;; => A MAT with a single value of about 10, the expected value of
;;; the instances in the dataset.

;;; The dataset can be a SEQUENCE in which case we'd better set
;;; TERMINATION else optimization would never finish. Note how a
;;; single epoch suffices.
(minimize (make-instance 'cg-optimizer :termination 6)
          *diff-fn-2*
          :weights (make-mat 1)
          :dataset '((0) (1) (2) (3) (4) (5)))
;;; => A MAT with a single value of about 2.5.
```


- [function] CG FN W &KEY (MAX-N-LINE-SEARCHES \*DEFAULT-MAX-N-LINE-SEARCHES\*) (MAX-N-EVALUATIONS-PER-LINE-SEARCH \*DEFAULT-MAX-N-EVALUATIONS-PER-LINE-SEARCH\*) (MAX-N-EVALUATIONS \*DEFAULT-MAX-N-EVALUATIONS\*) (SIG \*DEFAULT-SIG\*) (RHO \*DEFAULT-RHO\*) (INT \*DEFAULT-INT\*) (EXT \*DEFAULT-EXT\*) (RATIO \*DEFAULT-RATIO\*) SPARE-VECTORS

    CG-OPTIMIZER passes each batch of data to this function with its
    CG-ARGS passed on.
    
    Minimize a differentiable multivariate function with conjugate
    gradient. The Polak-Ribiere flavour of conjugate gradients is used
    to compute search directions, and a line search using quadratic and
    cubic polynomial approximations and the Wolfe-Powell stopping
    criteria is used together with the slope ratio method for guessing
    initial step sizes. Additionally a bunch of checks are made to make
    sure that exploration is taking place and that extrapolation will
    not be unboundedly large.
    
    FN is a function of two parameters: WEIGHTS and DERIVATIVES. WEIGHTS
    is a MAT of the same size as W that is where the search start from.
    DERIVATIVES is also a MAT of that size and it is where FN shall
    place the partial derivatives. FN returns the value of the function
    that is being minimized.
    
    CG performs a number of line searches and invokes FN at each step. A
    line search invokes FN at most MAX-N-EVALUATIONS-PER-LINE-SEARCH
    number of times and can succeed in improving the minimum by the
    sufficient margin or it can fail. Note, the even a failed line
    search may improve further and hence change the weights it's just
    that the improvement was deemed too small. CG stops when either:
    
    - two line searches fail in a row
    
    - MAX-N-LINE-SEARCHES is reached
    
    - MAX-N-EVALUATIONS is reached
    
    CG returns a MAT that contains the best weights, the minimum, the
    number of line searches performed, the number of succesful line
    searches and the number of evaluations.
    
    When using MAX-N-EVALUATIONS remember that there is an extra
    evaluation of FN before the first line search.
    
    SPARE-VECTORS is a list of preallocated MATs of the same size as W.
    Passing 6 of them covers the current need of the algorithm and it
    will not cons up vectors of size W at all.
    
    NOTE: If the function terminates within a few iterations, it could
    be an indication that the function values and derivatives are not
    consistent (ie, there may be a bug in the implementation of FN
    function).
    
    SIG and RHO are the constants controlling the Wolfe-Powell
    conditions. SIG is the maximum allowed absolute ratio between
    previous and new slopes (derivatives in the search direction), thus
    setting SIG to low (positive) values forces higher precision in the
    line-searches. RHO is the minimum allowed fraction of the
    expected (from the slope at the initial point in the linesearch).
    Constants must satisfy 0 < RHO < SIG < 1. Tuning of SIG (depending
    on the nature of the function to be optimized) may speed up the
    minimization; it is probably not worth playing much with RHO.

- [variable] *DEFAULT-INT* 0.1

    Don't reevaluate within INT of the limit of the current bracket.

- [variable] *DEFAULT-EXT* 3

    Extrapolate maximum EXT times the current step-size.

- [variable] *DEFAULT-SIG* 0.1

    SIG and RHO are the constants controlling the Wolfe-Powell
    conditions. SIG is the maximum allowed absolute ratio between
    previous and new slopes (derivatives in the search direction), thus
    setting SIG to low (positive) values forces higher precision in the
    line-searches.

- [variable] *DEFAULT-RHO* 0.05

    RHO is the minimum allowed fraction of the expected (from the slope
    at the initial point in the linesearch). Constants must satisfy 0 <
    RHO < SIG < 1.

- [variable] *DEFAULT-RATIO* 10

    Maximum allowed slope ratio.

- [variable] *DEFAULT-MAX-N-LINE-SEARCHES* NIL

- [variable] *DEFAULT-MAX-N-EVALUATIONS-PER-LINE-SEARCH* 20

- [variable] *DEFAULT-MAX-N-EVALUATIONS* NIL

- [class] CG-OPTIMIZER ITERATIVE-OPTIMIZER

    Updates all weights simultaneously after chewing
    through BATCH-SIZE inputs.

- [accessor] BATCH-SIZE CG-OPTIMIZER (:BATCH-SIZE)

    After having gone through BATCH-SIZE number of
    instances, weights are updated. Normally, CG operates on all
    available data, but it may be useful to introduce some noise into
    the optimization to reduce overfitting by using smaller batch
    sizes. If BATCH-SIZE is not set, it is initialized to the size of
    the dataset at the start of optimization.

- [accessor] CG-ARGS CG-OPTIMIZER (:CG-ARGS = 'NIL)

- [accessor] ON-CG-BATCH-DONE CG-OPTIMIZER (:ON-CG-BATCH-DONE = NIL)

    An event hook called when processing a conjugate
    gradient batch is done. The handlers on the hook are called with 8
    arguments:
    
        (optimizer gradient-source instances
         best-w best-f n-line-searches
         n-succesful-line-searches n-evaluations)
    
    The latter 5 of which are the return values of the CG function.

- [generic-function] LOG-CG-BATCH-DONE OPTIMIZER GRADIENT-SOURCE INSTANCES BEST-W BEST-F N-LINE-SEARCHES N-SUCCESFUL-LINE-SEARCHES N-EVALUATIONS

    This is a function can be added to
    ON-CG-BATCH-DONE. The default implementation simply logs the event
    arguments.

- [reader] SEGMENT-FILTER CG-OPTIMIZER (:SEGMENT-FILTER = (CONSTANTLY T))

    A predicate function on segments that filters out
    uninteresting segments. Called from INITIALIZE-OPTIMIZER\*.

### Extension API

#### Implementing Optimizers

The following generic functions must be specialized for new
optimizer types.

- [generic-function] MINIMIZE* OPTIMIZER GRADIENT-SOURCE WEIGHTS DATASET

    Called by MINIMIZE after INITIALIZE-OPTIMIZER\* and
    INITIALIZE-GRADIENT-SOURCE\*, this generic function is the main
    extension point for writing optimizers.

- [generic-function] INITIALIZE-OPTIMIZER* OPTIMIZER GRADIENT-SOURCE WEIGHTS DATASET

    Called automatically before training starts, this
    function sets up OPTIMIZER to be suitable for optimizing
    GRADIENT-SOURCE. It typically creates appropriately sized
    accumulators for the gradients.

- [generic-function] SEGMENTS OPTIMIZER

    Several weight matrices known as *segments* can be
    optimized by a single optimizer. This function returns them as a
    list.

The rest are just useful for utilities for implementing
optimizers.

- [function] TERMINATE-OPTIMIZATION-P N-INSTANCES TERMINATION

    Utility function for subclasses of ITERATIVE-OPTIMIZER. It returns
    whether optimization is to be terminated based on N-INSTANCES and
    TERMINATION that are values of the respective accessors of
    ITERATIVE-OPTIMIZER.

- [function] SET-N-INSTANCES OPTIMIZER GRADIENT-SOURCE N-INSTANCES

    Set N-INSTANCES of OPTIMIZER and
    fire ON-N-INSTANCES-CHANGED. ITERATIVE-OPTIMIZER subclasses must
    call this to increment N-INSTANCES.

- [class] SEGMENT-SET

    This is a utility class for optimizers that have a
    list of SEGMENTS and (the weights being optimized) is able to copy
    back and forth between those segments and a single MAT (the
    accumulator).

- [reader] SEGMENTS SEGMENT-SET (:SEGMENTS)

    A list of weight matrices.

- [reader] SIZE SEGMENT-SET

    The sum of the sizes of the weight matrices of
    SEGMENTS.

- [macro] DO-SEGMENT-SET (SEGMENT &OPTIONAL START) SEGMENT-SET &BODY BODY

    Iterate over SEGMENTS in SEGMENT-SET. If START is specified, the it
    is bound to the start index of SEGMENT within SEGMENT-SET. The start
    index is the sum of the sizes of previous segments.

- [function] SEGMENT-SET<-MAT SEGMENT-SET MAT

    Copy the values of MAT to the weight matrices of SEGMENT-SET as if
    they were concatenated into a single MAT.

- [function] SEGMENT-SET->MAT SEGMENT-SET MAT

    Copy the values of SEGMENT-SET to MAT as if they were concatenated
    into a single MAT.

#### Implementing Gradient Sources

Weights can be stored in a multitude of ways. Optimizers need to
update weights, so it is assumed that weights are stored in any
number of MAT objects called segments.

The generic functions in this section must all be specialized for
new gradient sources except where noted.

- [generic-function] MAP-SEGMENTS FN GRADIENT-SOURCE

    Apply FN to each segment of GRADIENT-SOURCE.

- [generic-function] MAP-SEGMENT-RUNS FN SEGMENT

    Call FN with start and end of intervals of
    consecutive indices that are not missing in SEGMENT. Called by
    optimizers that support partial updates. The default implementation
    assumes that all weights are present. This only needs to be
    specialized if one plans to use an optimizer that knows how to deal
    unused/missing weights such as MGL-GD:NORMALIZED-BATCH-GD-OPTIMIZER
    and OPTIMIZER MGL-GD:PER-WEIGHT-BATCH-GD-OPTIMIZER.

- [generic-function] SEGMENT-WEIGHTS SEGMENT

    Return the weight matrix of SEGMENT. A segment
    doesn't need to be a MAT object itself. For example, it may be a
    MGL-BM:CHUNK of a MGL-BM:BM or a MGL-BP:LUMP of a
    MGL-BP:BPN whose NODES slot holds the weights.

- [method] SEGMENT-WEIGHTS (MAT MAT)

    When the segment is really a MAT, then just return it.

- [generic-function] SEGMENT-DERIVATIVES SEGMENT

    Return the derivatives matrix of SEGMENT. A segment
    doesn't need to be a MAT object itself. For example, it may be a
    MGL-BM:CHUNK of a MGL-BM:BM or a MGL-BP:LUMP of a
    MGL-BP:BPN whose DERIVATIVES slot holds the gradient.

- [function] LIST-SEGMENTS GRADIENT-SOURCE

    A utility function that returns the list of segments from
    MAP-SEGMENTS on GRADIENT-SOURCE.

- [generic-function] INITIALIZE-GRADIENT-SOURCE* OPTIMIZER GRADIENT-SOURCE WEIGHTS DATASET

    Called automatically before MINIMIZE\* is called,
    this function may be specialized if GRADIENT-SOURCE needs some kind
    of setup.

- [method] INITIALIZE-GRADIENT-SOURCE* OPTIMIZER GRADIENT-SOURCE WEIGHTS DATASET

    The default method does nothing.

- [generic-function] ACCUMULATE-GRADIENTS* GRADIENT-SOURCE SINK BATCH MULTIPLIER VALUEP

    Add MULTIPLIER times the sum of first-order
    gradients to accumulators of SINK (normally accessed with
    DO-GRADIENT-SINK) and if VALUEP, return the sum of values of the
    function being optimized for a BATCH of instances. GRADIENT-SOURCE
    is the object representing the function being optimized, SINK is
    gradient sink.
    
    Note the number of instances in BATCH may be larger than what
    GRADIENT-SOURCE process in one go (in the sense of say,
    MAX-N-STRIPES), so DO-BATCHES-FOR-MODEL or something like (GROUP
    BATCH MAX-N-STRIPES) can be handy.

#### Implementing Gradient Sinks

Optimizers call ACCUMULATE-GRADIENTS\* on gradient sources. One
parameter of ACCUMULATE-GRADIENTS\* is the SINK. A gradient sink
knows what accumulator matrix (if any) belongs to a segment. Sinks
are defined entirely by MAP-GRADIENT-SINK.

- [generic-function] MAP-GRADIENT-SINK FN SINK

    Call FN of lambda list (SEGMENT ACCUMULATOR) on
    each segment and their corresponding accumulator MAT in SINK.

- [macro] DO-GRADIENT-SINK ((SEGMENT ACCUMULATOR) SINK) &BODY BODY

    A convenience macro on top of MAP-GRADIENT-SINK.

## Differentiable Functions

###### \[in package MGL-DIFFUN\]
- [class] DIFFUN

    DIFFUN dresses a lisp function (in its FN slot) as
    a gradient source (see MGL-OPT::@MGL-OPT-GRADIENT-SOURCE), which
    allows it to be used in MINIMIZE. See the examples in
    MGL-GD::@MGL-GD and MGL-CG::@MGL-CG.

- [reader] FN DIFFUN (:FN)

    A real valued lisp function. It may have any
    number of parameters.

- [reader] PARAMETER-INDICES DIFFUN (:PARAMETER-INDICES = NIL)

    The list of indices of parameters that we don't
    optimize. Values for these will come from the DATASET argument of
    MINIMIZE.

- [reader] WEIGHT-INDICES DIFFUN (:WEIGHT-INDICES = NIL)

    The list of indices of parameters to be optimized,
    the values of which will come from the WEIGHTS
    argument of MINIMIZE.

## Backpropagation Neural Networks

###### \[in package MGL-BP\]
### Backprop Overview

Backpropagation Neural Networks are just functions with lots of
parameters called *weights* and a layered structure when presented
as a [computational
graph](http://en.wikipedia.org/wiki/Automatic_differentiation). The
network is trained to MINIMIZE some kind of *loss function* whose
value the network computes.

In this implementation, a BPN is assembled from several
`LUMP`s (roughly corresponding to layers). Both feed-forward and
recurrent neural nets are supported (FNN and RNN, respectively).
`BPN`s can contain not only `LUMP`s but other `BPN`s, too. As we
see, networks are composite objects and the abstract base class for
composite and simple parts is called CLUMP.

- [class] CLUMP

    A CLUMP is a LUMP or a BPN. It represents
    a differentiable function. Arguments of clumps are given during
    instantiation. Some arguments are clumps themselves so they get
    permenantly wired together like this:
    
    ```commonlisp
    (->v*m (->input :size 10 :name 'input)
           (->weight :dimensions '(10 20) :name 'weight)
           :name 'activation)
    ```
    
    The above creates three clumps: the vector-matrix multiplication
    clumps called `ACTIVATION` which has a reference to its operands:
    INPUT and WEIGHT. Note that the example just defines a function, no
    actual computation has taken place, yet.
    
    This wiring of `CLUMP`s is how one builds feed-forward nets (FNN) or
    recurrent neural networks (RNN) that are `CLUMP`s themselves so one
    can build nets in a hiearchical style if desired. Non-composite
    `CLUMP`s are called LUMP (note the loss of `C` that stands for
    composite). The various LUMP subtypes correspond to different layer
    types (->SIGMOID, ->DROPOUT, ->RELU, ->TANH, etc).

At this point, you may want to jump ahead to get a feel for how
things work by reading the @MGL-FNN-TUTORIAL.

### Clump API

These are mostly for extension purposes. About the only thing
needed from here for normal operation is NODES when clamping inputs
or extracting predictions.

- [generic-function] STRIPEDP CLUMP

    For efficiency, forward and backprop phases do
    their stuff in batch mode: passing a number of instances through the
    network in batches. Thus clumps must be able to store values of and
    gradients for each of these instances. However, some clumps produce
    the same result for each instance in a batch. These clumps are the
    weights, the parameters of the network. STRIPEDP returns true iff
    CLUMP does not represent weights (i.e. it's not a ->WEIGHT).
    
    For striped clumps, their NODES and DERIVATIVES are MAT objects with
    a leading dimension (number of rows in the 2d case) equal to the
    number of instances in the batch. Non-striped clumps have no
    restriction on their shape apart from what their usage dictates.

- [generic-function] NODES OBJECT

    Returns a MAT object representing the state or
    result of OBJECT. The first dimension of the returned matrix is
    equal to the number of stripes.

`CLUMP`s' `NODES` holds the result computed by the most recent
FORWARD. For ->INPUT lumps, this is where input values shall be
placed (see SET-INPUT). Currently, the matrix is always two
dimensional but this restriction may go away in the future.

- [generic-function] DERIVATIVES CLUMP

    Return the MAT object representing the partial
    derivatives of the function CLUMP computes. The returned partial
    derivatives were accumulated by previous BACKWARD calls.
    
    This matrix is shaped like the matrix returned by NODES.

- [generic-function] FORWARD CLUMP

    Compute the values of the function represented by
    CLUMP for all stripes and place the results into NODES of CLUMP.

- [generic-function] BACKWARD CLUMP

    Compute the partial derivatives of the function
    represented by CLUMP and add them to DERIVATIVES of the
    corresponding argument clumps. The DERIVATIVES of CLUMP contains the
    sum of partial derivatives of all clumps by the corresponding
    output. This function is intended to be called after a FORWARD pass.
    
    Take the ->SIGMOID clump for example when the network is being
    applied to a batch of two instances `x1` and `x2`. `x1` and `x2` are
    set in the ->INPUT lump X. The sigmoid computes `1/(1+exp(-x))`
    where `X` is its only argument clump.
    
        f(x) = 1/(1+exp(-x))
    
    When BACKWARD is called on the sigmoid lump, its DERIVATIVES is a
    2x1 MAT object that contains the partial derivatives of the loss
    function:
    
        dL(x1)/df
        dL(x2)/df
    
    Now the BACKWARD method of the sigmoid needs to add `dL(x1)/dx1` and
    `dL(x2)/dx2` to DERIVATIVES of `X`. Now, `dL(x1)/dx1 = dL(x1)/df *
    df(x1)/dx1` and the first term is what we have in DERIVATIVES of the
    sigmoid so it only needs to calculate the second term.

In addition to the above, clumps also have to support SIZE,
N-STRIPES, MAX-N-STRIPES (and the SETF methods of the latter two)
which can be accomplished just by inheriting from BPN, FNN, RNN, or
a LUMP.

### BPNs

- [class] BPN CLUMP

    Abstract base class for FNN and RNN.

- [reader] N-STRIPES BPN (:N-STRIPES = 1)

    The current number of instances the network has.
    This is automatically set to the number of instances passed to
    SET-INPUT, so it rarely has to be manipulated directly although it
    can be set. When set N-STRIPES of all CLUMPS get set to the same
    value.

- [reader] MAX-N-STRIPES BPN (:MAX-N-STRIPES = NIL)

    The maximum number of instances the network can
    operate on in parallel. Within BUILD-FNN or BUILD-RNN, it defaults
    to MAX-N-STRIPES of that parent network, else it defaults to 1.
    When set MAX-N-STRIPES of all CLUMPS get set to the same value.

- [reader] CLUMPS BPN (:CLUMPS = (MAKE-ARRAY 0 :ELEMENT-TYPE 'CLUMP :ADJUSTABLE T :FILL-POINTER T))

    A topological sorted adjustable array with a fill
    pointer that holds the clumps that make up the network. Clumps are
    added to it by ADD-CLUMP or, more often, automatically when within
    a BUILD-FNN or BUILD-RNN. Rarely needed, FIND-CLUMP takes care of
    most uses.

- [function] FIND-CLUMP NAME BPN &KEY (ERRORP T)

    Find the clump with NAME among CLUMPS of BPN. As always, names are
    compared with EQUAL. If not found, then return NIL or signal and
    error depending on ERRORP.

- [function] ADD-CLUMP CLUMP BPN

    Add CLUMP to BPN. MAX-N-STRIPES of CLUMP gets set to that of BPN.
    It is an error to add a clump with a name already used by one of the
    CLUMPS of BPN.

#### Training

`BPN`s are trained to minimize the loss function they compute.
Before a BPN is passed to MINIMIZE (as its `GRADIENT-SOURCE`
argument), it must be wrapped in a BP-LEARNER object. BP-LEARNER has
MONITORS slot which is used for example by
RESET-OPTIMIZATION-MONITORS.

Without the bells an whistles, the basic shape of training is this:

```commonlisp
(minimize optimizer (make-instance 'bp-learner :bpn bpn)
          :dataset dataset)
```


- [class] BP-LEARNER

- [reader] BPN BP-LEARNER (:BPN)

    The BPN for which this BP-LEARNER provides the
    gradients.

- [accessor] MONITORS BP-LEARNER (:MONITORS = NIL)

    A list of `MONITOR`s.

#### Monitoring

- [function] MONITOR-BPN-RESULTS DATASET BPN MONITORS

    For every batch (of size MAX-N-STRIPES of BPN) of instances in
    DATASET, set the batch as the next input with SET-INPUT, perform a
    FORWARD pass and apply MONITORS to the BPN (with APPLY-MONITORS).
    Finally, return the counters of MONITORS. This is built on top of
    MONITOR-MODEL-RESULTS.

- [function] MAKE-STEP-MONITOR-MONITORS RNN &KEY (COUNTER-VALUES-FN #'COUNTER-RAW-VALUES) (MAKE-COUNTER #'MAKE-STEP-MONITOR-MONITOR-COUNTER)

    Return a list of monitors, one for every monitor in STEP-MONITORS
    of RNN. These monitors extract the results from their warp
    counterpairs with COUNTER-VALUES-FN and add them to their own
    counter that's created by MAKE-COUNTER. Wow. Ew. The idea is that
    one does something like this do monitor warped prediction:
    
    ```commonlisp
    (let ((*warp-time* t))
      (setf (step-monitors rnn)
            (make-cost-monitors rnn :attributes '(:event "warped pred.")))
      (monitor-bpn-results dataset rnn
                           ;; Just collect and reset the warp
                           ;; monitors after each batch of
                           ;; instances.
                           (make-step-monitor-monitors rnn)))
    ```


- [generic-function] MAKE-STEP-MONITOR-MONITOR-COUNTER STEP-COUNTER

    In an RNN, STEP-COUNTER aggregates results of all
    the time steps during the processing of instances in the current
    batch. Return a new counter into which results from STEP-COUNTER can
    be accumulated when the processing of the batch is finished. The
    default implementation creates a copy of STEP-COUNTER.

#### Feed-Forward Nets

FNN and RNN have a lot in common (see their common superclass, BPN).
There is very limited functionality that's specific to FNNs so let's
get them out of they way before we study a full example.

- [class] FNN BPN

    A feed-forward neural net (as opposed to a
    recurrent one, see RNN).

- [macro] BUILD-FNN (&KEY FNN (CLASS ''FNN) INITARGS MAX-N-STRIPES NAME) &BODY CLUMPS

    Syntactic sugar to assemble FNNs from CLUMPs. Like LET\*, it is a
    sequence of bindings (of symbols to CLUMPs). The names of the clumps
    created default to the symbol of the binding. In case a clump is not
    bound to a symbol (because it was created in a nested expression),
    the local function CLUMP can be used to find the clump with the
    given name in the fnn being built. Example:
    
        (build-fnn ()
          (features (->input :size n-features))
          (biases (->weight :size n-features))
          (weights (->weight :size (* n-hiddens n-features)))
          (activations0 (->v*m :weights weights :x (clump 'features)))
          (activations (->+ :args (list biases activations0)))
          (output (->sigmoid :x activations)))


##### FNN Tutorial

Hopefully this example from `example/digit-fnn.lisp` illustrates
the concepts involved. If it's too dense despite the comments, then
read up on MGL-DATASET::@MGL-DATASET, MGL-OPT::@MGL-OPT and come back.

```commonlisp
(cl:defpackage :mgl-example-digit-fnn
  (:use #:common-lisp #:mgl))

(in-package :mgl-example-digit-fnn)

;;; There are 10 possible digits used as inputs ...
(defparameter *n-inputs* 10)
;;; and we want to learn the rule that maps the input digit D to (MOD
;;; (1+ D) 3).
(defparameter *n-outputs* 3)

;;; We define a feed-forward net to be able to specialize how inputs
;;; are translated by adding a SET-INPUT method later.
(defclass digit-fnn (fnn)
  ())

;;; Build a DIGIT-FNN with a single hidden layer of rectified linear
;;; units and a softmax output.
(defun make-digit-fnn (&key (n-hiddens 5))
  (build-fnn (:class 'digit-fnn)
    (input (->input :size *n-inputs*))
    (hidden-activation (->activation input :size n-hiddens))
    (hidden (->relu hidden-activation))
    (output-activation (->activation hidden :size *n-outputs*))
    (output (->softmax-xe-loss output-activation))))

;;; This method is called with batches of 'instances' (input digits in
;;; this case) by MINIMIZE and also by MONITOR-BPN-RESULTS before
;;; performing a forward pass (i.e. computing the value of the
;;; function represented by the network). Its job is to encode the
;;; inputs by populating rows of the NODES matrix of the INPUT clump.
;;;
;;; Each input is encoded as a row of zeros with a single 1 at index
;;; determined by the input digit. This is called one-hot encoding.
;;; The TARGET could be encoded the same way, but instead we use the
;;; sparse option supported by TARGET of ->SOFTMAX-XE-LOSS.
(defmethod set-input (digits (fnn digit-fnn))
  (let* ((input (nodes (find-clump 'input fnn)))
         (output-lump (find-clump 'output fnn)))
    (fill! 0 input)
    (loop for i upfrom 0
          for digit in digits
          do (setf (mref input i digit) 1))
    (setf (target output-lump)
          (mapcar (lambda (digit)
                    (mod (1+ digit) *n-outputs*))
                  digits))))

;;; Train the network by minimizing the loss (cross-entropy here) with
;;; stochastic gradient descent.
(defun train-digit-fnn ()
  (let ((optimizer
          ;; First create the optimizer for MINIMIZE.
          (make-instance 'segmented-gd-optimizer
                         :segmenter
                         ;; We train each weight lump with the same
                         ;; parameters and, in fact, the same
                         ;; optimizer. But it need not be so, in
                         ;; general.
                         (constantly
                          (make-instance 'sgd-optimizer
                                         :learning-rate 1
                                         :momentum 0.9
                                         :batch-size 100))))
        (fnn (make-digit-fnn)))
    ;; The number of instances the FNN can work with in parallel. It's
    ;; usually equal to the batch size or is a its divisor.
    (setf (max-n-stripes fnn) 50)
    ;; Initialize all weights randomly.
    (map-segments (lambda (weights)
                    (gaussian-random! (nodes weights) :stddev 0.01))
                  fnn)
    ;; Arrange for training and test error to be logged.
    (monitor-optimization-periodically
     optimizer '((:fn log-test-error :period 10000)
                 (:fn reset-optimization-monitors :period 1000)))
    ;; Finally, start the optimization.
    (minimize optimizer
              ;; Dress FNN in a BP-LEARNER and attach monitors for the
              ;; cost to it. These monitors are going to be logged and
              ;; reset after every 100 training instance by
              ;; RESET-OPTIMIZATION-MONITORS above.
              (make-instance 'bp-learner
                             :bpn fnn
                             :monitors (make-cost-monitors
                                        fnn :attributes `(:event "train")))
              ;; Training stops when the sampler runs out (after 10000
              ;; instances).
              :dataset (make-sampler 10000))))

;;; Return a sampler object that produces MAX-N-SAMPLES number of
;;; random inputs (numbers between 0 and 9).
(defun make-sampler (max-n-samples)
  (make-instance 'function-sampler :max-n-samples max-n-samples
                 :generator (lambda () (random *n-inputs*))))

;;; Log the test error. Also, describe the optimizer and the bpn at
;;; the beginning of training. Called periodically during training
;;; (see above).
(defun log-test-error (optimizer learner)
  (when (zerop (n-instances optimizer))
    (describe optimizer)
    (describe (bpn learner)))
  (log-padded
   (monitor-bpn-results (make-sampler 1000) (bpn learner)
                        (make-cost-monitors
                         (bpn learner) :attributes `(:event "pred.")))))

#|

;;; Transcript follows:
(repeatably ()
  (let ((*log-time* nil))
    (train-digit-fnn)))
.. training at n-instances: 0
.. train cost: 0.000e+0 (0)
.. #<SEGMENTED-GD-OPTIMIZER {100E112E93}>
..  SEGMENTED-GD-OPTIMIZER description:
..    N-INSTANCES = 0
..    OPTIMIZERS = (#<SGD-OPTIMIZER
..                    #<SEGMENT-SET
..                      (#<->WEIGHT # :SIZE 15 1/1 :NORM 0.04473>
..                       #<->WEIGHT # :SIZE 3 1/1 :NORM 0.01850>
..                       #<->WEIGHT # :SIZE 50 1/1 :NORM 0.07159>
..                       #<->WEIGHT # :SIZE 5 1/1 :NORM 0.03056>)
..                      {100E335B73}>
..                    {100E06DF83}>)
..    SEGMENTS = (#<->WEIGHT (HIDDEN OUTPUT-ACTIVATION) :SIZE
..                  15 1/1 :NORM 0.04473>
..                #<->WEIGHT (:BIAS OUTPUT-ACTIVATION) :SIZE
..                  3 1/1 :NORM 0.01850>
..                #<->WEIGHT (INPUT HIDDEN-ACTIVATION) :SIZE
..                  50 1/1 :NORM 0.07159>
..                #<->WEIGHT (:BIAS HIDDEN-ACTIVATION) :SIZE
..                  5 1/1 :NORM 0.03056>)
..  
.. #<SGD-OPTIMIZER {100E06DF83}>
..  GD-OPTIMIZER description:
..    N-INSTANCES = 0
..    SEGMENT-SET = #<SEGMENT-SET
..                    (#<->WEIGHT (HIDDEN OUTPUT-ACTIVATION) :SIZE
..                       15 1/1 :NORM 0.04473>
..                     #<->WEIGHT (:BIAS OUTPUT-ACTIVATION) :SIZE
..                       3 1/1 :NORM 0.01850>
..                     #<->WEIGHT (INPUT HIDDEN-ACTIVATION) :SIZE
..                       50 1/1 :NORM 0.07159>
..                     #<->WEIGHT (:BIAS HIDDEN-ACTIVATION) :SIZE
..                       5 1/1 :NORM 0.03056>)
..                    {100E335B73}>
..    LEARNING-RATE = 1.00000e+0
..    MOMENTUM = 9.00000e-1
..    MOMENTUM-TYPE = :NORMAL
..    WEIGHT-DECAY = 0.00000e+0
..    WEIGHT-PENALTY = 0.00000e+0
..    N-AFTER-UPATE-HOOK = 0
..    BATCH-SIZE = 100
..  
..  BATCH-GD-OPTIMIZER description:
..    N-BEFORE-UPATE-HOOK = 0
..  #<DIGIT-FNN {100E11A423}>
..   BPN description:
..     CLUMPS = #(#<->INPUT INPUT :SIZE 10 1/50 :NORM 0.00000>
..                #<->ACTIVATION
..                  (HIDDEN-ACTIVATION :ACTIVATION) :STRIPES 1/50
..                  :CLUMPS 4>
..                #<->RELU HIDDEN :SIZE 5 1/50 :NORM 0.00000>
..                #<->ACTIVATION
..                  (OUTPUT-ACTIVATION :ACTIVATION) :STRIPES 1/50
..                  :CLUMPS 4>
..                #<->SOFTMAX-XE-LOSS OUTPUT :SIZE 3 1/50 :NORM 0.00000>)
..     N-STRIPES = 1
..     MAX-N-STRIPES = 50
..   pred. cost: 1.100d+0 (1000.00)
.. training at n-instances: 1000
.. train cost: 1.093d+0 (1000.00)
.. training at n-instances: 2000
.. train cost: 5.886d-1 (1000.00)
.. training at n-instances: 3000
.. train cost: 3.574d-3 (1000.00)
.. training at n-instances: 4000
.. train cost: 1.601d-7 (1000.00)
.. training at n-instances: 5000
.. train cost: 1.973d-9 (1000.00)
.. training at n-instances: 6000
.. train cost: 4.882d-10 (1000.00)
.. training at n-instances: 7000
.. train cost: 2.771d-10 (1000.00)
.. training at n-instances: 8000
.. train cost: 2.283d-10 (1000.00)
.. training at n-instances: 9000
.. train cost: 2.123d-10 (1000.00)
.. training at n-instances: 10000
.. train cost: 2.263d-10 (1000.00)
.. pred. cost: 2.210d-10 (1000.00)
..
==> (#<->WEIGHT (:BIAS HIDDEN-ACTIVATION) :SIZE 5 1/1 :NORM 2.94294>
-->  #<->WEIGHT (INPUT HIDDEN-ACTIVATION) :SIZE 50 1/1 :NORM 11.48995>
-->  #<->WEIGHT (:BIAS OUTPUT-ACTIVATION) :SIZE 3 1/1 :NORM 3.39103>
-->  #<->WEIGHT (HIDDEN OUTPUT-ACTIVATION) :SIZE 15 1/1 :NORM 11.39339>)

|#
```

#### Recurrent Neural Nets

##### RNN Tutorial

Hopefully this example from `example/sum-sign-fnn.lisp` illustrates
the concepts involved. Make sure you are comfortable with
@MGL-FNN-TUTORIAL before reading this.

```commonlisp
(cl:defpackage :mgl-example-sum-sign-rnn
  (:use #:common-lisp #:mgl))

(in-package :mgl-example-sum-sign-rnn)

;;; There is a single input at each time step...
(defparameter *n-inputs* 1)
;;; and we want to learn the rule that outputs the sign of the sum of
;;; inputs so far in the sequence.
(defparameter *n-outputs* 3)

;;; Generate a training example that's a sequence of random length
;;; between 1 and LENGTH. Elements of the sequence are lists of two
;;; elements:
;;;
;;; 1. The input for the network (a single random number).
;;;
;;; 2. The sign of the sum of inputs so far encoded as 0, 1, 2 (for
;;;    negative, zero and positive values). To add a twist, the sum is
;;;    reset whenever a negative input is seen.
(defun make-sum-sign-instance (&key (length 10))
  (let ((length (max 1 (random length)))
        (sum 0))
    (loop for i below length
          collect (let ((x (1- (* 2 (random 2)))))
                    (incf sum x)
                    (when (< x 0)
                      (setq sum x))
                    (list x (cond ((minusp sum) 0)
                                  ((zerop sum) 1)
                                  (t 2)))))))

;;; Build an RNN with a single lstm hidden layer and softmax output.
;;; For each time step, a SUM-SIGN-FNN will be instantiated.
(defun make-sum-sign-rnn (&key (n-hiddens 1))
  (build-rnn ()
    (build-fnn (:class 'sum-sign-fnn)
      (input (->input :size 1))
      (h (->lstm input :name 'h :size n-hiddens))
      (prediction (->softmax-xe-loss (->activation h :name 'prediction
                                                   :size *n-outputs*))))))

;;; We define this class to be able to specialize how inputs are
;;; translated by adding a SET-INPUT method later.
(defclass sum-sign-fnn (fnn)
  ())

;;; We have a batch of instances from MAKE-SUM-SIGN-INSTANCE for the
;;; RNN. This function is invoked with elements of these instances
;;; belonging to the same time step (i.e. at the same index) and sets
;;; the input and target up.
(defmethod set-input (instances (fnn sum-sign-fnn))
  (let ((input-nodes (nodes (find-clump 'input fnn))))
    (setf (target (find-clump 'prediction fnn))
          (loop for stripe upfrom 0
                for instance in instances
                collect
                ;; Sequences in the batch are not of equal length. The
                ;; RNN sends a NIL our way if a sequence has run out.
                (when instance
                  (destructuring-bind (input target) instance
                    (setf (mref input-nodes stripe 0) input)
                    target))))))

;;; Train the network by minimizing the loss (cross-entropy here) with
;;; the Adam optimizer.
(defun train-sum-sign-rnn ()
  (let ((rnn (make-sum-sign-rnn)))
    (setf (max-n-stripes rnn) 50)
    ;; Initialize the weights in the usual sqrt(1 / fan-in) style.
    (map-segments (lambda (weights)
                    (let* ((fan-in (mat-dimension (nodes weights) 0))
                           (limit (sqrt (/ 6 fan-in))))
                      (uniform-random! (nodes weights)
                                       :limit (* 2 limit))
                      (.+! (- limit) (nodes weights))))
                  rnn)
    (minimize (monitor-optimization-periodically
               (make-instance 'adam-optimizer
                              :learning-rate 0.2
                              :mean-decay 0.9
                              :mean-decay-decay 0.9
                              :variance-decay 0.9
                              :batch-size 100)
               '((:fn log-test-error :period 30000)
                 (:fn reset-optimization-monitors :period 3000)))
              (make-instance 'bp-learner
                             :bpn rnn
                             :monitors (make-cost-monitors rnn))
              :dataset (make-sampler 30000))))

;;; Return a sampler object that produces MAX-N-SAMPLES number of
;;; random inputs.
(defun make-sampler (max-n-samples &key (length 10))
  (make-instance 'function-sampler :max-n-samples max-n-samples
                 :generator (lambda ()
                              (make-sum-sign-instance :length length))))

;;; Log the test error. Also, describe the optimizer and the bpn at
;;; the beginning of training. Called periodically during training
;;; (see above).
(defun log-test-error (optimizer learner)
  (when (zerop (n-instances optimizer))
    (describe optimizer)
    (describe (bpn learner)))
  (let ((rnn (bpn learner)))
    (log-padded
     (append
      (monitor-bpn-results (make-sampler 1000) rnn
                           (make-cost-monitors
                            rnn :attributes '(:event "pred.")))
      ;; Same result in a different way: monitor predictions for
      ;; sequences up to length 20, but don't unfold the RNN
      ;; unnecessarily to save memory.
      (let ((*warp-time* t))
        (monitor-bpn-results (make-sampler 1000 :length 20) rnn
                             ;; Just collect and reset the warp
                             ;; monitors after each batch of
                             ;; instances.
                             (make-cost-monitors
                              rnn :attributes '(:event "warped pred."))))))
    ;; Verify that no further unfoldings took place.
    (assert (<= (length (clumps rnn)) 10)))
  (log-mat-room))

#|

;;; Transcript follows:
(let (;; Backprop nets do not need double float. Using single floats
      ;; is faster and needs less memory.
      (*default-mat-ctype* :float)
      ;; Enable moving data in and out of GPU memory so that the RNN
      ;; can work with sequences so long that the unfolded network
      ;; wouldn't otherwise fit in the GPU.
      (*cuda-window-start-time* 1)
      (*log-time* nil))
  ;; Seed the random number generators.
  (repeatably ()
    ;; Enable CUDA if available.
    (with-cuda* ()
      (train-sum-sign-rnn))))
.. training at n-instances: 0
.. cost: 0.000e+0 (0)
.. #<ADAM-OPTIMIZER {1006CD5663}>
..  GD-OPTIMIZER description:
..    N-INSTANCES = 0
..    SEGMENT-SET = #<SEGMENT-SET
..                    (#<->WEIGHT (H #) :SIZE 1 1/1 :NORM 1.73685>
..                     #<->WEIGHT (H #) :SIZE 1 1/1 :NORM 0.31893>
..                     #<->WEIGHT (#1=# #2=# :PEEPHOLE) :SIZE
..                       1 1/1 :NORM 1.81610>
..                     #<->WEIGHT (H #2#) :SIZE 1 1/1 :NORM 0.21965>
..                     #<->WEIGHT (#1# #3=# :PEEPHOLE) :SIZE
..                       1 1/1 :NORM 1.74939>
..                     #<->WEIGHT (H #3#) :SIZE 1 1/1 :NORM 0.40377>
..                     #<->WEIGHT (H PREDICTION) :SIZE
..                       3 1/1 :NORM 2.15898>
..                     #<->WEIGHT (:BIAS PREDICTION) :SIZE
..                       3 1/1 :NORM 2.94470>
..                     #<->WEIGHT (#1# #4=# :PEEPHOLE) :SIZE
..                       1 1/1 :NORM 0.97601>
..                     #<->WEIGHT (INPUT #4#) :SIZE 1 1/1 :NORM 0.65261>
..                     #<->WEIGHT (:BIAS #4#) :SIZE 1 1/1 :NORM 0.37653>
..                     #<->WEIGHT (INPUT #1#) :SIZE 1 1/1 :NORM 0.92334>
..                     #<->WEIGHT (:BIAS #1#) :SIZE 1 1/1 :NORM 0.01609>
..                     #<->WEIGHT (INPUT #5=#) :SIZE 1 1/1 :NORM 1.09995>
..                     #<->WEIGHT (:BIAS #5#) :SIZE 1 1/1 :NORM 1.41244>
..                     #<->WEIGHT (INPUT #6=#) :SIZE 1 1/1 :NORM 0.40475>
..                     #<->WEIGHT (:BIAS #6#) :SIZE 1 1/1 :NORM 1.75358>)
..                    {1006CD8753}>
..    LEARNING-RATE = 2.00000e-1
..    MOMENTUM = NONE
..    MOMENTUM-TYPE = :NONE
..    WEIGHT-DECAY = 0.00000e+0
..    WEIGHT-PENALTY = 0.00000e+0
..    N-AFTER-UPATE-HOOK = 0
..    BATCH-SIZE = 100
..  
..  BATCH-GD-OPTIMIZER description:
..    N-BEFORE-UPATE-HOOK = 0
..  
..  ADAM-OPTIMIZER description:
..    MEAN-DECAY-RATE = 1.00000e-1
..    MEAN-DECAY-RATE-DECAY = 9.00000e-1
..    VARIANCE-DECAY-RATE = 1.00000e-1
..    VARIANCE-ADJUSTMENT = 1.00000d-7
..  #<RNN {10047C77E3}>
..   BPN description:
..     CLUMPS = #(#<SUM-SIGN-FNN :STRIPES 1/50 :CLUMPS 4>
..                #<SUM-SIGN-FNN :STRIPES 1/50 :CLUMPS 4>)
..     N-STRIPES = 1
..     MAX-N-STRIPES = 50
..   
..   RNN description:
..     MAX-LAG = 1
..   pred.        cost: 1.223e+0 (4455.00)
.. warped pred. cost: 1.228e+0 (9476.00)
.. Foreign memory usage:
.. foreign arrays: 162 (used bytes: 39,600)
.. CUDA memory usage:
.. device arrays: 114 (used bytes: 220,892, pooled bytes: 19,200)
.. host arrays: 162 (used bytes: 39,600)
.. host->device copies: 6,164, device->host copies: 4,490
.. training at n-instances: 3000
.. cost: 3.323e-1 (13726.00)
.. training at n-instances: 6000
.. cost: 3.735e-2 (13890.00)
.. training at n-instances: 9000
.. cost: 1.012e-2 (13872.00)
.. training at n-instances: 12000
.. cost: 3.026e-3 (13953.00)
.. training at n-instances: 15000
.. cost: 9.267e-4 (13948.00)
.. training at n-instances: 18000
.. cost: 2.865e-4 (13849.00)
.. training at n-instances: 21000
.. cost: 8.893e-5 (13758.00)
.. training at n-instances: 24000
.. cost: 2.770e-5 (13908.00)
.. training at n-instances: 27000
.. cost: 8.514e-6 (13570.00)
.. training at n-instances: 30000
.. cost: 2.705e-6 (13721.00)
.. pred.        cost: 1.426e-6 (4593.00)
.. warped pred. cost: 1.406e-6 (9717.00)
.. Foreign memory usage:
.. foreign arrays: 216 (used bytes: 52,800)
.. CUDA memory usage:
.. device arrays: 148 (used bytes: 224,428, pooled bytes: 19,200)
.. host arrays: 216 (used bytes: 52,800)
.. host->device copies: 465,818, device->host copies: 371,990
..
==> (#<->WEIGHT (H (H :OUTPUT)) :SIZE 1 1/1 :NORM 0.10624>
-->  #<->WEIGHT (H (H :CELL)) :SIZE 1 1/1 :NORM 0.94460>
-->  #<->WEIGHT ((H :CELL) (H :FORGET) :PEEPHOLE) :SIZE 1 1/1 :NORM 0.61312>
-->  #<->WEIGHT (H (H :FORGET)) :SIZE 1 1/1 :NORM 0.38093>
-->  #<->WEIGHT ((H :CELL) (H :INPUT) :PEEPHOLE) :SIZE 1 1/1 :NORM 1.17956>
-->  #<->WEIGHT (H (H :INPUT)) :SIZE 1 1/1 :NORM 0.88011>
-->  #<->WEIGHT (H PREDICTION) :SIZE 3 1/1 :NORM 49.93808>
-->  #<->WEIGHT (:BIAS PREDICTION) :SIZE 3 1/1 :NORM 10.98112>
-->  #<->WEIGHT ((H :CELL) (H :OUTPUT) :PEEPHOLE) :SIZE 1 1/1 :NORM 0.67996>
-->  #<->WEIGHT (INPUT (H :OUTPUT)) :SIZE 1 1/1 :NORM 0.65251>
-->  #<->WEIGHT (:BIAS (H :OUTPUT)) :SIZE 1 1/1 :NORM 10.23003>
-->  #<->WEIGHT (INPUT (H :CELL)) :SIZE 1 1/1 :NORM 5.98116>
-->  #<->WEIGHT (:BIAS (H :CELL)) :SIZE 1 1/1 :NORM 0.10681>
-->  #<->WEIGHT (INPUT (H :FORGET)) :SIZE 1 1/1 :NORM 4.46301>
-->  #<->WEIGHT (:BIAS (H :FORGET)) :SIZE 1 1/1 :NORM 1.57195>
-->  #<->WEIGHT (INPUT (H :INPUT)) :SIZE 1 1/1 :NORM 0.36401>
-->  #<->WEIGHT (:BIAS (H :INPUT)) :SIZE 1 1/1 :NORM 8.63833>)

|#
```

- [class] RNN BPN

    A recurrent neural net (as opposed to a
    feed-forward one. It is typically built with BUILD-RNN that's no
    more than a shallow convenience macro.
    
    An RNN takes instances as inputs that are sequences of variable
    length. At each time step, the next unprocessed elements of these
    sequences are set as input until all input sequences in the batch
    run out. To be able to perform backpropagation, all intermediate
    `LUMP`s must be kept around, so the recursive connections are
    transformed out by
    [unfolding](http://en.wikipedia.org/wiki/Backpropagation_through_time)
    the network. Just how many lumps this means depends on the length of
    the sequences.
    
    When an RNN is created, `MAX-LAG + 1` BPNs are instantiated so
    that all weights are present and one can start training it.

- [reader] UNFOLDER RNN (:UNFOLDER)

    The UNFOLDER of an RNN is function of no arguments
    that builds and returns a BPN. The unfolder is allowed to create
    networks with arbitrary topology even different ones for different
    TIME-STEPs with the help of LAG, or nested RNNs. Weights of
    the same name are shared between the folds. That is, if a ->WEIGHT
    lump were to be created and a weight lump of the same name already
    exists, then the existing lump will be added to the BPN created by
    UNFOLDER.

- [reader] MAX-LAG RNN (:MAX-LAG = 1)

    The networks built by UNFOLDER may contain new
    weights up to time step MAX-LAG. Beyond that point, all weight
    lumps must be reappearances of weight lumps with the same name at
    previous time steps. Most recurrent networks reference only the
    state of lumps at the previous time step (with the function LAG),
    hence the default of 1. But it is possible to have connections to
    arbitrary time steps. The maximum connection lag must be specified
    when creating the RNN.

- [accessor] CUDA-WINDOW-START-TIME RNN (:CUDA-WINDOW-START-TIME = \*CUDA-WINDOW-START-TIME\*)

    Due to unfolding, the memory footprint of an RNN
    is almost linear in the number of time steps (i.e. the max
    sequence length). For prediction, this is addressed by
    @MGL-RNN-TIME-WARP. For training, we cannot discard results of
    previous time steps because they are needed for backpropagation,
    but we can at least move them out of GPU memory if they are not
    going to be used for a while and copy them back before they are
    needed. Obviously, this is only relevant if CUDA is being used.
    
    If CUDA-WINDOW-START-TIME is NIL, then this feature is turned off.
    Else, during training, at CUDA-WINDOW-START-TIME or later time
    steps, matrices belonging to non-weight lumps may be forced out of
    GPU memory and later brought back as neeeded.
    
    This feature is implemented in terms of
    MGL-MAT:WITH-SYNCING-CUDA-FACETS that uses CUDA host memory (also
    known as *page-locked* or *pinned memory*) to do asynchronous
    copies concurrently with normal computation. The consequence of
    this is that it is now main memory usage that's unbounded which
    toghether with page-locking makes it a potent weapon to bring a
    machine to a halt. You were warned.

- [variable] *CUDA-WINDOW-START-TIME* NIL

    The default for CUDA-WINDOW-START-TIME.

- [macro] BUILD-RNN (&KEY RNN (CLASS ''RNN) NAME INITARGS MAX-N-STRIPES (MAX-LAG 1)) &BODY BODY

    Create an RNN with MAX-N-STRIPES and MAX-LAG whose UNFOLDER is BODY
    wrapped in a lambda. Bind symbol given as the RNN argument to the
    RNN object so that BODY can see it.

- [function] LAG NAME &KEY (LAG 1) RNN PATH

    In RNN or if it's NIL the RNN being extended with another
    BPN (called *unfolding*), look up the CLUMP with NAME in the BPN
    that's LAG number of time steps before the BPN being added. If this
    function is called from UNFOLDER of an RNN (which is what happens
    behind the scene in the body of BUILD-RNN), then it returns an
    opaque object representing a lagged connection to a clump, else it
    returns the CLUMP itself.
    
    FIXDOC: PATH

- [function] TIME-STEP &KEY (RNN \*RNN\*)

    Return the time step RNN is currently executing or being unfolded for.
    It is 0 when the RNN is being unfolded for the first time.

- [method] SET-INPUT INSTANCES (RNN RNN)

    RNNs operate on batches of instances just like FNNs. But the
    instances here are like datasets: sequences or samplers and they are
    turned into sequences of batches of instances with
    MAP-DATASETS :IMPUTE NIL. The batch of instances at index 2 is
    clamped onto the BPN at time step 2 with SET-INPUT.
    
    When the input sequences in the batch are not of the same length,
    already exhausted sequences will produce NIL (due to :IMPUTE NIL)
    above. When such a NIL is clamped with SET-INPUT on a BPN of the
    RNN, SET-INPUT must set the IMPORTANCE of the ->ERROR lumps to 0
    else training would operate on the noise left there by previous
    invocations.

##### Time Warp

The unbounded memory usage of `RNN`s with one BPN allocated per
time step can become a problem. For training, where the gradients
often have to be backpropagated from the last time step to the very
beginning, this is hard to solve but with CUDA-WINDOW-START-TIME the
limit is no longer GPU memory.

For prediction on the other hand, one doesn't need to keep old steps
around indefinitely: they can be discarded when future time steps
will never reference them again.

- [variable] *WARP-TIME* NIL

    Controls whether warping is enabled (see @MGL-RNN-TIME-WARP). Don't
    enable it for training, as it would make backprop impossible.

- [function] WARPED-TIME &KEY (RNN \*RNN\*) (TIME (TIME-STEP :RNN RNN)) (LAG 0)

    Return the index of the BPN in CLUMPS of RNN whose task it is to
    execute computation at `(- (TIME-STEP RNN) LAG)`. This is normally
    the same as TIME-STEP (disregarding LAG). That is, CLUMPS can be
    indexed by TIME-STEP to get the BPN. However, when *WARP-TIME* is
    true, execution proceeds in a cycle as the structure of the network
    allows.
    
    Suppose we have a typical RNN that only ever references the previous
    time step so its MAX-LAG is 1. Its UNFOLDER returns `BPN`s of
    identical structure bar a shift in their time lagged connections
    except for the very first, so WARP-START and WARP-LENGTH are both 1.
    If *WARP-TIME* is NIL, then the mapping from TIME-STEP to the BPN in
    CLUMPS is straightforward:
    
        time:   |  0 |  1 |  2 |  3 |  4 |  5
        --------+----+----+----+----+----+----
        warped: |  0 |  1 |  2 |  3 |  4 |  5
        --------+----+----+----+----+----+----
        bpn:    | b0 | b1 | b2 | b3 | b4 | b5
    
    When *WARP-TIME* is true, we reuse the `B1` - `B2` bpns in a loop:
    
        time:   |  0 |  1 |  2 |  3 |  4 |  5
        --------+----+----+----+----+----+----
        warped: |  0 |  1 |  2 |  1 |  2 |  1
        --------+----+----+----+----+----+----
        bpn:    | b0 | b1 | b2 | b1*| b2 | b1*
    
    `B1*` is the same BPN as `B1`, but its connections created by LAG go
    through warped time and end up referencing `B2`. This way, memory
    consumption is independent of the number time steps needed to
    process a sequence or make predictions.
    
    To be able to pull this trick off WARP-START and WARP-LENGTH must be
    specified when the RNN is instantiated. In general, with
    *WARP-TIME* `(+ WARP-START (MAX 2 WARP-LENGTH))` bpns are needed.
    The 2 comes from the fact that with cycle length 1 a bpn would need
    to takes its input from itself which is problematic because it has
    NODES for only one set of values.

- [reader] WARP-START RNN (:WARP-START = 1)

    The TIME-STEP from which UNFOLDER will create
    `BPN`s that essentially repeat every WARP-LENGTH steps.

- [reader] WARP-LENGTH RNN (:WARP-LENGTH = 1)

    An integer such that the BPN UNFOLDER creates at
    time step `I` (where `(<= WARP-START I)`) is identical to the BPN
    created at time step `(+ WARP-START (MOD (- I WARP-START)
    WARP-LENGTH))` except for a shift in its time lagged
    connections.

- [accessor] STEP-MONITORS RNN (:STEP-MONITORS = NIL)

    During training, unfolded `BPN`s corresponding to
    previous time steps may be expensive to get at because they are no
    longer in GPU memory. This consideration also applies to making
    prediction with the additional caveat that with *WARP-TIME* true,
    previous states are discarded so it's not possible to gather
    statistics after FORWARD finished.
    
    Add monitor objects to this slot and they will be automatically
    applied to the RNN after each step when `FORWARD`ing the RNN
    during training or prediction. To be able to easily switch between
    sets of monitors, in addition to a list of monitors this can be a
    symbol or a function, too. If it's a symbol, then its a designator
    for its SYMBOL-VALUE. If it's a function, then it must have no
    arguments and it's a designator for its return value.

### Lumps

#### Lump Base Class

- [class] LUMP CLUMP

    A LUMP is a simple, layerlike component of a neural
    network. There are many kinds of lumps, each of which performs a
    specific operation or just stores inputs and weights. By convention,
    the names of lumps start with the prefix `->`. Defined as classes,
    they also have a function of the same name as the class to create
    them easily. These maker functions typically have keyword arguments
    corresponding to initargs of the class, with some (mainly the input
    lumps) turned into normal positional arguments. So instead of having
    to do
    
        (make-instance '->tanh :x some-input :name 'my-tanh)
    
    one can simply write
    
        (->tanh some-input :name 'my-tanh)
    
    Lumps instantiated in any way within a BUILD-FNN or BUILD-RNN are
    automatically added to the network being built.
    
    A lump has its own NODES and DERIVATIVES matrices allocated for it
    in which the results of the forward and backward passes are stored.
    This is in contrast to a BPN whose NODES and DERIVATIVES
    are those of its last constituent CLUMP.
    
    Since lumps almost always live within a BPN, their
    N-STRIPES and MAX-N-STRIPES are
    handled automagically behind the scenes.

- [reader] SIZE LUMP (:SIZE)

    The number of values in a single stripe.

- [reader] DEFAULT-VALUE LUMP (:DEFAULT-VALUE = 0)

    Upon creation or resize the lump's nodes get
    filled with this value.

- [generic-function] DEFAULT-SIZE LUMP

    Return a default for the SIZE of
    LUMP if one is not supplied at instantiation. The value is often
    computed based on the sizes of the inputs. This function is for
    implementing new lump types.

- [reader] NODES LUMP (= NIL)

    The values computed by the lump in the forward
    pass are stored here. It is an `N-STRIPES * SIZE` matrix that has
    storage allocated for `MAX-N-STRIPES * SIZE` elements for
    non-weight lumps. ->WEIGHT lumps have no stripes nor restrictions
    on their shape.

- [reader] DERIVATIVES LUMP

    The derivatives computed in the backward pass are
    stored here. This matrix is very much like NODES
    in shape and size.

#### Inputs

##### Input Lump

- [class] ->INPUT -\>DROPOUT

    A lump that has no input lumps, does not change its
    values in the forward pass (except when DROPOUT is non-zero), and does not compute derivatives. *Clamp*
    inputs on NODES of input lumps in SET-INPUT.
    
    For convenience, ->INPUT can perform dropout itself although it
    defaults to no dropout.
    
    ```common-lisp
    (->input :size 10 :name 'some-input)
    ==> #<->INPUT SOME-INPUT :SIZE 10 1/1 :NORM 0.00000>
    ```


- [accessor] DROPOUT -\>INPUT (= NIL)

    See DROPOUT.

##### Embedding Lump

This lump is like an input and a simple activation molded together
in the name of efficiency.

- [class] ->EMBEDDING LUMP

    Select rows of WEIGHTS, one row for each index in
    INPUT-ROW-INDICES. This lump is equivalent to adding an ->INPUT lump
    with a one hot encoding scheme and a ->V\*M lump on top of it, but it
    is more efficient in execution and in memory usage, because it works
    with a sparse representation of the input.
    
    The SIZE of this lump is the number of columns of WEIGHTS which is
    determined automatically.
    
    ```common-lisp
    (->embedding :weights (->weight :name 'embedding-weights
                                    :dimensions '(3 5))
                 :name 'embeddings)
    ==> #<->EMBEDDING EMBEDDINGS :SIZE 5 1/1 :NORM 0.00000>
    ```


- [reader] WEIGHTS -\>EMBEDDING (:WEIGHTS)

    A weight lump whose rows indexed by
    INPUT-ROW-INDICES are copied to the output of this lump.

- [reader] INPUT-ROW-INDICES -\>EMBEDDING (:INPUT-ROW-INDICES)

    A sequence of batch size length of row indices. To
    be set in SET-INPUT.

#### Weight Lump

- [class] ->WEIGHT LUMP

    A set of optimizable parameters of some kind. When
    a BPN is is trained (see @MGL-BP-TRAINING) the NODES of weight lumps
    will be changed. Weight lumps perform no computation.
    
    Weights can be created by specifying the total size or the
    dimensions:
    
    ```common-lisp
    (dimensions (->weight :size 10 :name 'w))
    => (1 10)
    (dimensions (->weight :dimensions '(5 10) :name 'w))
    => (5 10)
    ```


- [reader] DIMENSIONS -\>WEIGHT (:DIMENSIONS)

    NODES and DERIVATIVES of this lump will be
    allocated with these dimensions.

- [macro] WITH-WEIGHTS-COPIED (FROM-BPN) &BODY BODY

    In BODY ->WEIGHT will first look up if a weight lump of the same
    name exists in FROM-BPN and return that, or else create a weight
    lump normally. If FROM-BPN is NIL, then no weights are copied.

#### Activations

##### Activation Subnet

So we have some inputs. Usually the next step is to multiply the
input vector with a weight matrix and add biases. This can be done
directly with ->+, ->V\*M and ->WEIGHT, but it's more convenient to
use activation subnets to reduce the clutter.

- [class] ->ACTIVATION BPN

    Activation subnetworks are built by the function
    ->ACTIVATION and they have a number of lumps hidden inside them.
    Ultimately, this subnetwork computes a sum like `sum_i x_i * W_i +
    sum_j y_j .* V_j + biases` where `x_i` are input lumps, `W_i` are
    dense matrices representing connections, while `V_j` are peephole
    connection vectors that are mulitplied in an elementwise manner with
    their corresponding input `y_j`.

- [function] ->ACTIVATION INPUTS &KEY (NAME (GENSYM)) SIZE PEEPHOLES (ADD-BIAS-P T)

    Create a subnetwork of class ->ACTIVATION that computes the over
    activation from dense connection from lumps in INPUTS, and
    elementwise connection from lumps in PEEPHOLES. Create new ->WEIGHT
    lumps as necessary. INPUTS and PEEPHOLES can be a single lump or a
    list of lumps. Finally, if ADD-BIAS-P, then add an elementwise bias
    too. SIZE must be specified explicitly, because it is not possible
    to determine it unless there are peephole connections.
    
    ```common-lisp
    (->activation (->input :size 10 :name 'input) :name 'h1 :size 4)
    ==> #<->ACTIVATION (H1 :ACTIVATION) :STRIPES 1/1 :CLUMPS 4>
    ```
    
    This is the basic workhorse of neural networks which takes care of
    the linear transformation whose results and then fed to some
    non-linearity (->SIGMOID, ->TANH, etc).
    
    The name of the subnetwork clump is `(,NAME :ACTIVATION)`. The bias
    weight lump (if any) is named `(:BIAS ,NAME)`. Dense connection
    weight lumps are named are named after the input and NAME: `(,(NAME
    INPUT) ,NAME)`, while peepholes weight lumps are named `(,(NAME
    INPUT) ,NAME :PEEPHOLE)`. This is useful to know if, for example,
    they are to be initialized differently.

##### Batch-Normalization

- [class] ->BATCH-NORMALIZED LUMP

    This is an implementation of v3 of the [Batch
    Normalization paper](http://arxiv.org/abs/1502.03167). The output of
    ->BATCH-NORMALIZED is its input normalized so that for all elements
    the mean across stripes is zero and the variance is 1. That is, the
    mean of the batch is subtracted from the inputs and they are
    rescaled by their sample stddev. Actually, after the normalization
    step the values are rescaled and shifted (but this time with learnt
    parameters) in order to keep the representational power of the model
    the same. The primary purpose of this lump is to speed up learning,
    but it also acts as a regularizer. See the paper for the details.
    
    To normalize the output of LUMP without no additional
    regularizer effect:
    
    ```commonlisp
    (->batch-normalized lump :batch-size :use-population)
    ```
    
    The above uses an exponential moving average to estimate the mean
    and variance of batches and these estimations are used at both
    training and test time. In contrast to this, the published version
    uses the sample mean and variance of the current batch at training
    time which injects noise into the process. The noise is higher for
    lower batch sizes and has a regularizing effect. This is the default
    behavior (equivalent to `:BATCH-SIZE NIL`):
    
    ```commonlisp
    (->batch-normalized lump)
    ```
    
    For performance reasons one may wish to process a higher number of
    instances in a batch (in the sense of N-STRIPES) and get the
    regularization effect associated with a lower batch size. This is
    possible by setting :BATCH-SIZE to a divisor of the the number of
    stripes. Say, the number of stripes is 128, but we want as much
    regularization as we would get with 32:
    
    ```commonlisp
    (->batch-normalized lump :batch-size 32)
    ```
    
    The primary input of ->BATCH-NORMALIZED is often an ->ACTIVATION and
    its output is fed into an activation function (see
    @MGL-BP-ACTIVATION-FUNCTIONS).

- [reader] BATCH-NORMALIZATION -\>BATCH-NORMALIZED (:NORMALIZATION)

    The ->BATCH-NORMALIZATION of this lump. May be
    shared between multiple ->BATCH-NORMALIZED lumps.
    
    Batch normalization is special in that it has state apart from the
    computed results (NODES) and its derivatives (DERIVATIVES). This
    state is the estimated mean and variance of its inputs and they
    are encapsulated by ->BATCH-NORMALIZATION.
    
    If NORMALIZATION is not given at instantiation, then a new
    ->BATCH-NORMALIZATION object will be created automatically,
    passing :BATCH-SIZE, :VARIANCE-ADJUSTMENT, and :POPULATION-DECAY
    arguments on to ->BATCH-NORMALIZATION. See BATCH-SIZE, VARIANCE-ADJUSTMENT and POPULATION-DECAY. New scale and shift weight lumps will be
    created with names:
    
        `(,name :scale)
        `(,name :shift)
    
    where `NAME` is the NAME of this lump.
    
    This default behavior covers the use-case where the statistics
    kept by ->BATCH-NORMALIZATION are to be shared only between time
    steps of an RNN.

- [class] ->BATCH-NORMALIZATION -\>WEIGHT

    The primary purpose of this class is to hold the
    estimated mean and variance of the inputs to be normalized and allow
    them to be shared between multiple ->BATCH-NORMALIZED lumps that
    carry out the computation. These estimations are saved and loaded by
    SAVE-STATE and LOAD-STATE.
    
    ```commonlisp
    (->batch-normalization (->weight :name '(h1 :scale) :size 10)
                           (->weight :name '(h1 :shift) :size 10)
                           :name '(h1 :batch-normalization))
    ```


- [reader] SCALE -\>BATCH-NORMALIZATION (:SCALE)

    A weight lump of the same size as SHIFT. This is
    $\gamma$ in the paper.

- [reader] SHIFT -\>BATCH-NORMALIZATION (:SHIFT)

    A weight lump of the same size as SCALE. This is
    $\beta$ in the paper.

- [reader] BATCH-SIZE -\>BATCH-NORMALIZATION (:BATCH-SIZE = NIL)

    Normally all stripes participate in the batch.
    Lowering the number of stripes may increase the regularization
    effect, but it also makes the computation less efficient. By
    setting BATCH-SIZE to a divisor of N-STRIPES one can decouple the
    concern of efficiency from that of regularization. The default
    value, NIL, is equivalent to N-STRIPES. BATCH-SIZE only affects
    training.
    
    With the special value :USE-POPULATION, instead of the mean and
    the variance of the current batch, use the population statistics
    for normalization. This effectively cancels the regularization
    effect, leaving only the faster learning.

- [reader] VARIANCE-ADJUSTMENT -\>BATCH-NORMALIZATION (:VARIANCE-ADJUSTMENT = 1.0e-4)

    A small positive real number that's added to the
    sample variance. This is $\epsilon$ in the paper.

- [reader] POPULATION-DECAY -\>BATCH-NORMALIZATION (:POPULATION-DECAY = 0.99)

    While training, an exponential moving average of
    batch means and standard deviances (termed *population
    statistics*) is updated. When making predictions, normalization is
    performed using these statistics. These population statistics are
    persisted by SAVE-STATE.

- [function] ->BATCH-NORMALIZED-ACTIVATION INPUTS &KEY (NAME (GENSYM)) SIZE PEEPHOLES BATCH-SIZE VARIANCE-ADJUSTMENT POPULATION-DECAY

    A utility functions that creates and wraps an ->ACTIVATION in
    ->BATCH-NORMALIZED and with its BATCH-NORMALIZATION the two weight
    lumps for the scale and shift
    parameters. `(->BATCH-NORMALIZED-ACTIVATION INPUTS :NAME 'H1 :SIZE
    10)` is equivalent to:
    
    ```commonlisp
    (->batch-normalized (->activation inputs :name 'h1 :size 10 :add-bias-p nil)
                        :name '(h1 :batch-normalized-activation))
    ```
    
    Note how biases are turned off since normalization will cancel them
    anyway (but a shift is added which amounts to the same effect).

#### Activation Functions

Now we are moving on to the most important non-linearities to which
activations are fed.

##### Sigmoid Lump

- [class] ->SIGMOID -\>DROPOUT LUMP

    Applies the `1/(1 + e^{-x})` function elementwise
    to its inputs. This is one of the classic non-linearities for neural
    networks.
    
    For convenience, ->SIGMOID can perform dropout itself although it
    defaults to no dropout.
    
    ```common-lisp
    (->sigmoid (->activation (->input :size 10) :size 5) :name 'this)
    ==> #<->SIGMOID THIS :SIZE 5 1/1 :NORM 0.00000>
    ```
    
    The SIZE of this lump is the size of its input which is determined
    automatically.

- [accessor] DROPOUT -\>SIGMOID (= NIL)

    See DROPOUT.

##### Tanh Lump

- [class] ->TANH LUMP

    Applies the TANH function to its input in an
    elementwise manner. The SIZE of this lump is the size of its input
    which is determined automatically.

##### Scaled Tanh Lump

- [class] ->SCALED-TANH LUMP

    Pretty much like TANH but its input and output is
    scaled in such a way that the variance of its output is close to 1
    if the variance of its input is close to 1 which is a nice property
    to combat vanishing gradients. The actual function is `1.7159 *
    tanh(2/3 * x)`. The SIZE of this lump is the size of its input which
    is determined automatically.

##### Relu Lump

We are somewhere around year 2007 by now.

- [class] ->RELU LUMP

    `max(0,x)` activation function. Be careful, relu
    units can get stuck in the off state: if they move to far to
    negative territory it can be very difficult to get out of it. The
    SIZE of this lump is the size of its input which is determined
    automatically.

##### Max Lump

We are in about year 2011.

- [class] ->MAX LUMP

    This is basically maxout without dropout (see
    http://arxiv.org/abs/1302.4389). It groups its inputs by
    GROUP-SIZE, and outputs the maximum of each group.
    The SIZE of the output is automatically calculated, it is the size
    of the input divided by GROUP-SIZE.
    
    ```common-lisp
    (->max (->input :size 120) :group-size 3 :name 'my-max)
    ==> #<->MAX MY-MAX :SIZE 40 1/1 :NORM 0.00000 :GROUP-SIZE 3>
    ```
    
    The advantage of ->MAX over ->RELU is that flow gradient is never
    stopped so there is no problem of units getting stuck in off
    state.

- [reader] GROUP-SIZE -\>MAX (:GROUP-SIZE)

    The number of inputs in each group.

##### Min Lump

- [class] ->MIN LUMP

    Same as ->MAX, but it computes the MIN of groups.
    Rarely useful.

- [reader] GROUP-SIZE -\>MIN (:GROUP-SIZE)

    The number of inputs in each group.

##### Max-Channel Lump

- [class] ->MAX-CHANNEL LUMP

    Called LWTA (Local Winner Take All) or
    Channel-Out (see http://arxiv.org/abs/1312.1909) in the literature
    it is basically ->MAX, but instead of producing one output per
    group, it just produces zeros for all unit but the one with the
    maximum value in the group. This allows the next layer to get some
    information about the path along which information flowed. The SIZE
    of this lump is the size of its input which is determined
    automatically.

- [reader] GROUP-SIZE -\>MAX-CHANNEL (:GROUP-SIZE)

    The number of inputs in each group.

#### Losses

Ultimately, we need to tell the network what to learn which means
that the loss function to be minimized needs to be constructed as
part of the network.

##### Loss Lump

- [class] ->LOSS -\>SUM

    Calculate the loss for the instances in the batch.
    The main purpose of this lump is to provide a training signal.
    
    An error lump is usually a leaf in the graph of lumps (i.e. there
    are no other lumps whose input is this one). The special thing about
    error lumps is that 1 (but see IMPORTANCE) is added automatically to
    their derivatives. Error lumps have exactly one node (per stripe)
    whose value is computed as the sum of nodes in their input lump.

- [accessor] IMPORTANCE -\>LOSS (:IMPORTANCE = NIL)

    This is to support weighted instances. That is
    when not all training instances are equally important. If non-NIL,
    a 1d MAT with the importances of stripes of the batch. When
    IMPORTANCE is given (typically in SET-INPUT), then instead of
    adding 1 to the derivatives of all stripes, IMPORTANCE is added
    elemtwise.

##### Squared Difference Lump

In regression, the squared error loss is most common. The squared
error loss can be constructed by combining ->SQUARED-DIFFERENCE with
a ->LOSS.

- [class] ->SQUARED-DIFFERENCE LUMP

    This lump takes two input lumps and calculates
    their squared difference `(x - y)^2` in an elementwise manner. The
    SIZE of this lump is automatically determined from the size of its
    inputs. This lump is often fed into ->LOSS that sums the squared
    differences and makes it part of the function to be minimized.
    
    ```common-lisp
    (->loss (->squared-difference (->activation (->input :size 100)
                                                :size 10)
                                  (->input :name 'target :size 10))
            :name 'squared-error)
    ==> #<->LOSS SQUARED-ERROR :SIZE 1 1/1 :NORM 0.00000>
    ```
    
    Currently this lump is not CUDAized, but it will copy data from the
    GPU if it needs to.

##### Softmax Cross-Entropy Loss Lump

- [class] ->SOFTMAX-XE-LOSS LUMP

    A specialized lump that computes the softmax of its
    input in the forward pass and backpropagates a cross-entropy loss.
    The advantage of doing these together is numerical stability. The
    total cross-entropy is the sum of cross-entropies per group of
    GROUP-SIZE elements:
    
    $$
    XE(x) = - \sum\_\{i=1,g\} t\_i \ln(s\_i),
    $$
    
    where `g` is the number of classes (GROUP-SIZE), `t_i` are the targets (i.e. the true
    probabilities of the class, often all zero but one), `s_i` is the
    output of softmax calculated from input `X`:
    
    $$
    s\_i = \{softmax\}(x\_1, x\_2, ..., x\_g) =
      \frac\{e^x\_i\}\{\sum\_\{j=1,g\} e^x\_j\}
    $$
    
    In other words, in the forward phase this lump takes input `X`,
    computes its elementwise EXP, normalizes each group of
    GROUP-SIZE elements to sum to 1 to get
    the softmax which is the result that goes into NODES. In the
    backward phase, there are two sources of gradients: the lumps that
    use the output of this lump as their input (currently not
    implemented and would result in an error) and an implicit
    cross-entropy loss.
    
    One can get the cross-entropy calculated in the most recent forward
    pass by calling COST on this lump.
    
    This is the most common loss function for classification. In fact,
    it is nearly ubiquitous. See the @MGL-FNN-TUTORIAL and the
    @MGL-RNN-TUTORIAL for how this loss and SET-INPUT work together.

- [reader] GROUP-SIZE -\>SOFTMAX-XE-LOSS (:GROUP-SIZE)

    The number of elements in a softmax group. This is
    the number of classes for classification. Often GROUP-SIZE is
    equal to SIZE (it is the default), but in general the only
    constraint is that SIZE is a multiple of GROUP-SIZE.

- [accessor] TARGET -\>SOFTMAX-XE-LOSS (:TARGET = NIL)

    Set in SET-INPUT, this is either a MAT of the same
    size as the input lump `X` or if the target is very sparse, this
    can also be a sequence of batch size length that contains the
    index value pairs of non-zero entries:
    
        (;; first instance in batch has two non-zero targets
         (;; class 10 has 30% expected probability
          (10 . 0.3)
          ;; class 2 has 70% expected probability
          (2 .  0.7))
         ;; second instance in batch puts 100% on class 7
         7
         ;; more instances in the batch follow
         ...)
    
    Actually, in the rare case where GROUP-SIZE is not SIZE (i.e. there are several softmax
    normalization groups for every example), the length of the above
    target sequence is BATCH-SIZE \* N-GROUPS. Indices are always
    relative to the start of the group.
    
    If GROUP-SIZE is large (for example,
    in neural language models with a huge number of words), using
    sparse targets can make things go much faster, because calculation
    of the derivative is no longer quadratic.
    
    Giving different weights to training instances is implicitly
    supported. While target values in a group should sum to 1,
    multiplying all target values with a weight `W` is equivalent to
    training that `W` times on the same example.

- [function] ENSURE-SOFTMAX-TARGET-MATRIX SOFTMAX-XE-LOSS N

    Set TARGET of SOFTMAX-XE-LOSS to a MAT capable of holding the dense
    target values for N stripes.

#### Stochasticity

##### Dropout Lump

- [class] ->DROPOUT LUMP

    The output of this lump is identical to its input,
    except it randomly zeroes out some of them during training which act
    as a very strong regularizer. See Geoffrey Hinton's 'Improving
    neural networks by preventing co-adaptation of feature
    detectors'.
    
    The SIZE of this lump is the size of its input which is determined
    automatically.

- [accessor] DROPOUT -\>DROPOUT (:DROPOUT = 0.5)

    If non-NIL, then in the forward pass zero out each
    node in this chunk with DROPOUT probability.

##### Gaussian Random Lump

- [class] ->GAUSSIAN-RANDOM LUMP

    This lump has no input, it produces normally
    distributed independent random numbers with MEAN and VARIANCE (or
    VARIANCE-FOR-PREDICTION). This is useful building block for noise
    based regularization methods.
    
    ```common-lisp
    (->gaussian-random :size 10 :name 'normal :mean 1 :variance 2)
    ==> #<->GAUSSIAN-RANDOM NORMAL :SIZE 10 1/1 :NORM 0.00000>
    ```


- [accessor] MEAN -\>GAUSSIAN-RANDOM (:MEAN = 0)

    The mean of the normal distribution.

- [accessor] VARIANCE -\>GAUSSIAN-RANDOM (:VARIANCE = 1)

    The variance of the normal distribution.

- [accessor] VARIANCE-FOR-PREDICTION -\>GAUSSIAN-RANDOM (:VARIANCE-FOR-PREDICTION = 0)

    If not NIL, then this value overrides VARIANCE
    when not in training (i.e. when making predictions).

##### Binary Sampling Lump

- [class] ->SAMPLE-BINARY LUMP

    Treating values of its input as probabilities,
    sample independent binomials. Turn true into 1 and false into 0. The
    SIZE of this lump is determined automatically from the size of its
    input.
    
    ```common-lisp
    (->sample-binary (->input :size 10) :name 'binarized-input)
    ==> #<->SAMPLE-BINARY BINARIZED-INPUT :SIZE 10 1/1 :NORM 0.00000>
    ```


#### Arithmetic

##### Sum Lump

- [class] ->SUM LUMP

    Computes the sum of all nodes of its input per
    stripe. This SIZE of this lump is always 1.

##### Vector-Matrix Multiplication Lump

- [class] ->V*M LUMP

    Perform `X * WEIGHTS` where `X` (the input) is of
    size `M` and WEIGHTS is a ->WEIGHT whose single stripe is taken to
    be of dimensions `M x N` stored in row major order. `N` is the size
    of this lump. If TRANSPOSE-WEIGHTS-P then WEIGHTS is `N x M` and `X
    * WEIGHTS'` is computed.

- [reader] WEIGHTS -\>V\*M (:WEIGHTS)

    A ->WEIGHT lump.

- [reader] TRANSPOSE-WEIGHTS-P -\>V\*M (:TRANSPOSE-WEIGHTS-P = NIL)

    Determines whether the input is multiplied by
    WEIGHTS or its transpose.

##### Elementwise Addition Lump

- [class] ->+ LUMP

    Performs elementwise addition on its input lumps.
    The SIZE of this lump is automatically determined from the size of
    its inputs if there is at least one. If one of the inputs is a
    ->WEIGHT lump, then it is added to every stripe.
    
    ```common-lisp
    (->+ (list (->input :size 10) (->weight :size 10 :name 'bias))
         :name 'plus)
    ==> #<->+ PLUS :SIZE 10 1/1 :NORM 0.00000>
    ```


##### Elementwise Multiplication Lump

- [class] ->* LUMP

    Performs elementwise multiplication on its two
    input lumps. The SIZE of this lump is automatically determined from
    the size of its inputs. Either input can be a ->WEIGHT lump.
    
    ```common-lisp
    (->* (->input :size 10) (->weight :size 10 :name 'scale)
         :name 'mult)
    ==> #<->* MULT :SIZE 10 1/1 :NORM 0.00000>
    ```


##### Abs Lump

- [class] ->ABS LUMP

##### Exp Lump

- [class] ->EXP LUMP

##### Normalized Lump

- [class] ->NORMALIZED LUMP

#### Operations for RNNs

##### LSTM Subnet

- [class] ->LSTM BPN

    Long-Short Term Memory subnetworks are built by the
    function ->LSTM and they have many lumps hidden inside them. These
    lumps are packaged into a subnetwork to reduce clutter.

- [function] ->LSTM INPUTS &KEY NAME CELL-INIT OUTPUT-INIT SIZE (ACTIVATION-FN '-\>ACTIVATION) (GATE-FN '-\>SIGMOID) (INPUT-FN '-\>TANH) (OUTPUT-FN '-\>TANH) (PEEPHOLES T)

    Create an LSTM layer consisting of input, forget, output gates with
    which input, cell state and output are scaled. Lots of lumps are
    created, the final one representing to output of the LSTM has NAME.
    The rest of the lumps are named automatically based on NAME. This
    function returns only the output lump (`m`), but all created lumps
    are added automatically to the BPN being built.
    
    There are many papers and tutorials on LSTMs. This version is well
    described in "Long Short-Term Memory Recurrent Neural Network
    Architectures for Large Scale Acoustic Modeling" (2014, Hasim Sak,
    Andrew Senior, Francoise Beaufays). Using the notation from that
    paper:
    
    $$
    i\_t = s(W\_\{ix\} x\_t + W\_\{im\} m\_\{t-1\} + W\_\{ic\} \odot
    c\_\{t-1\} + b\_i)
    $$
    
    $$
    f\_t = s(W\_\{fx\} x\_t + W\_\{fm\} m\_\{t-1\} + W\_\{fc\} \odot
    c\_\{t-1\} + b\_f)
    $$
    
    $$
    c\_t = f\_t \odot c\_\{t-1\} + i\_t \odot g(W\_\{cx\} x\_t +
    W\_\{cm\} m\_\{t-1\} + b\_c)
    $$
    
    $$
    o\_t = s(W\_\{ox\} x\_t + W\_\{om\} m\_\{t-1\} + W\_\{oc\} \odot
    c\_t + b\_o)
    $$
    
    $$
    m\_t = o\_t \odot h(c\_t),
    $$
    
    where `i`, `f`, and `o` are the input, forget and output gates. `c`
    is the cell state and `m` is the actual output.
    
    Weight matrices for connections from `c` (`W_ic`, `W_fc` and `W_oc`)
    are diagonal and represented by just the vector of diagonal values.
    These connections are only added if PEEPHOLES is true.
    
    A notable difference from the paper is that in addition to being a
    single lump, `x_t` (INPUTS) can also be a list of lumps. Whenever
    some activation is to be calculated based on `x_t`, it is going to
    be the sum of individual activations. For example, `W_ix * x_t` is
    really `sum_j W_ijx * inputs_j`.
    
    If CELL-INIT is non-NIL, then it must be a CLUMP of SIZE form which
    stands for the initial state of the value cell (`c_{-1}`). CELL-INIT
    being NIL is equivalent to the state of all zeros.
    
    ACTIVATION-FN defaults to ->ACTIVATION, but it can be for example
    ->BATCH-NORMALIZED-ACTIVATION. In general, functions like the
    aforementioned two with signature like (INPUTS &KEY NAME SIZE
    PEEPHOLES) can be passed as ACTIVATION-FN.

##### Sequence Barrier Lump

- [class] ->SEQ-BARRIER LUMP

    In an RNN, processing of stripes (instances in the
    batch) may require different number of time step so the final state
    for stripe 0 is in stripe 0 of some lump L at time step 7, while for
    stripe 1 it is in stripe 1 of sump lump L at time step 42.
    
    This lump copies the per-stripe states from different lumps into a
    single lump so that further processing can take place (typically
    when the RNN is embedded in another network).
    
    The SIZE of this lump is automatically set to the size of the lump
    returned by `(FUNCALL SEQ-ELT-FN 0)`.

- [reader] SEQ-ELT-FN -\>SEQ-BARRIER (:SEQ-ELT-FN)

    A function of an INDEX argument that returns the
    lump with that index in some sequence.

- [accessor] SEQ-INDICES -\>SEQ-BARRIER

    A sequence of length batch size of indices. The
    element at index `I` is the index to be passed to SEQ-ELT-FN to
    find the lump whose stripe `I` is copied to stripe `I` of this
    this lump.

### Utilities

- [function] RENORMALIZE-ACTIVATIONS -\>V\*M-LUMPS L2-UPPER-BOUND

    If the l2 norm of the incoming weight vector of the a unit is
    larger than L2-UPPER-BOUND then renormalize it to L2-UPPER-BOUND.
    The list of ->V\*M-LUMPS is assumed to be eventually fed to the same
    lump.
    
    To use it, group the activation clumps into the same GD-OPTIMIZER
    and hang this function on AFTER-UPDATE-HOOK, that latter of which is
    done for you ARRANGE-FOR-RENORMALIZING-ACTIVATIONS.
    
    See "Improving neural networks by preventing co-adaptation of
    feature detectors (Hinton, 2012)",
    <http://arxiv.org/pdf/1207.0580.pdf>.

- [function] ARRANGE-FOR-RENORMALIZING-ACTIVATIONS BPN OPTIMIZER L2-UPPER-BOUND

    By pushing a lambda to AFTER-UPDATE-HOOK of OPTIMIZER arrange for
    all weights beings trained by OPTIMIZER to be renormalized (as in
    RENORMALIZE-ACTIVATIONS with L2-UPPER-BOUND).
    
    It is assumed that if the weights either belong to an activation
    lump or are simply added to the activations (i.e. they are biases).

## Boltzmann Machines


## Gaussian Processes


## Natural Language Processing

###### \[in package MGL-NLP\]
This in nothing more then a couple of utilities for now which may
grow into a more serious toolset for NLP eventually.

- [function] MAKE-N-GRAM-MAPPEE FUNCTION N

    Make a function of a single argument that's suitable as the
    function argument to a mapper function. It calls FUNCTION with every
    N element.
    
    ```common-lisp
    (map nil (make-n-gram-mappee #'print 3) '(a b c d e))
    ..
    .. (A B C) 
    .. (B C D) 
    .. (C D E) 
    ```


- [function] BLEU CANDIDATES REFERENCES &KEY CANDIDATE-KEY REFERENCE-KEY (N 4)

    Compute the [BLEU score](http://en.wikipedia.org/wiki/BLEU) for
    bilingual CORPUS. BLEU measures how good a translation is compared
    to human reference translations.
    
    CANDIDATES (keyed by CANDIDATE-KEY) and REFERENCES (keyed by
    REFERENCE-KEY) are sequences of sentences. A sentence is a sequence
    of words. Words are compared with EQUAL, and may be any kind of
    object (not necessarily strings).
    
    Currently there is no support for multiple reference translations. N
    determines the largest n-grams to consider.
    
    The first return value is the BLEU score (between 0 and 1, not as a
    percentage). The second value is the sum of the lengths of
    CANDIDATES divided by the sum of the lengths of REFERENCES (or NIL,
    if the denominator is 0). The third is a list of n-gram
    precisions (also between 0 and 1 or NIL), one for each element in
    \[1..`N`\].
    
    This is basically a reimplementation of
    [multi-bleu.perl](https://github.com/moses-smt/mosesdecoder/blob/master/scripts/generic/multi-bleu.perl).
    
    ```common-lisp
    (bleu '((1 2 3 4) (a b))
          '((1 2 3 4) (1 2)))
    => 0.8408964
    => 1
    => (;; 1-gram precision: 4/6
        2/3
        ;; 2-gram precision: 3/4
        3/4
        ;; 3-gram precision: 2/2
        1
        ;; 4-gram precision: 1/1
        1)
    ```


### Bag of Words

- [class] BAG-OF-WORDS-ENCODER

    ENCODE all features of a document with a sparse
    vector. Get the features of document from MAPPER, encode each
    feature with FEATURE-ENCODER. FEATURE-ENCODER may return NIL if the
    feature is not used. The result is a vector of encoded-feature/value
    conses. encoded-features are unique (under ENCODED-FEATURE-TEST)
    within the vector but are in no particular order.
    
    Depending on KIND, value is calculated in various ways:
    
    - For :FREQUENCY it is the number of times the corresponding feature
    was found in DOCUMENT.
    
    - For :BINARY it is always 1.
    
    - For :NORMALIZED-FREQUENCY and :NORMALIZED-BINARY are like the
      unnormalized counterparts except that as the final step values in
      the assembled sparse vector are normalized to sum to 1.
    
    - Finally, :COMPACTED-BINARY is like :BINARY but the return values
      is not a vector of conses, but a vector of element-type
      ENCODED-FEATURE-TYPE.
    
    ```common-lisp
    (let* ((feature-indexer
             (make-indexer
              (alexandria:alist-hash-table '(("I" . 3) ("me" . 2) ("mine" . 1)))
              2))
           (bag-of-words-encoder
             (make-instance 'bag-of-words-encoder
                            :feature-encoder feature-indexer
                            :feature-mapper (lambda (fn document)
                                              (map nil fn document))
                            :kind :frequency)))
      (encode bag-of-words-encoder '("All" "through" "day" "I" "me" "mine"
                                     "I" "me" "mine" "I" "me" "mine")))
    => #((0 . 3.0d0) (1 . 3.0d0))
    ```


- [reader] FEATURE-ENCODER BAG-OF-WORDS-ENCODER (:FEATURE-ENCODER)

- [reader] FEATURE-MAPPER BAG-OF-WORDS-ENCODER (:FEATURE-MAPPER)

- [reader] ENCODED-FEATURE-TEST BAG-OF-WORDS-ENCODER (:ENCODED-FEATURE-TEST = #'EQL)

- [reader] ENCODED-FEATURE-TYPE BAG-OF-WORDS-ENCODER (:ENCODED-FEATURE-TYPE = T)

- [reader] BAG-OF-WORDS-KIND BAG-OF-WORDS-ENCODER (:KIND = :BINARY)

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###### \[generated by [MGL-PAX](https://github.com/melisgl/mgl-pax)\]