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지난 섹션에서는 특정 카테고리에서 하나의 라벨을 이미지에 붙이는 문제인 이미지 분류에 대해 소개하였다. 또한, 학습 데이터셋에 있는 (라벨링된) 이미지들 중 가까이 있는 것들의 라벨을 활용하는 k-Nearest Neighbor (kNN) 분류기에 대해 설명하였다. 앞서 살펴보았듯이, kNN은 몇 가지 단점이 있다:
- 이 분류기는 모든 학습 데이터를 기억 해야 하고, 나중에 테스트 데이터와 비교하기 위해 저장해 두어야 한다. 이것은 메모리 공간 관점에서 매우 비효율적이고, 일반적인 데이터셋들은 용량이 기가바이트 단위를 쉽게 넘기는 것이 많기 때문에 문제가 된다.
- 테스트 이미지를 분류할 때 모든 학습 이미지와 다 비교를 해야 하기 때문에 매우 계산량/시간이 많이 소요된다.
Overview. 이번 노트에서는 이미지 분류를 위한 보다 강력한 방법들을 발전시켜나갈 것이고, 이는 나중에 뉴럴 네트워크와 컨볼루션 신경망으로 확장될 것이다. 이 방법들은 두 가지 중요한 요소가 있다: 데이터를 클래스 스코어로 매핑시키는 스코어 함수, 그리고 예측한 스코어와 실제(ground truth) 라벨과의 차이를 정량화해주는 손실 함수 가 그 두 가지이다. 우리는 이를 최적화 문제로 바꾸어서 스코어 함수의 파라미터들에 대한 손실 함수를 최소화할 것이다.
먼저, 이미지의 픽셀 값들을 각 클래스에 대한 신뢰도 점수 (confidence score)로 매핑시켜주는 스코어 함수를 정의한다. 여기서는 구체적인 예시를 통해 각 과정을 살펴볼 것이다. 이전 노트에서처럼, 학습 데이터셋 이미지들인 $$ x_i \in R^D
선형 분류기 (Linear Classifier). 이 파트에서는 가장 단순한 함수라고 할 수 있는 선형 매핑 함수로 시작할 것이다.
위 식에서, 우리는 각 이미지
여기서 몇 가지 짚고 넘어갈 점이 있다.
- 먼저, 한 번의 행렬곱
$$W x_i$$ 만으로 10 개의 로 다른 분류기(각 클래스마다 하나씩)를 병렬로 계산하는 효과를 나타내고 있다는 점을 살펴보자. 이 때 W 행렬의 각 열이 각각 하나의 분류기가 된다. - 또한, 여기서 입력 데이터 $$ (x_i, y_i) $$는 주어진 값이고 고정되어 있지만, 파라미터들인 W, b 의 세팅은 우리가 조절할 수 있다는 점을 생각하자. 우리의 최종 목표는 전체 학습 데이터에 대해서 우리가 계산할 스코어 값들이 실제 (ground truth) 라벨과 가장 잘 일치하도록 이 파라미터 값들을 정하는 것이다. 이후(아래)에 자세한 방법에 대해 다룰 것이지만, 직관적으로 간략하게 말하자면 올바르게 잘 맞춘 클래스가 틀린 클래스들보다 더 높은 스코어를 갖도록 조절할 것이다.
- 이러한 방식의 장점은, 학습 데이터가 파라미터들인 W, b 를 학습하는데 사용되지만 학습이 끝난 이후에는 학습된 파라미터들만 남기고, 학습에 사용된 데이터셋은 더 이상 필요가 없다는 (따라서 메모리에서 지워버려도 된다는) 점이다. 그 이유는, 새로운 테스트 이미지가 입력으로 들어올 때 위의 함수에 의해 스코어를 계산하고, 계산된 스코어를 통해 바로 분류되기 때문이다.
- 마지막으로, 테스트 이미지를 분류할 때 행렬곱 한 번과 덧셈 한 번을 하는 계산만 필요하다는 점을 주목하자. 이것은 테스트 이미지를 모든 학습 이미지와 비교하는 것에 비하면 매우 빠르다.
스포일러: 컨볼루션 신경망(Convolutional Neural Networks)은 정확히 위의 방식처럼 이미지 픽셀 값을 스코어 값으로 매핑시켜 주지만, 매핑시켜주는 함수 ( f ) 가 훨씬 더 복잡해지고 더 많은 수의 파라미터를 갖고 있을 것이다.
선형 분류기는 클래스 스코어를 이미지의 모든 픽셀 값들의 가중치 합으로 스코어를 계산하고, 이 때 각 픽셀의 3 개의 색 채널을 모두 고려하는 것에 주목하자. 이 때 각 가중치(파라미터, weights)에 어떤 값을 주느냐에 따라 스코어 함수는 이미지의 특정 위치에서 특정 색깔을 선호하거나 선호하지 않거나 (가중치 값의 부호에 따라) 할 수 있다. 예를 들어, "ship" 클래스는 이미지의 가장자리 부분에 파란색이 많은 경우에 (강, 바다 등의 물에 해당하는 색) 스코어 값이 더 높아질 것이라고 추측해 볼 수 있을 것이다. 즉, "ship" 분류기는 파란색 채널의 파라미터(weights)들이 양의 값을 갖고 (파란색이 존재하는 것이 ship의 스코어를 증가시키도록), 빨강/초록색 채널에는 음의 값을 갖는 파라미터들이 많을 것이라고 (빨간색/초록색의 존재는 ship의 스코어를 감소시키도록) 예상할 수 있다.
이미지와 고차원 공간 상의 점에 대한 비유. 이미지들을 고차원 열 벡터로 펼쳤기 때문에, 우리는 각 이미지를 이 고차원 공간 상의 하나의 점으로 생각할 수 있다 (e.g. CIFAR-10 데이터셋의 각 이미지는 32x32x3 개의 픽셀로 이루어진 3072-차원 공간 상의 한 점이 된다). 마찬가지로 생각하면, 전체 데이터셋은 라벨링된 고차원 공간 상의 점들의 집합이 될 것이다.
위에서 각 클래스에 대한 스코어를 이미지의 모든 픽셀에 대한 가중치 합으로 정의했기 때문에, 각 클래스 스코어는 이 공간 상에서의 선형 함수값이 된다. 3072-차원 공간은 시각화할 수 없지만, 2차원으로 축소시켰다고 상상해보면 우리의 분류기가 어떤 행동을 하는지를 시각화하려고 시도해볼 수 있을 것이다:
위에서 살펴보았듯이,
템플릿 매칭으로서의 선형 분류기 해석.
파라미터
추가적으로, horse 템플릿은 머리가 두 개인 말(horse)이 있는 것처럼 보이는데, 이것은 데이터셋 안에 왼쪽을 보고 있는 말과 오른쪽을 보고 있는 말이 섞여있기 때문이다. 선형 분류기는 말에 대한 이 두 가지 모드를 하나의 템플릿으로 합친 것을 확인할 수 있다. 이와 비슷한 현상으로, car 분류기는 모든 방향 및 색깔의 자동차 모양들을 하나의 템플릿으로 합쳐 놓았다. 특히, 이 템플릿이 결과적으로 붉은 색을 띄는 것으로 보아 CIFAR-10 데이터셋에는 다른 색깔에 비해 빨간색 자동차가 더 많다는 점을 알 수 있다. 선형 분류기는 여러 가지 색깔의 자동차를 제대로 분류하기에는 너무 모델이 단순하지만, 나중에 배울 뉴럴 네트워크는 이를 해결할 수 있다. 약간만 미리 살펴보자면, 뉴럴 네트워크는 히든 레이어의 각 뉴런들이 특정 자동차 타입 (e.g. 왼쪽을 바라보고 있는 초록색 자동차, 정면을 보고 있는 파란색 차, 등등)을 검출하도록 할 수 있고, 다음 레이어의 뉴런들이 이 정보들을 종합하여 각각의 자동차 타입 검출기의 점수의 가중치 합을 통해 보다 정확한 (자동차에 대한) 스코어를 계산할 수 있다.
Bias 트릭. 다음 내용으로 넘어가기 전에, 두 파라미터
앞으로 내용을 전개해 나갈 때 두 가지 파라미터를 (bias
With our CIFAR-10 example,
Image data preprocessing. As a quick note, in the examples above we used the raw pixel values (which range from [0...255]). In Machine Learning, it is a very common practice to always perform normalization of your input features (in the case of images, every pixel is thought of as a feature). In particular, it is important to center your data by subtracting the mean from every feature. In the case of images, this corresponds to computing a mean image across the training images and subtracting it from every image to get images where the pixels range from approximately [-127 ... 127]. Further common preprocessing is to scale each input feature so that its values range from [-1, 1]. Of these, zero mean centering is arguably more important but we will have to wait for its justification until we understand the dynamics of gradient descent.
In the previous section we defined a function from the pixel values to class scores, which was parameterized by a set of weights
For example, going back to the example image of a cat and its scores for the classes "cat", "dog" and "ship", we saw that the particular set of weights in that example was not very good at all: We fed in the pixels that depict a cat but the cat score came out very low (-96.8) compared to the other classes (dog score 437.9 and ship score 61.95). We are going to measure our unhappiness with outcomes such as this one with a loss function (or sometimes also referred to as the cost function or the objective). Intuitively, the loss will be high if we're doing a poor job of classifying the training data, and it will be low if we're doing well.
There are several ways to define the details of the loss function. As a first example we will first develop a commonly used loss called the Multiclass Support Vector Machine (SVM) loss. The SVM loss is set up so that the SVM "wants" the correct class for each image to a have a score higher than the incorrect classes by some fixed margin
Let's now get more precise. Recall that for the i-th example we are given the pixels of image $ x_i $ and the label $ y_i $ that specifies the index of the correct class. The score function takes the pixels and computes the vector $ f(x_i, W) $ of class scores, which we will abbreviate to
Example. Lets unpack this with an example to see how it works. Suppose that we have three classes that receive the scores $ s = [13, -7, 11]$, and that the first class is the true class (i.e.
You can see that the first term gives zero since [-7 - 13 + 10] gives a negative number, which is then thresholded to zero with the
Note that in this particular module we are working with linear score functions ( $ f(x_i; W) = W x_i $ ), so we can also rewrite the loss function in this equivalent form:
where
A last piece of terminology we'll mention before we finish with this section is that the threshold at zero
The loss function quantifies our unhappiness with predictions on the training set
Regularization. There is one bug with the loss function we presented above. Suppose that we have a dataset and a set of parameters W that correctly classify every example (i.e. all scores are so that all the margins are met, and
In other words, we wish to encode some preference for a certain set of weights W over others to remove this ambiguity. We can do so by extending the loss function with a regularization penalty
In the expression above, we are summing up all the squared elements of
$$ L = \underbrace{ \frac{1}{N} \sum_i L_i }\text{data loss} + \underbrace{ \lambda R(W) }\text{regularization loss} \\ $$
Or expanding this out in its full form:
$$ L = \frac{1}{N} \sum_i \sum_{j\neq y_i} \left[ \max(0, f(x_i; W)j - f(x_i; W){y_i} + \Delta) \right] + \lambda \sum_k\sum_l W_{k,l}^2 $$
Where
In addition to the motivation we provided above there are many desirable properties to include the regularization penalty, many of which we will come back to in later sections. For example, it turns out that including the L2 penalty leads to the appealing max margin property in SVMs (See CS229 lecture notes for full details if you are interested).
The most appealing property is that penalizing large weights tends to improve generalization, because it means that no input dimension can have a very large influence on the scores all by itself. For example, suppose that we have some input vector $x = [1,1,1,1] $ and two weight vectors
Note that biases do not have the same effect since, unlike the weights, they do not control the strength of influence of an input dimension. Therefore, it is common to only regularize the weights
Code. Here is the loss function (without regularization) implemented in Python, in both unvectorized and half-vectorized form:
def L_i(x, y, W):
"""
unvectorized version. Compute the multiclass svm loss for a single example (x,y)
- x is a column vector representing an image (e.g. 3073 x 1 in CIFAR-10)
with an appended bias dimension in the 3073-rd position (i.e. bias trick)
- y is an integer giving index of correct class (e.g. between 0 and 9 in CIFAR-10)
- W is the weight matrix (e.g. 10 x 3073 in CIFAR-10)
"""
delta = 1.0 # see notes about delta later in this section
scores = W.dot(x) # scores becomes of size 10 x 1, the scores for each class
correct_class_score = scores[y]
D = W.shape[0] # number of classes, e.g. 10
loss_i = 0.0
for j in xrange(D): # iterate over all wrong classes
if j == y:
# skip for the true class to only loop over incorrect classes
continue
# accumulate loss for the i-th example
loss_i += max(0, scores[j] - correct_class_score + delta)
return loss_i
def L_i_vectorized(x, y, W):
"""
A faster half-vectorized implementation. half-vectorized
refers to the fact that for a single example the implementation contains
no for loops, but there is still one loop over the examples (outside this function)
"""
delta = 1.0
scores = W.dot(x)
# compute the margins for all classes in one vector operation
margins = np.maximum(0, scores - scores[y] + delta)
# on y-th position scores[y] - scores[y] canceled and gave delta. We want
# to ignore the y-th position and only consider margin on max wrong class
margins[y] = 0
loss_i = np.sum(margins)
return loss_i
def L(X, y, W):
"""
fully-vectorized implementation :
- X holds all the training examples as columns (e.g. 3073 x 50,000 in CIFAR-10)
- y is array of integers specifying correct class (e.g. 50,000-D array)
- W are weights (e.g. 10 x 3073)
"""
# evaluate loss over all examples in X without using any for loops
# left as exercise to reader in the assignmentThe takeaway from this section is that the SVM loss takes one particular approach to measuring how consistent the predictions on training data are with the ground truth labels. Additionally, making good predictions on the training set is equivalent to minimizing the loss.
All we have to do now is to come up with a way to find the weights that minimize the loss.
Setting Delta. Note that we brushed over the hyperparameter
Relation to Binary Support Vector Machine. You may be coming to this class with previous experience with Binary Support Vector Machines, where the loss for the i-th example can be written as:
where
Aside: Optimization in primal. If you're coming to this class with previous knowledge of SVMs, you may have also heard of kernels, duals, the SMO algorithm, etc. In this class (as is the case with Neural Networks in general) we will always work with the optimization objectives in their unconstrained primal form. Many of these objectives are technically not differentiable (e.g. the max(x,y) function isn't because it has a kink when x=y), but in practice this is not a problem and it is common to use a subgradient.
Aside: Other Multiclass SVM formulations. It is worth noting that the Multiclass SVM presented in this section is one of few ways of formulating the SVM over multiple classes. Another commonly used form is the One-Vs-All (OVA) SVM which trains an independent binary SVM for each class vs. all other classes. Related, but less common to see in practice is also the All-vs-All (AVA) strategy. Our formulation follows the Weston and Watkins 1999 (pdf) version, which is a more powerful version than OVA (in the sense that you can construct multiclass datasets where this version can achieve zero data loss, but OVA cannot. See details in the paper if interested). The last formulation you may see is a Structured SVM, which maximizes the margin between the score of the correct class and the score of the highest-scoring incorrect runner-up class. Understanding the differences between these formulations is outside of the scope of the class. The version presented in these notes is a safe bet to use in practice, but the arguably simplest OVA strategy is likely to work just as well (as also argued by Rikin et al. 2004 in In Defense of One-Vs-All Classification (pdf)).
It turns out that the SVM is one of two commonly seen classifiers. The other popular choice is the Softmax classifier, which has a different loss function. If you've heard of the binary Logistic Regression classifier before, the Softmax classifier is its generalization to multiple classes. Unlike the SVM which treats the outputs
where we are using the notation
Information theory view. The cross-entropy between a "true" distribution
The Softmax classifier is hence minimizing the cross-entropy between the estimated class probabilities ( $q = e^{f_{y_i}} / \sum_j e^{f_j} $ as seen above) and the "true" distribution, which in this interpretation is the distribution where all probability mass is on the correct class (i.e.
Probabilistic interpretation. Looking at the expression, we see that
can be interpreted as the (normalized) probability assigned to the correct label
Practical issues: Numeric stability. When you're writing code for computing the Softmax function in practice, the intermediate terms
We are free to choose the value of
f = np.array([123, 456, 789]) # example with 3 classes and each having large scores
p = np.exp(f) / np.sum(np.exp(f)) # Bad: Numeric problem, potential blowup
# instead: first shift the values of f so that the highest number is 0:
f -= np.max(f) # f becomes [-666, -333, 0]
p = np.exp(f) / np.sum(np.exp(f)) # safe to do, gives the correct answerPossibly confusing naming conventions. To be precise, the SVM classifier uses the hinge loss, or also sometimes called the max-margin loss. The Softmax classifier uses the cross-entropy loss. The Softmax classifier gets its name from the softmax function, which is used to squash the raw class scores into normalized positive values that sum to one, so that the cross-entropy loss can be applied. In particular, note that technically it doesn't make sense to talk about the "softmax loss", since softmax is just the squashing function, but it is a relatively commonly used shorthand.
A picture might help clarify the distinction between the Softmax and SVM classifiers:
Softmax classifier provides "probabilities" for each class. Unlike the SVM which computes uncalibrated and not easy to interpret scores for all classes, the Softmax classifier allows us to compute "probabilities" for all labels. For example, given an image the SVM classifier might give you scores [12.5, 0.6, -23.0] for the classes "cat", "dog" and "ship". The softmax classifier can instead compute the probabilities of the three labels as [0.9, 0.09, 0.01], which allows you to interpret its confidence in each class. The reason we put the word "probabilities" in quotes, however, is that how peaky or diffuse these probabilities are depends directly on the regularization strength
Where the steps taken are to exponentiate and normalize to sum to one. Now, if the regularization strength
where the probabilites are now more diffuse. Moreover, in the limit where the weights go towards tiny numbers due to very strong regularization strength
In practice, SVM and Softmax are usually comparable. The performance difference between the SVM and Softmax are usually very small, and different people will have different opinions on which classifier works better. Compared to the Softmax classifier, the SVM is a more local objective, which could be thought of either as a bug or a feature. Consider an example that achieves the scores [10, -2, 3] and where the first class is correct. An SVM (e.g. with desired margin of
In summary,
- We defined a score function from image pixels to class scores (in this section, a linear function that depends on weights W and biases b).
- Unlike kNN classifier, the advantage of this parametric approach is that once we learn the parameters we can discard the training data. Additionally, the prediction for a new test image is fast since it requires a single matrix multiplication with W, not an exhaustive comparison to every single training example.
- We introduced the bias trick, which allows us to fold the bias vector into the weight matrix for convenience of only having to keep track of one parameter matrix.
- We defined a loss function (we introduced two commonly used losses for linear classifiers: the SVM and the Softmax) that measures how compatible a given set of parameters is with respect to the ground truth labels in the training dataset. We also saw that the loss function was defined in such way that making good predictions on the training data is equivalent to having a small loss.
We now saw one way to take a dataset of images and map each one to class scores based on a set of parameters, and we saw two examples of loss functions that we can use to measure the quality of the predictions. But how do we efficiently determine the parameters that give the best (lowest) loss? This process is optimization, and it is the topic of the next section.
These readings are optional and contain pointers of interest.
- Deep Learning using Linear Support Vector Machines from Charlie Tang 2013 presents some results claiming that the L2SVM outperforms Softmax.