Subspace quasi newton method

How to run numerical experiments

1. Select optimization problem and an algorithm and set the parameters in make_config.py.
2. Make config.json file using command python make_config.py.
3. Run numerical experiments using command python main.py path/to/config.json.
4. Check the results in results/problem_name/problem_parameters/constraints_name/constraints_parameters/algorithm_name/algorithm_parameters directory.
5. You can compare results using python result_show.py. with GUI interface.

problems

QUADRATIC

$$\min_{x\in \mathbb{R}^n} \frac{1}{2}x^\top A x + b^\top x$$ dim: n, if convex is True, then A become a semi definite matrix. data_name: only "random".

SPARSEQUADRATIC

not used

MATRIXFACTORIZATION

$$\min_{U,V} |UV - X|_F^2$$ data_name: only "movie", rank: the number of row of $U$, and column of $V$.

MATRIXFACTORIZATION_COMPLETION

$$\min_{U,V} |\mathcal{P}{\Omega}(UV) - \mathcal{P}{\Omega}(X)|_F^2$$ data_name: only "movie", rank: the number of row of $U$, and column of $V$.

LEASTSQUARE

not used

MLPNET

linear neural network: $$\min_{w} \sum_{i=1}^m \mathcal{L}(\mathcal{W}(w,x_i),y_i)$$ $(x_i,y_i)$:dataset, layers_size: [(in_features,out_feafures,use bias or not),], activation: activation function name (see utils/select.py), criterion: type of loss function (only 'CrossEntropy')

CNN

convolutional neural network: $$\min_{w} \sum_{i=1}^m \mathcal{L}(\mathcal{W}(w,x_i),y_i)$$ $(x_i,y_i)$:dataset, layers_size: [(input_channels,output_channelskernel_size,bias_flag)], activation: activation function name (see utils/select.py), criterion: type of loss function (only 'CrossEntropy')

SOFTMAX

minimizing softmax loss function.
data_name: "Scotus" or "news20"

LOGISTIC

minimizing logistic loss function
data_name:"rcv1" or "news20" or "random".

REGULARIZED

set problem_name = REGULARIZED + other_problem_name. minimizing regularized function $$\min_x f(x) + \lambda |x|_p^p$$ coeff: $\lambda$, ord: $p$, Fused: only False

constraints

POLYTOPE

$${ x| Ax-b \le 0}$$ data_name:only "random", dim: the dimension of $x$, constraints_num: the dimension of $b$

NONNEGATIVE

$${x | x_i \ge 0}$$ dim: the dimension of $x$

QUADRATIC

$${x| \frac{1}{2}x^\top A_i x + b_i^\top x + c_i \le 0, i = 1,...,m}$$ data_name: only "random", dim: the dimension of $x$, constraints_num: m

FUSEDLASSO

$${x| |x|1 \le s_1, \sum{i} |x_{i+1} - x_i|\le s_2}$$ threshold1: $s_1$, threshold2: $s_2$

BALL

$${x| |x|_p^p \le s}$$ ord: $p$, threshold: $s$

algorithms

backward parameters

True: use automatic differentiation (very fast, but memory leak sometimes happens (CNN)) DD: use directional derivative with automatic differentiation (efficiency depends on the dimension, no error) FD: use finite difference (efficiency depends on the dimension, error exists)

GD(Gradient descent)

lr: step size, eps: stop criteria, linesearch: if true, use armijo line search with $\alpha = 0.3, \beta = 0.8$.

SGD(Subspace gradient descent[https://arxiv.org/abs/2003.02684])

lr: step size, eps: stop criteria, reduced_dim: size of random matrix, mode: only "random", linesearch: if true, use armijo line search with $\alpha = 0.3, \beta = 0.8$.

AGD(Accelerated Gradient descent)

lr: step size, eps: stop criteria, restart: if true, use function value restart.

BFGS

alpha: parameter of line search, beta: parameter of line search, eps: stop criteria

LimitedMemoryBFGS[https://en.wikipedia.org/wiki/Limited-memory_BFGS]

alpha: parameter of line search, beta: parameter of line search, eps: stop criteria memory_size: the number of past data.

BacktrackingProximalGD(proximal gradient descent with line search)

alpha: parameter of line search, beta: parameter of line search, eps: stop criteria

BacktrackingAcceleratedProximalGD(accelerated proximal gradient descent with line search)

alpha: parameter of line search, beta: parameter of line search, eps: stop criteria, restart: if true, use function value restart.

Newton method

alpha: parameter of line search, beta: parameter of line search, eps: stop criteria,

SubspaceNewton[https://arxiv.org/abs/1905.10874]

dim: the dimension of problem, reduced_dim, the size of random matrix, mode: the type of random matrix(only "random"), alpha: parameter of line search, beta: parameter of line search, eps: stop criteria,

LimitedMemoryNewton[https://link.springer.com/article/10.1007/s12532-022-00219-z]

reduced_dim: the size of subspace matrix, threshold_eigenvalue: parameter of clipping eigenvalues, mode: the type of random matrix(only "LEESELECTION"), alpha: parameter of line search, beta: parameter of line search, eps: stop criteria,

SubspaceRNM(subspace regularized newton method[https://arxiv.org/abs/2209.04170])

reduced_dim:the size of random matrix, please refer to the paper for other parameters.

Proposed method

alpha: parameter of line search, beta: parameter of line search, eps: stop criteria, reduced_dim: the size of random matrix, matrix_size: the size of subspace matrix(not random), dim: the dimension of problem, lower_eigenvalue: the parameter of clipping eigenvalues, upper_eigenvalue: the parameter of clipping eigenvalues, mode: the type of random matrix(only "random")