mlstatpy.optim.sgd#

class mlstatpy.optim.sgd.BaseOptimizer(coef, learning_rate_init=0.1, min_threshold=None, max_threshold=None, l1=0.0, l2=0.0)[source][source]#

Base stochastic gradient descent optimizer.

Paramètres:

coef – array, initial coefficient
learning_rate_init – float The initial learning rate used. It controls the step-size in updating the weights.
min_threshold – coefficients must be higher than min_thresold
max_threshold – coefficients must be below than max_thresold

The class holds the following attributes:

learning_rate: float, the current learning rate
coef: optimized coefficients
min_threshold, max_threshold: coefficients thresholds
l2: L2 regularization
l1: L1 regularization

iteration_ends(time_step)[source][source]#: Performs update to learning rate and potentially other states at the end of an iteration.

train(X, y, fct_loss, fct_grad, max_iter=100, early_th=None, verbose=False)[source][source]#

Optimizes the coefficients.

Paramètres:

X – datasets (array)
y – expected target
fct_loss – loss function, signature: f(coef, X, y) -> float
fct_grad – gradient function, signature: g(coef, x, y, i) -> array
max_iter – number maximum of iteration
early_th – stops the training if the error goes below this threshold
verbose – display information

Renvoie:

loss

The method keeps the best coefficients for the minimal loss.

update_coef(grad)[source][source]#

Updates coefficients with given gradient.

Paramètres:: grad – array, gradient

class mlstatpy.optim.sgd.SGDOptimizer(coef, learning_rate_init=0.1, lr_schedule='invscaling', momentum=0.9, power_t=0.5, early_th=None, min_threshold=None, max_threshold=None, l1=0.0, l2=0.0)[source][source]#

Stochastic gradient descent optimizer with momentum.

Paramètres:

coef – array, initial coefficient
learning_rate_init – float The initial learning rate used. It controls the step-size in updating the weights,
lr_schedule – {“constant”, “adaptive”, “invscaling”}, learning rate schedule for weight updates, “constant” for a constant learning rate given by learning_rate_init. “invscaling” gradually decreases the learning rate learning_rate_ at each time step t using an inverse scaling exponent of power_t. learning_rate_ = learning_rate_init / pow(t, power_t), “adaptive”, keeps the learning rate constant to learning_rate_init as long as the training keeps decreasing. Each time 2 consecutive epochs fail to decrease the training loss by tol, or fail to increase validation score by tol if “early_stopping” is on, the current learning rate is divided by 5.
momentum – float Value of momentum used, must be larger than or equal to 0
power_t – double The exponent for inverse scaling learning rate.
early_th – stops if the error goes below that threshold
min_threshold – lower bound for parameters (can be None)
max_threshold – upper bound for parameters (can be None)
l1 – L1 regularization
l2 – L2 regularization

The class holds the following attributes:

learning_rate: float, the current learning rate
velocity*: array, velocity that are used to update params

Stochastic Gradient Descent applied to linear regression

The following example how to optimize a simple linear regression.

<<<

import numpy
from mlstatpy.optim import SGDOptimizer


def fct_loss(c, X, y):
    return numpy.linalg.norm(X @ c - y) ** 2


def fct_grad(c, x, y, i=0):
    return x * (x @ c - y) * 0.1


coef = numpy.array([0.5, 0.6, -0.7])
X = numpy.random.randn(10, 3)
y = X @ coef

sgd = SGDOptimizer(numpy.random.randn(3))
sgd.train(X, y, fct_loss, fct_grad, max_iter=15, verbose=True)
print("optimized coefficients:", sgd.coef)

>>>

    0/15: loss: 8.115 lr=0.1 max(coef): 1.3 l1=0/2.8 l2=0/2.9
    1/15: loss: 2.603 lr=0.0302 max(coef): 1.1 l1=0.079/2.2 l2=0.0021/1.8
    2/15: loss: 0.9904 lr=0.0218 max(coef): 0.87 l1=0.0089/1.9 l2=5e-05/1.2
    3/15: loss: 0.593 lr=0.018 max(coef): 0.77 l1=0.0025/1.7 l2=3.7e-06/1.1
    4/15: loss: 0.3241 lr=0.0156 max(coef): 0.69 l1=0.094/1.7 l2=0.0034/1
    5/15: loss: 0.1603 lr=0.014 max(coef): 0.66 l1=0.041/1.7 l2=0.00059/1
    6/15: loss: 0.07627 lr=0.0128 max(coef): 0.65 l1=0.0077/1.7 l2=3.8e-05/1
    7/15: loss: 0.03915 lr=0.0119 max(coef): 0.67 l1=0.0059/1.8 l2=2.2e-05/1.1
    8/15: loss: 0.02361 lr=0.0111 max(coef): 0.68 l1=0.023/1.8 l2=0.0002/1.1
    9/15: loss: 0.0149 lr=0.0105 max(coef): 0.69 l1=0.00051/1.8 l2=8.8e-08/1.1
    10/15: loss: 0.01025 lr=0.00995 max(coef): 0.69 l1=0.0082/1.8 l2=3.9e-05/1.1
    11/15: loss: 0.007828 lr=0.00949 max(coef): 0.69 l1=0.0059/1.8 l2=1.2e-05/1.1
    12/15: loss: 0.005864 lr=0.00909 max(coef): 0.7 l1=0.016/1.8 l2=8.7e-05/1.1
    13/15: loss: 0.004568 lr=0.00874 max(coef): 0.7 l1=0.0069/1.8 l2=2.7e-05/1.1
    14/15: loss: 0.003654 lr=0.00842 max(coef): 0.7 l1=0.00033/1.8 l2=6.2e-08/1.1
    15/15: loss: 0.003042 lr=0.00814 max(coef): 0.7 l1=0.0061/1.8 l2=1.3e-05/1.1
    optimized coefficients: [ 0.485  0.617 -0.702]

iteration_ends(time_step)[source][source]#

Performs updates to learning rate and potential other states at the end of an iteration.

Paramètres:: time_step – int number of training samples trained on so far, used to update learning rate for “invscaling”