Optimisation#

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: 12.77 lr=0.1 max(coef): 1.7 l1=0/3 l2=0/3.8
    1/15: loss: 9.599 lr=0.0302 max(coef): 1.6 l1=0.45/2.5 l2=0.075/2.9
    2/15: loss: 6.492 lr=0.0218 max(coef): 1.3 l1=0.035/1.8 l2=0.00044/1.8
    3/15: loss: 5.571 lr=0.018 max(coef): 1.2 l1=0.26/1.5 l2=0.028/1.4
    4/15: loss: 5.101 lr=0.0156 max(coef): 1.1 l1=0.19/1.4 l2=0.017/1.2
    5/15: loss: 4.68 lr=0.014 max(coef): 1 l1=0.35/1.4 l2=0.047/1.1
    6/15: loss: 4.357 lr=0.0128 max(coef): 0.96 l1=0.087/1.3 l2=0.0035/1
    7/15: loss: 4.137 lr=0.0119 max(coef): 0.91 l1=0.14/1.3 l2=0.0073/0.94
    8/15: loss: 3.988 lr=0.0111 max(coef): 0.87 l1=0.094/1.3 l2=0.0041/0.87
    9/15: loss: 3.817 lr=0.0105 max(coef): 0.84 l1=0.09/1.3 l2=0.0037/0.83
    10/15: loss: 3.689 lr=0.00995 max(coef): 0.81 l1=0.21/1.3 l2=0.02/0.8
    11/15: loss: 3.573 lr=0.00949 max(coef): 0.79 l1=0.007/1.2 l2=2e-05/0.76
    12/15: loss: 3.478 lr=0.00909 max(coef): 0.77 l1=0.2/1.2 l2=0.019/0.72
    13/15: loss: 3.383 lr=0.00874 max(coef): 0.75 l1=0.025/1.2 l2=0.00023/0.7
    14/15: loss: 3.278 lr=0.00842 max(coef): 0.74 l1=0.068/1.1 l2=0.0021/0.66
    15/15: loss: 3.201 lr=0.00814 max(coef): 0.72 l1=0.014/1.1 l2=6.6e-05/0.63
    optimized coefficients: [-0.723  0.325  0.044]

Optimisation#

Gradient#

Gradient #