momentum method yet another computationally cheap approach is the momentum methodin which the steepest descent iteration ( is modified to tht+ tht at ut dt where dt tht tht- and is tuning parameter this strategy frequently performs better than the "vanillasteepest descent methodbecause the search direction is less likely to change abruptly numerical experience suggests that the vanilla steepest-descent algorithm and the levenberg-marquardt algorithm are effective for networks with shallow architecturesbut not for networks with deep architectures in comparisonthe stochastic gradient descent methodthe limited-memory bfgs algorithm or any of the adaptive gradient methods in this sectioncan frequently handle networks with many hidden layers (provided that any tuning parameters and initialization values are carefully chosen via experimentation examples in python in this section we provide two numerical examples in python in the first examplewe train neural network with the stochastic gradient descent method using the polynomial regression data from example and without using any specialized python packages in the second examplewe consider realistic application of neural network to image recognition and classification here we use the specialized open-source python package pytorch simple polynomial regression consider again the polynomial regression data set depicted in figure we use network with architecture [ [ in other wordswe have two hidden layers with neuronsresulting in learner with total of dim(th parameters to implement such neural networkwe first import the numpy and the matplotlib packagesthen read the regression problem data and define the feed-forward neural network layers neuralnetpurepython py import numpy as np import matplotlib pyplot as plt #%import data data np genfromtxt ('polyreg csv ',delimiter =',' data [, reshape - , data [, reshape - , network setup [ shape [ , , , size of layers len( - number of layers