Detailed explanation of stochastic gradient descent algorithm in Python

The stochastic gradient descent algorithm is one of the commonly used optimization algorithms in machine learning. It is an optimized version of the gradient descent algorithm and can converge to the global optimal solution faster. This article will introduce the stochastic gradient descent algorithm in Python in detail, including its principles, application scenarios and code examples.

1. Principle of Stochastic Gradient Descent Algorithm

1. Gradient Descent Algorithm

Before introducing the stochastic gradient descent algorithm, let’s briefly introduce the gradient descent algorithm. . The gradient descent algorithm is one of the commonly used optimization algorithms in machine learning. Its idea is to move along the negative gradient direction of the loss function until it reaches the minimum value. Suppose there is a loss function f(x), x is a parameter, then the gradient descent algorithm can be expressed as:

x = x - learning_rate * gradient(f(x))

where learning_rate is the learning rate, gradient(f(x)) is the loss function f(x) gradient.

2. Stochastic gradient descent algorithm

The stochastic gradient descent algorithm is developed on the basis of the gradient descent algorithm. It only uses one sample at each update. gradients to update parameters instead of using the gradients of all samples, so it is faster. Specifically, the stochastic gradient descent algorithm can be expressed as:

x = x - learning_rate * gradient(f(x, y))

where (x, y) represents a sample, learning_rate is the learning rate, gradient(f(x, y)) is the loss function f(x, y) the gradient on the (x, y) sample.

The advantage of the stochastic gradient descent algorithm is that it is fast, but the disadvantage is that it is easy to fall into the local optimal solution. In order to solve this problem, people have developed some improved stochastic gradient descent algorithms, such as batch stochastic gradient descent (mini-batch SGD) and momentum gradient descent (momentum SGD).

3. Batch stochastic gradient descent algorithm

The batch stochastic gradient descent algorithm is an optimization algorithm between the gradient descent algorithm and the stochastic gradient descent algorithm. It uses the average gradient of a certain number of samples to update parameters at each update, so it is not as susceptible to the influence of a few samples as the stochastic gradient descent algorithm. Specifically, the batch stochastic gradient descent algorithm can be expressed as:

x = x - learning_rate * gradient(batch(f(x, y)))

where batch(f(x, y)) represents the calculation on the small batch data composed of (x, y) samples and their adjacent samples. The gradient of the loss function f(x, y).

4. Momentum gradient descent algorithm

The momentum gradient descent algorithm is a stochastic gradient descent algorithm that can accelerate convergence. It determines the next update by accumulating previous gradients. direction and step size. Specifically, the momentum gradient descent algorithm can be expressed as:

v = beta*v + (1-beta)*gradient(f(x, y))
x = x - learning_rate * v

where v is momentum and beta is the momentum parameter, usually taking a value of 0.9 or 0.99.

2. Stochastic Gradient Descent Algorithm Application Scenarios

The stochastic gradient descent algorithm is usually used in the training of large-scale data sets because it can converge to the global optimal solution faster. Its applicable scenarios include but are not limited to the following aspects:

1. Gradient-based optimization algorithms in deep learning.
2. Update parameters during online learning.
3. For high-dimensional data, the stochastic gradient descent algorithm can find the global optimal solution faster.
4. Processing of large-scale data sets, the stochastic gradient descent algorithm only needs to use part of the samples for training in each iteration, so it has great advantages when processing large-scale data sets.

3. Stochastic gradient descent algorithm code example

The following code is an example of using the stochastic gradient descent algorithm to train a linear regression model:

import numpy as np

class LinearRegression:
    def __init__(self, learning_rate=0.01, n_iter=100):
        self.learning_rate = learning_rate
        self.n_iter = n_iter
        self.weights = None
        self.bias = None

    def fit(self, X, y):
        n_samples, n_features = X.shape
        self.weights = np.zeros(n_features)
        self.bias = 0
        for _ in range(self.n_iter):
            for i in range(n_samples):
                y_pred = np.dot(X[i], self.weights) + self.bias
                error = y[i] - y_pred
                self.weights += self.learning_rate * error * X[i]
                self.bias += self.learning_rate * error

    def predict(self, X):
        return np.dot(X, self.weights) + self.bias

In the code, LinearRegression is a simple linear regression model that uses stochastic gradient descent algorithm to train parameters. In the fit function, only the gradient of one sample is used to update parameters for each iteration during training.

4. Summary

The stochastic gradient descent algorithm is one of the commonly used optimization algorithms in machine learning and has great advantages when training large-scale data sets. In addition to the stochastic gradient descent algorithm, there are also improved versions such as the batch stochastic gradient descent algorithm and the momentum gradient descent algorithm. In practical applications, it is necessary to select an appropriate optimization algorithm based on specific problems.

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