What is the gradient descent algorithm in Python?

What is the gradient descent algorithm in Python?

The gradient descent algorithm is a commonly used mathematical optimization technique used to find the minimum value of a function. The algorithm gradually updates the parameter values ​​of the function in an iterative manner, moving it toward the local minimum. In Python, the gradient descent algorithm is widely used in fields such as machine learning, deep learning, data science, and numerical optimization.

The principle of gradient descent algorithm

The basic principle of gradient descent algorithm is to update along the negative gradient direction of the objective function. On a two-dimensional plane, the objective function can be expressed as $f(x,y)=x^2 y^2$. We can get some example information about a function by visualizing its contours. Each contour represents a point where the function is horizontal at a fixed height. The rounder the contours of the function are, the flatter the gradient of the function is, and the steeper the gradient of the function.

In this particular example, the minimum value is at the $(0,0)$ point. We can iterate from any starting point to find a local optimal solution by gradually reducing the step size, moving in the opposite direction of the gradient. At each iteration step, we need to update our parameter values ​​by going in the opposite direction of the gradient. The variation of parameters is expressed as $ heta$:

$    heta =     heta - lpharac{partial}{partial    heta}J(    heta)$

where $ lpha$ is the step size, $J( heta)$ is the objective function, $ rac{partial}{partial heta}$ is the target Derivatives of functions. At each iteration step, the algorithm updates the value of $ heta $ until a satisfactory result is obtained.

Application of Gradient Descent Algorithm

The gradient descent algorithm is a general optimization technique that can be used to solve various problems. In machine learning, deep learning and data science, the gradient descent algorithm is widely used in the following fields:

Logistic regression: The gradient descent algorithm can be used to minimize the logistic regression loss function to obtain the best coefficient estimate value.

Linear regression: This algorithm can also be used for parameter optimization in linear regression.

Neural network: Gradient descent algorithm is the core algorithm for training neural networks. Typically, we use the backpropagation algorithm to calculate the error gradient and use it in the gradient descent optimizer.

PCA (Principal Component Analysis): The gradient descent algorithm can be used to optimize the objective function in principal component analysis to obtain a dimensionally reduced representation of the data.

Data Science: The gradient descent algorithm can be used to minimize error functions such as mean square error (MSE) to achieve modeling and prediction of data.

Summary

The gradient descent algorithm is an effective optimization technique that can be used to solve a variety of mathematical problems. In Python, the gradient descent algorithm is widely used in fields such as machine learning, deep learning, data science, and numerical optimization. When using the gradient descent algorithm, the step size parameters and the initial values ​​of the objective function need to be carefully chosen to ensure that the final result is optimal.

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