Chain derivation rule in machine learning

The derivation chain rule is one of the important mathematical tools in machine learning. It is widely used in algorithms such as linear regression, logistic regression, and neural networks. This rule is an application of the chain rule in calculus and helps us calculate the derivative of a function with respect to a variable.

The composite function f(x) consists of multiple simple functions, each of which has a derivative with respect to x. According to the chain rule, the derivative of f(x) with respect to x can be obtained by multiplying and adding the derivatives of simple functions.

The formal expression is: if y=f(u) and u=g(x), then the derivative of y with respect to x dy/dx=f'(u)*g '(x).

This formula shows that by knowing the derivatives of simple functions with respect to x and how they are combined, we can calculate the derivatives of composite functions with respect to x.

The derivation chain rule plays a key role in optimization algorithms, especially in optimization algorithms such as gradient descent. It is used to update model parameters to minimize the loss function. The core idea of ​​the chain rule is that if a function is composed of multiple simple functions, then the derivative of this function with respect to a variable can be obtained by multiplying the derivatives of each simple function with respect to the variable. In machine learning, this rule is widely used to calculate the gradient of the loss function with respect to the model parameters. The effectiveness of this approach allows us to efficiently train deep neural networks via the backpropagation algorithm.

In machine learning, we often need to optimize parameters, which involves solving the derivative of the loss function with respect to the parameters. The loss function is usually a composite function composed of multiple simple functions, so we need to use the chain rule to calculate the derivative of the loss function with respect to the parameters.

Suppose we have a simple linear regression model. The output y of the model is a linear combination of the input x, that is, y=Wx b, where W and b are the parameters of the model. If we have a loss function L(y,t), where t is the true label, we can calculate the gradient of the loss function with respect to the model parameters via the chain rule:

dL/dW= dL/dy*dy/dW

##dL/db=dL/dy*dy/db

where dL/dy is the loss function Derivatives of the output, dy/dW and dy/db are the derivatives of the model’s output with respect to the parameters. Through this formula, we can calculate the gradient of the loss function on the model parameters, and then use optimization algorithms such as gradient descent to update the parameters of the model to minimize the loss function.

In more complex models, such as neural networks, the chain rule is also widely used. Neural networks usually consist of multiple nonlinear and linear layers, each with its own parameters. In order to optimize the parameters of the model to minimize the loss function, we need to calculate the gradient of the loss function for each parameter using the chain rule.

In short, the derivation chain rule is one of the very important mathematical tools in machine learning. It can help us calculate the derivative of a composite function with respect to a certain variable, and then use it to optimize the model. parameters to minimize the loss function.

The above is the detailed content of Chain derivation rule in machine learning. For more information, please follow other related articles on the PHP Chinese website!