Application of chain rule in machine learning

The chain derivation rule is a commonly used derivation method in machine learning and is used to calculate the derivative of a composite function. The basic idea is to decompose a composite function into a combination of multiple simple functions, and then use the chain rule to derive its derivative layer by layer. Specifically, if y is a function of x, and z is a function of y, then the derivative of z with respect to x can be expressed as dz/dx=dz/dy·dy/dx. In the case of multiple nested functions, this rule can be applied layer by layer to obtain the derivative of the entire composite function. The advantage of the chain derivation rule is that it can decompose complex function derivative calculation problems into simple function derivative calculation problems. Through layer-by-layer derivation, the cumbersome calculation process can be avoided and the solution efficiency can be improved. In addition, the chain derivation rule also provides a theoretical basis for the backpropagation algorithm in machine learning, making it possible to train complex models such as neural networks. In short, the chain derivation rule is one of the indispensable tools in machine learning. It achieves efficient calculation of the derivatives of complex functions by decomposing composite functions into combinations of simple functions and using the chain rule to perform derivation layer by layer. .

More specifically, assuming that y=f(x), z=g(y) is a composite function from x to z, then the derivative of z with respect to x can be expressed as:

\frac{dz}{dx}=\frac{dz}{dy}\cdot\frac{dy}{dx}

Among them, \(\frac{dz}{dy}\) represents the derivative of function \(z\) with respect to variable \(y\), \(\frac{dy}{dx}\) represents function \(y\ ) with respect to the derivative of variable \(x\). In practical applications, we often need to apply the chain rule to more levels of function nesting, or combine it with other derivation rules to find the derivatives of more complex functions. Such a derivation process can help us study the change rules of functions, solve mathematical problems, and play an important role in the modeling and optimization process in physics, engineering and other fields.

In addition, it should be noted that the chain rule also applies to multiple variables. If y is a function of x_1, x_2, \ldots, x_n, and z is a function of y_1, y_2, \ldots, y_m, then the derivative of z with respect to x_i can be expressed in the following form:

\frac{\partial z}{\partial x_i}=\sum_{j=1}^m\frac{\partial z}{\partial y_j}\cdot\frac{\partial y_j}{\partial x_i}

Among them, \frac{\partial z}{\partial y_j} represents the partial derivative of z with respect to y_j, \frac{\partial y_j}{\partial x_i} represents the partial derivative of y_j with respect to x_i Partial derivative. This formula can be obtained by applying the chain rule layer by layer.

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