3.7 Gradients in a Deep Neural Network

In many machine learning applications, we find good model parameters by performing gradient descent, which relies on the fact that we can compute the gradient of a learning objective with respect to the parameters of the model. For a given objective function, we can obtain the gradient with respect to the model parameters using calculus and applying the chain rule. We already seen the gradient of a squared loss with respect to the parameters of a linear regression model.

Consider the function

$f(x)=\sqrt{(x^2+exp(x^2)}+cos(x^2+exp(x^2)$

By application of the chain rule, and noting that differentiation is linear,we compute the gradient

$\frac{\mathrm{d} f}{\mathrm{d} x}=\frac{2x + 2x\, exp(x^2)}{2\,\sqrt{x+exp(x^2)}}-sin(x^2+exp(x^2))(2x+exp(x^2)2x)$

Writing out the gradient in this explicit way is often impractical since it often results in a very lengthy expression for a derivative. In practice,it means that, if we are not careful, the implementation of the gradient could be significantly more expensive than computing the function, which imposes unnecessary overhead. For training deep neural network models, the back propagation algorithm (Kelley, 1960; Bryson, 1961; Dreyfus, 1962; Rumelhart et al., 1986) is an efficient way to compute the gradient of an error function with respect to the parameters of the model.

An area where the chain rule is used to an extreme is deep learning, where the function value $y$ is computed as a many-level function composition

$(f_k\circ f_{k-1}\circ \cdots f_1)(x)=f_k(f_{k-1}( \cdots f_1(x))\cdots))$

where $x$ are the inputs (e.g., images), $y$ are the observations (e.g., class labels), and every function $f_i, i = 1,\ldots,K$, possesses its own parameters.

In neural network with multipile layers we have functions $f_i(x_{i-1})=\sigma(A_{i-1}x_{i-1}+b_{i-1})$ in the $i$th layer.Here $x_{i-1}$ is the output of layer $i -1$and $\sigma$ an activation function, such as the logistic sigmoid $\frac{1}{1+e^x}$ , tanh or a rectified linear unit (ReLU). In order to train these models, we require the gradient of a loss function $L$ with respect to all model parameters $A_j , b_j$ for  $j = 1,\ldots,K$. This also requires us to compute the gradient of $L$ with respect to the inputs of each layer. For example, if we have inputs $x$ and observations $y$ and a network structure defined by

$f_0=x$

$f_i=\sigma_i(A_{i-1}f_{i-1}+b{i-1}), i =1,\ldots,K$

we may be interested in finding $A_j , b_j$  for$ j = 0,\ldots,K - 1$, such that the squared loss

$L(\theta)=\left \|y-f_k(\theta,x)\right\|^2$

is minimized , where $\theta=\{A_0,b_0,\ldots,A_{K-1},b_{K-1}\}$

To obtain the gradients with respect to the parameter set $\theta$, we require the partial derivatives of $L$ with respect to the parameters $\theta_j=\{A_j,b_j\}$ of each layer $j=0,\ldots,K-1$. The chain rule allows us to determine the partial derivatives as

The orange terms are partial derivatives of the output of a layer with respect to its inputs, whereas the blue terms are partial derivatives of the output of a layer with respect to its parameters. Assuming, we have already computed the partial derivatives $\frac{\partial L}{\partial \theta_{i+1}}$, then most of the computation can be reused to compute $\frac{\partial L}{\partial \theta_i}$,The additional terms that we need to compute are indicated by the boxes. These gradients are then passed backwards through the network.