3.8 Automatic Differentiation

It turns out that backpropagation is a special case of a general technique in numerical analysis called automatic differentiation. We can think of automatic differentiation as a set of techniques to numerically (in contrast to symbolically) evaluate the exact (up to machine precision) gradient of a function by working with intermediate variables and applying the chain rule. 

Automatic differentiation applies a series of elementary arithmetic operations, e.g., addition and multiplication and elementary functions, e.g., sin; cos; exp; log. By applying the chain rule to these operations, the gradient of quite complicated functions can be computed automatically.Automatic differentiation applies to general computer programs and has forward and reverse modes. Baydin et al. (2018) give a great overview of automatic differentiation in machine learning.

Figure shows a simple graph representing the data flow from inputs $x$ to outputs y via some intermediate variables $a,b$. If we were to compute the derivative $\frac{\mathrm{d} y}{\mathrm{d} x}$ we would apply the chain rule and obtain

$\frac{\mathrm{d} y}{\mathrm{d} x}=\frac{\mathrm{d} y}{\mathrm{d} b}\frac{\mathrm{d} b}{\mathrm{d} a}\frac{\mathrm{d} a}{\mathrm{d} x}$

Intuitively, the forward and reverse mode differ in the order of multiplication. Due to the associativity of matrix multiplication.

The following is the reverse mode where the gradients are propagated backward through the graph.

$\frac{\mathrm{d} y}{\mathrm{d} x}=\left ( \frac{\mathrm{d} y}{\mathrm{d} b}  \frac{\mathrm{d} b}{\mathrm{d} a}\right ) \frac{\mathrm{d} a}{\mathrm{d} x} $

In the forward mode, where the gradients flow with  the data from left to right through the graph.

$\frac{\mathrm{d} y}{\mathrm{d} x}= \frac{\mathrm{d} y}{\mathrm{d} b} \left ( \frac{\mathrm{d} b}{\mathrm{d} a} \frac{\mathrm{d} a}{\mathrm{d} x}\right ) $

The focus is on reverse mode automatic differentiation, which is backpropagation. In the context of neural networks, where the input dimensionality is often much higher than the dimensionality of the labels, the reverse mode is computationally significantly cheaper than the forward mode.

Example:

Note that the preceding set of equations requires fewer operations than the direct implementation of the function $f(x)$. The computation graph above shows the flow of data and computation required to obtain the function value $f$.The set of equations that include intermediate variables can be thought of as a computation graph, which is widely used in the implementation of neural network software libraries.

We can compute the derivative of elementary functions

$\frac{\partial a}{\partial x}=2x $

$\frac{\partial b}{\partial a}=exp(a) $

$\frac{\partial c}{\partial a}=\frac{\partial c}{\partial b}=1$

$\frac{\partial d}{\partial c}=\frac{1}{2\sqrt{c}} $

$\frac{\partial e}{\partial c}=-sin(c)$

$\frac{\partial f}{\partial d}=\frac{\partial f}{\partial e}=1$

By looking at the computation graph, we can compute $\frac{\partial f}{\partial x}$ by working backward from the output

$\frac{\partial f}{\partial c}=\frac{\partial f}{\partial d}\frac{\partial d}{\partial c}+\frac{\partial f}{\partial e}\frac{\partial e}{\partial c}$

$\frac{\partial f}{\partial b}=\frac{\partial f}{\partial c}\frac{\partial c}{\partial b} $

$\frac{\partial f}{\partial a}=\frac{\partial f}{\partial b}\frac{\partial b}{\partial a}+\frac{\partial f}{\partial c}\frac{\partial c}{\partial a} $

$\frac{\partial f}{\partial x}=\frac{\partial f}{\partial a}\frac{\partial a}{\partial x}$

By substituting the results of the derivatives of the elementary functions, we get

$\frac{\partial f}{\partial c}=1.\frac{1}{2\sqrt{c}}+1.(-sin(c))$

$\frac{\partial f}{\partial b}=\frac{\partial f}{\partial c}.1 $

$\frac{\partial f}{\partial a}=\frac{\partial f}{\partial b}.exp(a)+\frac{\partial f}{\partial c}.1 $

$\frac{\partial f}{\partial x}=\frac{\partial f}{\partial a}.2x$

The automatic differentiation approach above works, whenever we have a function that can be expressed as a computation graph, where the elementary functions are differentiable.