Gradient

A gradient is a vector that stores the partial derivatives of multi variable functions, often denoted by $\nabla$. It helps us calculate the slope at a specific point on a curve for functions with multiple independent variables.

Find a Gradient

Consider a function with two variables (x and y): $f(x,y) = x^2 + y^3$

1) Find partial derivative with respect to x (Treat y as a constant like a random number 12)

$$f'_x = \frac{\partial f}{\partial x} = 2x+0=2x$$

2) Find partial derivative with respect to y (Treat x as a constant)

$$f'_y =\frac{\partial f}{\partial y}= 0+3y^2=3y^2$$

3) Store partial derivatives in a gradient

$$\nabla f(x,y) = \begin{bmatrix}\frac{\partial f}{\partial x} \\ \frac{\partial f}{\partial y}\end{bmatrix} = \begin{bmatrix}2x \\ 3y^2\end{bmatrix}$$

Properties of Gradients

There are two additional properties of gradients that are especially useful in deep learning. A gradient:

1. Always points in the direction of greatest increase of a function (explained here)

2. Is zero at a local maximum or local minimum

Directional Derivative

The directional derivative $\nabla_{\vec v} f$ is the rate at which the function $f(x,y)$ changes at a point $(x_1,y_1)$ in the direction ${\vec v}$.

Directional derivative is computed by taking the dot product of the gradient of $f$ and a unit vector ${\vec v}$

Note: Directional derivative of a function is a scalar while gradient is a vector.

Find Directional Derivative

Consider a function with two variables (x and y): $f(x,y) = x^2 + y^3$

$${\vec v} = \begin{bmatrix}2 \\ 3\end{bmatrix}$$

As described above, we take the dot product of the gradient and the directional vector:

$$\begin{bmatrix}\frac{\partial f}{\partial x} \\ \frac{\partial f}{\partial y}\end{bmatrix} . \begin{bmatrix}2 \\ 3\end{bmatrix} \newline \newline \newline$$

We can rewrite the dot product as:

$$\nabla_{\vec v} f = 2\frac{\partial f}{\partial x} +3 \frac{\partial f}{\partial y} = 2(2x) + 3(3y^2)=4x+9y^2$$

Hence, the directional derivative $\nabla_{\vec v} f$at co-ordinates $(5, 4)$ is: $\nabla_{\vec v} f = 4x+9y^2 = 4(5)+9(4)^2 = 164$

Link: - http://wiki.fast.ai/index.php/Calculus_for_Deep_Learning