$\newcommand{\br}{\\}$ $\newcommand{\R}{\mathbb{R}}$ $\newcommand{\Q}{\mathbb{Q}}$ $\newcommand{\Z}{\mathbb{Z}}$ $\newcommand{\N}{\mathbb{N}}$ $\newcommand{\C}{\mathbb{C}}$ $\newcommand{\P}{\mathbb{P}}$ $\newcommand{\F}{\mathbb{F}}$ $\newcommand{\L}{\mathcal{L}}$ $\newcommand{\spa}[1]{\text{span}(#1)}$ $\newcommand{\dist}[1]{\text{dist}(#1)}$ $\newcommand{\max}[1]{\text{max}(#1)}$ $\newcommand{\min}[1]{\text{min}(#1)}$ $\newcommand{\supr}[1]{\text{sup}(#1)}$ $\newcommand{\infi}[1]{\text{inf}(#1)}$ $\newcommand{\set}[1]{\left\{#1\right\}}$ $\newcommand{\emptyset}{\varnothing}$ $\newcommand{\otherwise}{\text{ otherwise }}$ $\newcommand{\if}{\text{ if }}$ $\newcommand{\proj}{\text{proj}}$ $\newcommand{\union}{\cup}$ $\newcommand{\intercept}{\cap}$ $\newcommand{\abs}[1]{\left| #1 \right|}$ $\newcommand{\norm}[1]{\left\lVert#1\right\rVert}$ $\newcommand{\pare}[1]{\left(#1\right)}$ $\newcommand{\t}[1]{\text{ #1 }}$ $\newcommand{\head}{\text H}$ $\newcommand{\tail}{\text T}$ $\newcommand{\d}{\text d}$ $\newcommand{\limu}[2]{\underset{#1 \to #2}\lim}$ $\newcommand{\der}[2]{\frac{\d #1}{\d #2}}$ $\newcommand{\derw}[2]{\frac{\d #1^2}{\d^2 #2}}$ $\newcommand{\pder}[2]{\frac{\partial #1}{\partial #2}}$ $\newcommand{\pderw}[2]{\frac{\partial^2 #1}{\partial #2^2}}$ $\newcommand{\pderws}[3]{\frac{\partial^2 #1}{\partial #2 \partial #3}}$ $\newcommand{\inv}[1]{{#1}^{-1}}$ $\newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $\newcommand{\nullity}[1]{\text{nullity}(#1)}$ $\newcommand{\rank}[1]{\text{rank }#1}$ $\newcommand{\var}[1]{\text{var}(#1)}$ $\newcommand{\tr}[1]{\text{tr}(#1)}$ $\newcommand{\oto}{\text{ one-to-one }}$ $\newcommand{\ot}{\text{ onto }}$ $\newcommand{\ceil}[1]{\lceil#1\rceil}$ $\newcommand{\floor}[1]{\lfloor#1\rfloor}$ $\newcommand{\Re}[1]{\text{Re}(#1)}$ $\newcommand{\Im}[1]{\text{Im}(#1)}$ $\newcommand{\dom}[1]{\text{dom}(#1)}$ $\newcommand{\fnext}[1]{\overset{\sim}{#1}}$ $\newcommand{\transpose}[1]{#1^{\text{T}}}$ $\newcommand{\b}[1]{\boldsymbol{#1}}$ $\newcommand{\None}[1]{}$ $\newcommand{\Vcw}[2]{\begin{bmatrix} #1 \br #2 \end{bmatrix}}$ $\newcommand{\Vce}[3]{\begin{bmatrix} #1 \br #2 \br #3 \end{bmatrix}}$ $\newcommand{\Vcr}[4]{\begin{bmatrix} #1 \br #2 \br #3 \br #4 \end{bmatrix}}$ $\newcommand{\Vct}[5]{\begin{bmatrix} #1 \br #2 \br #3 \br #4 \br #5 \end{bmatrix}}$ $\newcommand{\Vcy}[6]{\begin{bmatrix} #1 \br #2 \br #3 \br #4 \br #5 \br #6 \end{bmatrix}}$ $\newcommand{\Vcu}[7]{\begin{bmatrix} #1 \br #2 \br #3 \br #4 \br #5 \br #6 \br #7 \end{bmatrix}}$ $\newcommand{\vcw}[2]{\begin{matrix} #1 \br #2 \end{matrix}}$ $\newcommand{\vce}[3]{\begin{matrix} #1 \br #2 \br #3 \end{matrix}}$ $\newcommand{\vcr}[4]{\begin{matrix} #1 \br #2 \br #3 \br #4 \end{matrix}}$ $\newcommand{\vct}[5]{\begin{matrix} #1 \br #2 \br #3 \br #4 \br #5 \end{matrix}}$ $\newcommand{\vcy}[6]{\begin{matrix} #1 \br #2 \br #3 \br #4 \br #5 \br #6 \end{matrix}}$ $\newcommand{\vcu}[7]{\begin{matrix} #1 \br #2 \br #3 \br #4 \br #5 \br #6 \br #7 \end{matrix}}$ $\newcommand{\Mqw}[2]{\begin{bmatrix} #1 & #2 \end{bmatrix}}$ $\newcommand{\Mqe}[3]{\begin{bmatrix} #1 & #2 & #3 \end{bmatrix}}$ $\newcommand{\Mqr}[4]{\begin{bmatrix} #1 & #2 & #3 & #4 \end{bmatrix}}$ $\newcommand{\Mqt}[5]{\begin{bmatrix} #1 & #2 & #3 & #4 & #5 \end{bmatrix}}$ $\newcommand{\Mwq}[2]{\begin{bmatrix} #1 \br #2 \end{bmatrix}}$ $\newcommand{\Meq}[3]{\begin{bmatrix} #1 \br #2 \br #3 \end{bmatrix}}$ $\newcommand{\Mrq}[4]{\begin{bmatrix} #1 \br #2 \br #3 \br #4 \end{bmatrix}}$ $\newcommand{\Mtq}[5]{\begin{bmatrix} #1 \br #2 \br #3 \br #4 \br #5 \end{bmatrix}}$ $\newcommand{\Mqw}[2]{\begin{bmatrix} #1 & #2 \end{bmatrix}}$ $\newcommand{\Mwq}[2]{\begin{bmatrix} #1 \br #2 \end{bmatrix}}$ $\newcommand{\Mww}[4]{\begin{bmatrix} #1 & #2 \br #3 & #4 \end{bmatrix}}$ $\newcommand{\Mqe}[3]{\begin{bmatrix} #1 & #2 & #3 \end{bmatrix}}$ $\newcommand{\Meq}[3]{\begin{bmatrix} #1 \br #2 \br #3 \end{bmatrix}}$ $\newcommand{\Mwe}[6]{\begin{bmatrix} #1 & #2 & #3\br #4 & #5 & #6 \end{bmatrix}}$ $\newcommand{\Mew}[6]{\begin{bmatrix} #1 & #2 \br #3 & #4 \br #5 & #6 \end{bmatrix}}$ $\newcommand{\Mee}[9]{\begin{bmatrix} #1 & #2 & #3 \br #4 & #5 & #6 \br #7 & #8 & #9 \end{bmatrix}}$
Definition: Level Set

A level set of a function $f: \R^n \to \R $ is the set of points $\b{x} $ satisfying $f(\b{x})$ for some constant $c $.

Note

$\nabla f(x_0)$, if it is not a zero vector, is orthogonal to the tangent vector to an arbitrary smooth curve passing through $\b{x}_0$ on the level set $f(\b{x}) = c$.

Thus, the direction of maximum rate of increase of a real-valued differentiable function at a point is orthogonal to the level set of the function through the point.

Note

In other words, the gradient acts in such a direction that for a given small displacement, the function $f $ increases more in the direction of the gradient than in any other direction.

Thus, the direction in which $\nabla f(\b{x})$ points is the direction of maximum rate of increase of $f$ at $\b{x}$. The direction in which $- \nabla f(\b{x})$ points is the direction of maximum rate of decrease of $f$ at $\b{x}$. Hence, the direction of negative gradient is a good direction to search if we want to find a function minimizer.

Proof 
    by the Cauchy-Schwarz inequality
    
  </span>
</span>
<span class="proof__expand"><a>[expand]</a></span>

By the Cauchy-Schwarz inequality, when $\norm{ d } = 1$, $\inner{ \nabla f(x)}{ \b{d}} \leq \norm{ \nabla f(\b{x})}$

where $\inner{ \nabla f(\b{x})}{ \b{d}}$ is the rate of increase of $f$ in the direction $\b{d}$ at the point $\b{x}$.

When $\b{d}$ is the direction of $\nabla f(\b{x}) $, that is $\b{d} = \frac{ \nabla f(\b{x})}{ \norm{ \nabla f(\b{x})}}$, then

$$\inner{ \nabla f(x)}{ \frac{ \nabla f(\b{x})}{ \norm{ \nabla f(\b{x})}}} = \norm{ \nabla f(\b{x})}$$

Q.E.D.

Note

Let $\b{x}_\pare{0} $ be a starting point, and consider the point $\b{x}^{(0)} - \alpha \nabla f(\b{x}^{(0)}) $. Then by Taylor’s theorem we obtain

$$f(\b{x}^{(0)} - \alpha \nabla f(\b{x}^{\pare{0}})) = f(x^{(0)}) - \alpha \norm{ \nabla f(\b{x}^{(0)})}^2 + o(\alpha)$$

Thus, if $\nabla f(\b{x}^{(0)}) \neq \b{0}$, then for sufficiently small $\alpha > 0 $, we have

$$f(\b{x}^{(0)} - \alpha \nabla f(\b{x}^{(0)})) < f(\b{x}^{(0)})$$

This means that the point $\b{x}^{(0)} - \alpha \nabla f(\b{x} ^{(0)})$ is an improvement over the point $\b{x}^{(0)}$ if we are searching for a minimizer.

Definition: Gradient Descent Algorithm

Suppose that we are given a point $\b{x}^{(k)} $. To find the next point $\b{x}^{(k+1)} $, we start at $\b{x}^{(k)} $ and move by an amount $- \alpha_k \nabla f(\b{x}^{(k)}) $, where $\alpha_k $ is a positive scalar called the step size. The above procedure leads to the following iterative algorithm:

$\b{x}^{(k+1)} = \b{x}^{(k)} - \alpha_k \nabla f(\b{x}^{(k)})$

We refer to the above as a gradient descent algorithm.