Gradient Descent Methods¶

This tour explores the use of gradient descent method for unconstrained and constrained optimization of a smooth function

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Gradient Descent for Unconstrained Problems¶

We consider the problem of finding a minimum of a function $f$, hence solving $$\umin{x \in \RR^d} f(x)$$ where $f : \RR^d \rightarrow \RR$ is a smooth function.

Note that the minimum is not necessarily unique. In the general case, $f$ might exhibit local minima, in which case the proposed algorithms is not expected to find a global minimizer of the problem. In this tour, we restrict our attention to convex function, so that the methods will converge to a global minimizer.

The simplest method is the gradient descent, that computes $$ x^{(k+1)} = x^{(k)} - \tau_k \nabla f(x^{(k)}), $$ where $\tau_k>0$ is a step size, and $\nabla f(x) \in \RR^d$ is the gradient of $f$ at the point $x$, and $x^{(0)} \in \RR^d$ is any initial point.

In the convex case, if $f$ is of class $C^2$, in order to ensure convergence, the step size should satisfy $$ 0 < \tau_k < \frac{2}{ \sup_x \norm{Hf(x)} } $$ where $Hf(x) \in \RR^{d \times d}$ is the Hessian of $f$ at $x$ and $\norm{\cdot}$ is the spectral operator norm (largest eigenvalue).

Gradient Descent in 2-D¶

We consider a simple problem, corresponding to the minimization of a 2-D quadratic form $$ f(x) = \frac{1}{2} \pa{ x_1^2 + \eta x_2^2 } ,$$ where $\eta>0$ controls the anisotropy, and hence the difficulty, of the problem.

Anisotropy parameter $\eta$.

Function $f$.

Background image of the function.

Display the function as a 2-D image.

Gradient.

The step size should satisfy $\tau_k < 2/\eta$. We use here a constant step size.

Exercice 1: Perform the gradient descent using a fixed step size $\tau_k=\tau$. Display the decay of the energy $f(x^{(k)})$ through the iteration. Save the iterates so that |X(:,k)| corresponds to $x^{(k)}$.

Display the iterations.

Display the iteration for several different step sizes.

Gradient and Divergence of Images¶

Local differential operators like gradient, divergence and laplacian are the building blocks for variational image processing.

Load an image $x_0 \in \RR^N$ of $N=n \times n$ pixels.

Display it.

For a continuous function $g$, the gradient reads $$\nabla g(s) = \pa{ \pd{g(s)}{s_1}, \pd{g(s)}{s_2} } \in \RR^2.$$ (note that here, the variable $d$ denotes the 2-D spacial position).

We discretize this differential operator on a discrete image $x \in \RR^N$ using first order finite differences. $$(\nabla x)_i = ( x_{i_1,i_2}-x_{i_1-1,i_2}, x_{i_1,i_2}-x_{i_1,i_2-1} ) \in \RR^2.$$ Note that for simplity we use periodic boundary conditions.

Compute its gradient, using finite differences.

One thus has $\nabla : \RR^N \mapsto \RR^{N \times 2}.$

One can display each of its components.

One can display its magnitude $\norm{(\nabla x)_i}$, which is large near edges.