The idea for what was later called Monte Carlo method occurred to me when I was playing solitaire during my illness.

Stanislaw Ulam, Adventures of a Mathematician

The Russian Roulette offers a simple way to construct an unbiased estimator for the limit of a sequence. It allows for example to …

Backtracking step-size strategies (also known as adaptive step-size or approximate line-search) that set the step-size based on a sufficient decrease condition are the standard way to set the step-size on gradient descent and quasi-Newton methods. However, these techniques are much less common for Frank-Wolfe-like algorithms. In this blog post I …

Six: All of this has happened before.
Baltar: But the question remains, does all of this have to happen again?
Six: This time I bet no.
Baltar: You know, I've never known you to play the optimist. Why the change of heart?
Six: Mathematics. Law of averages. Let a complex …

When an optimization problem has multiple global minima, different algorithms can find different solutions, a phenomenon often referred to as the implicit bias of optimization algorithms. In this post we'll characterize the implicit bias of gradient-based methods on a class of regression problems that includes linear least squares and Huber …

This is the first of a series of blog posts on short and beautiful proofs in optimization (let me know what you think in the comments!). For this first post in the series I'll show that stochastic gradient descent (SGD) converges exponentially fast to a neighborhood of the solution.

While the most common accelerated methods like Polyak and Nesterov incorporate a momentum term, a little known fact is that simple gradient descent –no momentum– can achieve the same rate through only a well-chosen sequence of step-sizes. In this post we'll derive this method and through simulations discuss its practical …

I've seen things you people wouldn't believe.
Valleys sculpted by trigonometric functions.
Rates on fire off the shoulder of divergence.
Beams glitter in the dark near the Polyak gate.
All those landscapes will be lost in time, like tears in rain.
Time to halt.

A momentum optimizer *

We can tighten the analysis of gradient descent with momentum through a cobination of Chebyshev polynomials of the first and second kind. Following this connection, we'll derive one of the most iconic methods in optimization: Polyak momentum.

There's a fascinating link between minimization of quadratic functions and polynomials. A link that goes deep and allows to phrase optimization problems in the language of polynomials and vice versa. Using this connection, we can tap into centuries of research in the theory of polynomials and shed new light on …

A naive implementation of the logistic regression loss can results in numerical indeterminacy even for moderate values. This post takes a closer look into the source of these instabilities and discusses more robust Python implementations.