Scaling laws have been one of the key achievements of theoretical analysis in various fields of applied mathematics and computer science, answering the following key question: How fast does my method or my algorithm converge as a function of (potentially partially) observable problem parameters. For supervised machine learning and statistics, probably the simplest and oldest…
Category: Machine learning
Machine learning concepts or tools
Unraveling spectral properties of kernel matrices – I
Since my early PhD years, I have plotted and studied eigenvalues of kernel matrices. In the simplest setting, take independent and identically distributed (i.i.d.) data, such as in the cube below in 2 dimensions, take your favorite kernels, such as the Gaussian or Abel kernels, plot eigenvalues in decreasing order, and see what happens. The…
Revisiting the classics: Jensen’s inequality
There are a few mathematical results that any researcher in applied mathematics uses on a daily basis. One of them is Jensen’s inequality, which allows bounding expectations of functions of random variables. This really happens a lot in any probabilistic arguments but also as a tool to generate inequalities and optimization algorithms. In this blog…
Rethinking SGD’s noise – II: Implicit Bias
In the previous post, we showed (or at least tried to!) how the inherent noise of the stochastic gradient descent algorithm (SGD), in the context of modern overparametrised architectures, is structured and carries two important features: (i) it vanishes for interpolating solutions and (ii) it belongs to a low-dimensional manifold spanned by the gradients. Building…
Rethinking SGD’s noise
It seemed a bit unfair to devote a blog to machine learning (ML) without talking about its current core algorithm: stochastic gradient descent (SGD). Indeed, SGD has become, year after year, the basic foundation of many algorithms used for large-scale ML problems. However, the history of stochastic approximation is much older than that of ML:…
Information theory with kernel methods
In last month blog post, I presented the von Neumann entropy. It is defined as a spectral function on positive semi-definite (PSD) matrices, and leads to a Bregman divergence called the von Neumann relative entropy (or matrix Kullback Leibler divergence), with interesting convexity properties and applications in optimization (mirror descent, or smoothing) and probability (concentration…
Playing with positive definite matrices – II: entropy edition
Symmetric positive semi-definite (PSD) matrices come up in a variety of places in machine learning, statistics, and optimization, and more generally in most domains of applied mathematics. When estimating or optimizing over the set of such matrices, several geometries can be used. The most direct one is to consider PSD matrices as a convex set…
Playing with positive definite matrices – I: matrix monotony and convexity
In a series of a few blog posts, I will present classical and non-classical results on symmetric positive definite matrices. Beyond being mathematically exciting, they arise naturally a lot in machine learning and optimization, as Hessians of twice continuously differentiable convex functions and through kernel methods. In this post, I will focus on the benefits…
Approximating integrals with Laplace’s method
Integrals appear everywhere in all scientific fields, and their numerical computation is an active area of research. In the playbook of approximation techniques, my personal favorite is “la méthode de Laplace”, a must-know for students that like to cut integrals into pieces, that comes with lots of applications. We will be concerned with integrals of…
The quest for adaptivity
Most machine learning classes and textbooks mention that there is no universal supervised learning algorithm that can do reasonably well on all learning problems. Indeed, a series of “no free lunch theorems” state that even in a simple input space, for any learning algorithm, there always exists a bad conditional distribution of outputs given inputs…