Which exceptional low-dimensional projections of a Gaussian point cloud can be found in polynomial time?

Given $d$-dimensional standard Gaussian vectors $\boldsymbol{x}_1,\dots, \boldsymbol{x}_n$, we consider the set of all empirical distributions of its $m$-dimensional projections, for $m$ a fixed constant. Diaconis and Freedman (1984) proved that, if $n/d\to \infty$, all such distributions converge to the standard Gaussian distribution. In contrast, we study the proportional asymptotics, whereby $n,d\to \infty$ with $n/d\to \alpha \in (0, \infty)$. In this case, the projection of the data points along a typical random subspace is again Gaussian, but the set $\mathscr{F}_{m,\alpha}$ of all probability distributions that are asymptotically feasible as $m$-dimensional projections contains non-Gaussian distributions corresponding to exceptional subspaces. Non-rigorous methods from statistical physics yield an indirect characterization of $\mathscr{F}_{m,\alpha}$ in terms of a generalized Parisi formula. Motivated by the goal of putting this formula on a rigorous basis, and to understand whether these projections can be found efficiently, we study the subset $\mathscr{F}^{\rm alg}_{m,\alpha}\subseteq \mathscr{F}_{m,\alpha}$ of distributions that can be realized by a class of iterative algorithms. We prove that this set is characterized by a certain stochastic optimal control problem, and obtain a dual characterization of this problem in terms of a variational principle that extends Parisi's formula. As a byproduct, we obtain computationally achievable values for a class of random optimization problems including `generalized spherical perceptron' models.

Scaling laws for learning with real and surrogate data

Collecting large quantities of high-quality data is often prohibitively expensive or impractical, and a crucial bottleneck in machine learning. One may instead augment a small set of $n$ data points from the target distribution with data from more accessible sources like public datasets, data collected under different circumstances, or synthesized by generative models. Blurring distinctions, we refer to such data as `surrogate data'. We define a simple scheme for integrating surrogate data into training and use both theoretical models and empirical studies to explore its behavior. Our main findings are: $(i)$ Integrating surrogate data can significantly reduce the test error on the original distribution; $(ii)$ In order to reap this benefit, it is crucial to use optimally weighted empirical risk minimization; $(iii)$ The test error of models trained on mixtures of real and surrogate data is well described by a scaling law. This can be used to predict the optimal weighting and the gain from surrogate data.

Universality of max-margin classifiers

Maximum margin binary classification is one of the most fundamental algorithms in machine learning, yet the role of featurization maps and the high-dimensional asymptotics of the misclassification error for non-Gaussian features are still poorly understood. We consider settings in which we observe binary labels $y_i$ and either $d$-dimensional covariates ${\boldsymbol z}_i$ that are mapped to a $p$-dimension space via a randomized featurization map ${\boldsymbol \phi}:\mathbb{R}^d \to\mathbb{R}^p$, or $p$-dimensional features of non-Gaussian independent entries. In this context, we study two fundamental questions: $(i)$ At what overparametrization ratio $p/n$ do the data become linearly separable? $(ii)$ What is the generalization error of the max-margin classifier? Working in the high-dimensional regime in which the number of features $p$, the number of samples $n$ and the input dimension $d$ (in the nonlinear featurization setting) diverge, with ratios of order one, we prove a universality result establishing that the asymptotic behavior is completely determined by the expected covariance of feature vectors and by the covariance between features and labels. In particular, the overparametrization threshold and generalization error can be computed within a simpler Gaussian model. The main technical challenge lies in the fact that max-margin is not the maximizer (or minimizer) of an empirical average, but the maximizer of a minimum over the samples. We address this by representing the classifier as an average over support vectors. Crucially, we find that in high dimensions, the support vector count is proportional to the number of samples, which ultimately yields universality.

Towards a statistical theory of data selection under weak supervision

Given a sample of size $N$, it is often useful to select a subsample of smaller size $n<N$ to be used for statistical estimation or learning. Such a data selection step is useful to reduce the requirements of data labeling and the computational complexity of learning. We assume to be given $N$ unlabeled samples $\{{\boldsymbol x}_i\}_{i\le N}$, and to be given access to a `surrogate model' that can predict labels $y_i$ better than random guessing. Our goal is to select a subset of the samples, to be denoted by $\{{\boldsymbol x}_i\}_{i\in G}$, of size $|G|=n<N$. We then acquire labels for this set and we use them to train a model via regularized empirical risk minimization. By using a mixture of numerical experiments on real and synthetic data, and mathematical derivations under low- and high- dimensional asymptotics, we show that: $(i)$~Data selection can be very effective, in particular beating training on the full sample in some cases; $(ii)$~Certain popular choices in data selection methods (e.g. unbiased reweighted subsampling, or influence function-based subsampling) can be substantially suboptimal.

Six Lectures on Linearized Neural Networks

In these six lectures, we examine what can be learnt about the behavior of multi-layer neural networks from the analysis of linear models. We first recall the correspondence between neural networks and linear models via the so-called lazy regime. We then review four models for linearized neural networks: linear regression with concentrated features, kernel ridge regression, random feature model and neural tangent model. Finally, we highlight the limitations of the linear theory and discuss how other approaches can overcome them.

Sampling, Diffusions, and Stochastic Localization

Diffusions are a successful technique to sample from high-dimensional distributions can be either explicitly given or learnt from a collection of samples. They implement a diffusion process whose endpoint is a sample from the target distribution and whose drift is typically represented as a neural network. Stochastic localization is a successful technique to prove mixing of Markov Chains and other functional inequalities in high dimension. An algorithmic version of stochastic localization was introduced in [EAMS2022], to obtain an algorithm that samples from certain statistical mechanics models. This notes have three objectives: (i) Generalize the construction [EAMS2022] to other stochastic localization processes; (ii) Clarify the connection between diffusions and stochastic localization. In particular we show that standard denoising diffusions are stochastic localizations but other examples that are naturally suggested by the proposed viewpoint; (iii) Describe some insights that follow from this viewpoint.

Learning time-scales in two-layers neural networks

Gradient-based learning in multi-layer neural networks displays a number of striking features. In particular, the decrease rate of empirical risk is non-monotone even after averaging over large batches. Long plateaus in which one observes barely any progress alternate with intervals of rapid decrease. These successive phases of learning often take place on very different time scales. Finally, models learnt in an early phase are typically `simpler' or `easier to learn' although in a way that is difficult to formalize. Although theoretical explanations of these phenomena have been put forward, each of them captures at best certain specific regimes. In this paper, we study the gradient flow dynamics of a wide two-layer neural network in high-dimension, when data are distributed according to a single-index model (i.e., the target function depends on a one-dimensional projection of the covariates). Based on a mixture of new rigorous results, non-rigorous mathematical derivations, and numerical simulations, we propose a scenario for the learning dynamics in this setting. In particular, the proposed evolution exhibits separation of timescales and intermittency. These behaviors arise naturally because the population gradient flow can be recast as a singularly perturbed dynamical system.

Dimension free ridge regression

Random matrix theory has become a widely useful tool in high-dimensional statistics and theoretical machine learning. However, random matrix theory is largely focused on the proportional asymptotics in which the number of columns grows proportionally to the number of rows of the data matrix. This is not always the most natural setting in statistics where columns correspond to covariates and rows to samples. With the objective to move beyond the proportional asymptotics, we revisit ridge regression ($\ell_2$-penalized least squares) on i.i.d. data $(x_i, y_i)$, $i\le n$, where $x_i$ is a feature vector and $y_i = \beta^\top x_i +\epsilon_i \in\mathbb{R}$ is a response. We allow the feature vector to be high-dimensional, or even infinite-dimensional, in which case it belongs to a separable Hilbert space, and assume either $z_i := \Sigma^{-1/2}x_i$ to have i.i.d. entries, or to satisfy a certain convex concentration property. Within this setting, we establish non-asymptotic bounds that approximate the bias and variance of ridge regression in terms of the bias and variance of an `equivalent' sequence model (a regression model with diagonal design matrix). The approximation is up to multiplicative factors bounded by $(1\pm \Delta)$ for some explicitly small $\Delta$. Previously, such an approximation result was known only in the proportional regime and only up to additive errors: in particular, it did not allow to characterize the behavior of the excess risk when this converges to $0$. Our general theory recovers earlier results in the proportional regime (with better error rates). As a new application, we obtain a completely explicit and sharp characterization of ridge regression for Hilbert covariates with regularly varying spectrum. Finally, we analyze the overparametrized near-interpolation setting and obtain sharp `benign overfitting' guarantees.

Overparametrized linear dimensionality reductions: From projection pursuit to two-layer neural networks

Given a cloud of $n$ data points in $\mathbb{R}^d$, consider all projections onto $m$-dimensional subspaces of $\mathbb{R}^d$ and, for each such projection, the empirical distribution of the projected points. What does this collection of probability distributions look like when $n,d$ grow large? We consider this question under the null model in which the points are i.i.d. standard Gaussian vectors, focusing on the asymptotic regime in which $n,d\to\infty$, with $n/d\to\alpha\in (0,\infty)$, while $m$ is fixed. Denoting by $\mathscr{F}_{m, \alpha}$ the set of probability distributions in $\mathbb{R}^m$ that arise as low-dimensional projections in this limit, we establish new inner and outer bounds on $\mathscr{F}_{m, \alpha}$. In particular, we characterize the Wasserstein radius of $\mathscr{F}_{m,\alpha}$ up to logarithmic factors, and determine it exactly for $m=1$. We also prove sharp bounds in terms of Kullback-Leibler divergence and R\'{e}nyi information dimension. The previous question has application to unsupervised learning methods, such as projection pursuit and independent component analysis. We introduce a version of the same problem that is relevant for supervised learning, and prove a sharp Wasserstein radius bound. As an application, we establish an upper bound on the interpolation threshold of two-layers neural networks with $m$ hidden neurons.

Adversarial Examples in Random Neural Networks with General Activations

A substantial body of empirical work documents the lack of robustness in deep learning models to adversarial examples. Recent theoretical work proved that adversarial examples are ubiquitous in two-layers networks with sub-exponential width and ReLU or smooth activations, and multi-layer ReLU networks with sub-exponential width. We present a result of the same type, with no restriction on width and for general locally Lipschitz continuous activations. More precisely, given a neural network $f(\,\cdot\,;{\boldsymbol \theta})$ with random weights ${\boldsymbol \theta}$, and feature vector ${\boldsymbol x}$, we show that an adversarial example ${\boldsymbol x}'$ can be found with high probability along the direction of the gradient $\nabla_{{\boldsymbol x}}f({\boldsymbol x};{\boldsymbol \theta})$. Our proof is based on a Gaussian conditioning technique. Instead of proving that $f$ is approximately linear in a neighborhood of ${\boldsymbol x}$, we characterize the joint distribution of $f({\boldsymbol x};{\boldsymbol \theta})$ and $f({\boldsymbol x}';{\boldsymbol \theta})$ for ${\boldsymbol x}' = {\boldsymbol x}-s({\boldsymbol x})\nabla_{{\boldsymbol x}}f({\boldsymbol x};{\boldsymbol \theta})$.