Asymptotics of Learning with Deep Structured (Random) Features

For a large class of feature maps we provide a tight asymptotic characterisation of the test error associated with learning the readout layer, in the high-dimensional limit where the input dimension, hidden layer widths, and number of training samples are proportionally large. This characterization is formulated in terms of the population covariance of the features. Our work is partially motivated by the problem of learning with Gaussian rainbow neural networks, namely deep non-linear fully-connected networks with random but structured weights, whose row-wise covariances are further allowed to depend on the weights of previous layers. For such networks we also derive a closed-form formula for the feature covariance in terms of the weight matrices. We further find that in some cases our results can capture feature maps learned by deep, finite-width neural networks trained under gradient descent.

Analysis of Bootstrap and Subsampling in High-dimensional Regularized Regression

We investigate popular resampling methods for estimating the uncertainty of statistical models, such as subsampling, bootstrap and the jackknife, and their performance in high-dimensional supervised regression tasks. We provide a tight asymptotic description of the biases and variances estimated by these methods in the context of generalized linear models, such as ridge and logistic regression, taking the limit where the number of samples $n$ and dimension $d$ of the covariates grow at a comparable fixed rate $\alpha\!=\! n/d$. Our findings are three-fold: i) resampling methods are fraught with problems in high dimensions and exhibit the double-descent-like behavior typical of these situations; ii) only when $\alpha$ is large enough do they provide consistent and reliable error estimations (we give convergence rates); iii) in the over-parametrized regime $\alpha\!<\!1$ relevant to modern machine learning practice, their predictions are not consistent, even with optimal regularization.

A High Dimensional Model for Adversarial Training: Geometry and Trade-Offs

This work investigates adversarial training in the context of margin-based linear classifiers in the high-dimensional regime where the dimension $d$ and the number of data points $n$ diverge with a fixed ratio $\alpha = n / d$. We introduce a tractable mathematical model where the interplay between the data and adversarial attacker geometries can be studied, while capturing the core phenomenology observed in the adversarial robustness literature. Our main theoretical contribution is an exact asymptotic description of the sufficient statistics for the adversarial empirical risk minimiser, under generic convex and non-increasing losses. Our result allow us to precisely characterise which directions in the data are associated with a higher generalisation/robustness trade-off, as defined by a robustness and a usefulness metric. In particular, we unveil the existence of directions which can be defended without penalising accuracy. Finally, we show the advantage of defending non-robust features during training, identifying a uniform protection as an inherently effective defence mechanism.

Asymptotics of feature learning in two-layer networks after one gradient-step

In this manuscript we investigate the problem of how two-layer neural networks learn features from data, and improve over the kernel regime, after being trained with a single gradient descent step. Leveraging a connection from (Ba et al., 2022) with a non-linear spiked matrix model and recent progress on Gaussian universality (Dandi et al., 2023), we provide an exact asymptotic description of the generalization error in the high-dimensional limit where the number of samples $n$, the width $p$ and the input dimension $d$ grow at a proportional rate. We characterize exactly how adapting to the data is crucial for the network to efficiently learn non-linear functions in the direction of the gradient -- where at initialization it can only express linear functions in this regime. To our knowledge, our results provides the first tight description of the impact of feature learning in the generalization of two-layer neural networks in the large learning rate regime $\eta=\Theta_{d}(d)$, beyond perturbative finite width corrections of the conjugate and neural tangent kernels.

High-dimensional robust regression under heavy-tailed data: Asymptotics and Universality

We investigate the high-dimensional properties of robust regression estimators in the presence of heavy-tailed contamination of both the covariates and response functions. In particular, we provide a sharp asymptotic characterisation of M-estimators trained on a family of elliptical covariate and noise data distributions including cases where second and higher moments do not exist. We show that, despite being consistent, the Huber loss with optimally tuned location parameter $\delta$ is suboptimal in the high-dimensional regime in the presence of heavy-tailed noise, highlighting the necessity of further regularisation to achieve optimal performance. This result also uncovers the existence of a curious transition in $\delta$ as a function of the sample complexity and contamination. Moreover, we derive the decay rates for the excess risk of ridge regression. We show that, while it is both optimal and universal for noise distributions with finite second moment, its decay rate can be considerably faster when the covariates' second moment does not exist. Finally, we show that our formulas readily generalise to a richer family of models and data distributions, such as generalised linear estimation with arbitrary convex regularisation trained on mixture models.

Learning Two-Layer Neural Networks, One (Giant) Step at a Time

We study the training dynamics of shallow neural networks, investigating the conditions under which a limited number of large batch gradient descent steps can facilitate feature learning beyond the kernel regime. We compare the influence of batch size and that of multiple (but finitely many) steps. Our analysis of a single-step process reveals that while a batch size of $n = O(d)$ enables feature learning, it is only adequate for learning a single direction, or a single-index model. In contrast, $n = O(d^2)$ is essential for learning multiple directions and specialization. Moreover, we demonstrate that ``hard'' directions, which lack the first $\ell$ Hermite coefficients, remain unobserved and require a batch size of $n = O(d^\ell)$ for being captured by gradient descent. Upon iterating a few steps, the scenario changes: a batch-size of $n = O(d)$ is enough to learn new target directions spanning the subspace linearly connected in the Hermite basis to the previously learned directions, thereby a staircase property. Our analysis utilizes a blend of techniques related to concentration, projection-based conditioning, and Gaussian equivalence that are of independent interest. By determining the conditions necessary for learning and specialization, our results highlight the interaction between batch size and number of iterations, and lead to a hierarchical depiction where learning performance exhibits a stairway to accuracy over time and batch size, shedding new light on feature learning in neural networks.

Escaping mediocrity: how two-layer networks learn hard single-index models with SGD

This study explores the sample complexity for two-layer neural networks to learn a single-index target function under Stochastic Gradient Descent (SGD), focusing on the challenging regime where many flat directions are present at initialization. It is well-established that in this scenario $n=O(d\log{d})$ samples are typically needed. However, we provide precise results concerning the pre-factors in high-dimensional contexts and for varying widths. Notably, our findings suggest that overparameterization can only enhance convergence by a constant factor within this problem class. These insights are grounded in the reduction of SGD dynamics to a stochastic process in lower dimensions, where escaping mediocrity equates to calculating an exit time. Yet, we demonstrate that a deterministic approximation of this process adequately represents the escape time, implying that the role of stochasticity may be minimal in this scenario.

Expectation consistency for calibration of neural networks

Despite their incredible performance, it is well reported that deep neural networks tend to be overoptimistic about their prediction confidence. Finding effective and efficient calibration methods for neural networks is therefore an important endeavour towards better uncertainty quantification in deep learning. In this manuscript, we introduce a novel calibration technique named expectation consistency (EC), consisting of a post-training rescaling of the last layer weights by enforcing that the average validation confidence coincides with the average proportion of correct labels. First, we show that the EC method achieves similar calibration performance to temperature scaling (TS) across different neural network architectures and data sets, all while requiring similar validation samples and computational resources. However, we argue that EC provides a principled method grounded on a Bayesian optimality principle known as the Nishimori identity. Next, we provide an asymptotic characterization of both TS and EC in a synthetic setting and show that their performance crucially depends on the target function. In particular, we discuss examples where EC significantly outperforms TS.

Universality laws for Gaussian mixtures in generalized linear models

Let $(x_{i}, y_{i})_{i=1,\dots,n}$ denote independent samples from a general mixture distribution $\sum_{c\in\mathcal{C}}\rho_{c}P_{c}^{x}$, and consider the hypothesis class of generalized linear models $\hat{y} = F(\Theta^{\top}x)$. In this work, we investigate the asymptotic joint statistics of the family of generalized linear estimators $(\Theta_{1}, \dots, \Theta_{M})$ obtained either from (a) minimizing an empirical risk $\hat{R}_{n}(\Theta;X,y)$ or (b) sampling from the associated Gibbs measure $\exp(-\beta n \hat{R}_{n}(\Theta;X,y))$. Our main contribution is to characterize under which conditions the asymptotic joint statistics of this family depends (on a weak sense) only on the means and covariances of the class conditional features distribution $P_{c}^{x}$. In particular, this allow us to prove the universality of different quantities of interest, such as the training and generalization errors, redeeming a recent line of work in high-dimensional statistics working under the Gaussian mixture hypothesis. Finally, we discuss the applications of our results to different machine learning tasks of interest, such as ensembling and uncertainty

Are Gaussian data all you need? Extents and limits of universality in high-dimensional generalized linear estimation

In this manuscript we consider the problem of generalized linear estimation on Gaussian mixture data with labels given by a single-index model. Our first result is a sharp asymptotic expression for the test and training errors in the high-dimensional regime. Motivated by the recent stream of results on the Gaussian universality of the test and training errors in generalized linear estimation, we ask ourselves the question: "when is a single Gaussian enough to characterize the error?". Our formula allow us to give sharp answers to this question, both in the positive and negative directions. More precisely, we show that the sufficient conditions for Gaussian universality (or lack of thereof) crucially depend on the alignment between the target weights and the means and covariances of the mixture clusters, which we precisely quantify. In the particular case of least-squares interpolation, we prove a strong universality property of the training error, and show it follows a simple, closed-form expression. Finally, we apply our results to real datasets, clarifying some recent discussion in the literature about Gaussian universality of the errors in this context.