We consider the problem of learning a target function corresponding to a deep, extensive-width, non-linear neural network with random Gaussian weights. We consider the asymptotic limit where the number of samples, the input dimension and the network width are proportionally large. We propose a closed-form expression for the Bayes-optimal test error, for regression and classification tasks. We further compute closed-form expressions for the test errors of ridge regression, kernel and random features regression. We find, in particular, that optimally regularized ridge regression, as well as kernel regression, achieve Bayes-optimal performances, while the logistic loss yields a near-optimal test error for classification. We further show numerically that when the number of samples grows faster than the dimension, ridge and kernel methods become suboptimal, while neural networks achieve test error close to zero from quadratically many samples.
Luca Arnaboldi, Ludovic Stephan, Florent Krzakala et al.
This manuscript investigates the one-pass stochastic gradient descent (SGD) dynamics of a two-layer neural network trained on Gaussian data and labels generated by a similar, though not necessarily identical, target function. We rigorously analyse the limiting dynamics via a deterministic and low-dimensional description in terms of the sufficient statistics for the population risk. Our unifying analysis bridges different regimes of interest, such as the classical gradient-flow regime of vanishing learning rate, the high-dimensional regime of large input dimension, and the overparameterised "mean-field" regime of large network width, covering as well the intermediate regimes where the limiting dynamics is determined by the interplay between these behaviours. In particular, in the high-dimensional limit, the infinite-width dynamics is found to remain close to a low-dimensional subspace spanned by the target principal directions. Our results therefore provide a unifying picture of the limiting SGD dynamics with synthetic data.
Recent years witnessed the development of powerful generative models based on flows, diffusion or autoregressive neural networks, achieving remarkable success in generating data from examples with applications in a broad range of areas. A theoretical analysis of the performance and understanding of the limitations of these methods remain, however, challenging. In this paper, we undertake a step in this direction by analysing the efficiency of sampling by these methods on a class of problems with a known probability distribution and comparing it with the sampling performance of more traditional methods such as the Monte Carlo Markov chain and Langevin dynamics. We focus on a class of probability distribution widely studied in the statistical physics of disordered systems that relate to spin glasses, statistical inference and constraint satisfaction problems. We leverage the fact that sampling via flow-based, diffusion-based or autoregressive networks methods can be equivalently mapped to the analysis of a Bayes optimal denoising of a modified probability measure. Our findings demonstrate that these methods encounter difficulties in sampling stemming from the presence of a first-order phase transition along the algorithm's denoising path. Our conclusions go both ways: we identify regions of parameters where these methods are unable to sample efficiently, while that is possible using standard Monte Carlo or Langevin approaches. We also identify regions where the opposite happens: standard approaches are inefficient while the discussed generative methods work well.
Cedric Gerbelot, Emanuele Troiani, Francesca Mignacco et al.
We prove closed-form equations for the exact high-dimensional asymptotics of a family of first order gradient-based methods, learning an estimator (e.g. M-estimator, shallow neural network, ...) from observations on Gaussian data with empirical risk minimization. This includes widely used algorithms such as stochastic gradient descent (SGD) or Nesterov acceleration. The obtained equations match those resulting from the discretization of dynamical mean-field theory (DMFT) equations from statistical physics when applied to gradient flow. Our proof method allows us to give an explicit description of how memory kernels build up in the effective dynamics, and to include non-separable update functions, allowing datasets with non-identity covariance matrices. Finally, we provide numerical implementations of the equations for SGD with generic extensive batch-size and with constant learning rates.
Luca Pesce, Florent Krzakala, Bruno Loureiro et al.
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.
Federica Gerace, Florent Krzakala, Bruno Loureiro et al.
While classical in many theoretical settings - and in particular in statistical physics-inspired works - the assumption of Gaussian i.i.d. input data is often perceived as a strong limitation in the context of statistics and machine learning. In this study, we redeem this line of work in the case of generalized linear classification, a.k.a. the perceptron model, with random labels. We argue that there is a large universality class of high-dimensional input data for which we obtain the same minimum training loss as for Gaussian data with corresponding data covariance. In the limit of vanishing regularization, we further demonstrate that the training loss is independent of the data covariance. On the theoretical side, we prove this universality for an arbitrary mixture of homogeneous Gaussian clouds. Empirically, we show that the universality holds also for a broad range of real datasets.
Hugo Cui, Florent Krzakala, Eric Vanden-Eijnden et al.
We study the problem of training a flow-based generative model, parametrized by a two-layer autoencoder, to sample from a high-dimensional Gaussian mixture. We provide a sharp end-to-end analysis of the problem. First, we provide a tight closed-form characterization of the learnt velocity field, when parametrized by a shallow denoising auto-encoder trained on a finite number $n$ of samples from the target distribution. Building on this analysis, we provide a sharp description of the corresponding generative flow, which pushes the base Gaussian density forward to an approximation of the target density. In particular, we provide closed-form formulae for the distance between the mean of the generated mixture and the mean of the target mixture, which we show decays as $Θ_n(\frac{1}{n})$. Finally, this rate is shown to be in fact Bayes-optimal.
Lucas Clarté, Bruno Loureiro, Florent Krzakala et al.
Uncertainty quantification is a central challenge in reliable and trustworthy machine learning. Naive measures such as last-layer scores are well-known to yield overconfident estimates in the context of overparametrized neural networks. Several methods, ranging from temperature scaling to different Bayesian treatments of neural networks, have been proposed to mitigate overconfidence, most often supported by the numerical observation that they yield better calibrated uncertainty measures. In this work, we provide a sharp comparison between popular uncertainty measures for binary classification in a mathematically tractable model for overparametrized neural networks: the random features model. We discuss a trade-off between classification accuracy and calibration, unveiling a double descent like behavior in the calibration curve of optimally regularized estimators as a function of overparametrization. This is in contrast with the empirical Bayes method, which we show to be well calibrated in our setting despite the higher generalization error and overparametrization.
In this paper, we study a spiked Wigner problem with an inhomogeneous noise profile. Our aim in this problem is to recover the signal passed through an inhomogeneous low-rank matrix channel. While the information-theoretic performances are well-known, we focus on the algorithmic problem. We derive an approximate message-passing algorithm (AMP) for the inhomogeneous problem and show that its rigorous state evolution coincides with the information-theoretic optimal Bayes fixed-point equations. We identify in particular the existence of a statistical-to-computational gap where known algorithms require a signal-to-noise ratio bigger than the information-theoretic threshold to perform better than random. Finally, from the adapted AMP iteration we deduce a simple and efficient spectral method that can be used to recover the transition for matrices with general variance profiles. This spectral method matches the conjectured optimal computational phase transition.
Lucas Clarté, Bruno Loureiro, Florent Krzakala et al.
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.
Luca Pesce, Bruno Loureiro, Florent Krzakala et al.
A simple model to study subspace clustering is the high-dimensional $k$-Gaussian mixture model where the cluster means are sparse vectors. Here we provide an exact asymptotic characterization of the statistically optimal reconstruction error in this model in the high-dimensional regime with extensive sparsity, i.e. when the fraction of non-zero components of the cluster means $ρ$, as well as the ratio $α$ between the number of samples and the dimension are fixed, while the dimension diverges. We identify the information-theoretic threshold below which obtaining a positive correlation with the true cluster means is statistically impossible. Additionally, we investigate the performance of the approximate message passing (AMP) algorithm analyzed via its state evolution, which is conjectured to be optimal among polynomial algorithm for this task. We identify in particular the existence of a statistical-to-computational gap between the algorithm that require a signal-to-noise ratio $λ_{\text{alg}} \ge k / \sqrtα $ to perform better than random, and the information theoretic threshold at $λ_{\text{it}} \approx \sqrt{-k ρ\logρ} / \sqrtα$. Finally, we discuss the case of sub-extensive sparsity $ρ$ by comparing the performance of the AMP with other sparsity-enhancing algorithms, such as sparse-PCA and diagonal thresholding.
Antoine Maillard, Emanuele Troiani, Simon Martin et al.
We consider the problem of learning a target function corresponding to a single hidden layer neural network, with a quadratic activation function after the first layer, and random weights. We consider the asymptotic limit where the input dimension and the network width are proportionally large. Recent work [Cui & al '23] established that linear regression provides Bayes-optimal test error to learn such a function when the number of available samples is only linear in the dimension. That work stressed the open challenge of theoretically analyzing the optimal test error in the more interesting regime where the number of samples is quadratic in the dimension. In this paper, we solve this challenge for quadratic activations and derive a closed-form expression for the Bayes-optimal test error. We also provide an algorithm, that we call GAMP-RIE, which combines approximate message passing with rotationally invariant matrix denoising, and that asymptotically achieves the optimal performance. Technically, our result is enabled by establishing a link with recent works on optimal denoising of extensive-rank matrices and on the ellipsoid fitting problem. We further show empirically that, in the absence of noise, randomly-initialized gradient descent seems to sample the space of weights, leading to zero training loss, and averaging over initialization leads to a test error equal to the Bayes-optimal one.
Ziao Wang, Kilian Müller, Matthew Filipovich et al.
Modern deep learning relies nearly exclusively on dedicated electronic hardware accelerators. Photonic approaches, with low consumption and high operation speed, are increasingly considered for inference but, to date, remain mostly limited to relatively basic tasks. Simultaneously, the problem of training deep and complex neural networks, overwhelmingly performed through backpropagation, remains a significant limitation to the size and, consequently, the performance of current architectures and a major compute and energy bottleneck. Here, we experimentally implement a versatile and scalable training algorithm, called direct feedback alignment, on a hybrid electronic-photonic platform. An optical processing unit performs large-scale random matrix multiplications, which is the central operation of this algorithm, at speeds up to 1500 TeraOPS under 30 Watts of power. We perform optical training of modern deep learning architectures, including Transformers, with more than 1B parameters, and obtain good performances on language, vision, and diffusion-based generative tasks. We study the scaling of the training time, and demonstrate a potential advantage of our hybrid opto-electronic approach for ultra-deep and wide neural networks, thus opening a promising route to sustain the exponential growth of modern artificial intelligence beyond traditional von Neumann approaches.
Alessandro Cappelli, Ruben Ohana, Julien Launay et al.
We propose a new defense mechanism against adversarial attacks inspired by an optical co-processor, providing robustness without compromising natural accuracy in both white-box and black-box settings. This hardware co-processor performs a nonlinear fixed random transformation, where the parameters are unknown and impossible to retrieve with sufficient precision for large enough dimensions. In the white-box setting, our defense works by obfuscating the parameters of the random projection. Unlike other defenses relying on obfuscated gradients, we find we are unable to build a reliable backward differentiable approximation for obfuscated parameters. Moreover, while our model reaches a good natural accuracy with a hybrid backpropagation - synthetic gradient method, the same approach is suboptimal if employed to generate adversarial examples. We find the combination of a random projection and binarization in the optical system also improves robustness against various types of black-box attacks. Finally, our hybrid training method builds robust features against transfer attacks. We demonstrate our approach on a VGG-like architecture, placing the defense on top of the convolutional features, on CIFAR-10 and CIFAR-100. Code is available at https://github.com/lightonai/adversarial-robustness-by-design.
Yatin Dandi, Emanuele Troiani, Luca Arnaboldi et al.
We investigate the training dynamics of two-layer neural networks when learning multi-index target functions. We focus on multi-pass gradient descent (GD) that reuses the batches multiple times and show that it significantly changes the conclusion about which functions are learnable compared to single-pass gradient descent. In particular, multi-pass GD with finite stepsize is found to overcome the limitations of gradient flow and single-pass GD given by the information exponent (Ben Arous et al., 2021) and leap exponent (Abbe et al., 2023) of the target function. We show that upon re-using batches, the network achieves in just two time steps an overlap with the target subspace even for functions not satisfying the staircase property (Abbe et al., 2021). We characterize the (broad) class of functions efficiently learned in finite time. The proof of our results is based on the analysis of the Dynamical Mean-Field Theory (DMFT). We further provide a closed-form description of the dynamical process of the low-dimensional projections of the weights, and numerical experiments illustrating the theory.
Many empirical studies have provided evidence for the emergence of algorithmic mechanisms (abilities) in the learning of language models, that lead to qualitative improvements of the model capabilities. Yet, a theoretical characterization of how such mechanisms emerge remains elusive. In this paper, we take a step in this direction by providing a tight theoretical analysis of the emergence of semantic attention in a solvable model of dot-product attention. More precisely, we consider a non-linear self-attention layer with trainable tied and low-rank query and key matrices. In the asymptotic limit of high-dimensional data and a comparably large number of training samples we provide a tight closed-form characterization of the global minimum of the non-convex empirical loss landscape. We show that this minimum corresponds to either a positional attention mechanism (with tokens attending to each other based on their respective positions) or a semantic attention mechanism (with tokens attending to each other based on their meaning), and evidence an emergent phase transition from the former to the latter with increasing sample complexity. Finally, we compare the dot-product attention layer to a linear positional baseline, and show that it outperforms the latter using the semantic mechanism provided it has access to sufficient data.
Luca Arnaboldi, Yatin Dandi, Florent Krzakala et al.
Neural networks can identify low-dimensional relevant structures within high-dimensional noisy data, yet our mathematical understanding of how they do so remains scarce. Here, we investigate the training dynamics of two-layer shallow neural networks trained with gradient-based algorithms, and discuss how they learn pertinent features in multi-index models, that is target functions with low-dimensional relevant directions. In the high-dimensional regime, where the input dimension $d$ diverges, we show that a simple modification of the idealized single-pass gradient descent training scenario, where data can now be repeated or iterated upon twice, drastically improves its computational efficiency. In particular, it surpasses the limitations previously believed to be dictated by the Information and Leap exponents associated with the target function to be learned. Our results highlight the ability of networks to learn relevant structures from data alone without any pre-processing. More precisely, we show that (almost) all directions are learned with at most $O(d \log d)$ steps. Among the exceptions is a set of hard functions that includes sparse parities. In the presence of coupling between directions, however, these can be learned sequentially through a hierarchical mechanism that generalizes the notion of staircase functions. Our results are proven by a rigorous study of the evolution of the relevant statistics for high-dimensional dynamics.
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 the insight from (Ba et al., 2022), we model the trained network by a spiked Random Features (sRF) model. Further building on recent progress on Gaussian universality (Dandi et al., 2023), we provide an exact asymptotic description of the generalization error of the sRF in the high-dimensional limit where the number of samples, the width, and the input dimension grow at a proportional rate. The resulting characterization for sRFs also captures closely the learning curves of the original network model. This enables us to understand 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.
Emanuele Troiani, Yatin Dandi, Leonardo Defilippis et al.
Multi-index models - functions which only depend on the covariates through a non-linear transformation of their projection on a subspace - are a useful benchmark for investigating feature learning with neural nets. This paper examines the theoretical boundaries of efficient learnability in this hypothesis class, focusing on the minimum sample complexity required for weakly recovering their low-dimensional structure with first-order iterative algorithms, in the high-dimensional regime where the number of samples $n\!=\!αd$ is proportional to the covariate dimension $d$. Our findings unfold in three parts: (i) we identify under which conditions a trivial subspace can be learned with a single step of a first-order algorithm for any $α\!>\!0$; (ii) if the trivial subspace is empty, we provide necessary and sufficient conditions for the existence of an easy subspace where directions that can be learned only above a certain sample complexity $α\!>\!α_c$, where $α_{c}$ marks a computational phase transition. In a limited but interesting set of really hard directions -- akin to the parity problem -- $α_c$ is found to diverge. Finally, (iii) we show that interactions between different directions can result in an intricate hierarchical learning phenomenon, where directions can be learned sequentially when coupled to easier ones. We discuss in detail the grand staircase picture associated to these functions (and contrast it with the original staircase one). Our theory builds on the optimality of approximate message-passing among first-order iterative methods, delineating the fundamental learnability limit across a broad spectrum of algorithms, including neural networks trained with gradient descent, which we discuss in this context.
In this manuscript, we study the learning of deep attention neural networks, defined as the composition of multiple self-attention layers, with tied and low-rank weights. We first establish a mapping of such models to sequence multi-index models, a generalization of the widely studied multi-index model to sequential covariates, for which we establish a number of general results. In the context of Bayesian-optimal learning, in the limit of large dimension $D$ and commensurably large number of samples $N$, we derive a sharp asymptotic characterization of the optimal performance as well as the performance of the best-known polynomial-time algorithm for this setting --namely approximate message-passing--, and characterize sharp thresholds on the minimal sample complexity required for better-than-random prediction performance. Our analysis uncovers, in particular, how the different layers are learned sequentially. Finally, we discuss how this sequential learning can also be observed in a realistic setup.
A key property of neural networks is their capacity of adapting to data during training. Yet, our current mathematical understanding of feature learning and its relationship to generalization remain limited. In this work, we provide a random matrix analysis of how fully-connected two-layer neural networks adapt to the target function after a single, but aggressive, gradient descent step. We rigorously establish the equivalence between the updated features and an isotropic spiked random feature model, in the limit of large batch size. For the latter model, we derive a deterministic equivalent description of the feature empirical covariance matrix in terms of certain low-dimensional operators. This allows us to sharply characterize the impact of training in the asymptotic feature spectrum, and in particular, provides a theoretical grounding for how the tails of the feature spectrum modify with training. The deterministic equivalent further yields the exact asymptotic generalization error, shedding light on the mechanisms behind its improvement in the presence of feature learning. Our result goes beyond standard random matrix ensembles, and therefore we believe it is of independent technical interest. Different from previous work, our result holds in the challenging maximal learning rate regime, is fully rigorous and allows for finitely supported second layer initialization, which turns out to be crucial for studying the functional expressivity of the learned features. This provides a sharp description of the impact of feature learning in the generalization of two-layer neural networks, beyond the random features and lazy training regimes.
We consider the task of estimating a low-rank matrix from non-linear and noisy observations. We prove a strong universality result showing that Bayes-optimal performances are characterized by an equivalent Gaussian model with an effective prior, whose parameters are entirely determined by an expansion of the non-linear function. In particular, we show that to reconstruct the signal accurately, one requires a signal-to-noise ratio growing as $N^{\frac 12 (1-1/k_F)}$, where $k_F$ is the first non-zero Fisher information coefficient of the function. We provide asymptotic characterization for the minimal achievable mean squared error (MMSE) and an approximate message-passing algorithm that reaches the MMSE under conditions analogous to the linear version of the problem. We also provide asymptotic errors achieved by methods such as principal component analysis combined with Bayesian denoising, and compare them with Bayes-optimal MMSE.
Lucas Clarté, Adrien Vandenbroucque, Guillaume Dalle et al.
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 $α\!=\! n/d$. 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 $α$ is large enough do they provide consistent and reliable error estimations (we give convergence rates); iii) in the over-parametrized regime $α\!<\!1$ relevant to modern machine learning practice, their predictions are not consistent, even with optimal regularization.
Matteo Vilucchio, Yatin Dandi, Cedric Gerbelot et al.
The analytic characterization of the high-dimensional behavior of optimization for Generalized Linear Models (GLMs) with Gaussian data has been a central focus in statistics and probability in recent years. While convex cases, such as the LASSO, ridge regression, and logistic regression, have been extensively studied using a variety of techniques, the non-convex case remains far less understood despite its significance. A non-rigorous statistical physics framework has provided remarkable predictions for the behavior of high-dimensional optimization problems, but rigorously establishing their validity for non-convex problems has remained a fundamental challenge. In this work, we address this challenge by developing a systematic framework that rigorously proves replica-symmetric formulas for non-convex GLMs and precisely determines the conditions under which these formulas are valid. Remarkably, the rigorous replica-symmetric predictions align exactly with the conjectures made by physicists, and the so-called replicon condition. The originality of our approach lies in connecting two powerful theoretical tools: the Gaussian Min-Max Theorem, which we use to provide precise lower bounds, and Approximate Message Passing (AMP), which is shown to achieve these bounds algorithmically. We demonstrate the utility of this framework through significant applications: (i) by proving the optimality of the Tukey loss over the more commonly used Huber loss under a $\varepsilon$ contaminated data model, (ii) establishing the optimality of negative regularization in high-dimensional non-convex regression and (iii) characterizing the performance limits of linearized AMP algorithms. By rigorously validating statistical physics predictions in non-convex settings, we aim to open new pathways for analyzing increasingly complex optimization landscapes beyond the convex regime.
We discuss the inhomogeneous spiked Wigner model, a theoretical framework recently introduced to study structured noise in various learning scenarios, through the prism of random matrix theory, with a specific focus on its spectral properties. Our primary objective is to find an optimal spectral method and to extend the celebrated \cite{BBP} (BBP) phase transition criterion -- well-known in the homogeneous case -- to our inhomogeneous, block-structured, Wigner model. We provide a thorough rigorous analysis of a transformed matrix and show that the transition for the appearance of 1) an outlier outside the bulk of the limiting spectral distribution and 2) a positive overlap between the associated eigenvector and the signal, occurs precisely at the optimal threshold, making the proposed spectral method optimal within the class of iterative methods for the inhomogeneous Wigner problem.
Luca Arnaboldi, Bruno Loureiro, Ludovic Stephan et al.
We study the dynamics of stochastic gradient descent (SGD) for a class of sequence models termed Sequence Single-Index (SSI) models, where the target depends on a single direction in input space applied to a sequence of tokens. This setting generalizes classical single-index models to the sequential domain, encompassing simplified one-layer attention architectures. We derive a closed-form expression for the population loss in terms of a pair of sufficient statistics capturing semantic and positional alignment, and characterize the induced high-dimensional SGD dynamics for these coordinates. Our analysis reveals two distinct training phases: escape from uninformative initialization and alignment with the target subspace, and demonstrates how the sequence length and positional encoding influence convergence speed and learning trajectories. These results provide a rigorous and interpretable foundation for understanding how sequential structure in data can be beneficial for learning with attention-based models.
Leonardo Defilippis, Yatin Dandi, Pierre Mergny et al.
We consider the problem of how many samples from a Gaussian multi-index model are required to weakly reconstruct the relevant index subspace. Despite its increasing popularity as a testbed for investigating the computational complexity of neural networks, results beyond the single-index setting remain elusive. In this work, we introduce spectral algorithms based on the linearization of a message passing scheme tailored to this problem. Our main contribution is to show that the proposed methods achieve the optimal reconstruction threshold. Leveraging a high-dimensional characterization of the algorithms, we show that above the critical threshold the leading eigenvector correlates with the relevant index subspace, a phenomenon reminiscent of the Baik-Ben Arous-Peche (BBP) transition in spiked models arising in random matrix theory. Supported by numerical experiments and a rigorous theoretical framework, our work bridges critical gaps in the computational limits of weak learnability in multi-index model.
Kasimir Tanner, Matteo Vilucchio, Bruno Loureiro et al.
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 $α= n / d$. 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 for a Block Feature Model. 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. We show that the the presence of multiple different feature types is crucial to the high sample complexity performances of adversarial training. 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.
Leonardo Defilippis, Yizhou Xu, Julius Girardin et al.
Neural scaling laws underlie many of the recent advances in deep learning, yet their theoretical understanding remains largely confined to linear models. In this work, we present a systematic analysis of scaling laws for quadratic and diagonal neural networks in the feature learning regime. Leveraging connections with matrix compressed sensing and LASSO, we derive a detailed phase diagram for the scaling exponents of the excess risk as a function of sample complexity and weight decay. This analysis uncovers crossovers between distinct scaling regimes and plateau behaviors, mirroring phenomena widely reported in the empirical neural scaling literature. Furthermore, we establish a precise link between these regimes and the spectral properties of the trained network weights, which we characterize in detail. As a consequence, we provide a theoretical validation of recent empirical observations connecting the emergence of power-law tails in the weight spectrum with network generalization performance, yielding an interpretation from first principles.
Hugo Tabanelli, Pierre Mergny, Lenka Zdeborova et al.
We study the recovery of multiple high-dimensional signals from two noisy, correlated modalities: a spiked matrix and a spiked tensor sharing a common low-rank structure. This setting generalizes classical spiked matrix and tensor models, unveiling intricate interactions between inference channels and surprising algorithmic behaviors. Notably, while the spiked tensor model is typically intractable at low signal-to-noise ratios, its correlation with the matrix enables efficient recovery via Bayesian Approximate Message Passing, inducing staircase-like phase transitions reminiscent of neural network phenomena. In contrast, empirical risk minimization for joint learning fails: the tensor component obstructs effective matrix recovery, and joint optimization significantly degrades performance, highlighting the limitations of naive multi-modal learning. We show that a simple Sequential Curriculum Learning strategy-first recovering the matrix, then leveraging it to guide tensor recovery-resolves this bottleneck and achieves optimal weak recovery thresholds. This strategy, implementable with spectral methods, emphasizes the critical role of structural correlation and learning order in multi-modal high-dimensional inference.
Understanding the advantages of deep neural networks trained by gradient descent (GD) compared to shallow models remains an open theoretical challenge. In this paper, we introduce a class of target functions (single and multi-index Gaussian hierarchical targets) that incorporate a hierarchy of latent subspace dimensionalities. This framework enables us to analytically study the learning dynamics and generalization performance of deep networks compared to shallow ones in the high-dimensional limit. Specifically, our main theorem shows that feature learning with GD successively reduces the effective dimensionality, transforming a high-dimensional problem into a sequence of lower-dimensional ones. This enables learning the target function with drastically less samples than with shallow networks. While the results are proven in a controlled training setting, we also discuss more common training procedures and argue that they learn through the same mechanisms.
Luca Arnaboldi, Yatin Dandi, Florent Krzakala et al.
We study the impact of the batch size $n_b$ on the iteration time $T$ of training two-layer neural networks with one-pass stochastic gradient descent (SGD) on multi-index target functions of isotropic covariates. We characterize the optimal batch size minimizing the iteration time as a function of the hardness of the target, as characterized by the information exponents. We show that performing gradient updates with large batches $n_b \lesssim d^{\frac{\ell}{2}}$ minimizes the training time without changing the total sample complexity, where $\ell$ is the information exponent of the target to be learned \citep{arous2021online} and $d$ is the input dimension. However, larger batch sizes than $n_b \gg d^{\frac{\ell}{2}}$ are detrimental for improving the time complexity of SGD. We provably overcome this fundamental limitation via a different training protocol, \textit{Correlation loss SGD}, which suppresses the auto-correlation terms in the loss function. We show that one can track the training progress by a system of low-dimensional ordinary differential equations (ODEs). Finally, we validate our theoretical results with numerical experiments.
Matteo Vilucchio, Emanuele Troiani, Vittorio Erba et al.
We study robust linear regression in high-dimension, when both the dimension $d$ and the number of data points $n$ diverge with a fixed ratio $α=n/d$, and study a data model that includes outliers. We provide exact asymptotics for the performances of the empirical risk minimisation (ERM) using $\ell_2$-regularised $\ell_2$, $\ell_1$, and Huber losses, which are the standard approach to such problems. We focus on two metrics for the performance: the generalisation error to similar datasets with outliers, and the estimation error of the original, unpolluted function. Our results are compared with the information theoretic Bayes-optimal estimation bound. For the generalization error, we find that optimally-regularised ERM is asymptotically consistent in the large sample complexity limit if one perform a simple calibration, and compute the rates of convergence. For the estimation error however, we show that due to a norm calibration mismatch, the consistency of the estimator requires an oracle estimate of the optimal norm, or the presence of a cross-validation set not corrupted by the outliers. We examine in detail how performance depends on the loss function and on the degree of outlier corruption in the training set and identify a region of parameters where the optimal performance of the Huber loss is identical to that of the $\ell_2$ loss, offering insights into the use cases of different loss functions.
Yatin Dandi, Florent Krzakala, Bruno Loureiro et al.
For high-dimensional Gaussian data, we investigate theoretically how the features of a two-layer neural network adapt to the structure of the target function through a few large batch gradient descent steps, leading to an improvement in the approximation capacity from initialization. First, we compare the influence of batch size to that of multiple steps. For a single step, a batch of size $n = \mathcal{O}(d)$ is both necessary and sufficient to align with the target function, although only a single direction can be learned. In contrast, $n = \mathcal{O}(d^2)$ is essential for neurons to specialize in multiple relevant directions of the target with a single gradient step. Even in this case, we show there might exist ``hard'' directions requiring $n = \mathcal{O}(d^\ell)$ samples to be learned, where $\ell$ is known as the leap index of the target. Second, we show that the picture drastically improves over multiple gradient steps: a batch size of $n = \mathcal{O}(d)$ is indeed sufficient to learn multiple target directions satisfying a staircase property, where more and more directions can be learned over time. Finally, we discuss how these directions allow for a drastic improvement in the approximation capacity and generalization error over the initialization, illustrating a separation of scale between the random features/lazy regime and the feature learning regime. Our technical analysis leverages a combination of techniques related to concentration, projection-based conditioning, and Gaussian equivalence, which we believe are of independent interest. By pinning down the conditions necessary for specialization and learning, our results highlight the intertwined role of the structure of the task to learn, the details of the algorithm, and the architecture, shedding new light on how neural networks adapt to the feature and learn complex task from data over time.
Luca Arnaboldi, Florent Krzakala, Bruno Loureiro et al.
This study explores the sample complexity for two-layer neural networks to learn a generalized linear 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.
Lucas Clarté, Bruno Loureiro, Florent Krzakala et al.
Being able to reliably assess not only the \emph{accuracy} but also the \emph{uncertainty} of models' predictions is an important endeavour in modern machine learning. Even if the model generating the data and labels is known, computing the intrinsic uncertainty after learning the model from a limited number of samples amounts to sampling the corresponding posterior probability measure. Such sampling is computationally challenging in high-dimensional problems and theoretical results on heuristic uncertainty estimators in high-dimensions are thus scarce. In this manuscript, we characterise uncertainty for learning from limited number of samples of high-dimensional Gaussian input data and labels generated by the probit model. In this setting, the Bayesian uncertainty (i.e. the posterior marginals) can be asymptotically obtained by the approximate message passing algorithm, bypassing the canonical but costly Monte Carlo sampling of the posterior. We then provide a closed-form formula for the joint statistics between the logistic classifier, the uncertainty of the statistically optimal Bayesian classifier and the ground-truth probit uncertainty. The formula allows us to investigate calibration of the logistic classifier learning from limited amount of samples. We discuss how over-confidence can be mitigated by appropriately regularising.
Rodrigo Veiga, Ludovic Stephan, Bruno Loureiro et al.
Despite the non-convex optimization landscape, over-parametrized shallow networks are able to achieve global convergence under gradient descent. The picture can be radically different for narrow networks, which tend to get stuck in badly-generalizing local minima. Here we investigate the cross-over between these two regimes in the high-dimensional setting, and in particular investigate the connection between the so-called mean-field/hydrodynamic regime and the seminal approach of Saad & Solla. Focusing on the case of Gaussian data, we study the interplay between the learning rate, the time scale, and the number of hidden units in the high-dimensional dynamics of stochastic gradient descent (SGD). Our work builds on a deterministic description of SGD in high-dimensions from statistical physics, which we extend and for which we provide rigorous convergence rates.
Bruno Loureiro, Cédric Gerbelot, Maria Refinetti et al.
From the sampling of data to the initialisation of parameters, randomness is ubiquitous in modern Machine Learning practice. Understanding the statistical fluctuations engendered by the different sources of randomness in prediction is therefore key to understanding robust generalisation. In this manuscript we develop a quantitative and rigorous theory for the study of fluctuations in an ensemble of generalised linear models trained on different, but correlated, features in high-dimensions. In particular, we provide a complete description of the asymptotic joint distribution of the empirical risk minimiser for generic convex loss and regularisation in the high-dimensional limit. Our result encompasses a rich set of classification and regression tasks, such as the lazy regime of overparametrised neural networks, or equivalently the random features approximation of kernels. While allowing to study directly the mitigating effect of ensembling (or bagging) on the bias-variance decomposition of the test error, our analysis also helps disentangle the contribution of statistical fluctuations, and the singular role played by the interpolation threshold that are at the roots of the "double-descent" phenomenon.
We consider the problem of kernel classification. While worst-case bounds on the decay rate of the prediction error with the number of samples are known for some classifiers, they often fail to accurately describe the learning curves of real data sets. In this work, we consider the important class of data sets satisfying the standard source and capacity conditions, comprising a number of real data sets as we show numerically. Under the Gaussian design, we derive the decay rates for the misclassification (prediction) error as a function of the source and capacity coefficients. We do so for two standard kernel classification settings, namely margin-maximizing Support Vector Machines (SVM) and ridge classification, and contrast the two methods. We find that our rates tightly describe the learning curves for this class of data sets, and are also observed on real data. Our results can also be seen as an explicit prediction of the exponents of a scaling law for kernel classification that is accurate on some real datasets.
Ali Bereyhi, Bruno Loureiro, Florent Krzakala et al.
Unlike the classical linear model, nonlinear generative models have been addressed sparsely in the literature of statistical learning. This work aims to bringing attention to these models and their secrecy potential. To this end, we invoke the replica method to derive the asymptotic normalized cross entropy in an inverse probability problem whose generative model is described by a Gaussian random field with a generic covariance function. Our derivations further demonstrate the asymptotic statistical decoupling of the Bayesian estimator and specify the decoupled setting for a given nonlinear model. The replica solution depicts that strictly nonlinear models establish an all-or-nothing phase transition: There exists a critical load at which the optimal Bayesian inference changes from perfect to an uncorrelated learning. Based on this finding, we design a new secure coding scheme which achieves the secrecy capacity of the wiretap channel. This interesting result implies that strictly nonlinear generative models are perfectly secured without any secure coding. We justify this latter statement through the analysis of an illustrative model for perfectly secure and reliable inference.
Bruno Loureiro, Gabriele Sicuro, Cédric Gerbelot et al.
Generalised linear models for multi-class classification problems are one of the fundamental building blocks of modern machine learning tasks. In this manuscript, we characterise the learning of a mixture of $K$ Gaussians with generic means and covariances via empirical risk minimisation (ERM) with any convex loss and regularisation. In particular, we prove exact asymptotics characterising the ERM estimator in high-dimensions, extending several previous results about Gaussian mixture classification in the literature. We exemplify our result in two tasks of interest in statistical learning: a) classification for a mixture with sparse means, where we study the efficiency of $\ell_1$ penalty with respect to $\ell_2$; b) max-margin multi-class classification, where we characterise the phase transition on the existence of the multi-class logistic maximum likelihood estimator for $K>2$. Finally, we discuss how our theory can be applied beyond the scope of synthetic data, showing that in different cases Gaussian mixtures capture closely the learning curve of classification tasks in real data sets.
In this manuscript we consider Kernel Ridge Regression (KRR) under the Gaussian design. Exponents for the decay of the excess generalization error of KRR have been reported in various works under the assumption of power-law decay of eigenvalues of the features co-variance. These decays were, however, provided for sizeably different setups, namely in the noiseless case with constant regularization and in the noisy optimally regularized case. Intermediary settings have been left substantially uncharted. In this work, we unify and extend this line of work, providing characterization of all regimes and excess error decay rates that can be observed in terms of the interplay of noise and regularization. In particular, we show the existence of a transition in the noisy setting between the noiseless exponents to its noisy values as the sample complexity is increased. Finally, we illustrate how this crossover can also be observed on real data sets.
Sebastian Goldt, Florent Krzakala, Lenka Zdeborová et al.
The advent of comprehensive synaptic wiring diagrams of large neural circuits has created the field of connectomics and given rise to a number of open research questions. One such question is whether it is possible to reconstruct the information stored in a recurrent network of neurons, given its synaptic connectivity matrix. Here, we address this question by determining when solving such an inference problem is theoretically possible in specific attractor network models and by providing a practical algorithm to do so. The algorithm builds on ideas from statistical physics to perform approximate Bayesian inference and is amenable to exact analysis. We study its performance on three different models, compare the algorithm to standard algorithms such as PCA, and explore the limitations of reconstructing stored patterns from synaptic connectivity.
Maria Refinetti, Sebastian Goldt, Florent Krzakala et al.
A recent series of theoretical works showed that the dynamics of neural networks with a certain initialisation are well-captured by kernel methods. Concurrent empirical work demonstrated that kernel methods can come close to the performance of neural networks on some image classification tasks. These results raise the question of whether neural networks only learn successfully if kernels also learn successfully, despite neural networks being more expressive. Here, we show theoretically that two-layer neural networks (2LNN) with only a few hidden neurons can beat the performance of kernel learning on a simple Gaussian mixture classification task. We study the high-dimensional limit where the number of samples is linearly proportional to the input dimension, and show that while small 2LNN achieve near-optimal performance on this task, lazy training approaches such as random features and kernel methods do not. Our analysis is based on the derivation of a closed set of equations that track the learning dynamics of the 2LNN and thus allow to extract the asymptotic performance of the network as a function of signal-to-noise ratio and other hyperparameters. We finally illustrate how over-parametrising the neural network leads to faster convergence, but does not improve its final performance.
Teacher-student models provide a framework in which the typical-case performance of high-dimensional supervised learning can be described in closed form. The assumptions of Gaussian i.i.d. input data underlying the canonical teacher-student model may, however, be perceived as too restrictive to capture the behaviour of realistic data sets. In this paper, we introduce a Gaussian covariate generalisation of the model where the teacher and student can act on different spaces, generated with fixed, but generic feature maps. While still solvable in a closed form, this generalization is able to capture the learning curves for a broad range of realistic data sets, thus redeeming the potential of the teacher-student framework. Our contribution is then two-fold: First, we prove a rigorous formula for the asymptotic training loss and generalisation error. Second, we present a number of situations where the learning curve of the model captures the one of a realistic data set learned with kernel regression and classification, with out-of-the-box feature maps such as random projections or scattering transforms, or with pre-learned ones - such as the features learned by training multi-layer neural networks. We discuss both the power and the limitations of the framework.
The scaling hypothesis motivates the expansion of models past trillions of parameters as a path towards better performance. Recent significant developments, such as GPT-3, have been driven by this conjecture. However, as models scale-up, training them efficiently with backpropagation becomes difficult. Because model, pipeline, and data parallelism distribute parameters and gradients over compute nodes, communication is challenging to orchestrate: this is a bottleneck to further scaling. In this work, we argue that alternative training methods can mitigate these issues, and can inform the design of extreme-scale training hardware. Indeed, using a synaptically asymmetric method with a parallelizable backward pass, such as Direct Feedback Alignement, communication needs are drastically reduced. We present a photonic accelerator for Direct Feedback Alignment, able to compute random projections with trillions of parameters. We demonstrate our system on benchmark tasks, using both fully-connected and graph convolutional networks. Our hardware is the first architecture-agnostic photonic co-processor for training neural networks. This is a significant step towards building scalable hardware, able to go beyond backpropagation, and opening new avenues for deep learning.
Antoine Baker, Indaco Biazzo, Alfredo Braunstein et al.
Contact-tracing is an essential tool in order to mitigate the impact of pandemic such as the COVID-19. In order to achieve efficient and scalable contact-tracing in real time, digital devices can play an important role. While a lot of attention has been paid to analyzing the privacy and ethical risks of the associated mobile applications, so far much less research has been devoted to optimizing their performance and assessing their impact on the mitigation of the epidemic. We develop Bayesian inference methods to estimate the risk that an individual is infected. This inference is based on the list of his recent contacts and their own risk levels, as well as personal information such as results of tests or presence of syndromes. We propose to use probabilistic risk estimation in order to optimize testing and quarantining strategies for the control of an epidemic. Our results show that in some range of epidemic spreading (typically when the manual tracing of all contacts of infected people becomes practically impossible, but before the fraction of infected people reaches the scale where a lock-down becomes unavoidable), this inference of individuals at risk could be an efficient way to mitigate the epidemic. Our approaches translate into fully distributed algorithms that only require communication between individuals who have recently been in contact. Such communication may be encrypted and anonymized and thus compatible with privacy preserving standards. We conclude that probabilistic risk estimation is capable to enhance performance of digital contact tracing and should be considered in the currently developed mobile applications.
Sebastian Goldt, Bruno Loureiro, Galen Reeves et al.
Understanding the impact of data structure on the computational tractability of learning is a key challenge for the theory of neural networks. Many theoretical works do not explicitly model training data, or assume that inputs are drawn component-wise independently from some simple probability distribution. Here, we go beyond this simple paradigm by studying the performance of neural networks trained on data drawn from pre-trained generative models. This is possible due to a Gaussian equivalence stating that the key metrics of interest, such as the training and test errors, can be fully captured by an appropriately chosen Gaussian model. We provide three strands of rigorous, analytical and numerical evidence corroborating this equivalence. First, we establish rigorous conditions for the Gaussian equivalence to hold in the case of single-layer generative models, as well as deterministic rates for convergence in distribution. Second, we leverage this equivalence to derive a closed set of equations describing the generalisation performance of two widely studied machine learning problems: two-layer neural networks trained using one-pass stochastic gradient descent, and full-batch pre-learned features or kernel methods. Finally, we perform experiments demonstrating how our theory applies to deep, pre-trained generative models. These results open a viable path to the theoretical study of machine learning models with realistic data.
Julien Launay, Iacopo Poli, François Boniface et al.
Despite being the workhorse of deep learning, the backpropagation algorithm is no panacea. It enforces sequential layer updates, thus preventing efficient parallelization of the training process. Furthermore, its biological plausibility is being challenged. Alternative schemes have been devised; yet, under the constraint of synaptic asymmetry, none have scaled to modern deep learning tasks and architectures. Here, we challenge this perspective, and study the applicability of Direct Feedback Alignment to neural view synthesis, recommender systems, geometric learning, and natural language processing. In contrast with previous studies limited to computer vision tasks, our findings show that it successfully trains a large range of state-of-the-art deep learning architectures, with performance close to fine-tuned backpropagation. At variance with common beliefs, our work supports that challenging tasks can be tackled in the absence of weight transport.
Jonathan Dong, Ruben Ohana, Mushegh Rafayelyan et al.
Reservoir Computing is a class of simple yet efficient Recurrent Neural Networks where internal weights are fixed at random and only a linear output layer is trained. In the large size limit, such random neural networks have a deep connection with kernel methods. Our contributions are threefold: a) We rigorously establish the recurrent kernel limit of Reservoir Computing and prove its convergence. b) We test our models on chaotic time series prediction, a classic but challenging benchmark in Reservoir Computing, and show how the Recurrent Kernel is competitive and computationally efficient when the number of data points remains moderate. c) When the number of samples is too large, we leverage the success of structured Random Features for kernel approximation by introducing Structured Reservoir Computing. The two proposed methods, Recurrent Kernel and Structured Reservoir Computing, turn out to be much faster and more memory-efficient than conventional Reservoir Computing.