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Researcher profile

Florent Krzakala

Florent Krzakala contributes to research discovery and scholarly infrastructure.

ResearcherAffiliation not importedOpen to collaborate
Machine Learningcond-mat.dis-nnmath.STStatistics TheoryInformation Theorymath.ITcond-mat.stat-mechmath.PReess.SPEmerging TechnologiesCryptography and SecurityData Structures and Algorithmseess.IVmath-phmath.MPNeural and Evolutionary Computingphysics.opticsSocial and Information Networks

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Published work

37 published item(s)

preprint2026arXiv

Asymptotics of Non-Convex Generalized Linear Models in High-Dimensions: A proof of the replica formula

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.

0 citations0 reviewsNo code linkedOpen access
preprint2026arXiv

Deep Learning as Neural Low-Degree Filtering: A Spectral Theory of Hierarchical Feature Learning

Understanding how deep neural networks learn useful internal representations from data remains a central open problem in the theory of deep learning. We introduce Neural Low-Degree Filtering (Neural LoFi), a stylized limit of gradient-based training in which hierarchical feature learning becomes an explicit iterative spectral procedure. In this limit, the dynamics at each layer decouple: given the current representation, the next layer selects directions with maximal accessible low-degree correlation to the label. This yields a tractable surrogate mechanism for deep learning, together with a natural kernel-space interpretation. Neural LoFi provides a mathematically explicit framework for studying multi-layer feature learning beyond the lazy regime. It predicts how representations are selected layer by layer, explains how emergence of concepts arises with given sample complexity,and gives a concrete mechanism by which depth progressively constructs new features from old ones through low-degree compositionality. We complement the theory with mechanistic experiments on fully connected and convolutional architectures, showing that Neural LoFi improves over lazy random-feature baselines, recovers meaningful structured filters, and predicts representations aligned with early gradient-descent feature discovery with real datasets.

0 citations0 reviewsNo code linkedOpen access
preprint2026arXiv

Scaling Laws from Sequential Feature Recovery: A Solvable Hierarchical Model

We propose a simple mechanism by which scaling laws emerge from feature learning in multi-layer networks. We study a high-dimensional hierarchical target that is a globally high-degree function, but that can be represented by a combination of latent compositional features whose weights decrease as a power law. We show that a layer-wise spectral algorithm adapted to this compositional structure achieves improved scaling relative to shallow, non-adaptive methods, and recovers the latent directions sequentially: strong features become detectable at small sample sizes, while weaker features require more data. We prove sharp feature-wise recovery thresholds and show that aggregating these transitions yields an explicit power-law decay of the prediction error. Technically, the analysis relies on random matrix methods and a resolvent-based perturbation argument, which gives matching upper and lower bounds for individual eigenvector recovery beyond what standard gap-based perturbation bounds provide. Numerical experiments confirm the predicted sequential recovery, finite-size smoothing of the thresholds, and separation from non-hierarchical kernel baselines. Together, these results show how smooth scaling laws can emerge from a cascade of sharp feature-learning transitions.

0 citations0 reviewsNo code linkedOpen access
preprint2024arXiv

A High Dimensional Statistical 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 $α= n / d$. 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.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Bayesian Inference with Nonlinear Generative Models: Comments on Secure Learning

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.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Perturbative construction of mean-field equations in extensive-rank matrix factorization and denoising

Factorization of matrices where the rank of the two factors diverges linearly with their sizes has many applications in diverse areas such as unsupervised representation learning, dictionary learning or sparse coding. We consider a setting where the two factors are generated from known component-wise independent prior distributions, and the statistician observes a (possibly noisy) component-wise function of their matrix product. In the limit where the dimensions of the matrices tend to infinity, but their ratios remain fixed, we expect to be able to derive closed form expressions for the optimal mean squared error on the estimation of the two factors. However, this remains a very involved mathematical and algorithmic problem. A related, but simpler, problem is extensive-rank matrix denoising, where one aims to reconstruct a matrix with extensive but usually small rank from noisy measurements. In this paper, we approach both these problems using high-temperature expansions at fixed order parameters. This allows to clarify how previous attempts at solving these problems failed at finding an asymptotically exact solution. We provide a systematic way to derive the corrections to these existing approximations, taking into account the structure of correlations particular to the problem. Finally, we illustrate our approach in detail on the case of extensive-rank matrix denoising. We compare our results with known optimal rotationally-invariant estimators, and show how exact asymptotic calculations of the minimal error can be performed using extensive-rank matrix integrals.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Secure Coding via Gaussian Random Fields

Inverse probability problems whose generative models are given by strictly nonlinear Gaussian random fields show the all-or-nothing behavior: There exists a critical rate at which Bayesian inference exhibits a phase transition. Below this rate, the optimal Bayesian estimator recovers the data perfectly, and above it the recovered data becomes uncorrelated. This study uses the replica method from the theory of spin glasses to show that this critical rate is the channel capacity. This interesting finding has a particular application to the problem of secure transmission: A strictly nonlinear Gaussian random field along with random binning can be used to securely encode a confidential message in a wiretap channel. Our large-system characterization demonstrates that this secure coding scheme asymptotically achieves the secrecy capacity of the Gaussian wiretap channel.

0 citations0 reviewsNo code linkedOpen access
preprint2021arXiv

Dynamical mean-field theory for stochastic gradient descent in Gaussian mixture classification

We analyze in a closed form the learning dynamics of stochastic gradient descent (SGD) for a single-layer neural network classifying a high-dimensional Gaussian mixture where each cluster is assigned one of two labels. This problem provides a prototype of a non-convex loss landscape with interpolating regimes and a large generalization gap. We define a particular stochastic process for which SGD can be extended to a continuous-time limit that we call stochastic gradient flow. In the full-batch limit, we recover the standard gradient flow. We apply dynamical mean-field theory from statistical physics to track the dynamics of the algorithm in the high-dimensional limit via a self-consistent stochastic process. We explore the performance of the algorithm as a function of the control parameters shedding light on how it navigates the loss landscape.

0 citations0 reviewsNo code linkedOpen access
preprint2021arXiv

The Gaussian equivalence of generative models for learning with shallow neural networks

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.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Asymptotic errors for convex penalized linear regression beyond Gaussian matrices

We consider the problem of learning a coefficient vector $x_{0}$ in $R^{N}$ from noisy linear observations $y=Fx_{0}+w$ in $R^{M}$ in the high dimensional limit $M,N$ to infinity with $α=M/N$ fixed. We provide a rigorous derivation of an explicit formula -- first conjectured using heuristic methods from statistical physics -- for the asymptotic mean squared error obtained by penalized convex regression estimators such as the LASSO or the elastic net, for a class of very generic random matrices corresponding to rotationally invariant data matrices with arbitrary spectrum. The proof is based on a convergence analysis of an oracle version of vector approximate message-passing (oracle-VAMP) and on the properties of its state evolution equations. Our method leverages on and highlights the link between vector approximate message-passing, Douglas-Rachford splitting and proximal descent algorithms, extending previous results obtained with i.i.d. matrices for a large class of problems. We illustrate our results on some concrete examples and show that even though they are asymptotic, our predictions agree remarkably well with numerics even for very moderate sizes.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Complex Dynamics in Simple Neural Networks: Understanding Gradient Flow in Phase Retrieval

Despite the widespread use of gradient-based algorithms for optimizing high-dimensional non-convex functions, understanding their ability of finding good minima instead of being trapped in spurious ones remains to a large extent an open problem. Here we focus on gradient flow dynamics for phase retrieval from random measurements. When the ratio of the number of measurements over the input dimension is small the dynamics remains trapped in spurious minima with large basins of attraction. We find analytically that above a critical ratio those critical points become unstable developing a negative direction toward the signal. By numerical experiments we show that in this regime the gradient flow algorithm is not trapped; it drifts away from the spurious critical points along the unstable direction and succeeds in finding the global minimum. Using tools from statistical physics we characterize this phenomenon, which is related to a BBP-type transition in the Hessian of the spurious minima.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Direct Feedback Alignment Scales to Modern Deep Learning Tasks and Architectures

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.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Double Trouble in Double Descent : Bias and Variance(s) in the Lazy Regime

Deep neural networks can achieve remarkable generalization performances while interpolating the training data perfectly. Rather than the U-curve emblematic of the bias-variance trade-off, their test error often follows a "double descent" - a mark of the beneficial role of overparametrization. In this work, we develop a quantitative theory for this phenomenon in the so-called lazy learning regime of neural networks, by considering the problem of learning a high-dimensional function with random features regression. We obtain a precise asymptotic expression for the bias-variance decomposition of the test error, and show that the bias displays a phase transition at the interpolation threshold, beyond which it remains constant. We disentangle the variances stemming from the sampling of the dataset, from the additive noise corrupting the labels, and from the initialization of the weights. Following up on Geiger et al. 2019, we first show that the latter two contributions are the crux of the double descent: they lead to the overfitting peak at the interpolation threshold and to the decay of the test error upon overparametrization. We then quantify how they are suppressed by ensemble averaging the outputs of K independently initialized estimators. When K is sent to infinity, the test error remains constant beyond the interpolation threshold. We further compare the effects of overparametrizing, ensembling and regularizing. Finally, we present numerical experiments on classic deep learning setups to show that our results hold qualitatively in realistic lazy learning scenarios.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Exact asymptotics for phase retrieval and compressed sensing with random generative priors

We consider the problem of compressed sensing and of (real-valued) phase retrieval with random measurement matrix. We derive sharp asymptotics for the information-theoretically optimal performance and for the best known polynomial algorithm for an ensemble of generative priors consisting of fully connected deep neural networks with random weight matrices and arbitrary activations. We compare the performance to sparse separable priors and conclude that generative priors might be advantageous in terms of algorithmic performance. In particular, while sparsity does not allow to perform compressive phase retrieval efficiently close to its information-theoretic limit, it is found that under the random generative prior compressed phase retrieval becomes tractable.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Generalisation error in learning with random features and the hidden manifold model

We study generalised linear regression and classification for a synthetically generated dataset encompassing different problems of interest, such as learning with random features, neural networks in the lazy training regime, and the hidden manifold model. We consider the high-dimensional regime and using the replica method from statistical physics, we provide a closed-form expression for the asymptotic generalisation performance in these problems, valid in both the under- and over-parametrised regimes and for a broad choice of generalised linear model loss functions. In particular, we show how to obtain analytically the so-called double descent behaviour for logistic regression with a peak at the interpolation threshold, we illustrate the superiority of orthogonal against random Gaussian projections in learning with random features, and discuss the role played by correlations in the data generated by the hidden manifold model. Beyond the interest in these particular problems, the theoretical formalism introduced in this manuscript provides a path to further extensions to more complex tasks.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Generalization error in high-dimensional perceptrons: Approaching Bayes error with convex optimization

We consider a commonly studied supervised classification of a synthetic dataset whose labels are generated by feeding a one-layer neural network with random iid inputs. We study the generalization performances of standard classifiers in the high-dimensional regime where $α=n/d$ is kept finite in the limit of a high dimension $d$ and number of samples $n$. Our contribution is three-fold: First, we prove a formula for the generalization error achieved by $\ell_2$ regularized classifiers that minimize a convex loss. This formula was first obtained by the heuristic replica method of statistical physics. Secondly, focussing on commonly used loss functions and optimizing the $\ell_2$ regularization strength, we observe that while ridge regression performance is poor, logistic and hinge regression are surprisingly able to approach the Bayes-optimal generalization error extremely closely. As $α\to \infty$ they lead to Bayes-optimal rates, a fact that does not follow from predictions of margin-based generalization error bounds. Third, we design an optimal loss and regularizer that provably leads to Bayes-optimal generalization error.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

High-temperature Expansions and Message Passing Algorithms

Improved mean-field technics are a central theme of statistical physics methods applied to inference and learning. We revisit here some of these methods using high-temperature expansions for disordered systems initiated by Plefka, Georges and Yedidia. We derive the Gibbs free entropy and the subsequent self-consistent equations for a generic class of statistical models with correlated matrices and show in particular that many classical approximation schemes, such as adaptive TAP, Expectation-Consistency, or the approximations behind the Vector Approximate Message Passing algorithm all rely on the same assumptions, that are also at the heart of high-temperature expansions. We focus on the case of rotationally invariant random coupling matrices in the `high-dimensional' limit in which the number of samples and the dimension are both large, but with a fixed ratio. This encapsulates many widely studied models, such as Restricted Boltzmann Machines or Generalized Linear Models with correlated data matrices. In this general setting, we show that all the approximation schemes described before are equivalent, and we conjecture that they are exact in the thermodynamic limit in the replica symmetric phases. We achieve this conclusion by resummation of the infinite perturbation series, which generalizes a seminal result of Parisi and Potters. A rigorous derivation of this conjecture is an interesting mathematical challenge. On the way to these conclusions, we uncover several diagrammatical results in connection with free probability and random matrix theory, that are interesting independently of the rest of our work.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Large-Scale Optical Reservoir Computing for Spatiotemporal Chaotic Systems Prediction

Reservoir computing is a relatively recent computational paradigm that originates from a recurrent neural network and is known for its wide range of implementations using different physical technologies. Large reservoirs are very hard to obtain in conventional computers, as both the computation complexity and memory usage grow quadratically. We propose an optical scheme performing reservoir computing over very large networks potentially being able to host several millions of fully connected photonic nodes thanks to its intrinsic properties of parallelism and scalability. Our experimental studies confirm that, in contrast to conventional computers, the computation time of our optical scheme is only linearly dependent on the number of photonic nodes of the network, which is due to electronic overheads, while the optical part of computation remains fully parallel and independent of the reservoir size. To demonstrate the scalability of our optical scheme, we perform for the first time predictions on large spatiotemporal chaotic datasets obtained from the Kuramoto-Sivashinsky equation using optical reservoirs with up to 50 000 optical nodes. Our results are extremely challenging for conventional von Neumann machines, and they significantly advance the state of the art of unconventional reservoir computing approaches, in general.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Light-in-the-loop: using a photonics co-processor for scalable training of neural networks

As neural networks grow larger and more complex and data-hungry, training costs are skyrocketing. Especially when lifelong learning is necessary, such as in recommender systems or self-driving cars, this might soon become unsustainable. In this study, we present the first optical co-processor able to accelerate the training phase of digitally-implemented neural networks. We rely on direct feedback alignment as an alternative to backpropagation, and perform the error projection step optically. Leveraging the optical random projections delivered by our co-processor, we demonstrate its use to train a neural network for handwritten digits recognition.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Marvels and Pitfalls of the Langevin Algorithm in Noisy High-dimensional Inference

Gradient-descent-based algorithms and their stochastic versions have widespread applications in machine learning and statistical inference. In this work we perform an analytic study of the performances of one of them, the Langevin algorithm, in the context of noisy high-dimensional inference. We employ the Langevin algorithm to sample the posterior probability measure for the spiked matrix-tensor model. The typical behaviour of this algorithm is described by a system of integro-differential equations that we call the Langevin state evolution, whose solution is compared with the one of the state evolution of approximate message passing (AMP). Our results show that, remarkably, the algorithmic threshold of the Langevin algorithm is sub-optimal with respect to the one given by AMP. We conjecture this phenomenon to be due to the residual glassiness present in that region of parameters. Finally we show how a landscape-annealing protocol, that uses the Langevin algorithm but violate the Bayes-optimality condition, can approach the performance of AMP.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Mutual Information and Optimality of Approximate Message-Passing in Random Linear Estimation

We consider the estimation of a signal from the knowledge of its noisy linear random Gaussian projections. A few examples where this problem is relevant are compressed sensing, sparse superposition codes, and code division multiple access. There has been a number of works considering the mutual information for this problem using the replica method from statistical physics. Here we put these considerations on a firm rigorous basis. First, we show, using a Guerra-Toninelli type interpolation, that the replica formula yields an upper bound to the exact mutual information. Secondly, for many relevant practical cases, we present a converse lower bound via a method that uses spatial coupling, state evolution analysis and the I-MMSE theorem. This yields a single letter formula for the mutual information and the minimal-mean-square error for random Gaussian linear estimation of all discrete bounded signals. In addition, we prove that the low complexity approximate message-passing algorithm is optimal outside of the so-called hard phase, in the sense that it asymptotically reaches the minimal-mean-square error. In this work spatial coupling is used primarily as a proof technique. However our results also prove two important features of spatially coupled noisy linear random Gaussian estimation. First there is no algorithmically hard phase. This means that for such systems approximate message-passing always reaches the minimal-mean-square error. Secondly, in a proper limit the mutual information associated to such systems is the same as the one of uncoupled linear random Gaussian estimation.

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preprint2020arXiv

Passed & Spurious: Descent Algorithms and Local Minima in Spiked Matrix-Tensor Models

In this work we analyse quantitatively the interplay between the loss landscape and performance of descent algorithms in a prototypical inference problem, the spiked matrix-tensor model. We study a loss function that is the negative log-likelihood of the model. We analyse the number of local minima at a fixed distance from the signal/spike with the Kac-Rice formula, and locate trivialization of the landscape at large signal-to-noise ratios. We evaluate in a closed form the performance of a gradient flow algorithm using integro-differential PDEs as developed in physics of disordered systems for the Langevin dynamics. We analyze the performance of an approximate message passing algorithm estimating the maximum likelihood configuration via its state evolution. We conclude by comparing the above results: while we observe a drastic slow down of the gradient flow dynamics even in the region where the landscape is trivial, both the analyzed algorithms are shown to perform well even in the part of the region of parameters where spurious local minima are present.

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preprint2020arXiv

Phase retrieval in high dimensions: Statistical and computational phase transitions

We consider the phase retrieval problem of reconstructing a $n$-dimensional real or complex signal $\mathbf{X}^{\star}$ from $m$ (possibly noisy) observations $Y_μ= | \sum_{i=1}^n Φ_{μi} X^{\star}_i/\sqrt{n}|$, for a large class of correlated real and complex random sensing matrices $\mathbfΦ$, in a high-dimensional setting where $m,n\to\infty$ while $α= m/n=Θ(1)$. First, we derive sharp asymptotics for the lowest possible estimation error achievable statistically and we unveil the existence of sharp phase transitions for the weak- and full-recovery thresholds as a function of the singular values of the matrix $\mathbfΦ$. This is achieved by providing a rigorous proof of a result first obtained by the replica method from statistical mechanics. In particular, the information-theoretic transition to perfect recovery for full-rank matrices appears at $α=1$ (real case) and $α=2$ (complex case). Secondly, we analyze the performance of the best-known polynomial time algorithm for this problem -- approximate message-passing -- establishing the existence of a statistical-to-algorithmic gap depending, again, on the spectral properties of $\mathbfΦ$. Our work provides an extensive classification of the statistical and algorithmic thresholds in high-dimensional phase retrieval for a broad class of random matrices.

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preprint2020arXiv

Rademacher complexity and spin glasses: A link between the replica and statistical theories of learning

Statistical learning theory provides bounds of the generalization gap, using in particular the Vapnik-Chervonenkis dimension and the Rademacher complexity. An alternative approach, mainly studied in the statistical physics literature, is the study of generalization in simple synthetic-data models. Here we discuss the connections between these approaches and focus on the link between the Rademacher complexity in statistical learning and the theories of generalization for typical-case synthetic models from statistical physics, involving quantities known as Gardner capacity and ground state energy. We show that in these models the Rademacher complexity is closely related to the ground state energy computed by replica theories. Using this connection, one may reinterpret many results of the literature as rigorous Rademacher bounds in a variety of models in the high-dimensional statistics limit. Somewhat surprisingly, we also show that statistical learning theory provides predictions for the behavior of the ground-state energies in some full replica symmetry breaking models.

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preprint2020arXiv

Reservoir Computing meets Recurrent Kernels and Structured Transforms

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.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Who is Afraid of Big Bad Minima? Analysis of Gradient-Flow in a Spiked Matrix-Tensor Model

Gradient-based algorithms are effective for many machine learning tasks, but despite ample recent effort and some progress, it often remains unclear why they work in practice in optimising high-dimensional non-convex functions and why they find good minima instead of being trapped in spurious ones. Here we present a quantitative theory explaining this behaviour in a spiked matrix-tensor model. Our framework is based on the Kac-Rice analysis of stationary points and a closed-form analysis of gradient-flow originating from statistical physics. We show that there is a well defined region of parameters where the gradient-flow algorithm finds a good global minimum despite the presence of exponentially many spurious local minima. We show that this is achieved by surfing on saddles that have strong negative direction towards the global minima, a phenomenon that is connected to a BBP-type threshold in the Hessian describing the critical points of the landscapes.

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preprint2019arXiv

Dynamics of stochastic gradient descent for two-layer neural networks in the teacher-student setup

Deep neural networks achieve stellar generalisation even when they have enough parameters to easily fit all their training data. We study this phenomenon by analysing the dynamics and the performance of over-parameterised two-layer neural networks in the teacher-student setup, where one network, the student, is trained on data generated by another network, called the teacher. We show how the dynamics of stochastic gradient descent (SGD) is captured by a set of differential equations and prove that this description is asymptotically exact in the limit of large inputs. Using this framework, we calculate the final generalisation error of student networks that have more parameters than their teachers. We find that the final generalisation error of the student increases with network size when training only the first layer, but stays constant or even decreases with size when training both layers. We show that these different behaviours have their root in the different solutions SGD finds for different activation functions. Our results indicate that achieving good generalisation in neural networks goes beyond the properties of SGD alone and depends on the interplay of at least the algorithm, the model architecture, and the data set.

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preprint2019arXiv

Kernel computations from large-scale random features obtained by Optical Processing Units

Approximating kernel functions with random features (RFs)has been a successful application of random projections for nonparametric estimation. However, performing random projections presents computational challenges for large-scale problems. Recently, a new optical hardware called Optical Processing Unit (OPU) has been developed for fast and energy-efficient computation of large-scale RFs in the analog domain. More specifically, the OPU performs the multiplication of input vectors by a large random matrix with complex-valued i.i.d. Gaussian entries, followed by the application of an element-wise squared absolute value operation - this last nonlinearity being intrinsic to the sensing process. In this paper, we show that this operation results in a dot-product kernel that has connections to the polynomial kernel, and we extend this computation to arbitrary powers of the feature map. Experiments demonstrate that the OPU kernel and its RF approximation achieve competitive performance in applications using kernel ridge regression and transfer learning for image classification. Crucially, thanks to the use of the OPU, these results are obtained with time and energy savings.

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preprint2019arXiv

On the Universality of Noiseless Linear Estimation with Respect to the Measurement Matrix

In a noiseless linear estimation problem, one aims to reconstruct a vector x* from the knowledge of its linear projections y=Phi x*. There have been many theoretical works concentrating on the case where the matrix Phi is a random i.i.d. one, but a number of heuristic evidence suggests that many of these results are universal and extend well beyond this restricted case. Here we revisit this problematic through the prism of development of message passing methods, and consider not only the universality of the l1 transition, as previously addressed, but also the one of the optimal Bayesian reconstruction. We observed that the universality extends to the Bayes-optimal minimum mean-squared (MMSE) error, and to a range of structured matrices.

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preprint2019arXiv

The spiked matrix model with generative priors

Using a low-dimensional parametrization of signals is a generic and powerful way to enhance performance in signal processing and statistical inference. A very popular and widely explored type of dimensionality reduction is sparsity; another type is generative modelling of signal distributions. Generative models based on neural networks, such as GANs or variational auto-encoders, are particularly performant and are gaining on applicability. In this paper we study spiked matrix models, where a low-rank matrix is observed through a noisy channel. This problem with sparse structure of the spikes has attracted broad attention in the past literature. Here, we replace the sparsity assumption by generative modelling, and investigate the consequences on statistical and algorithmic properties. We analyze the Bayes-optimal performance under specific generative models for the spike. In contrast with the sparsity assumption, we do not observe regions of parameters where statistical performance is superior to the best known algorithmic performance. We show that in the analyzed cases the approximate message passing algorithm is able to reach optimal performance. We also design enhanced spectral algorithms and analyze their performance and thresholds using random matrix theory, showing their superiority to the classical principal component analysis. We complement our theoretical results by illustrating the performance of the spectral algorithms when the spikes come from real datasets.

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preprint2018arXiv

Entropy and mutual information in models of deep neural networks

We examine a class of deep learning models with a tractable method to compute information-theoretic quantities. Our contributions are three-fold: (i) We show how entropies and mutual informations can be derived from heuristic statistical physics methods, under the assumption that weight matrices are independent and orthogonally-invariant. (ii) We extend particular cases in which this result is known to be rigorously exact by providing a proof for two-layers networks with Gaussian random weights, using the recently introduced adaptive interpolation method. (iii) We propose an experiment framework with generative models of synthetic datasets, on which we train deep neural networks with a weight constraint designed so that the assumption in (i) is verified during learning. We study the behavior of entropies and mutual informations throughout learning and conclude that, in the proposed setting, the relationship between compression and generalization remains elusive.

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preprint2018arXiv

Fundamental limits of detection in the spiked Wigner model

We study the fundamental limits of detecting the presence of an additive rank-one perturbation, or spike, to a Wigner matrix. When the spike comes from a prior that is i.i.d. across coordinates, we prove that the log-likelihood ratio of the spiked model against the non-spiked one is asymptotically normal below a certain reconstruction threshold which is not necessarily of a "spectral" nature, and that it is degenerate above. This establishes the maximal region of contiguity between the planted and null models. It is known that this threshold also marks a phase transition for estimating the spike: the latter task is possible above the threshold and impossible below. Therefore, both estimation and detection undergo the same transition in this random matrix model. We also provide further information about the performance of the optimal test. Our proofs are based on Gaussian interpolation methods and a rigorous incarnation of the cavity method, as devised by Guerra and Talagrand in their study of the Sherrington--Kirkpatrick spin-glass model.

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preprint2017arXiv

Decoding from Pooled Data: Phase Transitions of Message Passing

We consider the problem of decoding a discrete signal of categorical variables from the observation of several histograms of pooled subsets of it. We present an Approximate Message Passing (AMP) algorithm for recovering the signal in the random dense setting where each observed histogram involves a random subset of entries of size proportional to n. We characterize the performance of the algorithm in the asymptotic regime where the number of observations $m$ tends to infinity proportionally to n, by deriving the corresponding State Evolution (SE) equations and studying their dynamics. We initiate the analysis of the multi-dimensional SE dynamics by proving their convergence to a fixed point, along with some further properties of the iterates. The analysis reveals sharp phase transition phenomena where the behavior of AMP changes from exact recovery to weak correlation with the signal as m/n crosses a threshold. We derive formulae for the threshold in some special cases and show that they accurately match experimental behavior.

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preprint2017arXiv

Multi-Layer Generalized Linear Estimation

We consider the problem of reconstructing a signal from multi-layered (possibly) non-linear measurements. Using non-rigorous but standard methods from statistical physics we present the Multi-Layer Approximate Message Passing (ML-AMP) algorithm for computing marginal probabilities of the corresponding estimation problem and derive the associated state evolution equations to analyze its performance. We also give the expression of the asymptotic free energy and the minimal information-theoretically achievable reconstruction error. Finally, we present some applications of this measurement model for compressed sensing and perceptron learning with structured matrices/patterns, and for a simple model of estimation of latent variables in an auto-encoder.

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preprint2017arXiv

Statistical and computational phase transitions in spiked tensor estimation

We consider tensor factorizations using a generative model and a Bayesian approach. We compute rigorously the mutual information, the Minimal Mean Squared Error (MMSE), and unveil information-theoretic phase transitions. In addition, we study the performance of Approximate Message Passing (AMP) and show that it achieves the MMSE for a large set of parameters, and that factorization is algorithmically "easy" in a much wider region than previously believed. It exists, however, a "hard" region where AMP fails to reach the MMSE and we conjecture that no polynomial algorithm will improve on AMP.

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preprint2015arXiv

Phase Transitions in Sparse PCA

We study optimal estimation for sparse principal component analysis when the number of non-zero elements is small but on the same order as the dimension of the data. We employ approximate message passing (AMP) algorithm and its state evolution to analyze what is the information theoretically minimal mean-squared error and the one achieved by AMP in the limit of large sizes. For a special case of rank one and large enough density of non-zeros Deshpande and Montanari [1] proved that AMP is asymptotically optimal. We show that both for low density and for large rank the problem undergoes a series of phase transitions suggesting existence of a region of parameters where estimation is information theoretically possible, but AMP (and presumably every other polynomial algorithm) fails. The analysis of the large rank limit is particularly instructive.

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preprint2015arXiv

Spectral Detection in the Censored Block Model

We consider the problem of partially recovering hidden binary variables from the observation of (few) censored edge weights, a problem with applications in community detection, correlation clustering and synchronization. We describe two spectral algorithms for this task based on the non-backtracking and the Bethe Hessian operators. These algorithms are shown to be asymptotically optimal for the partial recovery problem, in that they detect the hidden assignment as soon as it is information theoretically possible to do so.

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