Marc Mézard

Marco Benedetti, Andrej Bogdanov, Enrico M. Malatesta et al.

We initiate the study of the algorithmic complexity of finding collisions in single-layer binary neural networks. Given a random matrix $\mathbf{A} \in \mathbb{R}^{m\times n}$, an input $\mathbf{x} \in \{-1,1\}^n$ is mapped to a binary output vector $φ(\mathbf{A}\mathbf{x})\in \{-1,1\}^m$, where $φ$ is an activation function with constant behavior on $[κ, \infty)$ for some threshold $κ\geq 0$. We identify the threshold scale $κ=Θ(1/\sqrtα)$, where $α=m/n$, as separating two complementary phenomena. When $κ\ll 1/\sqrtα$, we give a simple online algorithm that efficiently produces extensive collisions. When $κ\gg 1/\sqrtα$, for a natural \emph{randomized} non-periodic activation and suitable oscillation complexity, we prove that the extensive-collision space exhibits an overlap gap property (OGP), yielding an exponential lower bound against online algorithms. Ours is the first work to use the overlap gap property as a rigorous criterion for collision resistance. The key difference between collision finding and average-case search is that collision finding has a new ``worst-case'' aspect: the collision finder has full control over the choice of colliding pairs. Our lower bound is proved in the online model; extending such guarantees to broader classes of algorithms, including spectral, algebraic, lattice-based, or quantum methods, remains an open direction.

Francesco Camilli, Marc Mézard

Matrix factorization is an important mathematical problem encountered in the context of dictionary learning, recommendation systems and machine learning. We introduce a new `decimation' scheme that maps it to neural network models of associative memory and provide a detailed theoretical analysis of its performance, showing that decimation is able to factorize extensive-rank matrices and to denoise them efficiently. We introduce a decimation algorithm based on ground-state search of the neural network, which shows performances that match the theoretical prediction.

Francesco Camilli, Marc Mézard

Matrix factorization is an inference problem that has acquired importance due to its vast range of applications that go from dictionary learning to recommendation systems and machine learning with deep networks. The study of its fundamental statistical limits represents a true challenge, and despite a decade-long history of efforts in the community, there is still no closed formula able to describe its optimal performances in the case where the rank of the matrix scales linearly with its size. In the present paper, we study this extensive rank problem, extending the alternative 'decimation' procedure that we recently introduced, and carry out a thorough study of its performance. Decimation aims at recovering one column/line of the factors at a time, by mapping the problem into a sequence of neural network models of associative memory at a tunable temperature. Though being sub-optimal, decimation has the advantage of being theoretically analyzable. We extend its scope and analysis to two families of matrices. For a large class of compactly supported priors, we show that the replica symmetric free entropy of the neural network models takes a universal form in the low temperature limit. For sparse Ising prior, we show that the storage capacity of the neural network models diverges as sparsity in the patterns increases, and we introduce a simple algorithm based on a ground state search that implements decimation and performs matrix factorization, with no need of an informative initialization.

Jerome Garnier-Brun, Marc Mézard, Emanuele Moscato et al.

Understanding the learning process and the embedded computation in transformers is becoming a central goal for the development of interpretable AI. In the present study, we introduce a hierarchical filtering procedure for data models of sequences on trees, allowing us to hand-tune the range of positional correlations in the data. Leveraging this controlled setting, we provide evidence that vanilla encoder-only transformers can approximate the exact inference algorithm when trained on root classification and masked language modeling tasks, and study how this computation is discovered and implemented. We find that correlations at larger distances, corresponding to increasing layers of the hierarchy, are sequentially included by the network during training. By comparing attention maps from models trained with varying degrees of filtering and by probing the different encoder levels, we find clear evidence of a reconstruction of correlations on successive length scales corresponding to the various levels of the hierarchy, which we relate to a plausible implementation of the exact inference algorithm within the same architecture.

Giulio Biroli, Marc Mézard

This paper studies Kernel Density Estimation for a high-dimensional distribution $ρ(x)$. Traditional approaches have focused on the limit of large number of data points $n$ and fixed dimension $d$. We analyze instead the regime where both the number $n$ of data points $y_i$ and their dimensionality $d$ grow with a fixed ratio $α=(\log n)/d$. Our study reveals three distinct statistical regimes for the kernel-based estimate of the density $\hat ρ_h^{\mathcal {D}}(x)=\frac{1}{n h^d}\sum_{i=1}^n K\left(\frac{x-y_i}{h}\right)$, depending on the bandwidth $h$: a classical regime for large bandwidth where the Central Limit Theorem (CLT) holds, which is akin to the one found in traditional approaches. Below a certain value of the bandwidth, $h_{CLT}(α)$, we find that the CLT breaks down. The statistics of $\hatρ_h^{\mathcal {D}}(x)$ for a fixed $x$ drawn from $ρ(x)$ is given by a heavy-tailed distribution (an alpha-stable distribution). In particular below a value $h_G(α)$, we find that $\hatρ_h^{\mathcal {D}}(x)$ is governed by extreme value statistics: only a few points in the database matter and give the dominant contribution to the density estimator. We provide a detailed analysis for high-dimensional multivariate Gaussian data. We show that the optimal bandwidth threshold based on Kullback-Leibler divergence lies in the new statistical regime identified in this paper. As known by practitioners, when decreasing the bandwidth a Kernel-estimated estimated changes from a smooth curve to a collections of peaks centred on the data points. Our findings reveal that this general phenomenon is related to sharp transitions between phases characterized by different statistical properties, and offer new insights for Kernel density estimation in high-dimensional settings.

Jerome Garnier-Brun, Luca Biggio, Davide Beltrame et al.

Generalization in generative modeling is defined as the ability to learn an underlying distribution from a finite dataset and produce novel samples, with evaluation largely driven by held-out performance and perceived sample quality. In practice, training is often stopped at the minimum of the test loss, taken as an operational indicator of generalization. We challenge this viewpoint by identifying a phase of biased generalization during training, in which the model continues to decrease the test loss while favoring samples with anomalously high proximity to training data. By training the same network on two disjoint datasets and comparing the mutual distances of generated samples and their similarity to training data, we introduce a quantitative measure of bias and demonstrate its presence on real images. We then study the mechanism of bias, using a controlled hierarchical data model where access to exact scores and ground-truth statistics allows us to precisely characterize its onset. We attribute this phenomenon to the sequential nature of feature learning in deep networks, where coarse structure is learned early in a data-independent manner, while finer features are resolved later in a way that increasingly depends on individual training samples. Our results show that early stopping at the test loss minimum, while optimal under standard generalization criteria, may be insufficient for privacy-critical applications.

Clarissa Lauditi, Emanuele Troiani, Marc Mézard

In recent years statistical physics has proven to be a valuable tool to probe into large dimensional inference problems such as the ones occurring in machine learning. Statistical physics provides analytical tools to study fundamental limitations in their solutions and proposes algorithms to solve individual instances. In these notes, based on the lectures by Marc Mézard in 2022 at the summer school in Les Houches, we will present a general framework that can be used in a large variety of problems with weak long-range interactions, including the compressed sensing problem, or the problem of learning in a perceptron. We shall see how these problems can be studied at the replica symmetric level, using developments of the cavity methods, both as a theoretical tool and as an algorithm.

Beatrice Achilli, Marco Benedetti, Giulio Biroli et al.

Diffusion Models generate data by reversing a stochastic diffusion process, progressively transforming noise into structured samples drawn from a target distribution. Recent theoretical work has shown that this backward dynamics can undergo sharp qualitative transitions, known as speciation transitions, during which trajectories become dynamically committed to data classes. Existing theoretical analyses, however, are limited to settings where classes are identifiable through first moments, such as mixtures of Gaussians with well-separated means. In this work, we develop a general theory of speciation in diffusion models that applies to arbitrary target distributions admitting well-defined classes. We formalize the notion of class structure through Bayes classification and characterize speciation times in terms of free-entropy difference between classes. This criterion recovers known results in previously studied Gaussian-mixture models, while extending to situations in which classes are not distinguishable by first moments and may instead differ through higher-order or collective features. Our framework also accommodates multiple classes and predicts the existence of successive speciation times associated with increasingly fine-grained class commitment. We illustrate the theory on two analytically tractable examples: mixtures of one-dimensional Ising models at different temperatures and mixtures of zero-mean Gaussians with distinct covariance structures. In the Ising case, we obtain explicit expressions for speciation times by mapping the problem onto a random-field Ising model and solving it via the replica method. Our results provide a unified and broadly applicable description of speciation transitions in diffusion-based generative models.

Giulio Biroli, Tony Bonnaire, Valentin de Bortoli et al.

Using statistical physics methods, we study generative diffusion models in the regime where the dimension of space and the number of data are large, and the score function has been trained optimally. Our analysis reveals three distinct dynamical regimes during the backward generative diffusion process. The generative dynamics, starting from pure noise, encounters first a 'speciation' transition where the gross structure of data is unraveled, through a mechanism similar to symmetry breaking in phase transitions. It is followed at later time by a 'collapse' transition where the trajectories of the dynamics become attracted to one of the memorized data points, through a mechanism which is similar to the condensation in a glass phase. For any dataset, the speciation time can be found from a spectral analysis of the correlation matrix, and the collapse time can be found from the estimation of an 'excess entropy' in the data. The dependence of the collapse time on the dimension and number of data provides a thorough characterization of the curse of dimensionality for diffusion models. Analytical solutions for simple models like high-dimensional Gaussian mixtures substantiate these findings and provide a theoretical framework, while extensions to more complex scenarios and numerical validations with real datasets confirm the theoretical predictions.

Tony Bonnaire, Raphaël Urfin, Giulio Biroli et al.

Diffusion models have achieved remarkable success across a wide range of generative tasks. A key challenge is understanding the mechanisms that prevent their memorization of training data and allow generalization. In this work, we investigate the role of the training dynamics in the transition from generalization to memorization. Through extensive experiments and theoretical analysis, we identify two distinct timescales: an early time $τ_\mathrm{gen}$ at which models begin to generate high-quality samples, and a later time $τ_\mathrm{mem}$ beyond which memorization emerges. Crucially, we find that $τ_\mathrm{mem}$ increases linearly with the training set size $n$, while $τ_\mathrm{gen}$ remains constant. This creates a growing window of training times with $n$ where models generalize effectively, despite showing strong memorization if training continues beyond it. It is only when $n$ becomes larger than a model-dependent threshold that overfitting disappears at infinite training times. These findings reveal a form of implicit dynamical regularization in the training dynamics, which allow to avoid memorization even in highly overparameterized settings. Our results are supported by numerical experiments with standard U-Net architectures on realistic and synthetic datasets, and by a theoretical analysis using a tractable random features model studied in the high-dimensional limit.

Davide Badalotti, Carlo Baldassi, Marc Mézard et al.

We show that asymmetric deep recurrent neural networks, enhanced with additional sparse excitatory couplings, give rise to an exponentially large, dense accessible manifold of internal representations which can be found by different algorithms, including simple iterative dynamics. Building on the geometrical properties of the stable configurations, we propose a distributed learning scheme in which input-output associations emerge naturally from the recurrent dynamics, without any need of gradient evaluation. A critical feature enabling the learning process is the stability of the configurations reached at convergence, even after removal of the supervisory output signal. Extensive simulations demonstrate that this approach performs competitively on standard AI benchmarks. The model can be generalized in multiple directions, both computational and biological, potentially contributing to narrowing the gap between AI and computational neuroscience.

Bruno Loureiro, Cédric Gerbelot, Hugo Cui et al.

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.

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.

Federica Gerace, Bruno Loureiro, Florent Krzakala et al.

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.

Sebastian Goldt, Marc Mézard, Florent Krzakala et al.

Understanding the reasons for the success of deep neural networks trained using stochastic gradient-based methods is a key open problem for the nascent theory of deep learning. The types of data where these networks are most successful, such as images or sequences of speech, are characterised by intricate correlations. Yet, most theoretical work on neural networks does not explicitly model training data, or assumes that elements of each data sample are drawn independently from some factorised probability distribution. These approaches are thus by construction blind to the correlation structure of real-world data sets and their impact on learning in neural networks. Here, we introduce a generative model for structured data sets that we call the hidden manifold model (HMM). The idea is to construct high-dimensional inputs that lie on a lower-dimensional manifold, with labels that depend only on their position within this manifold, akin to a single layer decoder or generator in a generative adversarial network. We demonstrate that learning of the hidden manifold model is amenable to an analytical treatment by proving a "Gaussian Equivalence Property" (GEP), and we use the GEP to show how the dynamics of two-layer neural networks trained using one-pass stochastic gradient descent is captured by a set of integro-differential equations that track the performance of the network at all times. This permits us to analyse in detail how a neural network learns functions of increasing complexity during training, how its performance depends on its size and how it is impacted by parameters such as the learning rate or the dimension of the hidden manifold.

Andre Manoel, Florent Krzakala, Marc Mézard et al.

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.

Yoshiyuki Kabashima, Florent Krzakala, Marc Mézard et al.

We analyse the matrix factorization problem. Given a noisy measurement of a product of two matrices, the problem is to estimate back the original matrices. It arises in many applications such as dictionary learning, blind matrix calibration, sparse principal component analysis, blind source separation, low rank matrix completion, robust principal component analysis or factor analysis. It is also important in machine learning: unsupervised representation learning can often be studied through matrix factorization. We use the tools of statistical mechanics - the cavity and replica methods - to analyze the achievability and computational tractability of the inference problems in the setting of Bayes-optimal inference, which amounts to assuming that the two matrices have random independent elements generated from some known distribution, and this information is available to the inference algorithm. In this setting, we compute the minimal mean-squared-error achievable in principle in any computational time, and the error that can be achieved by an efficient approximate message passing algorithm. The computation is based on the asymptotic state-evolution analysis of the algorithm. The performance that our analysis predicts, both in terms of the achieved mean-squared-error, and in terms of sample complexity, is extremely promising and motivating for a further development of the algorithm.

Florent Krzakala, Marc Mézard, Lenka Zdeborová

We consider dictionary learning and blind calibration for signals and matrices created from a random ensemble. We study the mean-squared error in the limit of large signal dimension using the replica method and unveil the appearance of phase transitions delimiting impossible, possible-but-hard and possible inference regions. We also introduce an approximate message passing algorithm that asymptotically matches the theoretical performance, and show through numerical tests that it performs very well, for the calibration problem, for tractable system sizes.