Search Results for author:
Found 84 papers, 39 papers with code
Bayes-optimal learning of an extensive-width neural network from quadratically many samples
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.
Could ChatGPT get an Engineering Degree? Evaluating Higher Education Vulnerability to AI Assistants
We investigate the potential scale of this vulnerability by measuring the degree to which AI assistants can complete assessment questions in standard university-level STEM courses.
Understanding Counting in Small Transformers: The Interplay between Attention and Feed-Forward Layers
From a broader perspective, our analysis offers a framework to understand how the interaction of different architectural components of transformer models shapes diverse algorithmic solutions and approximations.
Optimal thresholds and algorithms for a model of multi-modal learning in high dimensions
This work explores multi-modal inference in a high-dimensional simplified model, analytically quantifying the performance gain of multi-modal inference over that of analyzing modalities in isolation.
Fundamental computational limits of weak learnability in high-dimensional multi-index models
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 networks.
Fundamental limits of Non-Linear Low-Rank Matrix Estimation
We consider the task of estimating a low-rank matrix from non-linear and noisy observations.
Analysis of Bootstrap and Subsampling in High-dimensional Regularized Regression
We investigate popular resampling methods for estimating the uncertainty of statistical models, such as subsampling, bootstrap and the jackknife, and their performance in high-dimensional supervised regression tasks.
Asymptotics of feature learning in two-layer networks after one gradient-step
In this manuscript, we investigate the problem of how two-layer neural networks learn features from data, and improve over the kernel regime, after being trained with a single gradient descent step.
A phase transition between positional and semantic learning in a solvable model of dot-product attention
We investigate how a dot-product attention layer learns a positional attention matrix (with tokens attending to each other based on their respective positions) and a semantic attention matrix (with tokens attending to each other based on their meaning).
The Benefits of Reusing Batches for Gradient Descent in Two-Layer Networks: Breaking the Curse of Information and Leap Exponents
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.
Analysis of learning a flow-based generative model from limited sample complexity
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.
Sampling with flows, diffusion and autoregressive neural networks: A spin-glass perspective
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.
Gibbs Sampling the Posterior of Neural Networks
In this paper, we study sampling from a posterior derived from a neural network.
Expectation consistency for calibration of neural networks
Despite their incredible performance, it is well reported that deep neural networks tend to be overoptimistic about their prediction confidence.
Bayes-optimal Learning of Deep Random Networks of Extensive-width
We consider the problem of learning a target function corresponding to a deep, extensive-width, non-linear neural network with random Gaussian weights.
On double-descent in uncertainty quantification in overparametrized models
Uncertainty quantification is a central challenge in reliable and trustworthy machine learning.
The planted XY model: thermodynamics and inference
We derive the replica symmetric (RS) phase diagram in the temperature, ferromagnetic bias plane using the approximate message passing (AMP) algorithm and its state evolution (SE).
Subspace clustering in high-dimensions: Phase transitions & Statistical-to-Computational gap
A simple model to study subspace clustering is the high-dimensional $k$-Gaussian mixture model where the cluster means are sparse vectors.
Gaussian Universality of Perceptrons 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.
Learning curves for the multi-class teacher-student perceptron
For Gaussian teacher weights, we investigate the performance of ERM with both cross-entropy and square losses, and explore the role of ridge regularisation in approaching Bayes-optimality.
Theoretical characterization of uncertainty in high-dimensional linear classification
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.
Phase diagram of Stochastic Gradient Descent in high-dimensional two-layer neural networks
Despite the non-convex optimization landscape, over-parametrized shallow networks are able to achieve global convergence under gradient descent.
Error Scaling Laws for Kernel Classification under Source and Capacity Conditions
We find that our rates tightly describe the learning curves for this class of data sets, and are also observed on real data.
Learning Gaussian Mixtures with Generalized Linear Models: Precise Asymptotics in High-dimensions
Generalised linear models for multi-class classification problems are one of the fundamental building blocks of modern machine learning tasks.
Probing transfer learning with a model of synthetic correlated datasets
Transfer learning can significantly improve the sample efficiency of neural networks, by exploiting the relatedness between a data-scarce target task and a data-abundant source task.
Learning Gaussian Mixtures with Generalised Linear Models: Precise Asymptotics in High-dimensions
Generalised linear models for multi-class classification problems are one of the fundamental building blocks of modern machine learning tasks.
Generalization Error Rates in Kernel Regression: The Crossover from the Noiseless to Noisy Regime
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.
Bayesian reconstruction of memories stored in neural networks from their connectivity
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.
Stochasticity helps to navigate rough landscapes: comparing gradient-descent-based algorithms in the phase retrieval problem
In this paper we investigate how gradient-based algorithms such as gradient descent, (multi-pass) stochastic gradient descent, its persistent variant, and the Langevin algorithm navigate non-convex loss-landscapes and which of them is able to reach the best generalization error at limited sample complexity.
Classifying high-dimensional Gaussian mixtures: Where kernel methods fail and neural networks succeed
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.
Learning curves of generic features maps for realistic datasets with a teacher-student model
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.
Construction of optimal spectral methods in phase retrieval
We consider the phase retrieval problem, in which the observer wishes to recover a $n$-dimensional real or complex signal $\mathbf{X}^\star$ from the (possibly noisy) observation of $|\mathbf{\Phi} \mathbf{X}^\star|$, in which $\mathbf{\Phi}$ is a matrix of size $m \times n$.
Information Theory Disordered Systems and Neural Networks Information Theory
Solvable Model for Inheriting the Regularization through Knowledge Distillation
In recent years the empirical success of transfer learning with neural networks has stimulated an increasing interest in obtaining a theoretical understanding of its core properties.
Epidemic mitigation by statistical inference from contact tracing data
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.
Optimization and Generalization of Shallow Neural Networks with Quadratic Activation Functions
We consider a teacher-student scenario where the teacher has the same structure as the student with a hidden layer of smaller width $m^*\le m$.
The Gaussian equivalence of generative models for learning with shallow neural networks
Here, we go beyond this simple paradigm by studying the performance of neural networks trained on data drawn from pre-trained generative models.
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.
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.
Dynamical mean-field theory for stochastic gradient descent in Gaussian mixture classification
We define a particular stochastic process for which SGD can be extended to a continuous-time limit that we call stochastic gradient flow.
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 $Y_\mu = | \sum_{i=1}^n \Phi_{\mu i} X^{\star}_i/\sqrt{n}|$, for a large class of correlated real and complex random sensing matrices $\mathbf{\Phi}$, in a high-dimensional setting where $m, n\to\infty$ while $\alpha = m/n=\Theta(1)$.
Tree-AMP: Compositional Inference with Tree Approximate Message Passing
We introduce Tree-AMP, standing for Tree Approximate Message Passing, a python package for compositional inference in high-dimensional tree-structured models.
The role of regularization in classification of high-dimensional noisy Gaussian mixture
We also illustrate the interpolation peak at low regularization, and analyze the role of the respective sizes of the two clusters.
Generalisation error in learning with random features and the hidden manifold model
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.
Large deviations for the perceptron model and consequences for active learning
These large deviations then provide optimal achievable performance boundaries for any active learning algorithm.
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.
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.
Who is Afraid of Big Bad Minima? Analysis of gradient-flow in spiked matrix-tensor models
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.
Modelling the influence of data structure on learning in neural networks: the hidden manifold model
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.
Multilayer Modularity Belief Propagation To Assess Detectability Of Community Structure
We compare our approach with a widely used community detection tool, GenLouvain, across a range of synthetic, multilayer benchmark networks, demonstrating that our method performs comparably to the state of the art.
Social and Information Networks Data Analysis, Statistics and Probability Physics and Society
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.
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.
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*.
The spiked matrix model with generative priors
Here, we replace the sparsity assumption by generative modelling, and investigate the consequences on statistical and algorithmic properties.
Machine learning and the physical sciences
Machine learning encompasses a broad range of algorithms and modeling tools used for a vast array of data processing tasks, which has entered most scientific disciplines in recent years.
Computational Physics Cosmology and Nongalactic Astrophysics Disordered Systems and Neural Networks High Energy Physics - Theory Quantum Physics
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.
Generalisation dynamics of online learning in over-parameterised neural networks
Deep neural networks achieve stellar generalisation on a variety of problems, despite often being large enough to easily fit all their training data.
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.
Rank-one matrix estimation: analysis of algorithmic and information theoretic limits by the spatial coupling method
We characterize the detectability phase transitions in a large set of estimation problems, where we show that there exists a gap between what currently known polynomial algorithms (in particular spectral methods and approximate message-passing) can do and what is expected information theoretically.
Approximate message-passing for convex optimization with non-separable penalties
We introduce an iterative optimization scheme for convex objectives consisting of a linear loss and a non-separable penalty, based on the expectation-consistent approximation and the vector approximate message-passing (VAMP) algorithm.
Approximate Survey Propagation for Statistical Inference
Approximate message passing algorithm enjoyed considerable attention in the last decade.
The committee machine: Computational to statistical gaps in learning a two-layers neural network
Heuristic tools from statistical physics have been used in the past to locate the phase transitions and compute the optimal learning and generalization errors in the teacher-student scenario in multi-layer neural networks.
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.
Glassy nature of the hard phase in inference problems
An algorithmically hard phase was described in a range of inference problems: even if the signal can be reconstructed with a small error from an information theoretic point of view, known algorithms fail unless the noise-to-signal ratio is sufficiently small.
Dense Limit of the Dawid-Skene Model for Crowdsourcing and Regions of Sub-optimality of Message Passing Algorithms
We further study numerically the performance of approximate message passing, derived in the dense limit, on sparse instances and carry out experiments on a real world dataset.
Optimal Errors and Phase Transitions in High-Dimensional Generalized Linear Models
Non-rigorous predictions for the optimal errors existed for special cases of GLMs, e. g. for the perceptron, in the field of statistical physics based on the so-called replica method.
Streaming Bayesian inference: theoretical limits and mini-batch approximate message-passing
In statistical learning for real-world large-scale data problems, one must often resort to "streaming" algorithms which operate sequentially on small batches of data.
Multi-Layer Generalized Linear Estimation
We consider the problem of reconstructing a signal from multi-layered (possibly) non-linear measurements.
Phase transitions and optimal algorithms in high-dimensional Gaussian mixture clustering
We consider the problem of Gaussian mixture clustering in the high-dimensional limit where the data consists of $m$ points in $n$ dimensions, $n, m \rightarrow \infty$ and $\alpha = m/n$ stays finite.
Fast Randomized Semi-Supervised Clustering
We consider the problem of clustering partially labeled data from a minimal number of randomly chosen pairwise comparisons between the items.
Clustering from Sparse Pairwise Measurements
We consider the problem of grouping items into clusters based on few random pairwise comparisons between the items.
Statistical physics of inference: Thresholds and algorithms
Many questions of fundamental interest in todays science can be formulated as inference problems: Some partial, or noisy, observations are performed over a set of variables and the goal is to recover, or infer, the values of the variables based on the indirect information contained in the measurements.
MMSE of probabilistic low-rank matrix estimation: Universality with respect to the output channel
This paper considers probabilistic estimation of a low-rank matrix from non-linear element-wise measurements of its elements.
Matrix Completion from Fewer Entries: Spectral Detectability and Rank Estimation
We propose a spectral algorithm for these two tasks called MaCBetH (for Matrix Completion with the Bethe Hessian).
Spectral Detection in the Censored Block Model
We describe two spectral algorithms for this task based on the non-backtracking and the Bethe Hessian operators.
Sparse Estimation with the Swept Approximated Message-Passing Algorithm
Approximate Message Passing (AMP) has been shown to be a superior method for inference problems, such as the recovery of signals from sets of noisy, lower-dimensionality measurements, both in terms of reconstruction accuracy and in computational efficiency.
Spectral Clustering of Graphs with the Bethe Hessian
We show that this approach combines the performances of the non-backtracking operator, thus detecting clusters all the way down to the theoretical limit in the stochastic block model, with the computational, theoretical and memory advantages of real symmetric matrices.
Phase transitions in semisupervised clustering of sparse networks
For larger $k$ where a hard but detectable regime exists, we find that the easy/hard transition (the point at which efficient algorithms can do better than chance) becomes a line of transitions where the accuracy jumps discontinuously at a critical value of $\alpha$.
Phase transitions and sample complexity in Bayes-optimal 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.
Blind Calibration in Compressed Sensing using Message Passing Algorithms
We study numerically the phase diagram of the blind calibration problem, and show that even in cases where convex relaxation is possible, our algorithm requires a smaller number of measurements and/or signals in order to perform well.
Spectral redemption: clustering sparse networks
Spectral algorithms are classic approaches to clustering and community detection in networks.
Probabilistic Reconstruction in Compressed Sensing: Algorithms, Phase Diagrams, and Threshold Achieving Matrices
We further develop the asymptotic analysis of the corresponding phase diagrams with and without measurement noise, for different distribution of signals, and discuss the best possible reconstruction performances regardless of the algorithm.
Statistical Mechanics Information Theory Information Theory
Statistical physics-based reconstruction in compressed sensing
Compressed sensing is triggering a major evolution in signal acquisition.
Statistical Mechanics Information Theory Information Theory
Asymptotic analysis of the stochastic block model for modular networks and its algorithmic applications
In this paper we extend our previous work on the stochastic block model, a commonly used generative model for social and biological networks, and the problem of inferring functional groups or communities from the topology of the network.
Statistical Mechanics Disordered Systems and Neural Networks Social and Information Networks Physics and Society
Phase transition in the detection of modules in sparse networks
We present an asymptotically exact analysis of the problem of detecting communities in sparse random networks.
