arXiv subjects

Marylou Gabrié

Publications and source records attributed to Marylou Gabrié.

32 records · Page 2Linked to original sources

arXiv2022

Local-Global MCMC kernels: the best of both worlds

Recent works leveraging learning to enhance sampling have shown promising results, in particular by designing effective non-local moves and global proposals. However, learning accuracy is inevitably limited in regions where little data is available such as in the tails of distributions as well as in high-dimensional problems. In the present paper we study an Explore-Exploit Markov chain Monte Carlo strategy ($Ex^2MCMC$) that combines local and global samplers showing that it enjoys the advantages of both approaches. We prove $V$-uniform geometric ergodicity of $Ex^2MCMC$ without requiring a uniform adaptation of the global sampler to the target distribution. We also compute explicit bounds on the mixing rate of the Explore-Exploit strategy under realistic conditions. Moreover, we also analyze an adaptive version of the strategy ($FlEx^2MCMC$) where a normalizing flow is trained while sampling to serve as a proposal for global moves. We illustrate the efficiency of $Ex^2MCMC$ and its adaptive version on classical sampling benchmarks as well as in sampling high-dimensional distributions defined by Generative Adversarial Networks seen as Energy Based Models. We provide the code to reproduce the experiments at the link: https://github.com/svsamsonov/ex2mcmc_new.

stat.ML↗

arXiv2022

Dual Training of Energy-Based Models with Overparametrized Shallow Neural Networks

Energy-based models (EBMs) are generative models that are usually trained via maximum likelihood estimation. This approach becomes challenging in generic situations where the trained energy is non-convex, due to the need to sample the Gibbs distribution associated with this energy. Using general Fenchel duality results, we derive variational principles dual to maximum likelihood EBMs with shallow overparametrized neural network energies, both in the feature-learning and lazy linearized regimes. In the feature-learning regime, this dual formulation justifies using a two time-scale gradient ascent-descent (GDA) training algorithm in which one updates concurrently the particles in the sample space and the neurons in the parameter space of the energy. We also consider a variant of this algorithm in which the particles are sometimes restarted at random samples drawn from the data set, and show that performing these restarts at every iteration step corresponds to score matching training. These results are illustrated in simple numerical experiments, which indicates that GDA performs best when features and particles are updated using similar time scales.

cs.LG↗

arXiv2022

Adaptive Monte Carlo augmented with normalizing flows

Many problems in the physical sciences, machine learning, and statistical inference necessitate sampling from a high-dimensional, multi-modal probability distribution. Markov Chain Monte Carlo (MCMC) algorithms, the ubiquitous tool for this task, typically rely on random local updates to propagate configurations of a given system in a way that ensures that generated configurations will be distributed according to a target probability distribution asymptotically. In high-dimensional settings with multiple relevant metastable basins, local approaches require either immense computational effort or intricately designed importance sampling strategies to capture information about, for example, the relative populations of such basins. Here we analyze an adaptive MCMC which augments MCMC sampling with nonlocal transition kernels parameterized with generative models known as normalizing flows. We focus on a setting where there is no preexisting data, as is commonly the case for problems in which MCMC is used. Our method uses: (i) a MCMC strategy that blends local moves obtained from any standard transition kernel with those from a generative model to accelerate the sampling and (ii) the data generated this way to adapt the generative model and improve its efficacy in the MCMC algorithm. We provide a theoretical analysis of the convergence properties of this algorithm, and investigate numerically its efficiency, in particular in terms of its propensity to equilibrate fast between metastable modes whose rough location is known \textit{a~priori} but respective probability weight is not. We show that our algorithm can sample effectively across large free energy barriers, providing dramatic accelerations relative to traditional MCMC algorithms.

physics.data-an↗

arXiv2021

Adaptation of the Independent Metropolis-Hastings Sampler with Normalizing Flow Proposals

Markov Chain Monte Carlo (MCMC) methods are a powerful tool for computation with complex probability distributions. However the performance of such methods is critically dependant on properly tuned parameters, most of which are difficult if not impossible to know a priori for a given target distribution. Adaptive MCMC methods aim to address this by allowing the parameters to be updated during sampling based on previous samples from the chain at the expense of requiring a new theoretical analysis to ensure convergence. In this work we extend the convergence theory of adaptive MCMC methods to a new class of methods built on a powerful class of parametric density estimators known as normalizing flows. In particular, we consider an independent Metropolis-Hastings sampler where the proposal distribution is represented by a normalizing flow whose parameters are updated using stochastic gradient descent. We explore the practical performance of this procedure on both synthetic settings and in the analysis of a physical field system and compare it against both adaptive and non-adaptive MCMC methods.

stat.CO↗

arXiv2021

On the interplay between data structure and loss function in classification problems

One of the central puzzles in modern machine learning is the ability of heavily overparametrized models to generalize well. Although the low-dimensional structure of typical datasets is key to this behavior, most theoretical studies of overparametrization focus on isotropic inputs. In this work, we instead consider an analytically tractable model of structured data, where the input covariance is built from independent blocks allowing us to tune the saliency of low-dimensional structures and their alignment with respect to the target function. Using methods from statistical physics, we derive a precise asymptotic expression for the train and test error achieved by random feature models trained to classify such data, which is valid for any convex loss function. We study in detail how the data structure affects the double descent curve, and show that in the over-parametrized regime, its impact is greater for logistic loss than for mean-squared loss: the easier the task, the wider the gap in performance at the advantage of the logistic loss. Our insights are confirmed by numerical experiments on MNIST and CIFAR10.

cs.LG↗

arXiv2021

Efficient Bayesian Sampling Using Normalizing Flows to Assist Markov Chain Monte Carlo Methods

Normalizing flows can generate complex target distributions and thus show promise in many applications in Bayesian statistics as an alternative or complement to MCMC for sampling posteriors. Since no data set from the target posterior distribution is available beforehand, the flow is typically trained using the reverse Kullback-Leibler (KL) divergence that only requires samples from a base distribution. This strategy may perform poorly when the posterior is complicated and hard to sample with an untrained normalizing flow. Here we explore a distinct training strategy, using the direct KL divergence as loss, in which samples from the posterior are generated by (i) assisting a local MCMC algorithm on the posterior with a normalizing flow to accelerate its mixing rate and (ii) using the data generated this way to train the flow. The method only requires a limited amount of \textit{a~priori} input about the posterior, and can be used to estimate the evidence required for model validation, as we illustrate on examples.

stat.ML↗

arXiv2021

Phase Retrieval with Holography and Untrained Priors: Tackling the Challenges of Low-Photon Nanoscale Imaging

Phase retrieval is the inverse problem of recovering a signal from magnitude-only Fourier measurements, and underlies numerous imaging modalities, such as Coherent Diffraction Imaging (CDI). A variant of this setup, known as holography, includes a reference object that is placed adjacent to the specimen of interest before measurements are collected. The resulting inverse problem, known as holographic phase retrieval, is well-known to have improved problem conditioning relative to the original. This innovation, i.e. Holographic CDI, becomes crucial at the nanoscale, where imaging specimens such as viruses, proteins, and crystals require low-photon measurements. This data is highly corrupted by Poisson shot noise, and often lacks low-frequency content as well. In this work, we introduce a dataset-free deep learning framework for holographic phase retrieval adapted to these challenges. The key ingredients of our approach are the explicit and flexible incorporation of the physical forward model into an automatic differentiation procedure, the Poisson log-likelihood objective function, and an optional untrained deep image prior. We perform extensive evaluation under realistic conditions. Compared to competing classical methods, our method recovers signal from higher noise levels and is more resilient to suboptimal reference design, as well as to large missing regions of low frequencies in the observations. Finally, we show that these properties carry over to experimental data acquired on optical wavelengths. To the best of our knowledge, this is the first work to consider a dataset-free machine learning approach for holographic phase retrieval.

cs.LG↗

arXiv2020

Blind calibration for compressed sensing: State evolution and an online algorithm

Compressed sensing, allows to acquire compressible signals with a small number of measurements. In applications, a hardware implementation often requires a calibration as the sensing process is not perfectly known. Blind calibration, that is performing at the same time calibration and compressed sensing is thus particularly appealing. A potential approach was suggested by Schülke and collaborators in Schülke et al. 2013 and 2015, using approximate message passing (AMP) for blind calibration (cal-AMP). Here, the algorithm is extended from the already proposed offline case to the online case, where the calibration is refined step by step as new measured samples are received. Furthermore, we show that the performance of both the offline and the online algorithms can be theoretically studied via the State Evolution (SE) formalism. Through numerical simulations, the efficiency of cal-AMP and the consistency of the theoretical predictions are confirmed.

cond-mat.stat-mech↗

arXiv2020

Mean-field inference methods for neural networks

Machine learning algorithms relying on deep neural networks recently allowed a great leap forward in artificial intelligence. Despite the popularity of their applications, the efficiency of these algorithms remains largely unexplained from a theoretical point of view. The mathematical description of learning problems involves very large collections of interacting random variables, difficult to handle analytically as well as numerically. This complexity is precisely the object of study of statistical physics. Its mission, originally pointed towards natural systems, is to understand how macroscopic behaviors arise from microscopic laws. Mean-field methods are one type of approximation strategy developed in this view. We review a selection of classical mean-field methods and recent progress relevant for inference in neural networks. In particular, we remind the principles of derivations of high-temperature expansions, the replica method and message passing algorithms, highlighting their equivalences and complementarities. We also provide references for past and current directions of research on neural networks relying on mean-field methods.

cond-mat.dis-nn↗

arXiv2018

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.

cs.LG↗

arXiv2018

A Deterministic and Generalized Framework for Unsupervised Learning with Restricted Boltzmann Machines

Restricted Boltzmann machines (RBMs) are energy-based neural-networks which are commonly used as the building blocks for deep architectures neural architectures. In this work, we derive a deterministic framework for the training, evaluation, and use of RBMs based upon the Thouless-Anderson-Palmer (TAP) mean-field approximation of widely-connected systems with weak interactions coming from spin-glass theory. While the TAP approach has been extensively studied for fully-visible binary spin systems, our construction is generalized to latent-variable models, as well as to arbitrarily distributed real-valued spin systems with bounded support. In our numerical experiments, we demonstrate the effective deterministic training of our proposed models and are able to show interesting features of unsupervised learning which could not be directly observed with sampling. Additionally, we demonstrate how to utilize our TAP-based framework for leveraging trained RBMs as joint priors in denoising problems.

cs.LG↗

arXiv2017

Phase transitions in the $q$-coloring of random hypergraphs

We study in this paper the structure of solutions in the random hypergraph coloring problem and the phase transitions they undergo when the density of constraints is varied. Hypergraph coloring is a constraint satisfaction problem where each constraint includes $K$ variables that must be assigned one out of $q$ colors in such a way that there are no monochromatic constraints, i.e. there are at least two distinct colors in the set of variables belonging to every constraint. This problem generalizes naturally coloring of random graphs ($K=2$) and bicoloring of random hypergraphs ($q=2$), both of which were extensively studied in past works. The study of random hypergraph coloring gives us access to a case where both the size $q$ of the domain of the variables and the arity $K$ of the constraints can be varied at will. Our work provides explicit values and predictions for a number of phase transitions that were discovered in other constraint satisfaction problems but never evaluated before in hypergraph coloring. Among other cases we revisit the hypergraph bicoloring problem ($q=2$) where we find that for $K=3$ and $K=4$ the colorability threshold is not given by the one-step-replica-symmetry-breaking analysis as the latter is unstable towards more levels of replica symmetry breaking. We also unveil and discuss the coexistence of two different 1RSB solutions in the case of $q=2$, $K \ge 4$. Finally we present asymptotic expansions for the density of constraints at which various phase transitions occur, in the limit where $q$ and/or $K$ diverge.

cond-mat.dis-nn↗

arXiv2016

Inferring Sparsity: Compressed Sensing using Generalized Restricted Boltzmann Machines

In this work, we consider compressed sensing reconstruction from $M$ measurements of $K$-sparse structured signals which do not possess a writable correlation model. Assuming that a generative statistical model, such as a Boltzmann machine, can be trained in an unsupervised manner on example signals, we demonstrate how this signal model can be used within a Bayesian framework of signal reconstruction. By deriving a message-passing inference for general distribution restricted Boltzmann machines, we are able to integrate these inferred signal models into approximate message passing for compressed sensing reconstruction. Finally, we show for the MNIST dataset that this approach can be very effective, even for $M < K$.

cs.IT↗

arXiv2015

Training Restricted Boltzmann Machines via the Thouless-Anderson-Palmer Free Energy

Restricted Boltzmann machines are undirected neural networks which have been shown to be effective in many applications, including serving as initializations for training deep multi-layer neural networks. One of the main reasons for their success is the existence of efficient and practical stochastic algorithms, such as contrastive divergence, for unsupervised training. We propose an alternative deterministic iterative procedure based on an improved mean field method from statistical physics known as the Thouless-Anderson-Palmer approach. We demonstrate that our algorithm provides performance equal to, and sometimes superior to, persistent contrastive divergence, while also providing a clear and easy to evaluate objective function. We believe that this strategy can be easily generalized to other models as well as to more accurate higher-order approximations, paving the way for systematic improvements in training Boltzmann machines with hidden units.

cond-mat.dis-nn↗