arXiv subjects

Eric Moulines

Publications and source records attributed to Eric Moulines.

At least 109 records · Page 6Linked to original sources

arXiv2022

AskewSGD : An Annealed interval-constrained Optimisation method to train Quantized Neural Networks

In this paper, we develop a new algorithm, Annealed Skewed SGD - AskewSGD - for training deep neural networks (DNNs) with quantized weights. First, we formulate the training of quantized neural networks (QNNs) as a smoothed sequence of interval-constrained optimization problems. Then, we propose a new first-order stochastic method, AskewSGD, to solve each constrained optimization subproblem. Unlike algorithms with active sets and feasible directions, AskewSGD avoids projections or optimization under the entire feasible set and allows iterates that are infeasible. The numerical complexity of AskewSGD is comparable to existing approaches for training QNNs, such as the straight-through gradient estimator used in BinaryConnect, or other state of the art methods (ProxQuant, LUQ). We establish convergence guarantees for AskewSGD (under general assumptions for the objective function). Experimental results show that the AskewSGD algorithm performs better than or on par with state of the art methods in classical benchmarks.

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arXiv2022

Federated Averaging Langevin Dynamics: Toward a unified theory and new algorithms

This paper focuses on Bayesian inference in a federated learning context (FL). While several distributed MCMC algorithms have been proposed, few consider the specific limitations of FL such as communication bottlenecks and statistical heterogeneity. Recently, Federated Averaging Langevin Dynamics (FALD) was introduced, which extends the Federated Averaging algorithm to Bayesian inference. We obtain a novel tight non-asymptotic upper bound on the Wasserstein distance to the global posterior for FALD. This bound highlights the effects of statistical heterogeneity, which causes a drift in the local updates that negatively impacts convergence. We propose a new algorithm VR-FALD* that uses control variates to correct the client drift. We establish non-asymptotic bounds showing that VR-FALD* is not affected by statistical heterogeneity. Finally, we illustrate our results on several FL benchmarks for Bayesian inference.

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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.

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arXiv2022

Optimistic Posterior Sampling for Reinforcement Learning with Few Samples and Tight Guarantees

We consider reinforcement learning in an environment modeled by an episodic, finite, stage-dependent Markov decision process of horizon $H$ with $S$ states, and $A$ actions. The performance of an agent is measured by the regret after interacting with the environment for $T$ episodes. We propose an optimistic posterior sampling algorithm for reinforcement learning (OPSRL), a simple variant of posterior sampling that only needs a number of posterior samples logarithmic in $H$, $S$, $A$, and $T$ per state-action pair. For OPSRL we guarantee a high-probability regret bound of order at most $\widetilde{\mathcal{O}}(\sqrt{H^3SAT})$ ignoring $\text{poly}\log(HSAT)$ terms. The key novel technical ingredient is a new sharp anti-concentration inequality for linear forms which may be of independent interest. Specifically, we extend the normal approximation-based lower bound for Beta distributions by Alfers and Dinges [1984] to Dirichlet distributions. Our bound matches the lower bound of order $Ω(\sqrt{H^3SAT})$, thereby answering the open problems raised by Agrawal and Jia [2017b] for the episodic setting.

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arXiv2022

Particle-based, rapid incremental smoother meets particle Gibbs

The particle-based, rapid incremental smoother (PARIS) is a sequential Monte Carlo technique allowing for efficient online approximation of expectations of additive functionals under Feynman--Kac path distributions. Under weak assumptions, the algorithm has linear computational complexity and limited memory requirements. It also comes with a number of non-asymptotic bounds and convergence results. However, being based on self-normalised importance sampling, the PARIS estimator is biased; its bias is inversely proportional to the number of particles but has been found to grow linearly with the time horizon under appropriate mixing conditions. In this work, we propose the Parisian particle Gibbs (PPG) sampler, whose complexity is essentially the same as that of the PARIS and which significantly reduces the bias for a given computational complexity at the price of a modest increase in the variance. This method is a wrapper in the sense that it uses the PARIS algorithm in the inner loop of particle Gibbs to form a bias-reduced version of the targeted quantities. We substantiate the PPG algorithm with theoretical results, including new bounds on bias and variance as well as deviation inequalities. We illustrate our theoretical results with numerical experiments supporting our claims.

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arXiv2022

BR-SNIS: Bias Reduced Self-Normalized Importance Sampling

Importance Sampling (IS) is a method for approximating expectations under a target distribution using independent samples from a proposal distribution and the associated importance weights. In many applications, the target distribution is known only up to a normalization constant, in which case self-normalized IS (SNIS) can be used. While the use of self-normalization can have a positive effect on the dispersion of the estimator, it introduces bias. In this work, we propose a new method, BR-SNIS, whose complexity is essentially the same as that of SNIS and which significantly reduces bias without increasing the variance. This method is a wrapper in the sense that it uses the same proposal samples and importance weights as SNIS, but makes clever use of iterated sampling--importance resampling (ISIR) to form a bias-reduced version of the estimator. We furnish the proposed algorithm with rigorous theoretical results, including new bias, variance and high-probability bounds, and these are illustrated by numerical examples.

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arXiv2022

Diffusion bridges vector quantized Variational AutoEncoders

Vector Quantized-Variational AutoEncoders (VQ-VAE) are generative models based on discrete latent representations of the data, where inputs are mapped to a finite set of learned embeddings.To generate new samples, an autoregressive prior distribution over the discrete states must be trained separately. This prior is generally very complex and leads to slow generation. In this work, we propose a new model to train the prior and the encoder/decoder networks simultaneously. We build a diffusion bridge between a continuous coded vector and a non-informative prior distribution. The latent discrete states are then given as random functions of these continuous vectors. We show that our model is competitive with the autoregressive prior on the mini-Imagenet and CIFAR dataset and is efficient in both optimization and sampling. Our framework also extends the standard VQ-VAE and enables end-to-end training.

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arXiv2022

Variational Inference of overparameterized Bayesian Neural Networks: a theoretical and empirical study

This paper studies the Variational Inference (VI) used for training Bayesian Neural Networks (BNN) in the overparameterized regime, i.e., when the number of neurons tends to infinity. More specifically, we consider overparameterized two-layer BNN and point out a critical issue in the mean-field VI training. This problem arises from the decomposition of the lower bound on the evidence (ELBO) into two terms: one corresponding to the likelihood function of the model and the second to the Kullback-Leibler (KL) divergence between the prior distribution and the variational posterior. In particular, we show both theoretically and empirically that there is a trade-off between these two terms in the overparameterized regime only when the KL is appropriately re-scaled with respect to the ratio between the the number of observations and neurons. We also illustrate our theoretical results with numerical experiments that highlight the critical choice of this ratio.

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arXiv2022

From Dirichlet to Rubin: Optimistic Exploration in RL without Bonuses

We propose the Bayes-UCBVI algorithm for reinforcement learning in tabular, stage-dependent, episodic Markov decision process: a natural extension of the Bayes-UCB algorithm by Kaufmann et al. (2012) for multi-armed bandits. Our method uses the quantile of a Q-value function posterior as upper confidence bound on the optimal Q-value function. For Bayes-UCBVI, we prove a regret bound of order $\widetilde{O}(\sqrt{H^3SAT})$ where $H$ is the length of one episode, $S$ is the number of states, $A$ the number of actions, $T$ the number of episodes, that matches the lower-bound of $Ω(\sqrt{H^3SAT})$ up to poly-$\log$ terms in $H,S,A,T$ for a large enough $T$. To the best of our knowledge, this is the first algorithm that obtains an optimal dependence on the horizon $H$ (and $S$) without the need for an involved Bernstein-like bonus or noise. Crucial to our analysis is a new fine-grained anti-concentration bound for a weighted Dirichlet sum that can be of independent interest. We then explain how Bayes-UCBVI can be easily extended beyond the tabular setting, exhibiting a strong link between our algorithm and Bayesian bootstrap (Rubin, 1981).

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arXiv2022

QLSD: Quantised Langevin stochastic dynamics for Bayesian federated learning

The objective of Federated Learning (FL) is to perform statistical inference for data which are decentralised and stored locally on networked clients. FL raises many constraints which include privacy and data ownership, communication overhead, statistical heterogeneity, and partial client participation. In this paper, we address these problems in the framework of the Bayesian paradigm. To this end, we propose a novel federated Markov Chain Monte Carlo algorithm, referred to as Quantised Langevin Stochastic Dynamics which may be seen as an extension to the FL setting of Stochastic Gradient Langevin Dynamics, which handles the communication bottleneck using gradient compression. To improve performance, we then introduce variance reduction techniques, which lead to two improved versions coined \texttt{QLSD}$^{\star}$ and \texttt{QLSD}$^{++}$. We give both non-asymptotic and asymptotic convergence guarantees for the proposed algorithms. We illustrate their performances using various Bayesian Federated Learning benchmarks.

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arXiv2021

Federated Expectation Maximization with heterogeneity mitigation and variance reduction

The Expectation Maximization (EM) algorithm is the default algorithm for inference in latent variable models. As in any other field of machine learning, applications of latent variable models to very large datasets make the use of advanced parallel and distributed architectures mandatory. This paper introduces FedEM, which is the first extension of the EM algorithm to the federated learning context. FedEM is a new communication efficient method, which handles partial participation of local devices, and is robust to heterogeneous distributions of the datasets. To alleviate the communication bottleneck, FedEM compresses appropriately defined complete data sufficient statistics. We also develop and analyze an extension of FedEM to further incorporate a variance reduction scheme. In all cases, we derive finite-time complexity bounds for smooth non-convex problems. Numerical results are presented to support our theoretical findings, as well as an application to federated missing values imputation for biodiversity monitoring.

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arXiv2021

NEO: Non Equilibrium Sampling on the Orbit of a Deterministic Transform

Sampling from a complex distribution $π$ and approximating its intractable normalizing constant Z are challenging problems. In this paper, a novel family of importance samplers (IS) and Markov chain Monte Carlo (MCMC) samplers is derived. Given an invertible map T, these schemes combine (with weights) elements from the forward and backward Orbits through points sampled from a proposal distribution $ρ$. The map T does not leave the target $π$ invariant, hence the name NEO, standing for Non-Equilibrium Orbits. NEO-IS provides unbiased estimators of the normalizing constant and self-normalized IS estimators of expectations under $π$ while NEO-MCMC combines multiple NEO-IS estimates of the normalizing constant and an iterated sampling-importance resampling mechanism to sample from $π$. For T chosen as a discrete-time integrator of a conformal Hamiltonian system, NEO-IS achieves state-of-the art performance on difficult benchmarks and NEO-MCMC is able to explore highly multimodal targets. Additionally, we provide detailed theoretical results for both methods. In particular, we show that NEO-MCMC is uniformly geometrically ergodic and establish explicit mixing time estimates under mild conditions.

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arXiv2021

Monte Carlo Variational Auto-Encoders

Variational auto-encoders (VAE) are popular deep latent variable models which are trained by maximizing an Evidence Lower Bound (ELBO). To obtain tighter ELBO and hence better variational approximations, it has been proposed to use importance sampling to get a lower variance estimate of the evidence. However, importance sampling is known to perform poorly in high dimensions. While it has been suggested many times in the literature to use more sophisticated algorithms such as Annealed Importance Sampling (AIS) and its Sequential Importance Sampling (SIS) extensions, the potential benefits brought by these advanced techniques have never been realized for VAE: the AIS estimate cannot be easily differentiated, while SIS requires the specification of carefully chosen backward Markov kernels. In this paper, we address both issues and demonstrate the performance of the resulting Monte Carlo VAEs on a variety of applications.

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arXiv2021

DG-LMC: A Turn-key and Scalable Synchronous Distributed MCMC Algorithm via Langevin Monte Carlo within Gibbs

Performing reliable Bayesian inference on a big data scale is becoming a keystone in the modern era of machine learning. A workhorse class of methods to achieve this task are Markov chain Monte Carlo (MCMC) algorithms and their design to handle distributed datasets has been the subject of many works. However, existing methods are not completely either reliable or computationally efficient. In this paper, we propose to fill this gap in the case where the dataset is partitioned and stored on computing nodes within a cluster under a master/slaves architecture. We derive a user-friendly centralised distributed MCMC algorithm with provable scaling in high-dimensional settings. We illustrate the relevance of the proposed methodology on both synthetic and real data experiments.

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arXiv2021

Tight High Probability Bounds for Linear Stochastic Approximation with Fixed Stepsize

This paper provides a non-asymptotic analysis of linear stochastic approximation (LSA) algorithms with fixed stepsize. This family of methods arises in many machine learning tasks and is used to obtain approximate solutions of a linear system $\bar{A}θ= \bar{b}$ for which $\bar{A}$ and $\bar{b}$ can only be accessed through random estimates $\{({\bf A}_n, {\bf b}_n): n \in \mathbb{N}^*\}$. Our analysis is based on new results regarding moments and high probability bounds for products of matrices which are shown to be tight. We derive high probability bounds on the performance of LSA under weaker conditions on the sequence $\{({\bf A}_n, {\bf b}_n): n \in \mathbb{N}^*\}$ than previous works. However, in contrast, we establish polynomial concentration bounds with order depending on the stepsize. We show that our conclusions cannot be improved without additional assumptions on the sequence of random matrices $\{{\bf A}_n: n \in \mathbb{N}^*\}$, and in particular that no Gaussian or exponential high probability bounds can hold. Finally, we pay a particular attention to establishing bounds with sharp order with respect to the number of iterations and the stepsize and whose leading terms contain the covariance matrices appearing in the central limit theorems.

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arXiv2021

The Perturbed Prox-Preconditioned SPIDER algorithm for EM-based large scale learning

Incremental Expectation Maximization (EM) algorithms were introduced to design EM for the large scale learning framework by avoiding the full data set to be processed at each iteration. Nevertheless, these algorithms all assume that the conditional expectations of the sufficient statistics are explicit. In this paper, we propose a novel algorithm named Perturbed Prox-Preconditioned SPIDER (3P-SPIDER), which builds on the Stochastic Path Integral Differential EstimatoR EM (SPIDER-EM) algorithm. The 3P-SPIDER algorithm addresses many intractabilities of the E-step of EM; it also deals with non-smooth regularization and convex constraint set. Numerical experiments show that 3P-SPIDER outperforms other incremental EM methods and discuss the role of some design parameters.

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arXiv2021

Nonreversible MCMC from conditional invertible transforms: a complete recipe with convergence guarantees

Markov Chain Monte Carlo (MCMC) is a class of algorithms to sample complex and high-dimensional probability distributions. The Metropolis-Hastings (MH) algorithm, the workhorse of MCMC, provides a simple recipe to construct reversible Markov kernels. Reversibility is a tractable property that implies a less tractable but essential property here, invariance. Reversibility is however not necessarily desirable when considering performance. This has prompted recent interest in designing kernels breaking this property. At the same time, an active stream of research has focused on the design of novel versions of the MH kernel, some nonreversible, relying on the use of complex invertible deterministic transforms. While standard implementations of the MH kernel are well understood, the aforementioned developments have not received the same systematic treatment to ensure their validity. This paper fills the gap by developing general tools to ensure that a class of nonreversible Markov kernels, possibly relying on complex transforms, has the desired invariance property and leads to convergent algorithms. This leads to a set of simple and practically verifiable conditions.

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arXiv2021

On the Stability of Random Matrix Product with Markovian Noise: Application to Linear Stochastic Approximation and TD Learning

This paper studies the exponential stability of random matrix products driven by a general (possibly unbounded) state space Markov chain. It is a cornerstone in the analysis of stochastic algorithms in machine learning (e.g. for parameter tracking in online learning or reinforcement learning). The existing results impose strong conditions such as uniform boundedness of the matrix-valued functions and uniform ergodicity of the Markov chains. Our main contribution is an exponential stability result for the $p$-th moment of random matrix product, provided that (i) the underlying Markov chain satisfies a super-Lyapunov drift condition, (ii) the growth of the matrix-valued functions is controlled by an appropriately defined function (related to the drift condition). Using this result, we give finite-time $p$-th moment bounds for constant and decreasing stepsize linear stochastic approximation schemes with Markovian noise on general state space. We illustrate these findings for linear value-function estimation in reinforcement learning. We provide finite-time $p$-th moment bound for various members of temporal difference (TD) family of algorithms.

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