arXiv subjects

Alain Durmus

Publications and source records attributed to Alain Durmus.

At least 73 records · Page 4Linked to original sources

arXiv2024

VITS : Variational Inference Thompson Sampling for contextual bandits

In this paper, we introduce and analyze a variant of the Thompson sampling (TS) algorithm for contextual bandits. At each round, traditional TS requires samples from the current posterior distribution, which is usually intractable. To circumvent this issue, approximate inference techniques can be used and provide samples with distribution close to the posteriors. However, current approximate techniques yield to either poor estimation (Laplace approximation) or can be computationally expensive (MCMC methods, Ensemble sampling...). In this paper, we propose a new algorithm, Varational Inference Thompson sampling VITS, based on Gaussian Variational Inference. This scheme provides powerful posterior approximations which are easy to sample from, and is computationally efficient, making it an ideal choice for TS. In addition, we show that VITS achieves a sub-linear regret bound of the same order in the dimension and number of round as traditional TS for linear contextual bandit. Finally, we demonstrate experimentally the effectiveness of VITS on both synthetic and real world datasets.

stat.ML↗

arXiv2024

Optimal Scaling Results for Moreau-Yosida Metropolis-adjusted Langevin Algorithms

We consider a recently proposed class of MCMC methods which uses proximity maps instead of gradients to build proposal mechanisms which can be employed for both differentiable and non-differentiable targets. These methods have been shown to be stable for a wide class of targets, making them a valuable alternative to Metropolis-adjusted Langevin algorithms (MALA); and have found wide application in imaging contexts. The wider stability properties are obtained by building the Moreau-Yosida envelope for the target of interest, which depends on a parameter $λ$. In this work, we investigate the optimal scaling problem for this class of algorithms, which encompasses MALA, and provide practical guidelines for the implementation of these methods.

stat.CO↗

arXiv2024

Theoretical Guarantees for Variational Inference with Fixed-Variance Mixture of Gaussians

Variational inference (VI) is a popular approach in Bayesian inference, that looks for the best approximation of the posterior distribution within a parametric family, minimizing a loss that is typically the (reverse) Kullback-Leibler (KL) divergence. Despite its empirical success, the theoretical properties of VI have only received attention recently, and mostly when the parametric family is the one of Gaussians. This work aims to contribute to the theoretical study of VI in the non-Gaussian case by investigating the setting of Mixture of Gaussians with fixed covariance and constant weights. In this view, VI over this specific family can be casted as the minimization of a Mollified relative entropy, i.e. the KL between the convolution (with respect to a Gaussian kernel) of an atomic measure supported on Diracs, and the target distribution. The support of the atomic measure corresponds to the localization of the Gaussian components. Hence, solving variational inference becomes equivalent to optimizing the positions of the Diracs (the particles), which can be done through gradient descent and takes the form of an interacting particle system. We study two sources of error of variational inference in this context when optimizing the mollified relative entropy. The first one is an optimization result, that is a descent lemma establishing that the algorithm decreases the objective at each iteration. The second one is an approximation error, that upper bounds the objective between an optimal finite mixture and the target distribution.

stat.ML↗

arXiv2024

Second order quantitative bounds for unadjusted generalized Hamiltonian Monte Carlo

This paper provides a convergence analysis for generalized Hamiltonian Monte Carlo samplers, a family of Markov Chain Monte Carlo methods based on leapfrog integration of Hamiltonian dynamics and kinetic Langevin diffusion, that encompasses the unadjusted Hamiltonian Monte Carlo method. Assuming that the target distribution $π$ satisfies a log-Sobolev inequality and mild conditions on the corresponding potential function, we establish quantitative bounds on the relative entropy of the iterates defined by the algorithm, with respect to $π$. Our approach is based on a perturbative and discrete version of the modified entropy method developed to establish hypocoercivity for the continuous-time kinetic Langevin process. As a corollary of our main result, we are able to derive complexity bounds for the class of algorithms at hand. In particular, we show that the total number of iterations to achieve a target accuracy $\varepsilon >0$ is of order $d/\varepsilon^{1/4}$, where $d$ is the dimension of the problem. This result can be further improved in the case of weakly interacting mean field potentials, for which we find a total number of iterations of order $(d/\varepsilon)^{1/4}$.

math.PR↗

arXiv2024

Incentivized Learning in Principal-Agent Bandit Games

This work considers a repeated principal-agent bandit game, where the principal can only interact with her environment through the agent. The principal and the agent have misaligned objectives and the choice of action is only left to the agent. However, the principal can influence the agent's decisions by offering incentives which add up to his rewards. The principal aims to iteratively learn an incentive policy to maximize her own total utility. This framework extends usual bandit problems and is motivated by several practical applications, such as healthcare or ecological taxation, where traditionally used mechanism design theories often overlook the learning aspect of the problem. We present nearly optimal (with respect to a horizon $T$) learning algorithms for the principal's regret in both multi-armed and linear contextual settings. Finally, we support our theoretical guarantees through numerical experiments.

stat.ML↗

arXiv2024

On Sampling with Approximate Transport Maps

Transport maps can ease the sampling of distributions with non-trivial geometries by transforming them into distributions that are easier to handle. The potential of this approach has risen with the development of Normalizing Flows (NF) which are maps parameterized with deep neural networks trained to push a reference distribution towards a target. NF-enhanced samplers recently proposed blend (Markov chain) Monte Carlo methods with either (i) proposal draws from the flow or (ii) a flow-based reparametrization. In both cases, the quality of the learned transport conditions performance. The present work clarifies for the first time the relative strengths and weaknesses of these two approaches. Our study concludes that multimodal targets can be reliably handled with flow-based proposals up to moderately high dimensions. In contrast, methods relying on reparametrization struggle with multimodality but are more robust otherwise in high-dimensional settings and under poor training. To further illustrate the influence of target-proposal adequacy, we also derive a new quantitative bound for the mixing time of the Independent Metropolis-Hastings sampler.

stat.ML↗

arXiv2024

Implicit Bias in Noisy-SGD: With Applications to Differentially Private Training

Training Deep Neural Networks (DNNs) with small batches using Stochastic Gradient Descent (SGD) yields superior test performance compared to larger batches. The specific noise structure inherent to SGD is known to be responsible for this implicit bias. DP-SGD, used to ensure differential privacy (DP) in DNNs' training, adds Gaussian noise to the clipped gradients. Surprisingly, large-batch training still results in a significant decrease in performance, which poses an important challenge because strong DP guarantees necessitate the use of massive batches. We first show that the phenomenon extends to Noisy-SGD (DP-SGD without clipping), suggesting that the stochasticity (and not the clipping) is the cause of this implicit bias, even with additional isotropic Gaussian noise. We theoretically analyse the solutions obtained with continuous versions of Noisy-SGD for the Linear Least Square and Diagonal Linear Network settings, and reveal that the implicit bias is indeed amplified by the additional noise. Thus, the performance issues of large-batch DP-SGD training are rooted in the same underlying principles as SGD, offering hope for potential improvements in large batch training strategies.

stat.ML↗

arXiv2023

Discrete sticky couplings of functional autoregressive processes

In this paper, we provide bounds in Wasserstein and total variation distances between the distributions of the successive iterates of two functional autoregressive processes with isotropic Gaussian noise of the form $Y_{k+1} = \mathrm{T}_γ(Y_k) + \sqrt{γσ^2} Z_{k+1}$ and $\tilde{Y}_{k+1} = \tilde{\mathrm{T}}_γ(\tilde{Y}_k) + \sqrt{γσ^2} \tilde{Z}_{k+1}$. More precisely, we give non-asymptotic bounds on $ρ(\mathcal{L}(Y_{k}),\mathcal{L}(\tilde{Y}_k))$, where $ρ$ is an appropriate weighted Wasserstein distance or a $V$-distance, uniformly in the parameter $γ$, and on $ρ(π_γ,\tildeπ_γ)$, where $π_γ$ and $\tildeπ_γ$ are the respective stationary measures of the two processes. The class of considered processes encompasses the Euler-Maruyama discretization of Langevin diffusions and its variants. The bounds we derive are of order $γ$ as $γ\to 0$. To obtain our results, we rely on the construction of a discrete sticky Markov chain $(W_k^{(γ)})_{k \in \mathbb{N}}$ which bounds the distance between an appropriate coupling of the two processes. We then establish stability and quantitative convergence results for this process uniformly on $γ$. In addition, we show that it converges in distribution to the continuous sticky process studied in previous work. Finally, we apply our result to Bayesian inference of ODE parameters and numerically illustrate them on two particular problems.

math.PR↗

arXiv2023

Unbiased constrained sampling with Self-Concordant Barrier Hamiltonian Monte Carlo

In this paper, we propose Barrier Hamiltonian Monte Carlo (BHMC), a version of the HMC algorithm which aims at sampling from a Gibbs distribution $π$ on a manifold $\mathrm{M}$, endowed with a Hessian metric $\mathfrak{g}$ derived from a self-concordant barrier. Our method relies on Hamiltonian dynamics which comprises $\mathfrak{g}$. Therefore, it incorporates the constraints defining $\mathrm{M}$ and is able to exploit its underlying geometry. However, the corresponding Hamiltonian dynamics is defined via non separable Ordinary Differential Equations (ODEs) in contrast to the Euclidean case. It implies unavoidable bias in existing generalization of HMC to Riemannian manifolds. In this paper, we propose a new filter step, called "involution checking step", to address this problem. This step is implemented in two versions of BHMC, coined continuous BHMC (c-BHMC) and numerical BHMC (n-BHMC) respectively. Our main results establish that these two new algorithms generate reversible Markov chains with respect to $π$ and do not suffer from any bias in comparison to previous implementations. Our conclusions are supported by numerical experiments where we consider target distributions defined on polytopes.

stat.ML↗

arXiv2023

Tree-Based Diffusion Schrödinger Bridge with Applications to Wasserstein Barycenters

Multi-marginal Optimal Transport (mOT), a generalization of OT, aims at minimizing the integral of a cost function with respect to a distribution with some prescribed marginals. In this paper, we consider an entropic version of mOT with a tree-structured quadratic cost, i.e., a function that can be written as a sum of pairwise cost functions between the nodes of a tree. To address this problem, we develop Tree-based Diffusion Schrödinger Bridge (TreeDSB), an extension of the Diffusion Schrödinger Bridge (DSB) algorithm. TreeDSB corresponds to a dynamic and continuous state-space counterpart of the multimarginal Sinkhorn algorithm. A notable use case of our methodology is to compute Wasserstein barycenters which can be recast as the solution of a mOT problem on a star-shaped tree. We demonstrate that our methodology can be applied in high-dimensional settings such as image interpolation and Bayesian fusion.

stat.ML↗

arXiv2023

Approximate Heavy Tails in Offline (Multi-Pass) Stochastic Gradient Descent

A recent line of empirical studies has demonstrated that SGD might exhibit a heavy-tailed behavior in practical settings, and the heaviness of the tails might correlate with the overall performance. In this paper, we investigate the emergence of such heavy tails. Previous works on this problem only considered, up to our knowledge, online (also called single-pass) SGD, in which the emergence of heavy tails in theoretical findings is contingent upon access to an infinite amount of data. Hence, the underlying mechanism generating the reported heavy-tailed behavior in practical settings, where the amount of training data is finite, is still not well-understood. Our contribution aims to fill this gap. In particular, we show that the stationary distribution of offline (also called multi-pass) SGD exhibits 'approximate' power-law tails and the approximation error is controlled by how fast the empirical distribution of the training data converges to the true underlying data distribution in the Wasserstein metric. Our main takeaway is that, as the number of data points increases, offline SGD will behave increasingly 'power-law-like'. To achieve this result, we first prove nonasymptotic Wasserstein convergence bounds for offline SGD to online SGD as the number of data points increases, which can be interesting on their own. Finally, we illustrate our theory on various experiments conducted on synthetic data and neural networks.

cs.LG↗

arXiv2023

Non-asymptotic convergence bounds for Sinkhorn iterates and their gradients: a coupling approach

Computational optimal transport (OT) has recently emerged as a powerful framework with applications in various fields. In this paper we focus on a relaxation of the original OT problem, the entropic OT problem, which allows to implement efficient and practical algorithmic solutions, even in high dimensional settings. This formulation, also known as the Schrödinger Bridge problem, notably connects with Stochastic Optimal Control (SOC) and can be solved with the popular Sinkhorn algorithm. In the case of discrete-state spaces, this algorithm is known to have exponential convergence; however, achieving a similar rate of convergence in a more general setting is still an active area of research. In this work, we analyze the convergence of the Sinkhorn algorithm for probability measures defined on the $d$-dimensional torus $\mathbb{T}_L^d$, that admit densities with respect to the Haar measure of $\mathbb{T}_L^d$. In particular, we prove pointwise exponential convergence of Sinkhorn iterates and their gradient. Our proof relies on the connection between these iterates and the evolution along the Hamilton-Jacobi-Bellman equations of value functions obtained from SOC-problems. Our approach is novel in that it is purely probabilistic and relies on coupling by reflection techniques for controlled diffusions on the torus.

math.PR↗

arXiv2023

Probability and moment inequalities for additive functionals of geometrically ergodic Markov chains

In this paper, we establish moment and Bernstein-type inequalities for additive functionals of geometrically ergodic Markov chains. These inequalities extend the corresponding inequalities for independent random variables. Our conditions cover Markov chains converging geometrically to the stationary distribution either in $V$-norms or in weighted Wasserstein distances. Our inequalities apply to unbounded functions and depend explicitly on constants appearing in the conditions that we consider.

math.PR↗

arXiv2023

Boost your favorite Markov Chain Monte Carlo sampler using Kac's theorem: the Kick-Kac teleportation algorithm

The present paper focuses on the problem of sampling from a given target distribution $π$ defined on some general state space. To this end, we introduce a novel class of non-reversible Markov chains, each chain being defined on an extended state space and having an invariant probability measure admitting $π$ as a marginal distribution. The proposed methodology is inspired by a new formulation of Kac's theorem and allows global and local dynamics to be smoothly combined. Under mild conditions, the corresponding Markov transition kernel can be shown to be irreducible and Harris recurrent. In addition, we establish that geometric ergodicity holds under appropriate conditions on the global and local dynamics. Finally, we illustrate numerically the use of the proposed method and its potential benefits in comparison to existing Markov chain Monte Carlo (MCMC) algorithms.

stat.CO↗

arXiv2023

Uniform minorization condition and convergence bounds for discretizations of kinetic Langevin dynamics

We study the convergence in total variation and $V$-norm of discretization schemes of the underdamped Langevin dynamics. Such algorithms are very popular and commonly used in molecular dynamics and computational statistics to approximatively sample from a target distribution of interest. We show first that, for a very large class of schemes, a minorization condition uniform in the stepsize holds. This class encompasses popular methods such as the Euler-Maruyama scheme and the schemes based on splitting strategies. Second, we provide mild conditions ensuring that the class of schemes that we consider satisfies a geometric Foster--Lyapunov drift condition, again uniform in the stepsize. This allows us to derive geometric convergence bounds, with a convergence rate scaling linearly with the stepsize. This kind of result is of prime interest to obtain estimates on norms of solutions to Poisson equations associated with a given numerical method.

math.PR↗

arXiv2023

Finite-time High-probability Bounds for Polyak-Ruppert Averaged Iterates of Linear Stochastic Approximation

This paper provides a finite-time analysis of linear stochastic approximation (LSA) algorithms with fixed step size, a core method in statistics and machine learning. LSA is used to compute approximate solutions of a $d$-dimensional linear system $\bar{\mathbf{A}} θ= \bar{\mathbf{b}}$ for which $(\bar{\mathbf{A}}, \bar{\mathbf{b}})$ can only be estimated by (asymptotically) unbiased observations $\{(\mathbf{A}(Z_n),\mathbf{b}(Z_n))\}_{n \in \mathbb{N}}$. We consider here the case where $\{Z_n\}_{n \in \mathbb{N}}$ is an i.i.d. sequence or a uniformly geometrically ergodic Markov chain. We derive $p$-th moment and high-probability deviation bounds for the iterates defined by LSA and its Polyak-Ruppert-averaged version. Our finite-time instance-dependent bounds for the averaged LSA iterates are sharp in the sense that the leading term we obtain coincides with the local asymptotic minimax limit. Moreover, the remainder terms of our bounds admit a tight dependence on the mixing time $t_{\operatorname{mix}}$ of the underlying chain and the norm of the noise variables. We emphasize that our result requires the SA step size to scale only with logarithm of the problem dimension $d$.

stat.ML↗

arXiv2023

FedPop: A Bayesian Approach for Personalised Federated Learning

Personalised federated learning (FL) aims at collaboratively learning a machine learning model taylored for each client. Albeit promising advances have been made in this direction, most of existing approaches works do not allow for uncertainty quantification which is crucial in many applications. In addition, personalisation in the cross-device setting still involves important issues, especially for new clients or those having small number of observations. This paper aims at filling these gaps. To this end, we propose a novel methodology coined FedPop by recasting personalised FL into the population modeling paradigm where clients' models involve fixed common population parameters and random effects, aiming at explaining data heterogeneity. To derive convergence guarantees for our scheme, we introduce a new class of federated stochastic optimisation algorithms which relies on Markov chain Monte Carlo methods. Compared to existing personalised FL methods, the proposed methodology has important benefits: it is robust to client drift, practical for inference on new clients, and above all, enables uncertainty quantification under mild computational and memory overheads. We provide non-asymptotic convergence guarantees for the proposed algorithms and illustrate their performances on various personalised federated learning tasks.

cs.LG↗

arXiv2022

Sticky nonlinear SDEs and convergence of McKean-Vlasov equations without confinement

We develop a new approach to study the long time behaviour of solutions to nonlinear stochastic differential equations in the sense of McKean, as well as propagation of chaos for the corresponding mean-field particle system approximations. Our approach is based on a sticky coupling between two solutions to the equation. We show that the distance process between the two copies is dominated by a solution to a one-dimensional nonlinear stochastic differential equation with a sticky boundary at zero. This new class of equations is then analyzed carefully. In particular, we show that the dominating equation has a phase transition. In the regime where the Dirac measure at zero is the only invariant probability measure, we prove exponential convergence to equilibrium both for the one-dimensional equation, and for the original nonlinear SDE. Similarly, propagation of chaos is shown by a componentwise sticky coupling and comparison with a system of one dimensional nonlinear SDEs with sticky boundaries at zero. The approach applies to equations without confinement potential and to interaction terms that are not of gradient type.

math.PR↗