In interactive systems, actions are often correlated, presenting an opportunity for more sample-efficient off-policy evaluation (OPE) and learning (OPL) in large action spaces. We introduce a unified Bayesian framework to capture these correlations through structured and informative priors. In this framework, we propose sDM, a generic Bayesian approach for OPE and OPL, grounded in both algorithmic and theoretical foundations. Notably, sDM leverages action correlations without compromising computational efficiency. Moreover, inspired by online Bayesian bandits, we introduce Bayesian metrics that assess the average performance of algorithms across multiple problem instances, deviating from the conventional worst-case assessments. We analyze sDM in OPE and OPL, highlighting the benefits of leveraging action correlations. Empirical evidence showcases the strong performance of sDM.

Deep Equilibrium Models (DEQs) are a class of implicit neural networks that solve for a fixed point of a neural network in their forward pass. Traditionally, DEQs take sequences as inputs, but have since been applied to a variety of data. In this work, we present Distributional Deep Equilibrium Models (DDEQs), extending DEQs to discrete measure inputs, such as sets or point clouds. We provide a theoretically grounded framework for DDEQs. Leveraging Wasserstein gradient flows, we show how the forward pass of the DEQ can be adapted to find fixed points of discrete measures under permutation-invariance, and derive adequate network architectures for DDEQs. In experiments, we show that they can compete with state-of-the-art models in tasks such as point cloud classification and point cloud completion, while being significantly more parameter-efficient.

Sampling and automatic differentiation are both ubiquitous in modern machine learning. At its intersection, differentiating through a sampling operation, with respect to the parameters of the sampling process, is a problem that is both challenging and broadly applicable. We introduce a general framework and a new algorithm for first-order optimization of parameterized stochastic diffusions, performing jointly, in a single loop, optimization and sampling steps. This approach is inspired by recent advances in bilevel optimization and automatic implicit differentiation, leveraging the point of view of sampling as optimization over the space of probability distributions. We provide theoretical and experimental results showcasing the performance of our method.

Geometric tempering is a popular approach to sampling from challenging multi-modal probability distributions by instead sampling from a sequence of distributions which interpolate, using the geometric mean, between an easier proposal distribution and the target distribution. In this paper, we theoretically investigate the soundness of this approach when the sampling algorithm is Langevin dynamics, proving both upper and lower bounds. Our upper bounds are the first analysis in the literature under functional inequalities. They assert the convergence of tempered Langevin in continuous and discrete-time, and their minimization leads to closed-form optimal tempering schedules for some pairs of proposal and target distributions. Our lower bounds demonstrate a simple case where the geometric tempering takes exponential time, and further reveal that the geometric tempering can suffer from poor functional inequalities and slow convergence, even when the target distribution is well-conditioned. Overall, our results indicate that the geometric tempering may not help, and can even be harmful for convergence.

Many applications in machine learning involve data represented as probability distributions. The emergence of such data requires radically novel techniques to design tractable gradient flows on probability distributions over this type of (infinite-dimensional) objects. For instance, being able to flow labeled datasets is a core task for applications ranging from domain adaptation to transfer learning or dataset distillation. In this setting, we propose to represent each class by the associated conditional distribution of features, and to model the dataset as a mixture distribution supported on these classes (which are themselves probability distributions), meaning that labeled datasets can be seen as probability distributions over probability distributions. We endow this space with a metric structure from optimal transport, namely the Wasserstein over Wasserstein (WoW) distance, derive a differential structure on this space, and define WoW gradient flows. The latter enables to design dynamics over this space that decrease a given objective functional. We apply our framework to transfer learning and dataset distillation tasks, leveraging our gradient flow construction as well as novel tractable functionals that take the form of Maximum Mean Discrepancies with Sliced-Wasserstein based kernels between probability distributions.

In this paper, we investigate the properties of the Sliced Wasserstein Distance (SW) when employed as an objective functional. The SW metric has gained significant interest in the optimal transport and machine learning literature, due to its ability to capture intricate geometric properties of probability distributions while remaining computationally tractable, making it a valuable tool for various applications, including generative modeling and domain adaptation. Our study aims to provide a rigorous analysis of the critical points arising from the optimization of the SW objective. By computing explicit perturbations, we establish that stable critical points of SW cannot concentrate on segments. This stability analysis is crucial for understanding the behaviour of optimization algorithms for models trained using the SW objective. Furthermore, we investigate the properties of the SW objective, shedding light on the existence and convergence behavior of critical points. We illustrate our theoretical results through numerical experiments.

Density ratio estimation in high dimensions can be reframed as integrating a certain quantity, the time score, over probability paths which interpolate between the two densities. In practice, the time score has to be estimated based on samples from the two densities. However, existing methods for this problem remain computationally expensive and can yield inaccurate estimates. Inspired by recent advances in generative modeling, we introduce a novel framework for time score estimation, based on a conditioning variable. Choosing the conditioning variable judiciously enables a closed-form objective function. We demonstrate that, compared to previous approaches, our approach results in faster learning of the time score and competitive or better estimation accuracies of the density ratio on challenging tasks. Furthermore, we establish theoretical guarantees on the error of the estimated density ratio.

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. In this paper, we focus on the following parametric family: mixtures of isotropic Gaussians (i.e., with diagonal covariance matrices proportional to the identity) and uniform weights. We develop a variational framework and provide efficient algorithms suited for this family. In contrast with mixtures of Gaussian with generic covariance matrices, this choice presents a balance between accurate approximations of multimodal Bayesian posteriors, while being memory and computationally efficient. Our algorithms implement gradient descent on the location of the mixture components (the modes of the Gaussians), and either (an entropic) Mirror or Bures descent on their variance parameters. We illustrate the performance of our algorithms on numerical experiments.

Even in low dimensions, sampling from multi-modal distributions is challenging. We provide the first sampling algorithm for a broad class of distributions --- including all Gaussian mixtures --- with a query complexity that is polynomial in the parameters governing multi-modality, assuming fixed dimension. Our sampling algorithm simulates a time-reversed diffusion process, using a self-normalized Monte Carlo estimator of the intermediate score functions. Unlike previous works, it avoids metastability, requires no prior knowledge of the mode locations, and relaxes the well-known log-smoothness assumption which excluded general Gaussian mixtures so far.

We introduce a (de)-regularization of the Maximum Mean Discrepancy (DrMMD) and its Wasserstein gradient flow. Existing gradient flows that transport samples from source distribution to target distribution with only target samples, either lack tractable numerical implementation ($f$-divergence flows) or require strong assumptions and modifications, such as noise injection, to ensure convergence (Maximum Mean Discrepancy flows). In contrast, DrMMD flow can simultaneously (i) guarantee near-global convergence for a broad class of targets in both continuous and discrete time, and (ii) be implemented in closed form using only samples. The former is achieved by leveraging the connection between the DrMMD and the $\chi^2$-divergence, while the latter comes by treating DrMMD as MMD with a de-regularized kernel. Our numerical scheme employs an adaptive de-regularization schedule throughout the flow to optimally balance the trade-off between discretization errors and deviations from the $\chi^2$ regime. The potential application of the DrMMD flow is demonstrated across several numerical experiments, including a large-scale setting of training student/teacher networks.

This paper explores the connections between tempering (for Sequential Monte Carlo; SMC) and entropic mirror descent to sample from a target probability distribution whose unnormalized density is known. We establish that tempering SMC corresponds to entropic mirror descent applied to the reverse Kullback-Leibler (KL) divergence and obtain convergence rates for the tempering iterates. Our result motivates the tempering iterates from an optimization point of view, showing that tempering can be seen as a descent scheme of the KL divergence with respect to the Fisher-Rao geometry, in contrast to Langevin dynamics that perform descent of the KL with respect to the Wasserstein-2 geometry. We exploit the connection between tempering and mirror descent iterates to justify common practices in SMC and derive adaptive tempering rules that improve over other alternative benchmarks in the literature.

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 recently received attention, and is restricted to the Gaussian case. This research paper 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. 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, where the support of the atomic measure correspond to the localization of the Gaussian components, and the target distribution. Hence, solving variational inference is 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 in variational inference in this context. The first is an optimization result that is a descent lemma establishing that the algorithm decreases the objective at each iteration. The second is an approximation error that upper bounds the mollified relative entropy between an optimal finite mixture and the target distribution.

As the problem of minimizing functionals on the Wasserstein space encompasses many applications in machine learning, different optimization algorithms on $\mathbb{R}^d$ have received their counterpart analog on the Wasserstein space. We focus here on lifting two explicit algorithms: mirror descent and preconditioned gradient descent. These algorithms have been introduced to better capture the geometry of the function to minimize and are provably convergent under appropriate (namely relative) smoothness and convexity conditions. Adapting these notions to the Wasserstein space, we prove guarantees of convergence of some Wasserstein-gradient-based discrete-time schemes for new pairings of objective functionals and regularizers. The difficulty here is to carefully select along which curves the functionals should be smooth and convex. We illustrate the advantages of adapting the geometry induced by the regularizer on ill conditioned optimization tasks, and showcase the improvement of choosing different discrepancies and geometries in a computational biology task of aligning single-cells.

This work considers the problem of sampling from a probability distribution known up to a normalization constant while satisfying a set of statistical constraints specified by the expected values of general nonlinear functions. This problem finds applications in, e.g., Bayesian inference, where it can constrain moments to evaluate counterfactual scenarios or enforce desiderata such as prediction fairness. Methods developed to handle support constraints, such as those based on mirror maps, barriers, and penalties, are not suited for this task. This work therefore relies on gradient descent-ascent dynamics in Wasserstein space to put forward a discrete-time primal-dual Langevin Monte Carlo algorithm (PD-LMC) that simultaneously constrains the target distribution and samples from it. We analyze the convergence of PD-LMC under standard assumptions on the target distribution and constraints, namely (strong) convexity and log-Sobolev inequalities. To do so, we bring classical optimization arguments for saddle-point algorithms to the geometry of Wasserstein space. We illustrate the relevance and effectiveness of PD-LMC in several applications.

In this paper, we study the statistical and geometrical properties of the Kullback-Leibler divergence with kernel covariance operators (KKL) introduced by [Bach, 2022, Information Theory with Kernel Methods]. Unlike the classical Kullback-Leibler (KL) divergence that involves density ratios, the KKL compares probability distributions through covariance operators (embeddings) in a reproducible kernel Hilbert space (RKHS), and compute the Kullback-Leibler quantum divergence. This novel divergence hence shares parallel but different aspects with both the standard Kullback-Leibler between probability distributions and kernel embeddings metrics such as the maximum mean discrepancy. A limitation faced with the original KKL divergence is its inability to be defined for distributions with disjoint supports. To solve this problem, we propose in this paper a regularised variant that guarantees that divergence is well defined for all distributions. We derive bounds that quantify the deviation of the regularised KKL to the original one, as well as concentration bounds. In addition, we provide a closed-form expression for the regularised KKL, specifically applicable when the distributions consist of finite sets of points, which makes it implementable. Furthermore, we derive a Wasserstein gradient descent scheme of the KKL divergence in the case of discrete distributions, and study empirically its properties to transport a set of points to a target distribution.

Off-policy learning (OPL) often involves minimizing a risk estimator based on importance weighting to correct bias from the logging policy used to collect data. However, this method can produce an estimator with a high variance. A common solution is to regularize the importance weights and learn the policy by minimizing an estimator with penalties derived from generalization bounds specific to the estimator. This approach, known as pessimism, has gained recent attention but lacks a unified framework for analysis. To address this gap, we introduce a comprehensive PAC-Bayesian framework to examine pessimism with regularized importance weighting. We derive a tractable PAC-Bayesian generalization bound that universally applies to common importance weight regularizations, enabling their comparison within a single framework. Our empirical results challenge common understanding, demonstrating the effectiveness of standard IW regularization techniques.

Off-policy learning (OPL) aims at finding improved policies from logged bandit data, often by minimizing the inverse propensity scoring (IPS) estimator of the risk. In this work, we investigate a smooth regularization for IPS, for which we derive a two-sided PAC-Bayes generalization bound. The bound is tractable, scalable, interpretable and provides learning certificates. In particular, it is also valid for standard IPS without making the assumption that the importance weights are bounded. We demonstrate the relevance of our approach and its favorable performance through a set of learning tasks. Since our bound holds for standard IPS, we are able to provide insight into when regularizing IPS is useful. Namely, we identify cases where regularization might not be needed. This goes against the belief that, in practice, clipped IPS often enjoys favorable performance than standard IPS in OPL.

Adaptive importance sampling is a widely spread Monte Carlo technique that uses a re-weighting strategy to iteratively estimate the so-called target distribution. A major drawback of adaptive importance sampling is the large variance of the weights which is known to badly impact the accuracy of the estimates. This paper investigates a regularization strategy whose basic principle is to raise the importance weights at a certain power. This regularization parameter, that might evolve between zero and one during the algorithm, is shown (i) to balance between the bias and the variance and (ii) to be connected to the mirror descent framework. Using a kernel density estimate to build the sampling policy, the uniform convergence is established under mild conditions. Finally, several practical ways to choose the regularization parameter are discussed and the benefits of the proposed approach are illustrated empirically.

Approximating a target probability distribution can be cast as an optimization problem where the objective functional measures the dissimilarity to the target. This optimization can be addressed by approximating Wasserstein and related gradient flows. In practice, these are simulated by interacting particle systems, whose stationary states define an empirical measure approximating the target distribution. This approach has been popularized recently to design sampling algorithms, e.g. Stein Variational Gradient Descent, or by minimizing the Maximum Mean or Kernel Stein Discrepancy. However, little is known about quantization properties of these approaches, i.e. how well is the target approximated by a finite number particles. We investigate this question theoretically and numerically. In particular, we prove general upper bounds on the quantization error of MMD and KSD at rates which significantly outperform quantization by i.i.d. samples. We conduct experiments which show that the particle systems at study achieve fast rates in practice, and notably outperform greedy algorithms, such as kernel herding. We compare different gradient flows and highlight their quantization rates. Furthermore we introduce a Normalized Stein Variational Gradient Descent and argue in favor of adaptive kernels, which exhibit faster convergence. Finally we compare the Gaussian and Laplace kernels and argue that the Laplace kernel provides a more robust quantization.

Many problems in machine learning can be formulated as optimizing a convex functional over a vector space of measures. This paper studies the convergence of the mirror descent algorithm in this infinite-dimensional setting. Defining Bregman divergences through directional derivatives, we derive the convergence of the scheme for relatively smooth and convex pairs of functionals. Such assumptions allow to handle non-smooth functionals such as the Kullback--Leibler (KL) divergence. Applying our result to joint distributions and KL, we show that Sinkhorn's primal iterations for entropic optimal transport in the continuous setting correspond to a mirror descent, and we obtain a new proof of its (sub)linear convergence. We also show that Expectation Maximization (EM) can always formally be written as a mirror descent. When optimizing only on the latent distribution while fixing the mixtures parameters -- which corresponds to the Richardson--Lucy deconvolution scheme in signal processing -- we derive sublinear rates of convergence.

Among dissimilarities between probability distributions, the Kernel Stein Discrepancy (KSD) has received much interest recently. We investigate the properties of its Wasserstein gradient flow to approximate a target probability distribution $\pi$ on $\mathbb{R}^d$, known up to a normalization constant. This leads to a straightforwardly implementable, deterministic score-based method to sample from $\pi$, named KSD Descent, which uses a set of particles to approximate $\pi$. Remarkably, owing to a tractable loss function, KSD Descent can leverage robust parameter-free optimization schemes such as L-BFGS; this contrasts with other popular particle-based schemes such as the Stein Variational Gradient Descent algorithm. We study the convergence properties of KSD Descent and demonstrate its practical relevance. However, we also highlight failure cases by showing that the algorithm can get stuck in spurious local minima.

We address the problem of causal effect estima-tion in the presence of unobserved confounding,but where proxies for the latent confounder(s) areobserved. We propose two kernel-based meth-ods for nonlinear causal effect estimation in thissetting: (a) a two-stage regression approach, and(b) a maximum moment restriction approach. Wefocus on the proximal causal learning setting, butour methods can be used to solve a wider classof inverse problems characterised by a Fredholmintegral equation. In particular, we provide a uni-fying view of two-stage and moment restrictionapproaches for solving this problem in a nonlin-ear setting. We provide consistency guaranteesfor each algorithm, and demonstrate that these ap-proaches achieve competitive results on syntheticdata and data simulating a real-world task. In par-ticular, our approach outperforms earlier methodsthat are not suited to leveraging proxy variables.

We study the Stein Variational Gradient Descent (SVGD) algorithm, which optimises a set of particles to approximate a target probability distribution $\pi\propto e^{-V}$ on $\R^d$. In the population limit, SVGD performs gradient descent in the space of probability distributions on the KL divergence with respect to $\pi$, where the gradient is smoothed through a kernel integral operator. In this paper, we provide a novel finite time analysis for the SVGD algorithm. We provide a descent lemma establishing that the algorithm decreases the objective at each iteration, and rates of convergence. We also provide a convergence result of the finite particle system corresponding to the practical implementation of SVGD to its population version.

Wasserstein gradient flows are continuous time dynamics that define curves of steepest descent to minimize an objective function over the space of probability measures (i.e., the Wasserstein space). This objective is typically a divergence w.r.t. a fixed target distribution. In recent years, these continuous time dynamics have been used to study the convergence of machine learning algorithms aiming at approximating a probability distribution. However, the discrete-time behavior of these algorithms might differ from the continuous time dynamics. Besides, although discretized gradient flows have been proposed in the literature, little is known about their minimization power. In this work, we propose a Forward Backward (FB) discretization scheme that can tackle the case where the objective function is the sum of a smooth and a nonsmooth geodesically convex terms. Using techniques from convex optimization and optimal transport, we analyze the FB scheme as a minimization algorithm on the Wasserstein space. More precisely, we show under mild assumptions that the FB scheme has convergence guarantees similar to the proximal gradient algorithm in Euclidean spaces (resp. similar to the associated Wasserstein gradient flow).

Whereas most dimensionality reduction techniques (\textit{e.g.} PCA, ICA, NMF) for multivariate data essentially rely on linear algebra to a certain extent, summarizing ranking data, viewed as realizations of a random permutation $\Sigma$ on a set of items indexed by $i\in \{1,\ldots,;{n}\}$, is a great statistical challenge, due to the absence of vector space structure for the set of permutations $\mathfrak{S}_n$. It is the goal of this article to develop an original framework for possibly reducing the number of parameters required to describe the distribution of a statistical population composed of rankings/permutations, on the premise that the collection of items under study can be partitioned into subsets/buckets, such that, with high probability, items in a certain bucket are either all ranked higher or else all ranked lower than items in another bucket. In this context, $\Sigma$’s distribution can be hopefully represented in a sparse manner by a \textit{bucket distribution}, \textit{i.e.} a bucket ordering plus the ranking distributions within each bucket. More precisely, we introduce a dedicated distortion measure, based on a mass transportation metric, in order to quantify the accuracy of such representations. The performance of buckets minimizing an empirical version of the distortion is investigated through a rate bound analysis. Complexity penalization techniques are also considered to select the shape of a bucket order with minimum expected distortion. Beyond theoretical concepts and results, numerical experiments on real ranking data are displayed in order to provide empirical evidence of the relevance of the approach promoted.

We construct a Wasserstein gradient flow of the maximum mean discrepancy (MMD) and study its convergence properties. The MMD is an integral probability metric defined for a reproducing kernel Hilbert space (RKHS), and serves as a metric on probability measures for a sufficiently rich RKHS. We obtain conditions for convergence of the gradient flow towards a global optimum, that can be related to particle transport when optimizing neural networks. We also propose a way to regularize this MMD flow, based on an injection of noise in the gradient. This algorithmic fix comes with theoretical and empirical evidence. The practical implementation of the flow is straightforward, since both the MMD and its gradient have simple closed-form expressions, which can be easily estimated with samples.

This article is devoted to the problem of predicting the value taken by a random permutation $Σ$, describing the preferences of an individual over a set of numbered items $\{1,; \ldots,;{n}\}$ say, based on the observation of an input/explanatory r.v. $X$ (\textit{e.g.} characteristics of the individual), when error is measured by the Kendall’s $τ$ distance. In the probabilistic formulation of the ’Learning to Order’ problem we propose, which extends the framework for statistical Kemeny ranking aggregation developped in \citet{CKS17}, this boils down to recovering conditional Kemeny medians of $Σ$ given $X$ from i.i.d. training examples $(X_1, \Sigma_1),; \ldots,; (X_N, \Sigma_N)$. For this reason, this statistical learning problem is referred to as \textit{ranking median regression} here. Our contribution is twofold. We first propose a probabilistic theory of ranking median regression: the set of optimal elements is characterized, the performance of empirical risk minimizers is investigated in this context and situations where fast learning rates can be achieved are also exhibited. Next we introduce the concept of local consensus/median, in order to derive efficient methods for ranking median regression. The major advantage of this local learning approach lies in its close connection with the widely studied Kemeny aggregation problem. From an algorithmic perspective, this permits to build predictive rules for ranking median regression by implementing efficient techniques for (approximate) Kemeny median computations at a local level in a tractable manner. In particular, versions of $k$-nearest neighbor and tree-based methods, tailored to ranking median regression, are investigated. Accuracy of piecewise constant ranking median regression rules is studied under a specific smoothness assumption for $Σ$’s conditional distribution given $X$. The results of various numerical experiments are also displayed for illustration purpose.

We propose to solve a label ranking problem as a structured output regression task. In this view, we adopt a least square surrogate loss approach that solves a supervised learning problem in two steps: a regression step in a well-chosen feature space and a pre-image (or decoding) step. We use specific feature maps/embeddings for ranking data, which convert any ranking/permutation into a vector representation. These embeddings are all well-tailored for our approach, either by resulting in consistent estimators, or by solving trivially the pre-image problem which is often the bottleneck in structured prediction. Their extension to the case of incomplete or partial rankings is also discussed. Finally, we provide empirical results on synthetic and real-world datasets showing the relevance of our method.

Originally formulated in Social Choice theory, Ranking Aggregation, also referred to as Consensus Ranking, has motivated the development of numerous statistical models since the middle of the 20th century. Recently, the analysis of ranking/preference data has been the subject of a renewed interest in machine-learning, boosted by modern applications such as meta-search engines, giving rise to the design of various scalable algorithmic approaches for approximately computing ranking medians, viewed as solutions of a discrete (generally NP-hard) minimization problem. This paper develops a statistical learning theory for ranking aggregation in a general probabilistic setting (avoiding any rigid ranking model assumptions), assessing the generalization ability of empirical ranking medians. Universal rate bounds are established and the situations where convergence occurs at an exponential rate are fully characterized. Minimax lower bounds are also proved, showing that the rate bounds we obtain are optimal.

Due to its numerous applications, rank aggregation has become a problem of major interest across many fields of the computer science literature. In the vast majority of situations, Kemeny consensus(es) are considered as the ideal solutions. It is however well known that their computation is NP-hard. Many contributions have thus established various results to apprehend this complexity. In this paper we introduce a practical method to predict, for a ranking and a dataset, how close the Kemeny consensus(es) are to this ranking. A major strength of this method is its generality: it does not require any assumption on the dataset nor the ranking. Furthermore, it relies on a new geometric interpretation of Kemeny aggregation that, we believe, could lead to many other results.