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Odalric-Ambrym Maillard

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43 papers
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43

EWRL Workshop 2025 Workshop Paper

Information-Based Exploration via Random Features

  • Waris Radji
  • Odalric-Ambrym Maillard

Representation learning has enabled classical exploration strategies to be extended to deep Reinforcement Learning (RL), but often makes algorithms more complex and theoretical guarantees harder to establish. We introduce Random Feature Information Gain (RFIG), grounded in Bayesian kernel methods theory, which uses random Fourier features to scalably approximate information gain and compute exploration bonuses in non-countable spaces. We provide error bounds on information gain approximation and avoid the black-box aspects of deep-based uncertainty estimation, for optimism-based exploration. We present practical details that make RFIG scalable to deep RL scenarios, enabling smooth integration with classical deep RL algorithms. Experimental evaluation across control and navigation tasks demonstrates that RFIG achieves competitive performance with well-established deep exploration methods while offering superior theoretical interpretation.

EWRL Workshop 2025 Workshop Paper

Leveraging priors on distribution functions for multi arm bandits

  • Sumit Vashishtha
  • Odalric-Ambrym Maillard

We introduce Dirichlet Process Posterior Sampling (DPPS), a Bayesian non-parametric al- gorithm for multi-arm bandits based on Dirichlet Process (DP) priors. Like Thompson- sampling, DPPS is a probability-matching algorithm, i. e. , it plays an arm based on its posterior- probability of being optimal. Instead of assuming a parametric class for the reward generating distribution of each arm, and then putting a prior on the parameters, in DPPS the reward gener- ating distribution is directly modeled using DP priors. DPPS provides a principled approach to incorporate prior belief about the bandit environment, and in the noninformative limit of the DP priors (i. e. Bayesian Bootstrap), we recover Non Parametric Thompson Sampling (NPTS), a popular non-parametric bandit algorithm, as a special case of DPPS. We employ stick-breaking representation of the DP priors, and show excellent empirical performance of DPPS in chal- lenging synthetic and real world bandit environments. Finally, using an information-theoretic analysis, we show non-asymptotic optimality of DPPS in the Bayesian regret setup.

RLC Conference 2025 Conference Paper

Leveraging priors on distribution functions for multi-arm bandits

  • Sumit Vashishtha
  • Odalric-Ambrym Maillard

We introduce Dirichlet Process Posterior Sampling (DPPS), a Bayesian non-parametric algorithm for multi-arm bandits based on Dirichlet Process (DP) priors. Like Thompson-sampling, DPPS is a probability-matching algorithm, i. e. , it plays an arm based on its posterior-probability of being optimal. However, instead of assuming a parametric class for the reward generating distribution of each arm, and then putting a prior on the parameters, in DPPS the reward generating distribution is directly modeled using DP priors. DPPS provides a principled approach to incorporate prior belief about the bandit environment, and in the noninformative limit of the DP posteriors (i. e. Bayesian Bootstrap), we recover Non Parametric Thompson Sampling (NPTS), a popular non-parametric bandit algorithm, as a special case of DPPS. We employ stick-breaking representation of the DP priors, and show excellent empirical performance of DPPS in challenging synthetic and real world bandit environments. Finally, using an information-theoretic analysis, we show non-asymptotic optimality of DPPS in the Bayesian regret setup.

RLJ Journal 2025 Journal Article

Leveraging priors on distribution functions for multi-arm bandits

  • Sumit Vashishtha
  • Odalric-Ambrym Maillard

We introduce Dirichlet Process Posterior Sampling (DPPS), a Bayesian non-parametric algorithm for multi-arm bandits based on Dirichlet Process (DP) priors. Like Thompson-sampling, DPPS is a probability-matching algorithm, i.e., it plays an arm based on its posterior-probability of being optimal. However, instead of assuming a parametric class for the reward generating distribution of each arm, and then putting a prior on the parameters, in DPPS the reward generating distribution is directly modeled using DP priors. DPPS provides a principled approach to incorporate prior belief about the bandit environment, and in the noninformative limit of the DP posteriors (i.e. Bayesian Bootstrap), we recover Non Parametric Thompson Sampling (NPTS), a popular non-parametric bandit algorithm, as a special case of DPPS. We employ stick-breaking representation of the DP priors, and show excellent empirical performance of DPPS in challenging synthetic and real world bandit environments. Finally, using an information-theoretic analysis, we show non-asymptotic optimality of DPPS in the Bayesian regret setup.

ICML Conference 2025 Conference Paper

Monte-Carlo Tree Search with Uncertainty Propagation via Optimal Transport

  • Tuan Dam
  • Pascal Stenger
  • Lukas Schneider
  • Joni Pajarinen
  • Carlo D'Eramo
  • Odalric-Ambrym Maillard

This paper introduces a novel backup strategy for Monte-Carlo Tree Search (MCTS) tailored for highly stochastic and partially observable Markov decision processes. We adopt a probabilistic approach, modeling both value and action-value nodes as Gaussian distributions, to introduce a novel backup operator that computes value nodes as the Wasserstein barycenter of their action-value children nodes; thus, propagating the uncertainty of the estimate across the tree to the root node. We study our novel backup operator when using a novel combination of $L^1$-Wasserstein barycenter with $\alpha$-divergence, by drawing a crucial connection to the generalized mean backup operator. We complement our probabilistic backup operator with two sampling strategies, based on optimistic selection and Thompson sampling, obtaining our Wasserstein MCTS algorithm. We provide theoretical guarantees of asymptotic convergence of $\mathcal{O}(n^{-1/2})$, with $n$ as the number of visited trajectories, to the optimal policy and an empirical evaluation on several stochastic and partially observable environments, where our approach outperforms well-known related baselines.

EWRL Workshop 2025 Workshop Paper

Offline Goal-Conditioned Reinforcement Learning with Projective Quasimetric Planning

  • Anthony Kobanda
  • Waris Radji
  • Mathieu Petitbois
  • Odalric-Ambrym Maillard
  • Rémy Portelas

Offline Goal-Conditioned Reinforcement Learning seeks to train agents to reach specified goals using previously collected reward-free data. Scaling that promise to long-horizon tasks with complex dynamics remains challenging, notably due to compounding value‑estimation errors. Principled geometric learning offers a potential solution to address these issues. Following this insight in our research, we introduce Projective Quasimetric Planning (ProQ), a compositional framework that learns a differentiable asymmetric distance and then repurposes it, firstly as a repulsive energy forcing a sparse set of keypoints to uniformly spread over learned latent space, secondly as a structured directional cost guiding towards proximal sub-goals. ProQ couples this geometry with a Lagrangian out-of-distribution detector to ensure the keypoints to stay within reachable areas. By unifying metric learning, keypoint coverage, and goal‑conditioned control, our approach produces meaningful sub‑goals and robustly drives long‑horizon goal‑reaching on diverse navigation and manipulation benchmarks.

RLC Conference 2025 Conference Paper

The Confusing Instance Principle for Online Linear Quadratic Control

  • Waris Radji
  • Odalric-Ambrym Maillard

We revisit the problem of controlling linear systems with quadratic cost under unknown dynamics with model-based reinforcement learning. Traditional methods like Optimism in the Face of Uncertainty and Thompson Sampling, rooted in multi-armed bandits (MABs), face practical limitations. In contrast, we propose an alternative based on the *Confusing Instance* (CI) principle, which underpins regret lower bounds in MABs and discrete Markov Decision Processes (MDPs) and is central to the *Minimum Empirical Divergence* (MED) family of algorithms, known for their asymptotic optimality in various settings. By leveraging the structure of LQR policies along with sensitivity and stability analysis, we develop `MED-LQ`. This novel control strategy extends the principles of CI and MED beyond small-scale settings. Our benchmarks on a comprehensive control suite demonstrate that `MED-LQ` achieves competitive performance in various scenarios while highlighting its potential for broader applications in large-scale MDPs.

RLJ Journal 2025 Journal Article

The Confusing Instance Principle for Online Linear Quadratic Control

  • Waris Radji
  • Odalric-Ambrym Maillard

We revisit the problem of controlling linear systems with quadratic cost under unknown dynamics with model-based reinforcement learning. Traditional methods like Optimism in the Face of Uncertainty and Thompson Sampling, rooted in multi-armed bandits (MABs), face practical limitations. In contrast, we propose an alternative based on the *Confusing Instance* (CI) principle, which underpins regret lower bounds in MABs and discrete Markov Decision Processes (MDPs) and is central to the *Minimum Empirical Divergence* (MED) family of algorithms, known for their asymptotic optimality in various settings. By leveraging the structure of LQR policies along with sensitivity and stability analysis, we develop `MED-LQ`. This novel control strategy extends the principles of CI and MED beyond small-scale settings. Our benchmarks on a comprehensive control suite demonstrate that `MED-LQ` achieves competitive performance in various scenarios while highlighting its potential for broader applications in large-scale MDPs.

TMLR Journal 2024 Journal Article

AdaStop: adaptive statistical testing for sound comparisons of Deep RL agents

  • Timothée Mathieu
  • Matheus Medeiros Centa
  • Riccardo Della Vecchia
  • Hector Kohler
  • Alena Shilova
  • Odalric-Ambrym Maillard
  • Philippe Preux

Recently, the scientific community has questioned the statistical reproducibility of many empirical results, especially in the field of machine learning. To contribute to the resolution of this reproducibility crisis, we propose a theoretically sound methodology for comparing the performance of a set of algorithms. We exemplify our methodology in Deep Reinforcement Learning (Deep RL). The performance of one execution of a Deep RL algorithm is a random variable. Therefore, several independent executions are needed to evaluate its performance. When comparing algorithms with random performance, a major question concerns the number of executions to perform to ensure that the result of the comparison is theoretically sound. Researchers in Deep RL often use less than 5 independent executions to compare algorithms: we claim that this is not enough in general. Moreover, when comparing more than 2 algorithms at once, we have to use a multiple tests procedure to preserve low error guarantees. We introduce AdaStop, a new statistical test based on multiple group sequential tests. When used to compare algorithms, AdaStop adapts the number of executions to stop as early as possible while ensuring that enough information has been collected to distinguish algorithms that have different score distributions. We prove theoretically that AdaStop has a low probability of making a (family-wise) error. We illustrate the effectiveness of AdaStop in various use-cases, including toy examples and Deep RL algorithms on challenging Mujoco environments. AdaStop is the first statistical test fitted to this sort of comparisons: it is both a significant contribution to statistics, and an important contribution to computational studies performed in reinforcement learning and in other domains.

TMLR Journal 2024 Journal Article

Bandits Corrupted by Nature: Lower Bounds on Regret and Robust Optimistic Algorithms

  • Timothée Mathieu
  • Debabrota Basu
  • Odalric-Ambrym Maillard

We study the corrupted bandit problem, i.e. a stochastic multi-armed bandit problem with $k$ unknown reward distributions, which are heavy-tailed and corrupted by a history-independent adversary or Nature. To be specific, the reward obtained by playing an arm comes from corresponding heavy-tailed reward distribution with probability $1-\varepsilon \in (0.5,1]$ and an arbitrary corruption distribution of unbounded support with probability $\varepsilon \in [0,0.5)$. First, we provide \textit{a problem-dependent lower bound on the regret} of any corrupted bandit algorithm. The lower bounds indicate that the corrupted bandit problem is harder than the classical stochastic bandit problem with subGaussian or heavy-tail rewards. Following that, we propose a novel UCB-type algorithm for corrupted bandits, namely \texttt{HubUCB}, that builds on Huber's estimator for robust mean estimation. Leveraging a novel concentration inequality of Huber's estimator, we prove that \texttt{HubUCB} achieves a near-optimal regret upper bound. Since computing Huber's estimator has quadratic complexity, we further introduce a sequential version of Huber's estimator that exhibits linear complexity. We leverage this sequential estimator to design \texttt{SeqHubUCB} that enjoys similar regret guarantees while reducing the computational burden. Finally, we experimentally illustrate the efficiency of \texttt{HubUCB} and \texttt{SeqHubUCB} in solving corrupted bandits for different reward distributions and different levels of corruptions.

RLC Conference 2024 Conference Paper

Bandits with Multimodal Structure

  • Hassan SABER
  • Odalric-Ambrym Maillard

We consider a multi-armed bandit problem specified by a set of one-dimensional exponential family distributions endowed with a multimodal structure. The multimodal structure naturally extends the unimodal structure and appears to be underlying in quite interesting ways popular structures such as linear or Lipschitz bandits. We introduce IMED-MB, an algorithm that optimally exploits the multimodal structure, by adapting to this setting the popular Indexed Minimum Empirical Divergence (IMED) algorithm. We provide instance-dependent regret analysis of this strategy. Numerical experiments show that \IMEDMB performs well in practice when assuming unimodal, polynomial or Lipschitz mean function.

RLJ Journal 2024 Journal Article

Bandits with Multimodal Structure

  • Hassan SABER
  • Odalric-Ambrym Maillard

We consider a multi-armed bandit problem specified by a set of one-dimensional exponential family distributions endowed with a multimodal structure. The multimodal structure naturally extends the unimodal structure and appears to be underlying in quite interesting ways popular structures such as linear or Lipschitz bandits. We introduce IMED-MB, an algorithm that optimally exploits the multimodal structure, by adapting to this setting the popular Indexed Minimum Empirical Divergence (IMED) algorithm. We provide instance-dependent regret analysis of this strategy. Numerical experiments show that \IMEDMB performs well in practice when assuming unimodal, polynomial or Lipschitz mean function.

UAI Conference 2024 Conference Paper

Power Mean Estimation in Stochastic Monte-Carlo Tree Search

  • Tuan Dam
  • Odalric-Ambrym Maillard
  • Emilie Kaufmann

Monte-Carlo Tree Search (MCTS) is a widely-used online planning strategy that effectively combines Monte-Carlo sampling with forward tree search to make optimal decisions in various real-world scenarios. Its success hinges on the application of the Upper Confidence bound for Trees (UCT) algorithm, an extension of the Upper Confidence bound (UCB) method used in multi-arm bandits. However, theoretical investigations of UCT have been found incomplete due to an error in the estimated "logarithmic" bonus term used for action selection in the tree. This issue was addressed by introducing a "polynomial" exploration bonus, which effectively balances the exploration-exploitation trade-off. However, most theoretical studies have focused on deterministic Markov Decision Processes (MDPs), neglecting stochastic settings. In this paper, we introduce Stochastic-Power-UCT, an MCTS algorithm explicitly designed for stochastic MDPs using power mean as the value estimator. We conduct a comprehensive study of the polynomial convergence assurance for selecting the optimal action and estimating the value at the root node within our methodology. Our findings demonstrate that Stochastic-Power-UCT shares the same convergence rate as UCT, with UCT being a special case of Stochastic-Power-UCT. Furthermore, we validate our theoretical results through empirical assessments across diverse stochastic MDP environments, providing empirical evidence to support our method’s theoretical claims

EWRL Workshop 2023 Workshop Paper

AdaStop: sequential testing for efficient and reliable comparisons of Deep RL Agents

  • Timothée Mathieu
  • Matheus Medeiros Centa
  • Riccardo Della Vecchia
  • Hector Kohler
  • Alena Shilova
  • Odalric-Ambrym Maillard
  • Philippe Preux

The reproducibility of many experimental results in Deep Reinforcement Learning (RL) is under question. To solve this reproducibility crisis, we propose a theoretically sound methodology to compare multiple Deep RL algorithms. The performance of one execution of a Deep RL algorithm is random so that independent executions are needed to assess it precisely. When comparing several RL algorithms, a major question is how many executions must be made and how can we assure that the results of such a comparison is theoretically sound. Researchers in Deep RL often use less than 5 independent executions to compare algorithms: we claim that this is not enough in general. Moreover, when comparing several algorithms at once, the error of each comparison accumulates and must be taken into account with a multiple tests procedure to preserve low error guarantees. To address this problem in a statistically sound way, we introduce AdaStop, a new statistical test based on multiple group sequential tests. When comparing algorithms, AdaStop adapts the number of executions to stop as early as possible while ensuring that we have enough information to distinguish algorithms that perform better than the others in a statistical significant way. We prove both theoretically and empirically that AdaStop has a low probability of making an error (Family-Wise Error). Finally, we illustrate the effectiveness of AdaStop in multiple use-cases, including toy examples and difficult cases such as Mujoco environments.

NeurIPS Conference 2023 Conference Paper

Fast Asymptotically Optimal Algorithms for Non-Parametric Stochastic Bandits

  • Dorian Baudry
  • Fabien Pesquerel
  • Rémy Degenne
  • Odalric-Ambrym Maillard

We consider the problem of regret minimization in non-parametric stochastic bandits. When the rewards are known to be bounded from above, there exists asymptotically optimal algorithms, with asymptotic regret depending on an infimum of Kullback-Leibler divergences (KL). These algorithms are computationally expensive and require storing all past rewards, thus simpler but non-optimal algorithms are often used instead. We introduce several methods to approximate the infimum KL which reduce drastically the computational and memory costs of existing optimal algorithms, while keeping their regret guaranties. We apply our findings to design new variants of the MED and IMED algorithms, and demonstrate their interest with extensive numerical simulations.

TMLR Journal 2022 Journal Article

Collaborative Algorithms for Online Personalized Mean Estimation

  • Mahsa Asadi
  • Aurélien Bellet
  • Odalric-Ambrym Maillard
  • Marc Tommasi

We consider an online estimation problem involving a set of agents. Each agent has access to a (personal) process that generates samples from a real-valued distribution and seeks to estimate its mean. We study the case where some of the distributions have the same mean, and the agents are allowed to actively query information from other agents. The goal is to design an algorithm that enables each agent to improve its mean estimate thanks to communication with other agents. The means as well as the number of distributions with same mean are unknown, which makes the task nontrivial. We introduce a novel collaborative strategy to solve this online personalized mean estimation problem. We analyze its time complexity and introduce variants that enjoy good performance in numerical experiments. We also extend our approach to the setting where clusters of agents with similar means seek to estimate the mean of their cluster.

JMLR Journal 2022 Journal Article

Efficient Change-Point Detection for Tackling Piecewise-Stationary Bandits

  • Lilian Besson
  • Emilie Kaufmann
  • Odalric-Ambrym Maillard
  • Julien Seznec

We introduce GLRklUCB, a novel algorithm for the piecewise iid non-stationary bandit problem with bounded rewards. This algorithm combines an efficient bandit algorithm, klUCB, with an efficient, parameter-free, change-point detector, the Bernoulli Generalized Likelihood Ratio Test, for which we provide new theoretical guarantees of independent interest. Unlike previous non-stationary bandit algorithms using a change-point detector, GLRklUCB does not need to be calibrated based on prior knowledge on the arms' means. We prove that this algorithm can attain a $O(\sqrt{TA\Upsilon_T\log(T)})$ regret in $T$ rounds on some “easy” instances in which there is sufficient delay between two change-points, where $A$ is the number of arms and $\Upsilon_T$ the number of change-points, without prior knowledge of $\Upsilon_T$. In contrast with recently proposed algorithms that are agnostic to $\Upsilon_T$, we perform a numerical study showing that GLRklUCB is also very efficient in practice, beyond easy instances. [abs] [ pdf ][ bib ] [ code ] &copy JMLR 2022. ( edit, beta )

NeurIPS Conference 2022 Conference Paper

IMED-RL: Regret optimal learning of ergodic Markov decision processes

  • Fabien Pesquerel
  • Odalric-Ambrym Maillard

We consider reinforcement learning in a discrete, undiscounted, infinite-horizon Markov decision problem (MDP) under the average reward criterion, and focus on the minimization of the regret with respect to an optimal policy, when the learner does not know the rewards nor transitions of the MDP. In light of their success at regret minimization in multi-armed bandits, popular bandit strategies, such as the optimistic \texttt{UCB}, \texttt{KL-UCB} or the Bayesian Thompson sampling strategy, have been extended to the MDP setup. Despite some key successes, existing strategies for solving this problem either fail to be provably asymptotically optimal, or suffer from prohibitive burn-in phase and computational complexity when implemented in practice. In this work, we shed a novel light on regret minimization strategies, by extending to reinforcement learning the computationally appealing Indexed Minimum Empirical Divergence (\texttt{IMED}) bandit algorithm. Traditional asymptotic problem-dependent lower bounds on the regret are known under the assumption that the MDP is \emph{ergodic}. Under this assumption, we introduce \texttt{IMED-RL} and prove that its regret upper bound asymptotically matches the regret lower bound. We discuss both the case when the supports of transitions are unknown, and the more informative but a priori harder-to-exploit-optimally case when they are known. Rewards are assumed light-tailed, semi-bounded from above. Last, we provide numerical illustrations on classical tabular MDPs, \textit{ergodic} and \textit{communicative} only, showing the competitiveness of \texttt{IMED-RL} in finite-time against state-of-the-art algorithms. \texttt{IMED-RL} also benefits from a lighter complexity.

NeurIPS Conference 2021 Conference Paper

From Optimality to Robustness: Adaptive Re-Sampling Strategies in Stochastic Bandits

  • Dorian Baudry
  • Patrick Saux
  • Odalric-Ambrym Maillard

The stochastic multi-arm bandit problem has been extensively studied under standard assumptions on the arm's distribution (e. g bounded with known support, exponential family, etc). These assumptions are suitable for many real-world problems but sometimes they require knowledge (on tails for instance) that may not be precisely accessible to the practitioner, raising the question of the robustness of bandit algorithms to model misspecification. In this paper we study a generic \emph{Dirichlet Sampling} (DS) algorithm, based on pairwise comparisons of empirical indices computed with \textit{re-sampling} of the arms' observations and a data-dependent \textit{exploration bonus}. We show that different variants of this strategy achieve provably optimal regret guarantees when the distributions are bounded and logarithmic regret for semi-bounded distributions with a mild quantile condition. We also show that a simple tuning achieve robustness with respect to a large class of unbounded distributions, at the cost of slightly worse than logarithmic asymptotic regret. We finally provide numerical experiments showing the merits of DS in a decision-making problem on synthetic agriculture data.

NeurIPS Conference 2021 Conference Paper

Indexed Minimum Empirical Divergence for Unimodal Bandits

  • Hassan SABER
  • Pierre Ménard
  • Odalric-Ambrym Maillard

We consider a stochastic multi-armed bandit problem specified by a set of one-dimensional family exponential distributions endowed with a unimodal structure. The unimodal structure is of practical relevance for several applications. We introduce IMED-UB, an algorithm that exploits provably optimally the unimodal-structure, by adapting to this setting the Indexed Minimum Empirical Divergence (IMED) algorithm introduced by Honda and Takemura (2015). Owing to our proof technique, we are able to provide a concise finite-time analysis of the IMED-UB algorithm, that is simple and yet yields asymptotic optimality. We finally provide numerical experiments showing that IMED-UB competes favorably with the recently introduced state-of-the-art algorithms.

ICLR Conference 2021 Conference Paper

Learning Value Functions in Deep Policy Gradients using Residual Variance

  • Yannis Flet-Berliac
  • Reda Ouhamma
  • Odalric-Ambrym Maillard
  • Philippe Preux

Policy gradient algorithms have proven to be successful in diverse decision making and control tasks. However, these methods suffer from high sample complexity and instability issues. In this paper, we address these challenges by providing a different approach for training the critic in the actor-critic framework. Our work builds on recent studies indicating that traditional actor-critic algorithms do not succeed in fitting the true value function, calling for the need to identify a better objective for the critic. In our method, the critic uses a new state-value (resp. state-action-value) function approximation that learns the value of the states (resp. state-action pairs) relative to their mean value rather than the absolute value as in conventional actor-critic. We prove the theoretical consistency of the new gradient estimator and observe dramatic empirical improvement across a variety of continuous control tasks and algorithms. Furthermore, we validate our method in tasks with sparse rewards, where we provide experimental evidence and theoretical insights.

ICML Conference 2021 Conference Paper

Optimal Thompson Sampling strategies for support-aware CVaR bandits

  • Dorian Baudry
  • Romain Gautron
  • Emilie Kaufmann
  • Odalric-Ambrym Maillard

In this paper we study a multi-arm bandit problem in which the quality of each arm is measured by the Conditional Value at Risk (CVaR) at some level alpha of the reward distribution. While existing works in this setting mainly focus on Upper Confidence Bound algorithms, we introduce a new Thompson Sampling approach for CVaR bandits on bounded rewards that is flexible enough to solve a variety of problems grounded on physical resources. Building on a recent work by Riou & Honda (2020), we introduce B-CVTS for continuous bounded rewards and M-CVTS for multinomial distributions. On the theoretical side, we provide a non-trivial extension of their analysis that enables to theoretically bound their CVaR regret minimization performance. Strikingly, our results show that these strategies are the first to provably achieve asymptotic optimality in CVaR bandits, matching the corresponding asymptotic lower bounds for this setting. Further, we illustrate empirically the benefit of Thompson Sampling approaches both in a realistic environment simulating a use-case in agriculture and on various synthetic examples.

NeurIPS Conference 2021 Conference Paper

Stochastic bandits with groups of similar arms.

  • Fabien Pesquerel
  • Hassan SABER
  • Odalric-Ambrym Maillard

We consider a variant of the stochastic multi-armed bandit problem where arms are known to be organized into different groups having the same mean. The groups are unknown but a lower bound $q$ on their size is known. This situation typically appears when each arm can be described with a list of categorical attributes, and the (unknown) mean reward function only depends on a subset of them, the others being redundant. In this case, $q$ is linked naturally to the number of attributes considered redundant, and the number of categories of each attribute. For this structured problem of practical relevance, we first derive the asymptotic regret lower bound and corresponding constrained optimization problem. They reveal the achievable regret can be substantially reduced when compared to the unstructured setup, possibly by a factor $q$. However, solving exactly the exact constrained optimization problem involves a combinatorial problem. We introduce a lower-bound inspired strategy involving a computationally efficient relaxation that is based on a sorting mechanism. We further prove it achieves a lower bound close to the optimal one up to a controlled factor, and achieves an asymptotic regret $q$ times smaller than the unstructured one. We believe this shows it is a valuable strategy for the practitioner. Last, we illustrate the performance of the considered strategy on numerical experiments involving a large number of arms.

NeurIPS Conference 2021 Conference Paper

Stochastic Online Linear Regression: the Forward Algorithm to Replace Ridge

  • Reda Ouhamma
  • Odalric-Ambrym Maillard
  • Vianney Perchet

We consider the problem of online linear regression in the stochastic setting. We derive high probability regret bounds for online $\textit{ridge}$ regression and the $\textit{forward}$ algorithm. This enables us to compare online regression algorithms more accurately and eliminate assumptions of bounded observations and predictions. Our study advocates for the use of the forward algorithm in lieu of ridge due to its enhanced bounds and robustness to the regularization parameter. Moreover, we explain how to integrate it in algorithms involving linear function approximation to remove a boundedness assumption without deteriorating theoretical bounds. We showcase this modification in linear bandit settings where it yields improved regret bounds. Last, we provide numerical experiments to illustrate our results and endorse our intuitions.

ICML Conference 2020 Conference Paper

Restarted Bayesian Online Change-point Detector achieves Optimal Detection Delay

  • Réda Alami
  • Odalric-Ambrym Maillard
  • Raphaël Féraud

we consider the problem of sequential change-point detection where both the change-points and the distributions before and after the change are assumed to be unknown. For this problem of primary importance in statistical and sequential learning theory, we derive a variant of the Bayesian Online Change Point Detector proposed by \cite{fearnhead2007line} which is easier to analyze than the original version while keeping its powerful message-passing algorithm. We provide a non-asymptotic analysis of the false-alarm rate and the detection delay that matches the existing lower-bound. We further provide the first explicit high-probability control of the detection delay for such approach. Experiments on synthetic and real-world data show that this proposal outperforms the state-of-art change-point detection strategy, namely the Improved Generalized Likelihood Ratio (Improved GLR) while compares favorably with the original Bayesian Online Change Point Detection strategy.

NeurIPS Conference 2020 Conference Paper

Robust-Adaptive Control of Linear Systems: beyond Quadratic Costs

  • Edouard Leurent
  • Odalric-Ambrym Maillard
  • Denis Efimov

We consider the problem of robust and adaptive model predictive control (MPC) of a linear system, with unknown parameters that are learned along the way (adaptive), in a critical setting where failures must be prevented (robust). This problem has been studied from different perspectives by different communities. However, the existing theory deals only with the case of quadratic costs (the LQ problem), which limits applications to stabilisation and tracking tasks only. In order to handle more general (non-convex) costs that naturally arise in many practical problems, we carefully select and bring together several tools from different communities, namely non-asymptotic linear regression, recent results in interval prediction, and tree-based planning. Combining and adapting the theoretical guarantees at each layer is non trivial, and we provide the first end-to-end suboptimality analysis for this setting. Interestingly, our analysis naturally adapts to handle many models and combines with a data-driven robust model selection strategy, which enables to relax the modelling assumptions. Last, we strive to preserve tractability at any stage of the method, that we illustrate on two challenging simulated environments.

NeurIPS Conference 2020 Conference Paper

Sub-sampling for Efficient Non-Parametric Bandit Exploration

  • Dorian Baudry
  • Emilie Kaufmann
  • Odalric-Ambrym Maillard

In this paper we propose the first multi-armed bandit algorithm based on re-sampling that achieves asymptotically optimal regret simultaneously for different families of arms (namely Bernoulli, Gaussian and Poisson distributions). Unlike Thompson Sampling which requires to specify a different prior to be optimal in each case, our proposal RB-SDA does not need any distribution-dependent tuning. RB-SDA belongs to the family of Sub-sampling Duelling Algorithms (SDA) which combines the sub-sampling idea first used by the BESA and SSMC algorithms with different sub-sampling schemes. In particular, RB-SDA uses Random Block sampling. We perform an experimental study assessing the flexibility and robustness of this promising novel approach for exploration in bandit models.

ICML Conference 2020 Conference Paper

Tightening Exploration in Upper Confidence Reinforcement Learning

  • Hippolyte Bourel
  • Odalric-Ambrym Maillard
  • Mohammad Sadegh Talebi

The upper confidence reinforcement learning (UCRL2) algorithm introduced in \citep{jaksch2010near} is a popular method to perform regret minimization in unknown discrete Markov Decision Processes under the average-reward criterion. Despite its nice and generic theoretical regret guarantees, this algorithm and its variants have remained until now mostly theoretical as numerical experiments in simple environments exhibit long burn-in phases before the learning takes place. In pursuit of practical efficiency, we present UCRL3, following the lines of UCRL2, but with two key modifications: First, it uses state-of-the-art time-uniform concentration inequalities to compute confidence sets on the reward and (component-wise) transition distributions for each state-action pair. Furthermore, to tighten exploration, it uses an adaptive computation of the support of each transition distribution, which in turn enables us to revisit the extended value iteration procedure of UCRL2 to optimize over distributions with reduced support by disregarding low probability transitions, while still ensuring near-optimism. We demonstrate, through numerical experiments in standard environments, that reducing exploration this way yields a substantial numerical improvement compared to UCRL2 and its variants. On the theoretical side, these key modifications enable us to derive a regret bound for UCRL3 improving on UCRL2, that for the first time makes appear notions of local diameter and local effective support, thanks to variance-aware concentration bounds.

NeurIPS Conference 2019 Conference Paper

Budgeted Reinforcement Learning in Continuous State Space

  • Nicolas Carrara
  • Edouard Leurent
  • Romain Laroche
  • Tanguy Urvoy
  • Odalric-Ambrym Maillard
  • Olivier Pietquin

A Budgeted Markov Decision Process (BMDP) is an extension of a Markov Decision Process to critical applications requiring safety constraints. It relies on a notion of risk implemented in the shape of an upper bound on a constrains violation signal that -- importantly -- can be modified in real-time. So far, BMDPs could only be solved in the case of finite state spaces with known dynamics. This work extends the state-of-the-art to continuous spaces environments and unknown dynamics. We show that the solution to a BMDP is the fixed point of a novel Budgeted Bellman Optimality operator. This observation allows us to introduce natural extensions of Deep Reinforcement Learning algorithms to address large-scale BMDPs. We validate our approach on two simulated applications: spoken dialogue and autonomous driving.

NeurIPS Conference 2019 Conference Paper

Learning Multiple Markov Chains via Adaptive Allocation

  • Mohammad Sadegh Talebi
  • Odalric-Ambrym Maillard

We study the problem of learning the transition matrices of a set of Markov chains from a single stream of observations on each chain. We assume that the Markov chains are ergodic but otherwise unknown. The learner can sample Markov chains sequentially to observe their states. The goal of the learner is to sequentially select various chains to learn transition matrices uniformly well with respect to some loss function. We introduce a notion of loss that naturally extends the squared loss for learning distributions to the case of Markov chains, and further characterize the notion of being \emph{uniformly good} in all problem instances. We present a novel learning algorithm that efficiently balances \emph{exploration} and \emph{exploitation} intrinsic to this problem, without any prior knowledge of the chains. We provide finite-sample PAC-type guarantees on the performance of the algorithm. Further, we show that our algorithm asymptotically attains an optimal loss.

NeurIPS Conference 2019 Conference Paper

Regret Bounds for Learning State Representations in Reinforcement Learning

  • Ronald Ortner
  • Matteo Pirotta
  • Alessandro Lazaric
  • Ronan Fruit
  • Odalric-Ambrym Maillard

We consider the problem of online reinforcement learning when several state representations (mapping histories to a discrete state space) are available to the learning agent. At least one of these representations is assumed to induce a Markov decision process (MDP), and the performance of the agent is measured in terms of cumulative regret against the optimal policy giving the highest average reward in this MDP representation. We propose an algorithm (UCB-MS) with O(sqrt(T)) regret in any communicating Markov decision process. The regret bound shows that UCB-MS automatically adapts to the Markov model. This improves over the currently known best results in the literature that gave regret bounds of order O(T^(2/3)).

EWRL Workshop 2018 Workshop Paper

KL-UCRL Revisited: Variance-Aware Regret Bounds

  • M. Sadegh Talebi
  • Odalric-Ambrym Maillard

The problem of reinforcement learning in an unknown and discrete Markov Decision Process (MDP) under the average-reward criterion is considered, when the learner interacts with the system in a single stream of observations, starting from an initial state without any reset. We provide a novel analysis of the KL-Ucrl algorithm establishing a high-probability regret bound scaling as e O q S P s, a Vp(·|s, a)(b?)T for this algorithm for ergodic MDPs, where S denotes the number of states and where Vp(·|s, a)(b? ) is the variance of the bias function of an optimal policy, with respect to the next-state distribution following action a in state s. The resulting bound improves upon the best previously known regret bound e O(DS √ AT) for that algorithm, where A and D respectively denote the maximum number of actions (per state) and the diameter of MDP. We finally compare the leading terms of the two bounds in some benchmark MDPs indicating that the derived bound can provide an order of magnitude improvement in some cases. Our analysis leverages novel variations of the transportation lemma combined with Kullback-Leibler concentration inequalities, that we believe to be of independent interest.

JMLR Journal 2018 Journal Article

Streaming kernel regression with provably adaptive mean, variance, and regularization

  • Audrey Durand
  • Odalric-Ambrym Maillard
  • Joelle Pineau

We consider the problem of streaming kernel regression, when the observations arrive sequentially and the goal is to recover the underlying mean function, assumed to belong to an RKHS. The variance of the noise is not assumed to be known. In this context, we tackle the problem of tuning the regularization parameter adaptively at each time step, while maintaining tight confidence bounds estimates on the value of the mean function at each point. To this end, we first generalize existing results for finite-dimensional linear regression with fixed regularization and known variance to the kernel setup with a regularization parameter allowed to be a measurable function of past observations. Then, using appropriate self-normalized inequalities we build upper and lower bound estimates for the variance, leading to Bernstein-like concentration bounds. The latter is used in order to define the adaptive regularization. The bounds resulting from our technique are valid uniformly over all observation points and all time steps, and are compared against the literature with numerical experiments. Finally, the potential of these tools is illustrated by an application to kernelized bandits, where we revisit the Kernel UCB and Kernel Thompson Sampling procedures, and show the benefits of the novel adaptive kernel tuning strategy. [abs] [ pdf ][ bib ] &copy JMLR 2018. ( edit, beta )

ICML Conference 2017 Conference Paper

Spectral Learning from a Single Trajectory under Finite-State Policies

  • Borja Balle
  • Odalric-Ambrym Maillard

We present spectral methods of moments for learning sequential models from a single trajectory, in stark contrast with the classical literature that assumes the availability of multiple i. i. d. trajectories. Our approach leverages an efficient SVD-based learning algorithm for weighted automata and provides the first rigorous analysis for learning many important models using dependent data. We state and analyze the algorithm under three increasingly difficult scenarios: probabilistic automata, stochastic weighted automata, and reactive predictive state representations controlled by a finite-state policy. Our proofs include novel tools for studying mixing properties of stochastic weighted automata.

ICML Conference 2016 Conference Paper

Pliable Rejection Sampling

  • Akram Erraqabi
  • Michal Valko
  • Alexandra Carpentier
  • Odalric-Ambrym Maillard

Rejection sampling is a technique for sampling from difficult distributions. However, its use is limited due to a high rejection rate. Common adaptive rejection sampling methods either work only for very specific distributions or without performance guarantees. In this paper, we present pliable rejection sampling (PRS), a new approach to rejection sampling, where we learn the sampling proposal using a kernel estimator. Since our method builds on rejection sampling, the samples obtained are with high probability i. i. d. and distributed according to f. Moreover, PRS comes with a guarantee on the number of accepted samples.

NeurIPS Conference 2014 Conference Paper

How hard is my MDP?" The distribution-norm to the rescue"

  • Odalric-Ambrym Maillard
  • Timothy Mann
  • Shie Mannor

In Reinforcement Learning (RL), state-of-the-art algorithms require a large number of samples per state-action pair to estimate the transition kernel $p$. In many problems, a good approximation of $p$ is not needed. For instance, if from one state-action pair $(s, a)$, one can only transit to states with the same value, learning $p(\cdot|s, a)$ accurately is irrelevant (only its support matters). This paper aims at capturing such behavior by defining a novel hardness measure for Markov Decision Processes (MDPs) we call the {\em distribution-norm}. The distribution-norm w. r. t. ~a measure $\nu$ is defined on zero $\nu$-mean functions $f$ by the standard variation of $f$ with respect to $\nu$. We first provide a concentration inequality for the dual of the distribution-norm. This allows us to replace the generic but loose $||\cdot||_1$ concentration inequalities used in most previous analysis of RL algorithms, to benefit from this new hardness measure. We then show that several common RL benchmarks have low hardness when measured using the new norm. The distribution-norm captures finer properties than the number of states or the diameter and can be used to assess the difficulty of MDPs.

ICML Conference 2014 Conference Paper

Latent Bandits

  • Odalric-Ambrym Maillard
  • Shie Mannor

We consider a multi-armed bandit problem where the reward distributions are indexed by two sets –one for arms, one for type– and can be partitioned into a small number of clusters according to the type. First, we consider the setting where all reward distributions are known and all types have the same underlying cluster, the type’s identity is, however, unknown. Second, we study the case where types may come from different classes, which is significantly more challenging. Finally, we tackle the case where the reward distributions are completely unknown. In each setting, we introduce specific algorithms and derive non-trivial regret performance. Numerical experiments show that, in the most challenging agnostic case, the proposed algorithm achieves excellent performance in several difficult scenarios.

ICML Conference 2013 Conference Paper

Optimal Regret Bounds for Selecting the State Representation in Reinforcement Learning

  • Odalric-Ambrym Maillard
  • Phuong Nguyen
  • Ronald Ortner
  • Daniil Ryabko

We consider an agent interacting with an environment in a single stream of actions, observations, and rewards, with no reset. This process is not assumed to be a Markov Decision Process (MDP). Rather, the agent has several representations (mapping histories of past interactions to a discrete state space) of the environment with unknown dynamics, only some of which result in an MDP. The goal is to minimize the average regret criterion against an agent who knows an MDP representation giving the highest optimal reward, and acts optimally in it. Recent regret bounds for this setting are of order O(T^2/3) with an additive term constant yet exponential in some characteristics of the optimal MDP. We propose an algorithm whose regret after T time steps is O(\sqrtT), with all constants reasonably small. This is optimal in T since O(\sqrtT) is the optimal regret in the setting of learning in a (single discrete) MDP.

NeurIPS Conference 2012 Conference Paper

Hierarchical Optimistic Region Selection driven by Curiosity

  • Odalric-Ambrym Maillard

This paper aims to take a step forwards making the term ``intrinsic motivation'' from reinforcement learning theoretically well founded, focusing on curiosity-driven learning. To that end, we consider the setting where, a fixed partition P of a continuous space X being given, and a process \nu defined on X being unknown, we are asked to sequentially decide which cell of the partition to select as well as where to sample \nu in that cell, in order to minimize a loss function that is inspired from previous work on curiosity-driven learning. The loss on each cell consists of one term measuring a simple worst case quadratic sampling error, and a penalty term proportional to the range of the variance in that cell. The corresponding problem formulation extends the setting known as active learning for multi-armed bandits to the case when each arm is a continuous region, and we show how an adaptation of recent algorithms for that problem and of hierarchical optimistic sampling algorithms for optimization can be used in order to solve this problem. The resulting procedure, called Hierarchical Optimistic Region SElection driven by Curiosity (HORSE. C) is provided together with a finite-time regret analysis.

JMLR Journal 2012 Journal Article

Linear Regression With Random Projections

  • Odalric-Ambrym Maillard
  • Rémi Munos

We investigate a method for regression that makes use of a randomly generated subspace G P ⊂F (of finite dimension P ) of a given large (possibly infinite) dimensional function space F, for example, L 2 ([0,1] d;ℜ). G P is defined as the span of P random features that are linear combinations of a basis functions of F weighted by random Gaussian i.i.d. coefficients. We show practical motivation for the use of this approach, detail the link that this random projections method share with RKHS and Gaussian objects theory and prove, both in deterministic and random design, approximation error bounds when searching for the best regression function in G P rather than in F, and derive excess risk bounds for a specific regression algorithm (least squares regression in G P ). This paper stresses the motivation to study such methods, thus the analysis developed is kept simple for explanations purpose and leaves room for future developments. [abs] [ pdf ][ bib ] &copy JMLR 2012. ( edit, beta )

NeurIPS Conference 2012 Conference Paper

Online allocation and homogeneous partitioning for piecewise constant mean-approximation

  • Alexandra Carpentier
  • Odalric-Ambrym Maillard

In the setting of active learning for the multi-armed bandit, where the goal of a learner is to estimate with equal precision the mean of a finite number of arms, recent results show that it is possible to derive strategies based on finite-time confidence bounds that are competitive with the best possible strategy. We here consider an extension of this problem to the case when the arms are the cells of a finite partition P of a continuous sampling space X \subset \Real^d. Our goal is now to build a piecewise constant approximation of a noisy function (where each piece is one region of P and P is fixed beforehand) in order to maintain the local quadratic error of approximation on each cell equally low. Although this extension is not trivial, we show that a simple algorithm based on upper confidence bounds can be proved to be adaptive to the function itself in a near-optimal way, when |P| is chosen to be of minimax-optimal order on the class of \alpha-Hölder functions.

NeurIPS Conference 2011 Conference Paper

Selecting the State-Representation in Reinforcement Learning

  • Odalric-Ambrym Maillard
  • Daniil Ryabko
  • Rémi Munos

The problem of selecting the right state-representation in a reinforcement learning problem is considered. Several models (functions mapping past observations to a finite set) of the observations are given, and it is known that for at least one of these models the resulting state dynamics are indeed Markovian. Without knowing neither which of the models is the correct one, nor what are the probabilistic characteristics of the resulting MDP, it is required to obtain as much reward as the optimal policy for the correct model (or for the best of the correct models, if there are several). We propose an algorithm that achieves that, with a regret of order T^{2/3} where T is the horizon time.

NeurIPS Conference 2011 Conference Paper

Sparse Recovery with Brownian Sensing

  • Alexandra Carpentier
  • Odalric-Ambrym Maillard
  • Rémi Munos

We consider the problem of recovering the parameter alpha in R^K of a sparse function f, i. e. the number of non-zero entries of alpha is small compared to the number K of features, given noisy evaluations of f at a set of well-chosen sampling points. We introduce an additional randomisation process, called Brownian sensing, based on the computation of stochastic integrals, which produces a Gaussian sensing matrix, for which good recovery properties are proven independently on the number of sampling points N, even when the features are arbitrarily non-orthogonal. Under the assumption that f is Hölder continuous with exponent at least 1/2, we provide an estimate a of the parameter such that ||\alpha - a||_2 = O(||eta||_2\sqrt{N}), where eta is the observation noise. The method uses a set of sampling points uniformly distributed along a one-dimensional curve selected according to the features. We report numerical experiments illustrating our method.

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