Pierre Ménard

Tadashi Kozuno, Wenhao Yang, Nino Vieillard et al. · deepmind

In this work, we consider and analyze the sample complexity of model-free reinforcement learning with a generative model. Particularly, we analyze mirror descent value iteration (MDVI) by Geist et al. (2019) and Vieillard et al. (2020a), which uses the Kullback-Leibler divergence and entropy regularization in its value and policy updates. Our analysis shows that it is nearly minimax-optimal for finding an $\varepsilon$-optimal policy when $\varepsilon$ is sufficiently small. This is the first theoretical result that demonstrates that a simple model-free algorithm without variance-reduction can be nearly minimax-optimal under the considered setting.

Côme Fiegel, Pierre Ménard, Tadashi Kozuno et al.

Imperfect information games (IIG) are games in which each player only partially observes the current game state. We study how to learn $ε$-optimal strategies in a zero-sum IIG through self-play with trajectory feedback. We give a problem-independent lower bound $\widetilde{\mathcal{O}}(H(A_{\mathcal{X}}+B_{\mathcal{Y}})/ε^2)$ on the required number of realizations to learn these strategies with high probability, where $H$ is the length of the game, $A_{\mathcal{X}}$ and $B_{\mathcal{Y}}$ are the total number of actions for the two players. We also propose two Follow the Regularized leader (FTRL) algorithms for this setting: Balanced FTRL which matches this lower bound, but requires the knowledge of the information set structure beforehand to define the regularization; and Adaptive FTRL which needs $\widetilde{\mathcal{O}}(H^2(A_{\mathcal{X}}+B_{\mathcal{Y}})/ε^2)$ realizations without this requirement by progressively adapting the regularization to the observations.

Julien Seznec, Pierre Ménard, Alessandro Lazaric et al.

In many application domains (e.g., recommender systems, intelligent tutoring systems), the rewards associated to the actions tend to decrease over time. This decay is either caused by the actions executed in the past (e.g., a user may get bored when songs of the same genre are recommended over and over) or by an external factor (e.g., content becomes outdated). These two situations can be modeled as specific instances of the rested and restless bandit settings, where arms are rotting (i.e., their value decrease over time). These problems were thought to be significantly different, since Levine et al. (2017) showed that state-of-the-art algorithms for restless bandit perform poorly in the rested rotting setting. In this paper, we introduce a novel algorithm, Rotting Adaptive Window UCB (RAW-UCB), that achieves near-optimal regret in both rotting rested and restless bandit, without any prior knowledge of the setting (rested or restless) and the type of non-stationarity (e.g., piece-wise constant, bounded variation). This is in striking contrast with previous negative results showing that no algorithm can achieve similar results as soon as rewards are allowed to increase. We confirm our theoretical findings on a number of synthetic and dataset-based experiments.

Côme Fiegel, Pierre Ménard, Tadashi Kozuno et al.

We study the problem of learning in zero-sum matrix games with repeated play and bandit feedback. Specifically, we focus on developing uncoupled algorithms that guarantee, without communication between players, the convergence of the last-iterate to a Nash equilibrium. Although the non-bandit case has been studied extensively, this setting has only been explored recently, with a bound of $\mathcal{O}(T^{-1/8})$ on the exploitability gap. We show that, for uncoupled algorithms, guaranteeing convergence of the policy profiles to a Nash equilibrium is detrimental to the performance, with the best attainable rate being $Ω(T^{-1/4})$ in contrast to the usual $Ω(T^{-1/2})$ rate for convergence of the average iterates. We then propose two algorithms that achieve this optimal rate up to constant and logarithmic factors. The first algorithm leverages a straightforward trade-off between exploration and exploitation, while the second employs a regularization technique based on a two-step mirror descent approach.

Romain Froger, Pierre Andrews, Matteo Bettini et al.

We introduce Gaia2, a benchmark for evaluating large language model agents in realistic, asynchronous environments. Unlike prior static or synchronous evaluations, Gaia2 introduces scenarios where environments evolve independently of agent actions, requiring agents to operate under temporal constraints, adapt to noisy and dynamic events, resolve ambiguity, and collaborate with other agents. Each scenario is paired with a write-action verifier, enabling fine-grained, action-level evaluation and making Gaia2 directly usable for reinforcement learning from verifiable rewards. Our evaluation of state-of-the-art proprietary and open-source models shows that no model dominates across capabilities: GPT-5 (high) reaches the strongest overall score of 42% pass@1 but fails on time-sensitive tasks, Claude-4 Sonnet trades accuracy and speed for cost, Kimi-K2 leads among open-source models with 21% pass@1. These results highlight fundamental trade-offs between reasoning, efficiency, robustness, and expose challenges in closing the "sim2real" gap. Gaia2 is built on a consumer environment with the open-source Agents Research Environments platform and designed to be easy to extend. By releasing Gaia2 alongside the foundational ARE framework, we aim to provide the community with a flexible infrastructure for developing, benchmarking, and training the next generation of practical agent systems.

Mariana Vargas Vieyra, Pierre Ménard

We present a novel, alternative framework for learning generative models with goal-conditioned reinforcement learning. We define two agents, a goal conditioned agent (GC-agent) and a supervised agent (S-agent). Given a user-input initial state, the GC-agent learns to reconstruct the training set. In this context, elements in the training set are the goals. During training, the S-agent learns to imitate the GC-agent while remaining agnostic of the goals. At inference we generate new samples with the S-agent. Following a similar route as in variational auto-encoders, we derive an upper bound on the negative log-likelihood that consists of a reconstruction term and a divergence between the GC-agent policy and the (goal-agnostic) S-agent policy. We empirically demonstrate that our method is able to generate diverse and high quality samples in the task of image synthesis.

Jean-Bastien Grill, Omar Darwiche Domingues, Pierre Ménard et al.

We propose SmoothCruiser, a new planning algorithm for estimating the value function in entropy-regularized Markov decision processes and two-player games, given a generative model of the environment. SmoothCruiser makes use of the smoothness of the Bellman operator promoted by the regularization to achieve problem-independent sample complexity of order O~(1/epsilon^4) for a desired accuracy epsilon, whereas for non-regularized settings there are no known algorithms with guaranteed polynomial sample complexity in the worst case.

Antoine Scheid, Etienne Boursier, Alain Durmus et al.

Reinforcement Learning from Human Feedback (RLHF) has become a popular approach to align language models (LMs) with human preferences. This method involves collecting a large dataset of human pairwise preferences across various text generations and using it to infer (implicitly or explicitly) a reward model. Numerous methods have been proposed to learn the reward model and align a LM with it. However, the costly process of collecting human preferences has received little attention and could benefit from theoretical insights. This paper addresses this issue and aims to formalize the reward training model in RLHF. We frame the selection of an effective dataset as a simple regret minimization task, using a linear contextual dueling bandit method. Given the potentially large number of arms, this approach is more coherent than the best-arm identification setting. We then propose an offline framework for solving this problem. Under appropriate assumptions - linearity of the reward model in the embedding space, and boundedness of the reward parameter - we derive bounds on the simple regret. Finally, we provide a lower bound that matches our upper bound up to constant and logarithmic terms. To our knowledge, this is the first theoretical contribution in this area to provide an offline approach as well as worst-case guarantees.

Pierre Andrews, Amine Benhalloum, Gerard Moreno-Torres Bertran et al.

We introduce Meta Agents Research Environments (ARE), a research platform for scalable creation of environments, integration of synthetic or real applications, and execution of agentic orchestrations. ARE provides simple abstractions to build complex and diverse environments, each with their own rules, tools, content, and verifiers, helping to bridge the gap between model development and real-world deployment. We also propose Gaia2, a benchmark built in ARE and designed to measure general agent capabilities. Beyond search and execution, Gaia2 requires agents to handle ambiguities and noise, adapt to dynamic environments, collaborate with other agents, and operate under temporal constraints. Unlike prior benchmarks, Gaia2 runs asynchronously, surfacing new failure modes that are invisible in static settings. Our experiments show that no system dominates across the intelligence spectrum: stronger reasoning often comes at the cost of efficiency, and budget scaling curves plateau, highlighting the need for new architectures and adaptive compute strategies. Perhaps more importantly, ARE abstractions enable continuous extension of Gaia2 to other environments, empowering the community to rapidly create new benchmarks tailored to their domains. In AI's second half, progress increasingly depends on defining meaningful tasks and robust evaluations to drive frontier capabilities forward.

Côme Fiegel, Pierre Ménard, Tadashi Kozuno et al.

We study how to learn $ε$-optimal strategies in zero-sum imperfect information games (IIG) with trajectory feedback. In this setting, players update their policies sequentially based on their observations over a fixed number of episodes, denoted by $T$. Existing procedures suffer from high variance due to the use of importance sampling over sequences of actions (Steinberger et al., 2020; McAleer et al., 2022). To reduce this variance, we consider a fixed sampling approach, where players still update their policies over time, but with observations obtained through a given fixed sampling policy. Our approach is based on an adaptive Online Mirror Descent (OMD) algorithm that applies OMD locally to each information set, using individually decreasing learning rates and a regularized loss. We show that this approach guarantees a convergence rate of $\tilde{\mathcal{O}}(T^{-1/2})$ with high probability and has a near-optimal dependence on the game parameters when applied with the best theoretical choices of learning rates and sampling policies. To achieve these results, we generalize the notion of OMD stabilization, allowing for time-varying regularization with convex increments.

Toshinori Kitamura, Tadashi Kozuno, Yunhao Tang et al.

Mirror descent value iteration (MDVI), an abstraction of Kullback-Leibler (KL) and entropy-regularized reinforcement learning (RL), has served as the basis for recent high-performing practical RL algorithms. However, despite the use of function approximation in practice, the theoretical understanding of MDVI has been limited to tabular Markov decision processes (MDPs). We study MDVI with linear function approximation through its sample complexity required to identify an $\varepsilon$-optimal policy with probability $1-δ$ under the settings of an infinite-horizon linear MDP, generative model, and G-optimal design. We demonstrate that least-squares regression weighted by the variance of an estimated optimal value function of the next state is crucial to achieving minimax optimality. Based on this observation, we present Variance-Weighted Least-Squares MDVI (VWLS-MDVI), the first theoretical algorithm that achieves nearly minimax optimal sample complexity for infinite-horizon linear MDPs. Furthermore, we propose a practical VWLS algorithm for value-based deep RL, Deep Variance Weighting (DVW). Our experiments demonstrate that DVW improves the performance of popular value-based deep RL algorithms on a set of MinAtar benchmarks.

Hassan Saber, Pierre Ménard, Odalric-Ambrym Maillard

We consider a multi-armed bandit problem specified by a set of one-dimensional family exponential distributions endowed with a unimodal structure. We introduce IMED-UB, a algorithm that optimally exploits 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 IMED-UB algorithm. Numerical experiments show that IMED-UB competes with the state-of-the-art algorithms.

Jean Tarbouriech, Omar Darwiche Domingues, Pierre Ménard et al.

We introduce a generic strategy for provably efficient multi-goal exploration. It relies on AdaGoal, a novel goal selection scheme that leverages a measure of uncertainty in reaching states to adaptively target goals that are neither too difficult nor too easy. We show how AdaGoal can be used to tackle the objective of learning an $ε$-optimal goal-conditioned policy for the (initially unknown) set of goal states that are reachable within $L$ steps in expectation from a reference state $s_0$ in a reward-free Markov decision process. In the tabular case with $S$ states and $A$ actions, our algorithm requires $\tilde{O}(L^3 S A ε^{-2})$ exploration steps, which is nearly minimax optimal. We also readily instantiate AdaGoal in linear mixture Markov decision processes, yielding the first goal-oriented PAC guarantee with linear function approximation. Beyond its strong theoretical guarantees, we anchor AdaGoal in goal-conditioned deep reinforcement learning, both conceptually and empirically, by connecting its idea of selecting "uncertain" goals to maximizing value ensemble disagreement.

James Cheshire, Pierre Ménard, Alexandra Carpentier

We investigate the problem dependent regime in the stochastic Thresholding Bandit problem (TBP) under several shape constraints. In the TBP, the objective of the learner is to output, at the end of a sequential game, the set of arms whose means are above a given threshold. The vanilla, unstructured, case is already well studied in the literature. Taking $K$ as the number of arms, we consider the case where (i) the sequence of arm's means $(μ_k)_{k=1}^K$ is monotonically increasing (MTBP) and (ii) the case where $(μ_k)_{k=1}^K$ is concave (CTBP). We consider both cases in the problem dependent regime and study the probability of error - i.e. the probability to mis-classify at least one arm. In the fixed budget setting, we provide upper and lower bounds for the probability of error in both the concave and monotone settings, as well as associated algorithms. In both settings the bounds match in the problem dependent regime up to universal constants in the exponential.

Tadashi Kozuno, Pierre Ménard, Rémi Munos et al.

We study the problem of learning a Nash equilibrium (NE) in an imperfect information game (IIG) through self-play. Precisely, we focus on two-player, zero-sum, episodic, tabular IIG under the perfect-recall assumption where the only feedback is realizations of the game (bandit feedback). In particular, the dynamic of the IIG is not known -- we can only access it by sampling or interacting with a game simulator. For this learning setting, we provide the Implicit Exploration Online Mirror Descent (IXOMD) algorithm. It is a model-free algorithm with a high-probability bound on the convergence rate to the NE of order $1/\sqrt{T}$ where $T$ is the number of played games. Moreover, IXOMD is computationally efficient as it needs to perform the updates only along the sampled trajectory.

Rianne de Heide, James Cheshire, Pierre Ménard et al.

We consider a stochastic bandit problem with a possibly infinite number of arms. We write $p^*$ for the proportion of optimal arms and $Δ$ for the minimal mean-gap between optimal and sub-optimal arms. We characterize the optimal learning rates both in the cumulative regret setting, and in the best-arm identification setting in terms of the problem parameters $T$ (the budget), $p^*$ and $Δ$. For the objective of minimizing the cumulative regret, we provide a lower bound of order $Ω(\log(T)/(p^*Δ))$ and a UCB-style algorithm with matching upper bound up to a factor of $\log(1/Δ)$. Our algorithm needs $p^*$ to calibrate its parameters, and we prove that this knowledge is necessary, since adapting to $p^*$ in this setting is impossible. For best-arm identification we also provide a lower bound of order $Ω(\exp(-cTΔ^2 p^*))$ on the probability of outputting a sub-optimal arm where $c>0$ is an absolute constant. We also provide an elimination algorithm with an upper bound matching the lower bound up to a factor of order $\log(T)$ in the exponential, and that does not need $p^*$ or $Δ$ as parameter. Our results apply directly to the three related problems of competing against the $j$-th best arm, identifying an $ε$ good arm, and finding an arm with mean larger than a quantile of a known order.

Omar Darwiche Domingues, Pierre Ménard, Emilie Kaufmann et al.

In this paper, we propose new problem-independent lower bounds on the sample complexity and regret in episodic MDPs, with a particular focus on the non-stationary case in which the transition kernel is allowed to change in each stage of the episode. Our main contribution is a novel lower bound of $Ω((H^3SA/ε^2)\log(1/δ))$ on the sample complexity of an $(\varepsilon,δ)$-PAC algorithm for best policy identification in a non-stationary MDP. This lower bound relies on a construction of "hard MDPs" which is different from the ones previously used in the literature. Using this same class of MDPs, we also provide a rigorous proof of the $Ω(\sqrt{H^3SAT})$ regret bound for non-stationary MDPs. Finally, we discuss connections to PAC-MDP lower bounds.

Pierre Ménard, Omar Darwiche Domingues, Anders Jonsson et al.

Realistic environments often provide agents with very limited feedback. When the environment is initially unknown, the feedback, in the beginning, can be completely absent, and the agents may first choose to devote all their effort on exploring efficiently. The exploration remains a challenge while it has been addressed with many hand-tuned heuristics with different levels of generality on one side, and a few theoretically-backed exploration strategies on the other. Many of them are incarnated by intrinsic motivation and in particular explorations bonuses. A common rule of thumb for exploration bonuses is to use $1/\sqrt{n}$ bonus that is added to the empirical estimates of the reward, where $n$ is a number of times this particular state (or a state-action pair) was visited. We show that, surprisingly, for a pure-exploration objective of reward-free exploration, bonuses that scale with $1/n$ bring faster learning rates, improving the known upper bounds with respect to the dependence on the horizon $H$. Furthermore, we show that with an improved analysis of the stopping time, we can improve by a factor $H$ the sample complexity in the best-policy identification setting, which is another pure-exploration objective, where the environment provides rewards but the agent is not penalized for its behavior during the exploration phase.

Omar Darwiche Domingues, Pierre Ménard, Matteo Pirotta et al.

In this work, we propose KeRNS: an algorithm for episodic reinforcement learning in non-stationary Markov Decision Processes (MDPs) whose state-action set is endowed with a metric. Using a non-parametric model of the MDP built with time-dependent kernels, we prove a regret bound that scales with the covering dimension of the state-action space and the total variation of the MDP with time, which quantifies its level of non-stationarity. Our method generalizes previous approaches based on sliding windows and exponential discounting used to handle changing environments. We further propose a practical implementation of KeRNS, we analyze its regret and validate it experimentally.

Hassan Saber, Pierre Ménard, Odalric-Ambrym Maillard

We study a structured variant of the multi-armed bandit problem specified by a set of Bernoulli distributions $ ν\!= \!(ν\_{a,b})\_{a \in \mathcal{A}, b \in \mathcal{B}}$ with means $(μ\_{a,b})\_{a \in \mathcal{A}, b \in \mathcal{B}}\!\in\![0,1]^{\mathcal{A}\times\mathcal{B}}$ and by a given weight matrix $ω\!=\! (ω\_{b,b'})\_{b,b' \in \mathcal{B}}$, where $ \mathcal{A}$ is a finite set of arms and $ \mathcal{B} $ is a finite set of users. The weight matrix $ω$ is such that for any two users $b,b'\!\in\!\mathcal{B}, \text{max}\_{a\in\mathcal{A}}|μ\_{a,b} \!-\! μ\_{a,b'}| \!\leq\! ω\_{b,b'} $. This formulation is flexible enough to capture various situations, from highly-structured scenarios ($ω\!\in\!\{0,1\}^{\mathcal{B}\times\mathcal{B}}$) to fully unstructured setups ($ω\!\equiv\! 1$).We consider two scenarios depending on whether the learner chooses only the actions to sample rewards from or both users and actions. We first derive problem-dependent lower bounds on the regret for this generic graph-structure that involves a structure dependent linear programming problem. Second, we adapt to this setting the Indexed Minimum Empirical Divergence (IMED) algorithm introduced by Honda and Takemura (2015), and introduce the IMED-GS$^\star$ algorithm. Interestingly, IMED-GS$^\star$ does not require computing the solution of the linear programming problem more than about $\log(T)$ times after $T$ steps, while being provably asymptotically optimal. Also, unlike existing bandit strategies designed for other popular structures, IMED-GS$^\star$ does not resort to an explicit forced exploration scheme and only makes use of local counts of empirical events. We finally provide numerical illustration of our results that confirm the performance of IMED-GS$^\star$.

Rémy Degenne, Pierre Ménard, Xuedong Shang et al.

We investigate an active pure-exploration setting, that includes best-arm identification, in the context of linear stochastic bandits. While asymptotically optimal algorithms exist for standard multi-arm bandits, the existence of such algorithms for the best-arm identification in linear bandits has been elusive despite several attempts to address it. First, we provide a thorough comparison and new insight over different notions of optimality in the linear case, including G-optimality, transductive optimality from optimal experimental design and asymptotic optimality. Second, we design the first asymptotically optimal algorithm for fixed-confidence pure exploration in linear bandits. As a consequence, our algorithm naturally bypasses the pitfall caused by a simple but difficult instance, that most prior algorithms had to be engineered to deal with explicitly. Finally, we avoid the need to fully solve an optimal design problem by providing an approach that entails an efficient implementation.

Hassan Saber, Pierre Ménard, Odalric-Ambrym Maillard

We consider a multi-armed bandit problem specified by a set of Gaussian or Bernoulli distributions endowed with a unimodal structure. Although this problem has been addressed in the literature (Combes and Proutiere, 2014), the state-of-the-art algorithms for such structure make appear a forced-exploration mechanism. We introduce IMED-UB, the first forced-exploration free strategy that exploits the unimodal-structure, by adapting to this setting the Indexed Minimum Empirical Divergence (IMED) strategy introduced by Honda and Takemura (2015). This strategy is proven optimal. We then derive KLUCB-UB, a KLUCB version of IMED-UB, which is also proven optimal. Owing to our proof technique, we are further able to provide a concise finite-time analysis of both strategies in an unified way. Numerical experiments show that both IMED-UB and KLUCB-UB perform similarly in practice and outperform the state-of-the-art algorithms.

Emilie Kaufmann, Pierre Ménard, Omar Darwiche Domingues et al.

Reward-free exploration is a reinforcement learning setting studied by Jin et al. (2020), who address it by running several algorithms with regret guarantees in parallel. In our work, we instead give a more natural adaptive approach for reward-free exploration which directly reduces upper bounds on the maximum MDP estimation error. We show that, interestingly, our reward-free UCRL algorithm can be seen as a variant of an algorithm of Fiechter from 1994, originally proposed for a different objective that we call best-policy identification. We prove that RF-UCRL needs of order $({SAH^4}/{\varepsilon^2})(\log(1/δ) + S)$ episodes to output, with probability $1-δ$, an $\varepsilon$-approximation of the optimal policy for any reward function. This bound improves over existing sample-complexity bounds in both the small $\varepsilon$ and the small $δ$ regimes. We further investigate the relative complexities of reward-free exploration and best-policy identification.

Anders Jonsson, Emilie Kaufmann, Pierre Ménard et al.

We propose MDP-GapE, a new trajectory-based Monte-Carlo Tree Search algorithm for planning in a Markov Decision Process in which transitions have a finite support. We prove an upper bound on the number of calls to the generative models needed for MDP-GapE to identify a near-optimal action with high probability. This problem-dependent sample complexity result is expressed in terms of the sub-optimality gaps of the state-action pairs that are visited during exploration. Our experiments reveal that MDP-GapE is also effective in practice, in contrast with other algorithms with sample complexity guarantees in the fixed-confidence setting, that are mostly theoretical.

Omar Darwiche Domingues, Pierre Ménard, Matteo Pirotta et al.

We consider the exploration-exploitation dilemma in finite-horizon reinforcement learning problems whose state-action space is endowed with a metric. We introduce Kernel-UCBVI, a model-based optimistic algorithm that leverages the smoothness of the MDP and a non-parametric kernel estimator of the rewards and transitions to efficiently balance exploration and exploitation. For problems with $K$ episodes and horizon $H$, we provide a regret bound of $\widetilde{O}\left( H^3 K^{\frac{2d}{2d+1}}\right)$, where $d$ is the covering dimension of the joint state-action space. This is the first regret bound for kernel-based RL using smoothing kernels, which requires very weak assumptions on the MDP and has been previously applied to a wide range of tasks. We empirically validate our approach in continuous MDPs with sparse rewards.

Xuedong Shang, Rianne de Heide, Emilie Kaufmann et al.

We investigate and provide new insights on the sampling rule called Top-Two Thompson Sampling (TTTS). In particular, we justify its use for fixed-confidence best-arm identification. We further propose a variant of TTTS called Top-Two Transportation Cost (T3C), which disposes of the computational burden of TTTS. As our main contribution, we provide the first sample complexity analysis of TTTS and T3C when coupled with a very natural Bayesian stopping rule, for bandits with Gaussian rewards, solving one of the open questions raised by Russo (2016). We also provide new posterior convergence results for TTTS under two models that are commonly used in practice: bandits with Gaussian and Bernoulli rewards and conjugate priors.

Rémy Degenne, Wouter M. Koolen, Pierre Ménard

Pure exploration (aka active testing) is the fundamental task of sequentially gathering information to answer a query about a stochastic environment. Good algorithms make few mistakes and take few samples. Lower bounds (for multi-armed bandit models with arms in an exponential family) reveal that the sample complexity is determined by the solution to an optimisation problem. The existing state of the art algorithms achieve asymptotic optimality by solving a plug-in estimate of that optimisation problem at each step. We interpret the optimisation problem as an unknown game, and propose sampling rules based on iterative strategies to estimate and converge to its saddle point. We apply no-regret learners to obtain the first finite confidence guarantees that are adapted to the exponential family and which apply to any pure exploration query and bandit structure. Moreover, our algorithms only use a best response oracle instead of fully solving the optimisation problem.

Pierre Ménard

We present a new algorithm based on an gradient ascent for a general Active Exploration bandit problem in the fixed confidence setting. This problem encompasses several well studied problems such that the Best Arm Identification or Thresholding Bandits. It consists of a new sampling rule based on an online lazy mirror ascent. We prove that this algorithm is asymptotically optimal and, most importantly, computationally efficient.

Aurélien Garivier, Pierre Ménard, Laurent Rossi et al.

We analyze the sample complexity of the thresholding bandit problem, with and without the assumption that the mean values of the arms are increasing. In each case, we provide a lower bound valid for any risk $δ$ and any $δ$-correct algorithm; in addition, we propose an algorithm whose sample complexity is of the same order of magnitude for small risks. This work is motivated by phase 1 clinical trials, a practically important setting where the arm means are increasing by nature, and where no satisfactory solution is available so far.

Pierre Ménard, Aurélien Garivier

We propose the kl-UCB ++ algorithm for regret minimization in stochastic bandit models with exponential families of distributions. We prove that it is simultaneously asymptotically optimal (in the sense of Lai and Robbins' lower bound) and minimax optimal. This is the first algorithm proved to enjoy these two properties at the same time. This work thus merges two different lines of research with simple and clear proofs.

Aurélien Garivier, Pierre Ménard, Gilles Stoltz

We revisit lower bounds on the regret in the case of multi-armed bandit problems. We obtain non-asymptotic, distribution-dependent bounds and provide straightforward proofs based only on well-known properties of Kullback-Leibler divergences. These bounds show in particular that in an initial phase the regret grows almost linearly, and that the well-known logarithmic growth of the regret only holds in a final phase. The proof techniques come to the essence of the information-theoretic arguments used and they are deprived of all unnecessary complications.