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Pierre Ménard

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19 papers

2 author rows

ICML Conference 2025 Conference Paper

The Harder Path: Last Iterate Convergence for Uncoupled Learning in Zero-Sum Games with Bandit Feedback

Côme Fiegel
Pierre Ménard
Tadashi Kozuno
Michal Valko
Vianney Perchet

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, 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 performances, with the best attainable rate being $\mathcal{O}(T^{-1/4})$ in contrast to the usual $\mathcal{O}(T^{-1/2})$ rate for convergence of the average iterates. We then propose two algorithms that achieve this optimal rate. The first algorithm leverages a straightforward tradeoff between exploration and exploitation, while the second employs a regularization technique based on a two-step mirror descent approach.

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ICLR Conference 2024 Conference Paper

Demonstration-Regularized RL

Daniil Tiapkin
Denis Belomestny
Daniele Calandriello
Eric Moulines
Alexey Naumov
Pierre Perrault
Michal Valko
Pierre Ménard

Incorporating expert demonstrations has empirically helped to improve the sample efficiency of reinforcement learning (RL). This paper quantifies theoretically to what extent this extra information reduces RL's sample complexity. In particular, we study the demonstration-regularized reinforcement learning framework that leverages the expert demonstrations by $\mathrm{KL}$-regularization for a policy learned by behavior cloning. Our findings reveal that using $N^{\mathrm{E}}$ expert demonstrations enables the identification of an optimal policy at a sample complexity of order $\widetilde{\mathcal{O}}(\mathrm{Poly}(S,A,H)/(\varepsilon^2 N^{\mathrm{E}}))$ in finite and $\widetilde{\mathcal{O}}(\mathrm{Poly}(d,H)/(\varepsilon^2 N^{\mathrm{E}}))$ in linear Markov decision processes, where $\varepsilon$is the target precision, $H$ the horizon, $A$ the number of action, $S$ the number of states in the finite case and $d$ the dimension of the feature space in the linear case. As a by-product, we provide tight convergence guarantees for the behavior cloning procedure under general assumptions on the policy classes. Additionally, we establish that demonstration-regularized methods are provably efficient for reinforcement learning from human feedback (RLHF). In this respect, we provide theoretical evidence showing the benefits of KL-regularization for RLHF in tabular and linear MDPs. Interestingly, we avoid pessimism injection by employing computationally feasible regularization to handle reward estimation uncertainty, thus setting our approach apart from the prior works.

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NeurIPS Conference 2024 Conference Paper

Local and Adaptive Mirror Descents in Extensive-Form Games

Côme Fiegel
Pierre Ménard
Tadashi Kozuno
Rémi Munos
Vianney Perchet
Michal Valko

We study how to learn $\epsilon$-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 $T$. Most existing procedures suffer from high variance due to the use of importance sampling over sequences of actions. 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.

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ICML Conference 2023 Conference Paper

Adapting to game trees in zero-sum imperfect information games

Côme Fiegel
Pierre Ménard
Tadashi Kozuno
Rémi Munos
Vianney Perchet
Michal Valko

Imperfect information games (IIG) are games in which each player only partially observes the current game state. We study how to learn $\epsilon$-optimal strategies in a zero-sum IIG through self-play with trajectory feedback. We give a problem-independent lower bound $\widetilde{\mathcal{O}}(H(A_{\mathcal{X}}+B_{\mathcal{Y}})/\epsilon^2)$ 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 $\widetilde{\mathcal{O}}(H^2(A_{\mathcal{X}}+B_{\mathcal{Y}})/\epsilon^2)$ realizations without this requirement by progressively adapting the regularization to the observations.

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ICML Conference 2023 Conference Paper

Fast Rates for Maximum Entropy Exploration

Daniil Tiapkin
Denis Belomestny
Daniele Calandriello
Eric Moulines
Rémi Munos
Alexey Naumov
Pierre Perrault
Yunhao Tang

We address the challenge of exploration in reinforcement learning (RL) when the agent operates in an unknown environment with sparse or no rewards. In this work, we study the maximum entropy exploration problem of two different types. The first type is visitation entropy maximization previously considered by Hazan et al. (2019) in the discounted setting. For this type of exploration, we propose a game-theoretic algorithm that has $\widetilde{\mathcal{O}}(H^3S^2A/\varepsilon^2)$ sample complexity thus improving the $\varepsilon$-dependence upon existing results, where $S$ is a number of states, $A$ is a number of actions, $H$ is an episode length, and $\varepsilon$ is a desired accuracy. The second type of entropy we study is the trajectory entropy. This objective function is closely related to the entropy-regularized MDPs, and we propose a simple algorithm that has a sample complexity of order $\widetilde{\mathcal{O}}(\mathrm{poly}(S, A, H)/\varepsilon)$. Interestingly, it is the first theoretical result in RL literature that establishes the potential statistical advantage of regularized MDPs for exploration. Finally, we apply developed regularization techniques to reduce sample complexity of visitation entropy maximization to $\widetilde{\mathcal{O}}(H^2SA/\varepsilon^2)$, yielding a statistical separation between maximum entropy exploration and reward-free exploration.

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NeurIPS Conference 2023 Conference Paper

Model-free Posterior Sampling via Learning Rate Randomization

Daniil Tiapkin
Denis Belomestny
Daniele Calandriello
Eric Moulines
Remi Munos
Alexey Naumov
Pierre Perrault
Michal Valko

In this paper, we introduce Randomized Q-learning (RandQL), a novel randomized model-free algorithm for regret minimization in episodic Markov Decision Processes (MDPs). To the best of our knowledge, RandQL is the first tractable model-free posterior sampling-based algorithm. We analyze the performance of RandQL in both tabular and non-tabular metric space settings. In tabular MDPs, RandQL achieves a regret bound of order $\widetilde{\mathcal{O}}(\sqrt{H^{5}SAT})$, where $H$ is the planning horizon, $S$ is the number of states, $A$ is the number of actions, and $T$ is the number of episodes. For a metric state-action space, RandQL enjoys a regret bound of order $\widetilde{\mathcal{O}}(H^{5/2} T^{(d_z+1)/(d_z+2)})$, where $d_z$ denotes the zooming dimension. Notably, RandQL achieves optimistic exploration without using bonuses, relying instead on a novel idea of learning rate randomization. Our empirical study shows that RandQL outperforms existing approaches on baseline exploration environments.

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ICML Conference 2023 Conference Paper

Regularization and Variance-Weighted Regression Achieves Minimax Optimality in Linear MDPs: Theory and Practice

Toshinori Kitamura
Tadashi Kozuno
Yunhao Tang
Nino Vieillard
Michal Valko
Wenhao Yang
Jincheng Mei
Pierre Ménard

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 $1-\delta$ 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.

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ICML Conference 2022 Conference Paper

From Dirichlet to Rubin: Optimistic Exploration in RL without Bonuses

Daniil Tiapkin
Denis Belomestny
Eric Moulines
Alexey Naumov
Sergey Samsonov
Yunhao Tang
Michal Valko
Pierre Ménard

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

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JMLR Journal 2022 Journal Article

KL-UCB-Switch: Optimal Regret Bounds for Stochastic Bandits from Both a Distribution-Dependent and a Distribution-Free Viewpoints

Aurélien Garivier
Hédi Hadiji
Pierre Ménard
Gilles Stoltz

We consider $K$-armed stochastic bandits and consider cumulative regret bounds up to time $T$. We are interested in strategies achieving simultaneously a distribution-free regret bound of optimal order $\sqrt{KT}$ and a distribution-dependent regret that is asymptotically optimal, that is, matching the $\kappa \ln T$ lower bound by Lai and Robbins (1985) and Burnetas and Katehakis (1996), where $\kappa$ is the optimal problem-dependent constant. This constant $\kappa$ depends on the model $\mathcal{D}$ considered (the family of possible distributions over the arms). Ménard and Garivier (2017) provided strategies achieving such a bi-optimality in the parametric case of models given by one-dimensional exponential families, while Lattimore (2016, 2018) did so for the family of (sub)Gaussian distributions with variance less than $1$. We extend this result to the non-parametric case of all distributions over $[0,1]$. We do so by combining the MOSS strategy by Audibert and Bubeck (2009), which enjoys a distribution-free regret bound of optimal order $\sqrt{KT}$, and the KL-UCB strategy by Cappé et al. (2013), for which we provide in passing the first analysis of an optimal distribution-dependent $\kappa\ln T$ regret bound in the model of all distributions over $[0,1]$. We were able to obtain this non-parametric bi-optimality result while working hard to streamline the proofs (of previously known regret bounds and thus of the new analyses carried out); a second merit of the present contribution is therefore to provide a review of proofs of classical regret bounds for index-based strategies for $K$-armed stochastic bandits. [abs] [ pdf ][ bib ] &copy JMLR 2022. ( edit, beta )

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NeurIPS Conference 2022 Conference Paper

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

Daniil Tiapkin
Denis Belomestny
Daniele Calandriello
Eric Moulines
Remi Munos
Alexey Naumov
Mark Rowland
Michal Valko

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

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NeurIPS Conference 2021 Conference Paper

Bandits with many optimal arms

Rianne de Heide
James Cheshire
Pierre Ménard
Alexandra Carpentier

We consider a stochastic bandit problem with a possibly infinite number of arms. We write $p^*$ for the proportion of optimal arms and $\Delta$ 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 $\Delta$. For the objective of minimizing the cumulative regret, we provide a lower bound of order $\Omega(\log(T)/(p^*\Delta))$ and a UCB-style algorithm with matching upper bound up to a factor of $\log(1/\Delta)$. 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 $\Omega(\exp(-cT\Delta^2p^*))$ 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 $\Delta$ as parameter. Our results apply directly to the three related problems of competing against the $j$-th best arm, identifying an $\epsilon$ good arm, and finding an arm with mean larger than a quantile of a known order.

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ICML Conference 2021 Conference Paper

Fast active learning for pure exploration in reinforcement learning

Pierre Ménard
Omar Darwiche Domingues
Anders Jonsson 0001
Emilie Kaufmann
Edouard Leurent
Michal Valko

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 \emph{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 \emph{intrinsic motivation} and in particular \emph{explorations bonuses}. A common choice is to use $1/\sqrt{n}$ bonus, where $n$ is a number of times this particular state-action pair was visited. We show that, surprisingly, for a pure-exploration objective of \emph{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 \emph{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.

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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.

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ICML Conference 2021 Conference Paper

Kernel-Based Reinforcement Learning: A Finite-Time Analysis

Omar Darwiche Domingues
Pierre Ménard
Matteo Pirotta
Emilie Kaufmann
Michal Valko

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 applies to a wide range of tasks. We empirically validate our approach in continuous MDPs with sparse rewards.

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NeurIPS Conference 2021 Conference Paper

Learning in two-player zero-sum partially observable Markov games with perfect recall

Tadashi Kozuno
Pierre Ménard
Remi Munos
Michal Valko

We study the problem of learning a Nash equilibrium (NE) in an extensive game with imperfect information (EGII) through self-play. Precisely, we focus on two-player, zero-sum, episodic, tabular EGII under the \textit{perfect-recall} assumption where the only feedback is realizations of the game (bandit feedback). In particular the \textit{dynamics of the EGII 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 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.

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ICML Conference 2021 Conference Paper

Problem Dependent View on Structured Thresholding Bandit Problems

James Cheshire
Pierre Ménard
Alexandra Carpentier

We investigate the \textit{problem dependent regime} in the stochastic \emph{Thresholding Bandit problem} (\tbp) under several \emph{shape constraints}. In the \tbp the objective of the learner is to output, after interacting with the environment, 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 $(\mu_k){k=1}^K$ is monotonically increasing (\textit{MTBP}) and (ii) the case where $(\mu_k){k=1}^K$ is concave (\textit{CTBP}). We consider both cases in the \emph{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 nearly matching upper and lower bounds for the probability of error in both the concave and monotone settings, as well as associated algorithms. Of interest, is that for both the monotone and concave cases, optimal bounds on probability of error are of the same order as those for the two armed bandit problem.

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ICML Conference 2021 Conference Paper

UCB Momentum Q-learning: Correcting the bias without forgetting

Pierre Ménard
Omar Darwiche Domingues
Xuedong Shang
Michal Valko

We propose UCBMQ, Upper Confidence Bound Momentum Q-learning, a new algorithm for reinforcement learning in tabular and possibly stage-dependent, episodic Markov decision process. UCBMQ is based on Q-learning where we add a momentum term and rely on the principle of optimism in face of uncertainty to deal with exploration. Our new technical ingredient of UCBMQ is the use of momentum to correct the bias that Q-learning suffers while, \emph{at the same time}, limiting the impact it has on the second-order term of the regret. For UCBMQ, we are able to guarantee a regret of at most $\tilde{O}(\sqrt{H^3SAT}+ H^4 S A)$ where $H$ is the length of an episode, $S$ the number of states, $A$ the number of actions, $T$ the number of episodes and ignoring terms in poly$\log(SAHT)$. Notably, UCBMQ is the first algorithm that simultaneously matches the lower bound of $\Omega(\sqrt{H^3SAT})$ for large enough $T$ and has a second-order term (with respect to $T$) that scales \emph{only linearly} with the number of states $S$.

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ICML Conference 2020 Conference Paper

Gamification of Pure Exploration for Linear Bandits

Rémy Degenne
Pierre Ménard
Xuedong Shang
Michal Valko

We investigate an active \emph{pure-exploration} setting, that includes \emph{best-arm identification}, in the context of \emph{linear stochastic bandits}. While asymptotically optimal algorithms exist for standard \emph{multi-armed 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.

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NeurIPS Conference 2019 Conference Paper

Non-Asymptotic Pure Exploration by Solving Games

Rémy Degenne
Wouter 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.

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