arXiv subjects

Pierre Ménard

Publications and source records attributed to Pierre Ménard.

33 records · Page 2Linked to original sources

arXiv2021

Model-Free Learning for Two-Player Zero-Sum Partially Observable Markov Games with Perfect Recall

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.

stat.ML↗

arXiv2020

Fast active learning for pure exploration in reinforcement learning

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.

cs.LG↗

arXiv2020

Adaptive Reward-Free Exploration

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.

cs.LG↗

arXiv2020

Episodic Reinforcement Learning in Finite MDPs: Minimax Lower Bounds Revisited

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.

cs.LG↗

arXiv2020

Optimal Strategies for Graph-Structured Bandits

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$^\star$.

cs.IT↗

arXiv2020

Gamification of Pure Exploration for Linear Bandits

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.

stat.ML↗

arXiv2020

Forced-exploration free Strategies for Unimodal Bandits

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.

cs.LG↗

arXiv2020

Planning in Markov Decision Processes with Gap-Dependent Sample Complexity

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.

cs.LG↗

arXiv2019

Fixed-Confidence Guarantees for Bayesian Best-Arm Identification

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.

cs.LG↗

arXiv2019

Non-Asymptotic Pure Exploration by Solving Games

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.

stat.ML↗

arXiv2019

Fano's inequality for random variables

We extend Fano's inequality, which controls the average probability of events in terms of the average of some $f$--divergences, to work with arbitrary events (not necessarily forming a partition) and even with arbitrary $[0,1]$--valued random variables, possibly in continuously infinite number. We provide two applications of these extensions, in which the consideration of random variables is particularly handy: we offer new and elegant proofs for existing lower bounds, on Bayesian posterior concentration (minimax or distribution-dependent) rates and on the regret in non-stochastic sequential learning.

math.ST↗

arXiv2019

Gradient Ascent for Active Exploration in Bandit Problems

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.

stat.ML↗

arXiv2018

Explore First, Exploit Next: The True Shape of Regret in Bandit Problems

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.

math.ST↗

arXiv2018

Thresholding Bandit for Dose-ranging: The Impact of Monotonicity

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.

math.ST↗

arXiv2017

A minimax and asymptotically optimal algorithm for stochastic bandits

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.

stat.ML↗