arXiv subjects

Aurélien Garivier

Publications and source records attributed to Aurélien Garivier.

At least 55 records · Page 3Linked to original sources

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↗

arXiv2017

Max K-armed bandit: On the ExtremeHunter algorithm and beyond

This paper is devoted to the study of the max K-armed bandit problem, which consists in sequentially allocating resources in order to detect extreme values. Our contribution is twofold. We first significantly refine the analysis of the ExtremeHunter algorithm carried out in Carpentier and Valko (2014), and next propose an alternative approach, showing that, remarkably, Extreme Bandits can be reduced to a classical version of the bandit problem to a certain extent. Beyond the formal analysis, these two approaches are compared through numerical experiments.

stat.ML↗

arXiv2016

On the Complexity of Best Arm Identification in Multi-Armed Bandit Models

The stochastic multi-armed bandit model is a simple abstraction that has proven useful in many different contexts in statistics and machine learning. Whereas the achievable limit in terms of regret minimization is now well known, our aim is to contribute to a better understanding of the performance in terms of identifying the m best arms. We introduce generic notions of complexity for the two dominant frameworks considered in the literature: fixed-budget and fixed-confidence settings. In the fixed-confidence setting, we provide the first known distribution-dependent lower bound on the complexity that involves information-theoretic quantities and holds when m is larger than 1 under general assumptions. In the specific case of two armed-bandits, we derive refined lower bounds in both the fixed-confidence and fixed-budget settings, along with matching algorithms for Gaussian and Bernoulli bandit models. These results show in particular that the complexity of the fixed-budget setting may be smaller than the complexity of the fixed-confidence setting, contradicting the familiar behavior observed when testing fully specified alternatives. In addition, we also provide improved sequential stopping rules that have guaranteed error probabilities and shorter average running times. The proofs rely on two technical results that are of independent interest : a deviation lemma for self-normalized sums (Lemma 19) and a novel change of measure inequality for bandit models (Lemma 1).

stat.ML↗

arXiv2016

On Explore-Then-Commit Strategies

We study the problem of minimising regret in two-armed bandit problems with Gaussian rewards. Our objective is to use this simple setting to illustrate that strategies based on an exploration phase (up to a stopping time) followed by exploitation are necessarily suboptimal. The results hold regardless of whether or not the difference in means between the two arms is known. Besides the main message, we also refine existing deviation inequalities, which allow us to design fully sequential strategies with finite-time regret guarantees that are (a) asymptotically optimal as the horizon grows and (b) order-optimal in the minimax sense. Furthermore we provide empirical evidence that the theory also holds in practice and discuss extensions to non-gaussian and multiple-armed case.

math.ST↗

arXiv2016

Optimal Best Arm Identification with Fixed Confidence

We give a complete characterization of the complexity of best-arm identification in one-parameter bandit problems. We prove a new, tight lower bound on the sample complexity. We propose the `Track-and-Stop' strategy, which we prove to be asymptotically optimal. It consists in a new sampling rule (which tracks the optimal proportions of arm draws highlighted by the lower bound) and in a stopping rule named after Chernoff, for which we give a new analysis.

math.ST↗

arXiv2016

Maximin Action Identification: A New Bandit Framework for Games

We study an original problem of pure exploration in a strategic bandit model motivated by Monte Carlo Tree Search. It consists in identifying the best action in a game, when the player may sample random outcomes of sequentially chosen pairs of actions. We propose two strategies for the fixed-confidence setting: Maximin-LUCB, based on lower-and upper-confidence bounds; and Maximin-Racing, which operates by successively eliminating the sub-optimal actions. We discuss the sample complexity of both methods and compare their performance empirically. We sketch a lower bound analysis, and possible connections to an optimal algorithm.

math.ST↗

arXiv2016

Bregman superquantiles. Estimation methods and applications

In this work, we extend some quantities introduced in "Optimization of conditional value-at-risk" of R.T Rockafellar and S. Uryasev to the case where the proximity between real numbers is measured by using a Bregman divergence. This leads to the definition of the Bregman superquantile. Axioms of a coherent measure of risk discussed in "Coherent approches to risk in optimization under uncertainty" of R.T Rockafellar are studied in the case of Bregman superquantile. Furthermore, we deal with asymptotic properties of a Monte Carlo estimator of the Bregman superquantile.

math.ST↗

arXiv2015

On the Complexity of A/B Testing

A/B testing refers to the task of determining the best option among two alternatives that yield random outcomes. We provide distribution-dependent lower bounds for the performance of A/B testing that improve over the results currently available both in the fixed-confidence (or delta-PAC) and fixed-budget settings. When the distribution of the outcomes are Gaussian, we prove that the complexity of the fixed-confidence and fixed-budget settings are equivalent, and that uniform sampling of both alternatives is optimal only in the case of equal variances. In the common variance case, we also provide a stopping rule that terminates faster than existing fixed-confidence algorithms. In the case of Bernoulli distributions, we show that the complexity of fixed-budget setting is smaller than that of fixed-confidence setting and that uniform sampling of both alternatives -though not optimal- is advisable in practice when combined with an appropriate stopping criterion.

math.ST↗

arXiv2014

Big Data Analytics - Retour vers le Futur 3; De Statisticien à Data Scientist

The rapid evolution of information systems managing more and more voluminous data has caused profound paradigm shifts in the job of statistician, becoming successively data miner, bioinformatician and now data scientist. Without the sake of completeness and after having illustrated these successive mutations, this article briefly introduced the new research issues that quickly rise in Statistics, and more generally in Mathematics, in order to integrate the characteristics: volume, variety and velocity, of big data.

math.ST↗

arXiv2013

Perfect Simulation Of Processes With Long Memory: A `Coupling Into And From The Past' Algorithm

We describe a new algorithm for the perfect simulation of variable length Markov chains and random systems with perfect connections. This algorithm, which generalizes Propp and Wilson's simulation scheme, is based on the idea of coupling into and from the past. It improves on existing algorithms by relaxing the conditions on the kernel and by accelerating convergence, even in the simple case of finite order Markov chains. Although chains of variable or infinite order have been widely investigated for decades, their use in applied probability, from information theory to bio-informatics and linguistics, has recently led to considerable renewed interest.

math.PR↗

arXiv2013

Informational Confidence Bounds for Self-Normalized Averages and Applications

We present deviation bounds for self-normalized averages and applications to estimation with a random number of observations. The results rely on a peeling argument in exponential martingale techniques that represents an alternative to the method of mixture. The motivating examples of bandit problems and context tree estimation are detailed.

math.ST↗

arXiv2013

The KL-UCB Algorithm for Bounded Stochastic Bandits and Beyond

This paper presents a finite-time analysis of the KL-UCB algorithm, an online, horizon-free index policy for stochastic bandit problems. We prove two distinct results: first, for arbitrary bounded rewards, the KL-UCB algorithm satisfies a uniformly better regret bound than UCB or UCB2; second, in the special case of Bernoulli rewards, it reaches the lower bound of Lai and Robbins. Furthermore, we show that simple adaptations of the KL-UCB algorithm are also optimal for specific classes of (possibly unbounded) rewards, including those generated from exponential families of distributions. A large-scale numerical study comparing KL-UCB with its main competitors (UCB, UCB2, UCB-Tuned, UCB-V, DMED) shows that KL-UCB is remarkably efficient and stable, including for short time horizons. KL-UCB is also the only method that always performs better than the basic UCB policy. Our regret bounds rely on deviations results of independent interest which are stated and proved in the Appendix. As a by-product, we also obtain an improved regret bound for the standard UCB algorithm.

math.ST↗

arXiv2013

Kullback-Leibler upper confidence bounds for optimal sequential allocation

We consider optimal sequential allocation in the context of the so-called stochastic multi-armed bandit model. We describe a generic index policy, in the sense of Gittins [J. R. Stat. Soc. Ser. B Stat. Methodol. 41 (1979) 148-177], based on upper confidence bounds of the arm payoffs computed using the Kullback-Leibler divergence. We consider two classes of distributions for which instances of this general idea are analyzed: the kl-UCB algorithm is designed for one-parameter exponential families and the empirical KL-UCB algorithm for bounded and finitely supported distributions. Our main contribution is a unified finite-time analysis of the regret of these algorithms that asymptotically matches the lower bounds of Lai and Robbins [Adv. in Appl. Math. 6 (1985) 4-22] and Burnetas and Katehakis [Adv. in Appl. Math. 17 (1996) 122-142], respectively. We also investigate the behavior of these algorithms when used with general bounded rewards, showing in particular that they provide significant improvements over the state-of-the-art.

math.PR↗

arXiv2012

Joint estimation of intersecting context tree models

We study a problem of model selection for data produced by two different context tree sources. Motivated by linguistic questions, we consider the case where the probabilistic context trees corresponding to the two sources are finite and share many of their contexts. In order to understand the differences between the two sources, it is important to identify which contexts and which transition probabilities are specific to each source. We consider a class of probabilistic context tree models with three types of contexts: those which appear in one, the other, or both sources. We use a BIC penalized maximum likelihood procedure that jointly estimates the two sources. We propose a new algorithm which efficiently computes the estimated context trees. We prove that the procedure is strongly consistent. We also present a simulation study showing the practical advantage of our procedure over a procedure that works separately on each dataset.

math.ST↗

arXiv2012

Sequential Monte Carlo smoothing for general state space hidden Markov models

Computing smoothing distributions, the distributions of one or more states conditional on past, present, and future observations is a recurring problem when operating on general hidden Markov models. The aim of this paper is to provide a foundation of particle-based approximation of such distributions and to analyze, in a common unifying framework, different schemes producing such approximations. In this setting, general convergence results, including exponential deviation inequalities and central limit theorems, are established. In particular, time uniform bounds on the marginal smoothing error are obtained under appropriate mixing conditions on the transition kernel of the latent chain. In addition, we propose an algorithm approximating the joint smoothing distribution at a cost that grows only linearly with the number of particles.

math.PR↗

arXiv2011

Optimal discovery with probabilistic expert advice

We consider an original problem that arises from the issue of security analysis of a power system and that we name optimal discovery with probabilistic expert advice. We address it with an algorithm based on the optimistic paradigm and the Good-Turing missing mass estimator. We show that this strategy uniformly attains the optimal discovery rate in a macroscopic limit sense, under some assumptions on the probabilistic experts. We also provide numerical experiments suggesting that this optimal behavior may still hold under weaker assumptions.

math.OC↗

arXiv2011

Context Tree Selection: A Unifying View

The present paper investigates non-asymptotic properties of two popular procedures of context tree (or Variable Length Markov Chains) estimation: Rissanen's algorithm Context and the Penalized Maximum Likelihood criterion. First showing how they are related, we prove finite horizon bounds for the probability of over- and under-estimation. Concerning overestimation, no boundedness or loss-of-memory conditions are required: the proof relies on new deviation inequalities for empirical probabilities of independent interest. The underestimation properties rely on loss-of-memory and separation conditions of the process. These results improve and generalize the bounds obtained previously. Context tree models have been introduced by Rissanen as a parsimonious generalization of Markov models. Since then, they have been widely used in applied probability and statistics.

math.ST↗

arXiv2010

Optimism in Reinforcement Learning and Kullback-Leibler Divergence

We consider model-based reinforcement learning in finite Markov De- cision Processes (MDPs), focussing on so-called optimistic strategies. In MDPs, optimism can be implemented by carrying out extended value it- erations under a constraint of consistency with the estimated model tran- sition probabilities. The UCRL2 algorithm by Auer, Jaksch and Ortner (2009), which follows this strategy, has recently been shown to guarantee near-optimal regret bounds. In this paper, we strongly argue in favor of using the Kullback-Leibler (KL) divergence for this purpose. By studying the linear maximization problem under KL constraints, we provide an ef- ficient algorithm, termed KL-UCRL, for solving KL-optimistic extended value iteration. Using recent deviation bounds on the KL divergence, we prove that KL-UCRL provides the same guarantees as UCRL2 in terms of regret. However, numerical experiments on classical benchmarks show a significantly improved behavior, particularly when the MDP has reduced connectivity. To support this observation, we provide elements of com- parison between the two algorithms based on geometric considerations.

cs.LG↗