Dorian Baudry

Marc Jourdan, Rémy Degenne, Dorian Baudry et al.

Top Two algorithms arose as an adaptation of Thompson sampling to best arm identification in multi-armed bandit models (Russo, 2016), for parametric families of arms. They select the next arm to sample from by randomizing among two candidate arms, a leader and a challenger. Despite their good empirical performance, theoretical guarantees for fixed-confidence best arm identification have only been obtained when the arms are Gaussian with known variances. In this paper, we provide a general analysis of Top Two methods, which identifies desirable properties of the leader, the challenger, and the (possibly non-parametric) distributions of the arms. As a result, we obtain theoretically supported Top Two algorithms for best arm identification with bounded distributions. Our proof method demonstrates in particular that the sampling step used to select the leader inherited from Thompson sampling can be replaced by other choices, like selecting the empirical best arm.

Dorian Baudry, Yoan Russac, Emilie Kaufmann

In this paper, we contribute to the Extreme Bandit problem, a variant of Multi-Armed Bandits in which the learner seeks to collect the largest possible reward. We first study the concentration of the maximum of i.i.d random variables under mild assumptions on the tail of the rewards distributions. This analysis motivates the introduction of Quantile of Maxima (QoMax). The properties of QoMax are sufficient to build an Explore-Then-Commit (ETC) strategy, QoMax-ETC, achieving strong asymptotic guarantees despite its simplicity. We then propose and analyze a more adaptive, anytime algorithm, QoMax-SDA, which combines QoMax with a subsampling method recently introduced by Baudry et al. (2021). Both algorithms are more efficient than existing approaches in two aspects (1) they lead to better empirical performance (2) they enjoy a significant reduction of the memory and time complexities.

Romain Gautron, Dorian Baudry, Myriam Adam et al.

Identification of best performing fertilizer practices among a set of contrasting practices with field trials is challenging as crop losses are costly for farmers. To identify best management practices, an ''intuitive strategy'' would be to set multi-year field trials with equal proportion of each practice to test. Our objective was to provide an identification strategy using a bandit algorithm that was better at minimizing farmers' losses occurring during the identification, compared with the ''intuitive strategy''. We used a modification of the Decision Support Systems for Agro-Technological Transfer (DSSAT) crop model to mimic field trial responses, with a case-study in Southern Mali. We compared fertilizer practices using a risk-aware measure, the Conditional Value-at-Risk (CVaR), and a novel agronomic metric, the Yield Excess (YE). YE accounts for both grain yield and agronomic nitrogen use efficiency. The bandit-algorithm performed better than the intuitive strategy: it increased, in most cases, farmers' protection against worst outcomes. This study is a methodological step which opens up new horizons for risk-aware ensemble identification of the performance of contrasting crop management practices in real conditions.

Dorian Baudry, Kazuya Suzuki, Junya Honda

In this paper we propose a general methodology to derive regret bounds for randomized multi-armed bandit algorithms. It consists in checking a set of sufficient conditions on the sampling probability of each arm and on the family of distributions to prove a logarithmic regret. As a direct application we revisit two famous bandit algorithms, Minimum Empirical Divergence (MED) and Thompson Sampling (TS), under various models for the distributions including single parameter exponential families, Gaussian distributions, bounded distributions, or distributions satisfying some conditions on their moments. In particular, we prove that MED is asymptotically optimal for all these models, but also provide a simple regret analysis of some TS algorithms for which the optimality is already known. We then further illustrate the interest of our approach, by analyzing a new Non-Parametric TS algorithm (h-NPTS), adapted to some families of unbounded reward distributions with a bounded h-moment. This model can for instance capture some non-parametric families of distributions whose variance is upper bounded by a known constant.

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 Dirichlet Sampling (DS) algorithm, based on pairwise comparisons of empirical indices computed with re-sampling of the arms' observations and a data-dependent 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.

Dorian Baudry, Yoan Russac, Olivier Cappé

There has been a recent surge of interest in nonparametric bandit algorithms based on subsampling. One drawback however of these approaches is the additional complexity required by random subsampling and the storage of the full history of rewards. Our first contribution is to show that a simple deterministic subsampling rule, proposed in the recent work of Baudry et al. (2020) under the name of ''last-block subsampling'', is asymptotically optimal in one-parameter exponential families. In addition, we prove that these guarantees also hold when limiting the algorithm memory to a polylogarithmic function of the time horizon. These findings open up new perspectives, in particular for non-stationary scenarios in which the arm distributions evolve over time. We propose a variant of the algorithm in which only the most recent observations are used for subsampling, achieving optimal regret guarantees under the assumption of a known number of abrupt changes. Extensive numerical simulations highlight the merits of this approach, particularly when the changes are not only affecting the means of the rewards.

Dorian Baudry, Romain Gautron, Emilie Kaufmann et al.

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.

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 [1] and SSMC [2] 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.