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Register a new userPure Exploration in Infinitely-Armed Bandit Models with Fixed-Confidence
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We consider the problem of near-optimal arm identification in the fixed confidence setting of the infinitely armed bandit problem when nothing is known about the arm reservoir distribution. We (1) introduce a PAC-like framework within which to derive and cast results; (2) derive a sample complexity lower bound for near-optimal arm identification; (3) propose an algorithm that identifies a nearly-optimal arm with high probability and derive an upper bound on its sample complexity which is within a log factor of our lower bound; and (4) discuss whether our log^2(1/delta) dependence is inescapable for two-phase (select arms first, identify the best later) algorithms in the infinite setting. This work permits the application of bandit models to a broader class of problems where fewer assumptions hold.
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We consider the framework of stochastic multi-armed bandit problems and study the possibilities and limitations of forecasters that perform an on-line exploration of the arms. These forecasters are assessed in terms of their simple regret, a regret notion that captures the fact that exploration is only constrained by the number of available rounds (not necessarily known in advance), in contrast to the case when the cumulative regret is considered and when exploitation needs to be performed at the same time. We believe that this performance criterion is suited to situations when the cost of pulling an arm is expressed in terms of resources rather than rewards. We discuss the links between the simple and the cumulative regret. One of the main results in the case of a finite number of arms is a general lower bound on the simple regret of a forecaster in terms of its cumulative regret: the smaller the latter, the larger the former. Keeping this result in mind, we then exhibit upper bounds on the simple regret of some forecasters. The paper ends with a study devoted to continuous-armed bandit problems; we show that the simple regret can be minimized with respect to a family of probability distributions if and only if the cumulative regret can be minimized for it. Based on this equivalence, we are able to prove that the separable metric spaces are exactly the metric spaces on which these regrets can be minimized with respect to the family of all probability distributions with continuous mean-payoff functions.
We discuss a multiple-play multi-armed bandit (MAB) problem in which several arms are selected at each round. Recently, Thompson sampling (TS), a randomized algorithm with a Bayesian spirit, has attracted much attention for its empirically excellent performance, and it is revealed to have an optimal regret bound in the standard single-play MAB problem. In this paper, we propose the multiple-play Thompson sampling (MP-TS) algorithm, an extension of TS to the multiple-play MAB problem, and discuss its regret analysis. We prove that MP-TS for binary rewards has the optimal regret upper bound that matches the regret lower bound provided by Anantharam et al. (1987). Therefore, MP-TS is the first computationally efficient algorithm with optimal regret. A set of computer simulations was also conducted, which compared MP-TS with state-of-the-art algorithms. We also propose a modification of MP-TS, which is shown to have better empirical performance.
We study pure exploration in bandits, where the dimension of the feature representation can be much larger than the number of arms. To overcome the curse of dimensionality, we propose to adaptively embed the feature representation of each arm into a lower-dimensional space and carefully deal with the induced model misspecifications. Our approach is conceptually very different from existing works that can either only handle low-dimensional linear bandits or passively deal with model misspecifications. We showcase the application of our approach to two pure exploration settings that were previously under-studied: (1) the reward function belongs to a possibly infinite-dimensional Reproducing Kernel Hilbert Space, and (2) the reward function is nonlinear and can be approximated by neural networks. Our main results provide sample complexity guarantees that only depend on the effective dimension of the feature spaces in the kernel or neural representations. Extensive experiments conducted on both synthetic and real-world datasets demonstrate the efficacy of our methods.
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.
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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