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Register a new userA Finite-Time Analysis of Multi-armed Bandits Problems with Kullback-Leibler Divergences
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Added by
Gilles Stoltz
Publication date
2011
fields
Mathematical Statistics
and research's language is
English
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We consider a Kullback-Leibler-based algorithm for the stochastic multi-armed bandit problem in the case of distributions with finite supports (not necessarily known beforehand), whose asymptotic regret matches the lower bound of cite{Burnetas96}. Our contribution is to provide a finite-time analysis of this algorithm; we get bounds whose main terms are smaller than the ones of previously known algorithms with finite-time analyses (like UCB-type algorithms).
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We consider nonstationary multi-armed bandit problems where the model parameters of the arms change over time. We introduce the adaptive resetting bandit (ADR-bandit), which is a class of bandit algorithms that leverages adaptive windowing techniques from the data stream community. We first provide new guarantees on the quality of estimators resulting from adaptive windowing techniques, which are of independent interest in the data mining community. Furthermore, we conduct a finite-time analysis of ADR-bandit in two typical environments: an abrupt environment where changes occur instantaneously and a gradual environment where changes occur progressively. We demonstrate that ADR-bandit has nearly optimal performance when the abrupt or global changes occur in a coordinated manner that we call global changes. We demonstrate that forced exploration is unnecessary when we restrict the interest to the global changes. Unlike the existing nonstationary bandit algorithms, ADR-bandit has optimal performance in stationary environments as well as nonstationary environments with global changes. Our experiments show that the proposed algorithms outperform the existing approaches in synthetic and real-world environments.
Bayesian nonparametric statistics is an area of considerable research interest. While recently there has been an extensive concentration in developing Bayesian nonparametric procedures for model checking, the use of the Dirichlet process, in its simplest form, along with the Kullback-Leibler divergence is still an open problem. This is mainly attributed to the discreteness property of the Dirichlet process and that the Kullback-Leibler divergence between any discrete distribution and any continuous distribution is infinity. The approach proposed in this paper, which is based on incorporating the Dirichlet process, the Kullback-Leibler divergence and the relative belief ratio, is considered the first concrete solution to this issue. Applying the approach is simple and does not require obtaining a closed form of the relative belief ratio. A Monte Carlo study and real data examples show that the developed approach exhibits excellent performance.
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 study a variant of the classical multi-armed bandit problem (MABP) which we call as Multi-Armed Bandits with dependent arms. More specifically, multiple arms are grouped together to form a cluster, and the reward distributions of arms belonging to the same cluster are known functions of an unknown parameter that is a characteristic of the cluster. Thus, pulling an arm $i$ not only reveals information about its own reward distribution, but also about all those arms that share the same cluster with arm $i$. This correlation amongst the arms complicates the exploration-exploitation trade-off that is encountered in the MABP because the observation dependencies allow us to test simultaneously multiple hypotheses regarding the optimality of an arm. We develop learning algorithms based on the UCB principle which utilize these additional side observations appropriately while performing exploration-exploitation trade-off. We show that the regret of our algorithms grows as $O(Klog T)$, where $K$ is the number of clusters. In contrast, for an algorithm such as the vanilla UCB that is optimal for the classical MABP and does not utilize these dependencies, the regret scales as $O(Mlog T)$ where $M$ is the number of arms.
We introduce a new class of reinforcement learning methods referred to as {em episodic multi-armed bandits} (eMAB). In eMAB the learner proceeds in {em episodes}, each composed of several {em steps}, in which it chooses an action and observes a feedback signal. Moreover, in each step, it can take a special action, called the $stop$ action, that ends the current episode. After the $stop$ action is taken, the learner collects a terminal reward, and observes the costs and terminal rewards associated with each step of the episode. The goal of the learner is to maximize its cumulative gain (i.e., the terminal reward minus costs) over all episodes by learning to choose the best sequence of actions based on the feedback. First, we define an {em oracle} benchmark, which sequentially selects the actions that maximize the expected immediate gain. Then, we propose our online learning algorithm, named {em FeedBack Adaptive Learning} (FeedBAL), and prove that its regret with respect to the benchmark is bounded with high probability and increases logarithmically in expectation. Moreover, the regret only has polynomial dependence on the number of steps, actions and states. eMAB can be used to model applications that involve humans in the loop, ranging from personalized medical screening to personalized web-based education, where sequences of actions are taken in each episode, and optimal behavior requires adapting the chosen actions based on the feedback.
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