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During online decision making in Multi-Armed Bandits (MAB), one needs to conduct inference on the true mean reward of each arm based on data collected so far at each step. However, since the arms are adaptively selected--thereby yielding non-iid data--conducting inference accurately is not straightforward. In particular, sample averaging, which is used in the family of UCB and Thompson sampling (TS) algorithms, does not provide a good choice as it suffers from bias and a lack of good statistical properties (e.g. asymptotic normality). Our thesis in this paper is that more sophisticated inference schemes that take into account the adaptive nature of the sequentially collected data can unlock further performance gains, even though both UCB and TS type algorithms are optimal in the worst case. In particular, we propose a variant of TS-style algorithms--which we call doubly adaptive TS--that leverages recent advances in causal inference and adaptively reweights the terms of a doubly robust estimator on the true mean reward of each arm. Through 20 synthetic domain experiments and a semi-synthetic experiment based on data from an A/B test of a web service, we demonstrate that using an adaptive inferential scheme (while still retaining the exploration efficacy of TS) provides clear benefits in online decision making: the proposed DATS algorithm has superior empirical performance to existing baselines (UCB and TS) in terms of regret and sample complexity in identifying the best arm. In addition, we also provide a finite-time regret bound of doubly adaptive TS that matches (up to log factors) those of UCB and TS algorithms, thereby establishing that its improved practical benefits do not come at the expense of worst-case suboptimality.
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We introduce a framework for decentralized online learning for multi-armed bandits (MAB) with multiple cooperative players. The reward obtained by the players in each round depends on the actions taken by all the players. Its a team setting, and the objective is common. Information asymmetry is what makes the problem interesting and challenging. We consider three types of information asymmetry: action information asymmetry when the actions of the players cant be observed but the rewards received are common; reward information asymmetry when the actions of the other players are observable but rewards received are IID from the same distribution; and when we have both action and reward information asymmetry. For the first setting, we propose a UCB-inspired algorithm that achieves $O(log T)$ regret whether the rewards are IID or Markovian. For the second section, we offer an environment such that the algorithm given for the first setting gives linear regret. For the third setting, we show that a variation of the `explore then commit algorithm achieves almost log regret.
This paper studies a new variant of the stochastic multi-armed bandits problem, where the learner has access to auxiliary information about the arms. The auxiliary information is correlated with the arm rewards, which we treat as control variates. In many applications, the arm rewards are a function of some exogenous values, whose mean value is known a priori from historical data and hence can be used as control variates. We use the control variates to obtain mean estimates with smaller variance and tighter confidence bounds. We then develop an algorithm named UCB-CV that uses improved estimates. We characterize the regret bounds in terms of the correlation between the rewards and control variates. The experiments on synthetic data validate the performance guarantees of our proposed algorithm.
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.
We propose a generalization of the best arm identification problem in stochastic multi-armed bandits (MAB) to the setting where every pull of an arm is associated with delayed feedback. The delay in feedback increases the effective sample complexity of standard algorithms, but can be offset if we have access to partial feedback received before a pull is completed. We propose a general framework to model the relationship between partial and delayed feedback, and as a special case we introduce efficient algorithms for settings where the partial feedback are biased or unbiased estimators of the delayed feedback. Additionally, we propose a novel extension of the algorithms to the parallel MAB setting where an agent can control a batch of arms. Our experiments in real-world settings, involving policy search and hyperparameter optimization in computational sustainability domains for fast charging of batteries and wildlife corridor construction, demonstrate that exploiting the structure of partial feedback can lead to significant improvements over baselines in both sequential and parallel MAB.
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