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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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We propose an online algorithm for cumulative regret minimization in a stochastic multi-armed bandit. The algorithm adds $O(t)$ i.i.d. pseudo-rewards to its history in round $t$ and then pulls the arm with the highest average reward in its perturbed history. Therefore, we call it perturbed-history exploration (PHE). The pseudo-rewards are carefully designed to offset potentially underestimated mean rewards of arms with a high probability. We derive near-optimal gap-dependent and gap-free bounds on the $n$-round regret of PHE. The key step in our analysis is a novel argument that shows that randomized Bernoulli rewards lead to optimism. Finally, we empirically evaluate PHE and show that it is competitive with state-of-the-art baselines.
We consider a resource-aware variant of the classical multi-armed bandit problem: In each round, the learner selects an arm and determines a resource limit. It then observes a corresponding (random) reward, provided the (random) amount of consumed resources remains below the limit. Otherwise, the observation is censored, i.e., no reward is obtained. For this problem setting, we introduce a measure of regret, which incorporates the actual amount of allocated resources of each learning round as well as the optimality of realizable rewards. Thus, to minimize regret, the learner needs to set a resource limit and choose an arm in such a way that the chance to realize a high reward within the predefined resource limit is high, while the resource limit itself should be kept as low as possible. We derive the theoretical lower bound on the cumulative regret and propose a learning algorithm having a regret upper bound that matches the lower bound. In a simulation study, we show that our learning algorithm outperforms straightforward extensions of standard multi-armed bandit algorithms.
We study incentivized exploration for the multi-armed bandit (MAB) problem where the players receive compensation for exploring arms other than the greedy choice and may provide biased feedback on reward. We seek to understand the impact of this drifted reward feedback by analyzing the performance of three instantiations of the incentivized MAB algorithm: UCB, $varepsilon$-Greedy, and Thompson Sampling. Our results show that they all achieve $mathcal{O}(log T)$ regret and compensation under the drifted reward, and are therefore effective in incentivizing exploration. Numerical examples are provided to complement the theoretical analysis.
We propose a bandit algorithm that explores by randomizing its history of rewards. Specifically, it pulls the arm with the highest mean reward in a non-parametric bootstrap sample of its history with pseudo rewards. We design the pseudo rewards such that the bootstrap mean is optimistic with a sufficiently high probability. We call our algorithm Giro, which stands for garbage in, reward out. We analyze Giro in a Bernoulli bandit and derive a $O(K Delta^{-1} log n)$ bound on its $n$-round regret, where $Delta$ is the difference in the expected rewards of the optimal and the best suboptimal arms, and $K$ is the number of arms. The main advantage of our exploration design is that it easily generalizes to structured problems. To show this, we propose contextual Giro with an arbitrary reward generalization model. We evaluate Giro and its contextual variant on multiple synthetic and real-world problems, and observe that it performs well.
We consider the stochastic bandit problem with a continuous set of arms, with the expected reward function over the arms assumed to be fixed but unknown. We provide two new Gaussian process-based algorithms for continuous bandit optimization-Improved GP-UCB (IGP-UCB) and GP-Thomson sampling (GP-TS), and derive corresponding regret bounds. Specifically, the bounds hold when the expected reward function belongs to the reproducing kernel Hilbert space (RKHS) that naturally corresponds to a Gaussian process kernel used as input by the algorithms. Along the way, we derive a new self-normalized concentration inequality for vector- valued martingales of arbitrary, possibly infinite, dimension. Finally, experimental evaluation and comparisons to existing algorithms on synthetic and real-world environments are carried out that highlight the favorable gains of the proposed strategies in many cases.
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