We study the effect of impairment on stochastic multi-armed bandits and develop new ways to mitigate it. Impairment effect is the phenomena where an agent only accrues reward for an action if they have played it at least a few times in the recent past. It is practically motivated by repetition and recency effects in domains such as advertising (here consumer behavior may require repeat actions by advertisers) and vocational training (here actions are complex skills that can only be mastered with repetition to get a payoff). Impairment can be naturally modelled as a temporal constraint on the strategy space, and we provide two novel algorithms that achieve sublinear regret, each working with different assumptions on the impairment effect. We introduce a new notion called bucketing in our algorithm design, and show how it can effectively address impairment as well as a broader class of temporal constraints. Our regret bounds explicitly capture the cost of impairment and show that it scales (sub-)linearly with the degree of impairment. Our work complements recent work on modeling delays and corruptions, and we provide experimental evidence supporting our claims.
The notion of bucketing incentivises exploration and exploitation within each phase (see ). And this is an important attribute when the environment and specifications are changing. For instance, @cite_21 introduce the sleeping bandits problem, where the set of available arms changes with time and the algorithm needs to balance exploration and exploitation when new actions are made available. Similar models have also been considered in . In , the authors assume piece-wise stationarity of the rewards distributions (i.e., the reward distributions abruptly change at certain breakpoints in the time horizon).Under this model, they propose algorithms that estimate mean rewards of arms using only recent history (e.g., using a sliding-window or by discounting). However, none of these overlap with or subsume the impairment effect considered in this work.
@cite_21
2742049221 1761637522 2040774021 1587673378
The multi-armed bandit (MAB) problem features the classical tradeoff between exploration and exploitation. The input specifies several stochastic arms which evolve with each pull, and the goal is to maximize the expected reward after a fixed budget of pulls. The celebrated work of , surveyed in [8], presumes a condition on the arms called the martingale assumption. In [9], A. obtained an LP-based 1 48-approximation for the problem with the martingale assumption removed. We improve the algorithm to a 4 27-approximation, with simpler analysis. Our algorithm also generalizes to the case of MAB superprocesses with (stochastic) multi-period actions. This generalization captures the framework introduced by Guha and Munagala in [11], and yields new results for their budgeted learning problems. Also, we obtain a (1 2 -- e)-approximation for the variant of MAB where preemption (playing an arm, switching to another arm, then coming back to the first arm) is not allowed. This contains the stochastic knapsack problem of Dean, Goemans, and Vondrak in [6] with correlated rewards, where we are given a knapsack of fixed size, and a set of jobs each with a joint distribution for its size and reward. The actual size and reward of a job can only be discovered in real-time as it is being scheduled, and the objective is to maximize expected reward before the knapsack size is exhausted. Our (1 2 -- e)-approximation improves the 1 16 and 1 8 approximations in [9] for correlated stochastic knapsack with cancellation and no cancellation, respectively, providing to our knowledge the first tight algorithm for these problems that matches the integrality gap of 2. We sample probabilities from an exponential-sized dynamic programming solution, whose existence is guaranteed by an LP projection argument. We hope this technique can also be applied to other dynamic programming problems which can be projected down onto a small LP. We introduce the budget-limited multi-armed bandit (MAB), which captures situations where a learner's actions are costly and constrained by a fixed budget that is incommensurable with the rewards earned from the bandit machine, and then describe a first algorithm for solving it. Since the learner has a budget, the problem's duration is finite. Consequently an optimal exploitation policy is not to pull the optimal arm repeatedly, but to pull the combination of arms that maximises the agent's total reward within the budget. As such, the rewards for all arms must be estimated, because any of them may appear in the optimal combination. This difference from existing MABs means that new approaches to maximising the total reward are required. To this end, we propose an ∊-first algorithm, in which the first ∊ of the budget is used solely to learn the arms' rewards (exploration), while the remaining 1 - ∊ is used to maximise the received reward based on those estimates (exploitation). We derive bounds on the algorithm's loss for generic and uniform exploration methods, and compare its performance with traditional MAB algorithms under various distributions of rewards and costs, showing that it outperforms the others by up to 50 . The restless bandit problem is one of the most well-studied generalizations of the celebrated stochastic multi-armed bandit (MAB) problem in decision theory. In its ultimate generality, the restless bandit problem is known to be PSPACE-Hard to approximate to any nontrivial factor, and little progress has been made on this problem despite its significance in modeling activity allocation under uncertainty. In this article, we consider the Feedback MAB problem, where the reward obtained by playing each of n independent arms varies according to an underlying on off Markov process whose exact state is only revealed when the arm is played. The goal is to design a policy for playing the arms in order to maximize the infinite horizon time average expected reward. This problem is also an instance of a Partially Observable Markov Decision Process (POMDP), and is widely studied in wireless scheduling and unmanned aerial vehicle (UAV) routing. Unlike the stochastic MAB problem, the Feedback MAB problem does not admit to greedy index-based optimal policies. We develop a novel duality-based algorithmic technique that yields a surprisingly simple and intuitive (2+e)-approximate greedy policy to this problem. We show that both in terms of approximation factor and computational efficiency, our policy is closely related to the Whittle index, which is widely used for its simplicity and efficiency of computation. Subsequently we define a multi-state generalization, that we term Monotone bandits, which remains subclass of the restless bandit problem. We show that our policy remains a 2-approximation in this setting, and further, our technique is robust enough to incorporate various side-constraints such as blocking plays, switching costs, and even models where determining the state of an arm is a separate operation from playing it. Our technique is also of independent interest for other restless bandit problems, and we provide an example in nonpreemptive machine replenishment. Interestingly, in this case, our policy provides a constant factor guarantee, whereas the Whittle index is provably polynomially worse. By presenting the first O(1) approximations for nontrivial instances of restless bandits as well as of POMDPs, our work initiates the study of approximation algorithms in both these contexts. The multiarmed bandit is often used as an analogy for the tradeoff between exploration and exploitation in search problems. The classic problem involves allocating trials to the arms of a multiarmed slot machine to maximize the expected sum of rewards. We pose a new variation of the multiarmed bandit--the Max K-Armed Bandit--in which trials must be allocated among the arms to maximize the expected best single sample reward of the series of trials. Motivation for the Max K-Armed Bandit is the allocation of restarts among a set of multistart stochastic search algorithms. We present an analysis of this Max K-Armed Bandit showing under certain assumptions that the optimal strategy allocates trials to the observed best arm at a rate increasing double exponentially relative to the other arms. This motivates an exploration strategy that follows a Boltzmann distribution with an exponentially decaying temperature parameter. We compare this exploration policy to policies that allocate trials to the observed best arm at rates faster (and slower) than double exponentially. The results confirm, for two scheduling domains, that the double exponential increase in the rate of allocations to the observed best heuristic outperfonns the other approaches.