We introduce the E$^4$ algorithm for the batched linear bandit problem, incorporating an Explore-Estimate-Eliminate-Exploit framework. With a proper choice of exploration rate, we prove E$^4$ achieves the finite-time minimax optimal regret with only $O(\log\log T)$ batches, and the asymptotically optimal regret with only $3$ batches as $T\rightarrow\infty$, where $T$ is the time horizon. We further prove a lower bound on the batch complexity of linear contextual bandits showing that any asymptotically optimal algorithm must require at least $3$ batches in expectation as $T\rightarrow\infty$, which indicates E$^4$ achieves the asymptotic optimality in regret and batch complexity simultaneously. To the best of our knowledge, E$^4$ is the first algorithm for linear bandits that simultaneously achieves the minimax and asymptotic optimality in regret with the corresponding optimal batch complexities. In addition, we show that with another choice of exploration rate E$^4$ achieves an instance-dependent regret bound requiring at most $O(\log T)$ batches, and maintains the minimax optimality and asymptotic optimality. We conduct thorough experiments to evaluate our algorithm on randomly generated instances and the challenging \textit{End of Optimism} instances \citep{lattimore2017end} which were shown to be hard to learn for optimism based algorithms. Empirical results show that E$^4$ consistently outperforms baseline algorithms with respect to regret minimization, batch complexity, and computational efficiency.
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Smooth Contextual Bandits: Bridging the Parametric and Nondifferentiable Regret Regimes
We study a nonparametric contextual bandit problem in which the expected reward functions belong to a Hölder class with smoothness parameter β. We show how this interpolates between two extremes that were previously studied in isolation: nondifferentiable bandits (β at most 1), with which rate-optimal regret is achieved by running separate noncontextual bandits in different context regions, and parametric-response bandits (infinite [Formula: see text]), with which rate-optimal regret can be achieved with minimal or no exploration because of infinite extrapolatability. We develop a novel algorithm that carefully adjusts to all smoothness settings, and we prove its regret is rate-optimal by establishing matching upper and lower bounds, recovering the existing results at the two extremes. In this sense, our work bridges the gap between the existing literature on parametric and nondifferentiable contextual bandit problems and between bandit algorithms that exclusively use global or local information, shedding light on the crucial interplay of complexity and regret in contextual bandits.
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- Award ID(s):
- 1846210
- PAR ID:
- 10320785
- Date Published:
- Journal Name:
- Operations Research
- ISSN:
- 0030-364X
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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In this paper, we propose and study opportunistic contextual bandits - a special case of contextual bandits where the exploration cost varies under different environmental conditions, such as network load or return variation in recommendations. When the exploration cost is low, so is the actual regret of pulling a sub-optimal arm (e.g., trying a suboptimal recommendation). Therefore, intuitively, we could explore more when the exploration cost is relatively low and exploit more when the exploration cost is relatively high. Inspired by this intuition, for opportunistic contextual bandits with Linear payoffs, we propose an Adaptive Upper-Confidence-Bound algorithm (AdaLinUCB) to adaptively balance the exploration-exploitation trade-off for opportunistic learning. We prove that AdaLinUCB achieves O((log T)^2) problem-dependent regret upper bound, which has a smaller coefficient than that of the traditional LinUCB algorithm. Moreover, based on both synthetic and real-world dataset, we show that AdaLinUCB significantly outperforms other contextual bandit algorithms, under large exploration cost fluctuations.
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Non-stationary bandits and clustered bandits lift the restrictive assumptions in contextual bandits and provide solutions to many important real-world scenarios. Though they have been studied independently so far, we point out the essence in solving these two problems overlaps considerably. In this work, we connect these two strands of bandit research under the notion of test of homogeneity, which seamlessly addresses change detection for non-stationary bandit and cluster identification for clustered bandit in a unified solution framework. Rigorous regret analysis and extensive empirical evaluations demonstrate the value of our proposed solution, especially its flexibility in handling various environment assumptions, e.g., a clustered non-stationary environment.more » « less
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null (Ed.)Non-stationary bandits and clustered bandits lift the restrictive assumptions in contextual bandits and provide solutions to many important real-world scenarios. Though they have been studied independently so far, we point out the essence in solving these two problems overlaps considerably. In this work, we connect these two strands of bandit research under the notion of test of homogeneity, which seamlessly addresses change detection for non-stationary bandit and cluster identification for clustered bandit in a unified solution framework. Rigorous regret analysis and extensive empirical evaluations demonstrate the value of our proposed solution, especially its flexibility in handling various environment assumptions, e.g., a clustered non-stationary environment.more » « less
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Data that is gathered adaptively --- via bandit algorithms, for example --- exhibits bias. This is true both when gathering simple numeric valued data --- the empirical means kept track of by stochastic bandit algorithms are biased downwards --- and when gathering more complicated data --- running hypothesis tests on complex data gathered via contextual bandit algorithms leads to false discovery. In this paper, we show that this problem is mitigated if the data collection procedure is differentially private. This lets us both bound the bias of simple numeric valued quantities (like the empirical means of stochastic bandit algorithms), and correct the p-values of hypothesis tests run on the adaptively gathered data. Moreover, there exist differentially private bandit algorithms with near optimal regret bounds: we apply existing theorems in the simple stochastic case, and give a new analysis for linear contextual bandits. We complement our theoretical results with experiments validating our theory.more » « less
- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1287/opre.2021.2237
