Stochastic Multi-armed Bandit Problem Research Articles

A Simple and Optimal Policy Design with Safety Against Heavy-Tailed Risk for Stochastic Bandits

We study the stochastic multi-armed bandit problem and design new policies that enjoy both optimal regret expectation and light-tailed risk for regret distribution. We first find that any policy that obtains the optimal instance-dependent expected regret could incur a heavy-tailed regret tail risk that decays slowly with T. We then focus on policies that achieve optimal worst-case expected regret. We design a novel policy that (i) enjoys the worst-case optimality for regret expectation and (ii) has the worst-case tail probability of incurring a regret larger than any regret threshold that decays exponentially with respect to T. The decaying rate is proved to be optimal for all worst-case optimal policies. Our proposed policy achieves a delicate balance between doing more exploration at the beginning of the time horizon and doing more exploitation when approaching the end, compared with standard confidence-bound-based policies. We also enhance the policy design to accommodate the “any-time” setting where T is unknown a priori, highlighting “lifelong exploration”, and prove equivalently desired policy performances as compared with the “fixed-time” setting with known T. From a managerial perspective, we show through numerical experiments that our new policy design yields similar efficiency and better safety compared to celebrated policies. Our policy design is preferable especially when (i) there is a risk of underestimating the volatility profile, or (ii) there is a challenge of tuning policy hyper-parameters. We conclude by extending our proposed policy design to the stochastic linear bandit setting that leads to both worst-case optimality in terms of regret expectation and light-tailed risk on regret distribution. This paper was accepted by J. George Shanthikumar, data science. Funding: The work of D. Simchi-Levi and F. Zhu is partially supported by the MIT Data Science Laboratory. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2022.03512 .

The Choice of Noninformative Priors for Thompson Sampling in Multiparameter Bandit Models

Thompson sampling (TS) has been known for its outstanding empirical performance supported by theoretical guarantees across various reward models in the classical stochastic multi-armed bandit problems. Nonetheless, its optimality is often restricted to specific priors due to the common observation that TS is fairly insensitive to the choice of the prior when it comes to asymptotic regret bounds. However, when the model contains multiple parameters, the optimality of TS highly depends on the choice of priors, which casts doubt on the generalizability of previous findings to other models. To address this gap, this study explores the impact of selecting noninformative priors, offering insights into the performance of TS when dealing with new models that lack theoretical understanding. We first extend the regret analysis of TS to the model of uniform distributions with unknown supports, which would be the simplest non-regular model. Our findings reveal that changing noninformative priors can significantly affect the expected regret, aligning with previously known results in other multiparameter bandit models. Although the uniform prior is shown to be optimal, we highlight the inherent limitation of its optimality, which is limited to specific parameterizations and emphasizes the significance of the invariance property of priors. In light of this limitation, we propose a slightly modified TS-based policy, called TS with Truncation (TS-T), which can achieve the asymptotic optimality for the Gaussian models and the uniform models by using the reference prior and the Jeffreys prior that are invariant under one-to-one reparameterizations. This policy provides an alternative approach to achieving optimality by employing fine-tuned truncation, which would be much easier than hunting for optimal priors in practice.

Maximal Objectives in the Multiarmed Bandit with Applications

In several applications of the stochastic multiarmed bandit problem, the traditional objective of maximizing the expected total reward can be inappropriate. In this paper, we study a new objective in the classic setup. Given K arms, instead of maximizing the expected total reward from T pulls (the traditional “sum” objective), we consider the vector of total rewards earned from each of the K arms at the end of T pulls and aim to maximize the expected highest total reward across arms (the “max” objective). For this objective, we show that any policy must incur an instance-dependent asymptotic regret of [Formula: see text] (with a higher instance-dependent constant compared with the traditional objective) and a worst case regret of [Formula: see text]. We then design an adaptive explore-then-commit policy featuring exploration based on appropriately tuned confidence bounds on the mean reward and an adaptive stopping criterion, which adapts to the problem difficulty and simultaneously achieves these bounds (up to logarithmic factors). We then generalize our algorithmic insights to the problem of maximizing the expected value of the average total reward of the top m arms with the highest total rewards. Our numerical experiments demonstrate the efficacy of our policies compared with several natural alternatives in practical parameter regimes. We discuss applications of these new objectives to the problem of conditioning an adequate supply of value-providing market entities (workers/sellers/service providers) in online platforms and marketplaces. This paper was accepted by Vivek Farias, data science. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2022.00801 .

Model-Based Thompson Sampling for Frequency and Rate Selection in Underwater Acoustic Communications

Due to the harsh propagation environment, limited bandwidth, and constrained battery life, transmission efficiency is a crucial issue for underwater acoustic (UWA) communications. This paper studies the link adaptation problem of a single UWA link by jointly selecting the transmission frequency and data rate. Since the current UWA channel lacks a universal model, we formulate this joint selection problem as a model-based stochastic multi-armed bandit (SMAB) problem. Thereafter, we propose three algorithms to solve this model-based SMAB problem under the settings of the stationary channel, non-stationary channel, and large arm (i.e., frequency and rate pair) space. For the stationary channel, we propose a unimodal objective-based Thompson sampling (UO-TS) algorithm by exploiting the unimodal feature of the objective function. For the non-stationary channel, we put forth a hybrid change detection UO-TS (HCD-UO-TS) algorithm based on the features of the unimodal objective function and non-stationary channel. For the large arm space, we propose an iterative boundary-shrinking TS (IBS-TS) algorithm by using the logistic regression-based arm classification model. These algorithms are all data-driven and have low complexity and a fast convergence rate. In addition, we derive an upper regret bound for the UO-TS algorithm. Numerical results show that the proposed algorithms outperform the state-of-the-art bandit algorithms and are not sensitive to the arm space.

Phase Transitions in Bandits with Switching Constraints

We consider the classic stochastic multiarmed bandit problem with a constraint that limits the total cost incurred by switching between actions to be no larger than a given switching budget. For this problem, we prove matching upper and lower bounds on the optimal (i.e., minimax) regret and provide efficient rate-optimal algorithms. Surprisingly, the optimal regret of this problem exhibits a nonconventional growth rate in terms of the time horizon and the number of arms. Consequently, we discover surprising “phase transitions” regarding how the optimal regret rate changes with respect to the switching budget: when the number of arms is fixed, there are equal-length phases, in which the optimal regret rate remains (almost) the same within each phase and exhibits abrupt changes between phases; when the number of arms grows with the time horizon, such abrupt changes become subtler and may disappear, but a generalized notion of phase transitions involving certain new measurements still exists. The results enable us to fully characterize the trade-off between the regret rate and the incurred switching cost in the stochastic multiarmed bandit problem, contributing new insights to this fundamental problem. Under the general switching cost structure, the results reveal interesting connections between bandit problems and graph traversal problems, such as the shortest Hamiltonian path problem. This paper was accepted by J. George Shanthikumar, data science. Supplemental Material: The online appendix and data are available at https://doi.org/10.1287/mnsc.2023.4755 .

Learning Equilibria in Matching Markets with Bandit Feedback

Large-scale, two-sided matching platforms must find market outcomes that align with user preferences while simultaneously learning these preferences from data. Classical notions of stability (Gale and Shapley, 1962; Shapley and Shubik, 1971) are, unfortunately, of limited value in the learning setting, given that preferences are inherently uncertain and destabilizing while they are being learned. To bridge this gap, we develop a framework and algorithms for learning stable market outcomes under uncertainty. Our primary setting is matching with transferable utilities, where the platform both matches agents and sets monetary transfers between them. We design an incentive-aware learning objective that captures the distance of a market outcome from equilibrium. Using this objective, we analyze the complexity of learning as a function of preference structure, casting learning as a stochastic multi-armed bandit problem. Algorithmically, we show that “optimism in the face of uncertainty,” the principle underlying many bandit algorithms, applies to a primal-dual formulation of matching with transfers and leads to near-optimal regret bounds. Our work takes a first step toward elucidating when and how stable matchings arise in large, data-driven marketplaces.

A General Framework for Bandit Problems Beyond Cumulative Objectives

The stochastic multiarmed bandit (MAB) problem is a common model for sequential decision problems. In the standard setup, a decision maker has to choose at every instant between several competing arms; each of them provides a scalar random variable, referred to as a “reward.” Nearly all research on this topic considers the total cumulative reward as the criterion of interest. This work focuses on other natural objectives that cannot be cast as a sum over rewards but rather, more involved functions of the reward stream. Unlike the case of cumulative criteria, in the problems we study here, the oracle policy, which knows the problem parameters a priori and is used to “center” the regret, is not trivial. We provide a systematic approach to such problems and derive general conditions under which the oracle policy is sufficiently tractable to facilitate the design of optimism-based (upper confidence bound) learning policies. These conditions elucidate an interesting interplay between the arm reward distributions and the performance metric. Our main findings are illustrated for several commonly used objectives, such as conditional value-at-risk, mean-variance trade-offs, Sharpe ratio, and more. Funding: This work was partially funded by the Israel Science Foundation [Contract 2199/20] and by the European Community’s Seventh Framework Programme FP7/2007–2013 [Grant 306638 (Scaling Up Reinforcement Learning: Structure Learning, Skill Acquisition, and Reward Shaping)].

Multiplayer Bandits Without Observing Collision Information

We study multiplayer stochastic multiarmed bandit problems in which the players cannot communicate, and if two or more players pull the same arm, a collision occurs and the involved players receive zero reward. We consider two feedback models: a model in which the players can observe whether a collision has occurred and a more difficult setup in which no collision information is available. We give the first theoretical guarantees for the second model: an algorithm with a logarithmic regret and an algorithm with a square-root regret that does not depend on the gaps between the means. For the first model, we give the first square-root regret bounds that do not depend on the gaps. Building on these ideas, we also give an algorithm for reaching approximate Nash equilibria quickly in stochastic anticoordination games.

Minimax Policy for Heavy-Tailed Bandits

We study the stochastic Multi-Armed Bandit (MAB) problem under worst-case regret and heavy-tailed reward distribution. We modify the minimax policy MOSS for the sub-Gaussian reward distribution by using saturated empirical mean to design a new algorithm called Robust MOSS. We show that if the moment of order $1+\epsilon $ for the reward distribution exists, then the refined strategy has a worst-case regret matching the lower bound while maintaining a distribution-dependent logarithm regret.

DUCT: An Upper Confidence Bound Approach to Distributed Constraint Optimization Problems

The Upper Confidence Bounds (UCB) algorithm is a well-known near-optimal strategy for the stochastic multi-armed bandit problem. Its extensions to trees, such as the Upper Confidence Tree (UCT) algorithm, have resulted in good solutions to the problem of Go. This paper introduces DUCT, a distributed algorithm inspired by UCT, for solving Distributed Constraint Optimization Problems (DCOP). Bounds on the solution quality are provided, and experiments show that, compared to existing DCOP approaches, DUCT is able to solve very large problems much more efficiently, or to find significantly higher quality solutions.

Robust risk-averse multi-armed bandits with application in social engagement behavior of children with autism spectrum disorder while imitating a humanoid robot

The stochastic multi-armed bandit problem is a standard model to solve the exploration–exploitation trade-off in sequential decision problems. In clinical trials, which are sensitive to outlier data, the goal is to learn a risk-averse policy to provide a trade-off between exploration, exploitation, and safety. In this paper, we present a risk-averse multi-armed bandit algorithm to solve a decision-making problem based on the social engagement behaviors of children with Autism Spectrum Disorder (ASD). The algorithm is carried out when children interact with a humanoid robot and imitate a sequence of the robot's movements. The proposed algorithm is based on the Best Empirical Sampled Average algorithm under Entropic Value-at-Risk as a risk measure to decide on the best sequence of movements that can improve the social engagement behaviors of the children with ASD while imitating the robot's movements. We provide a detailed experimental analysis to compare the performance of our proposed algorithm to some well-known risk-averse multi-armed bandit algorithms on some artificial scenarios and our real-world problem. The experimental results report that the proposed algorithm outperforms its competitors in terms of robustness, risk avoidance, and cumulative regret, promoting the social engagement behaviors of children with ASD when imitating a robot's movements.

An Optimal Algorithm for the Stochastic Bandits While Knowing the Near-Optimal Mean Reward.

This brief studies a variation of the stochastic multiarmed bandit (MAB) problems, where the agent knows the a priori knowledge named the near-optimal mean reward (NoMR). In common MAB problems, an agent tries to find the optimal arm without knowing the optimal mean reward. However, in more practical applications, the agent can usually get an estimation of the optimal mean reward defined as NoMR. For instance, in an online Web advertising system based on MAB methods, a user's near-optimal average click rate (NoMR) can be roughly estimated from his/her demographic characteristics. As a result, application of the NoMR is efficient at improving the algorithm's performance. First, we formalize the stochastic MAB problem by knowing the NoMR that is in between the suboptimal mean reward and the optimal mean reward. Second, we use the cumulative regret as the performance metric for our problem, and we get that this problem's lower bound of the cumulative regret is Ω(1/∆) , where ∆ is the difference between the suboptimal mean reward and the optimal mean reward. Compared with the conventional MAB problem with the increasing logarithmic lower bound of the regret, our regret lower bound is uniform with the learning step. Third, a novel algorithm, NoMR-BANDIT, is set forth to solve this problem. In NoMR-BANDIT, the NoMR is used to design an efficient exploration strategy. In addition, we analyzed the regret's upper bound in NoMR-BANDIT and concluded that it also has a uniform upper bound of O(1/∆) , which is in the same order as the lower bound. Consequently, NoMR-BANDIT is an optimal algorithm of this problem. To enhance our method's generalization, CASCADE-BANDIT based on NoMR-BANDIT is proposed to solve the problem, where NoMR is less than the suboptimal mean reward. CASCADE-BANDIT has an upper bound of O(∆logn) , where n represents the learning step, and the order of O(∆logn) is the same with that of the conventional MAB methods. Finally, extensive experimental results demonstrated that the established NoMR-BANDIT is more efficient than the compared bandit solutions. After sufficient iterations, NOMR-BANDIT saved 10%-80% more cumulative regret than the state of the art.

Non-stationary Stochastic Multi-armed Bandit Problems with External Information on Stationarity

Multi-armed bandit problem is a fundamental mathematical problem in sequential optimization and reinforcementlearning that has a variety of application such as online recommendation system and clinical trial design. Multiarmedbandit problem can describe a situation in which a player tries to select a good choice sequentially from givencandidate choices to maximize the cumulative reward. In this paper, we consider the non-stationary multi-armed banditproblems. Non-stationary means the reward distribution of each arm varies with time. We point out that in somereal application, we can utilize information on the change of reward distribution. Especially we consider the type ofinformation that may restrict the rounds at which the reward distribution changes. Against such scenario, we proposea novel strategy called PM policy. The proposed policy is based on existing CUSUM-UCB policy and M-UCB policythat do not consider external information. Though such existing policies monitor all arms to detect the change ofreward distribution, our policy monitors only important arms and rounds. As a result, the ratio of unnecessary monitoringis reduced, and an efficient search can be performed. The regret bound of the proposed policy is described. Wealso show the effectiveness of the proposed method by numerical experiments.

Learning Algorithms for Minimizing Queue Length Regret

We consider a system consisting of a single transmitter/receiver pair and N channels over which they may communicate. Packets randomly arrive to the transmitter’s queue and wait to be successfully sent to the receiver. The transmitter may attempt a frame transmission on one channel at a time, where each frame includes a packet if one is in the queue. For each channel, an attempted transmission is successful with an unknown probability. The transmitter’s objective is to quickly identify the best channel to minimize the number of packets in the queue over T time slots. To analyze system performance, we introduce queue length regret, which is the expected difference between the total queue length of a learning policy and a controller that knows the rates, a priori. One approach to designing a transmission policy would be to apply algorithms from the literature that solve the closely-related stochastic multi-armed bandit problem. These policies would focus on maximizing the number of successful frame transmissions over time. However, we show that these methods have $\Omega (\log {{T}})$ queue length regret. On the other hand, we show that there exists a set of queue-length based policies that can obtain order optimal ${O}(1)$ queue length regret. We use our theoretical analysis to devise heuristic methods that are shown to perform well in simulation.

Waiting But Not Aging: Optimizing Information Freshness Under the Pull Model

The Age-of-Information is an important metric for investigating the timeliness performance in information-update systems. In this paper, we study the AoI minimization problem under a new Pull model with replication schemes, where a user proactively sends a replicated request to multiple servers to “pull” the information of interest. Interestingly, we find that under this new Pull model, replication schemes capture a novel tradeoff between different values of the AoI across the servers (due to the random updating processes) and different response times across the servers, which can be exploited to minimize the expected AoI at the user's side. Specifically, assuming Poisson updating process for the servers and exponentially distributed response time, we derive a closed-form formula for computing the expected AoI and obtain the optimal number of responses to wait for to minimize the expected AoI. Then, we extend our analysis to the setting where the user aims to maximize the AoI-based utility, which represents the user's satisfaction level with respect to freshness of the received information. Furthermore, we consider a more realistic scenario where the user has no prior knowledge of the system. In this case, we reformulate the utility maximization problem as a stochastic Multi-Armed Bandit problem with side observations and leverage a special linear structure of side observations to design learning algorithms with improved performance guarantees. Finally, we conduct extensive simulations to elucidate our theoretical results and compare the performance of different algorithms. Our findings reveal that under the Pull model, waiting does not necessarily lead to aging; waiting for more than one response can often significantly reduce the AoI and improve the AoI-based utility in most scenarios.

Multi-Armed Bandits on Partially Revealed Unit Interval Graphs

A stochastic multi-armed bandit problem with side information on the similarity and dissimilarity across different arms is considered. The action space of the problem can be represented by a unit interval graph (UIG) where each node represents an arm and the presence (absence) of an edge between two nodes indicates similarity (dissimilarity) between their mean rewards. Two settings of complete and partial side information based on whether the UIG is fully revealed are studied and a general two-step learning structure consisting of an offline reduction of the action space and online aggregation of reward observations from similar arms is proposed to fully exploit the topological structure of the side information. In both cases, the computation efficiency and the order optimality of the proposed learning policies in terms of both the size of the action space and the time length are established.

Achieving Fairness in the Stochastic Multi-Armed Bandit Problem

We study an interesting variant of the stochastic multi-armed bandit problem, which we call the Fair-MAB problem, where, in addition to the objective of maximizing the sum of expected rewards, the algorithm also needs to ensure that at any time, each arm is pulled at least a pre-specified fraction of times. We investigate the interplay between learning and fairness in terms of a pre-specified vector denoting the fractions of guaranteed pulls. We define a fairness-aware regret, which we call r-Regret, that takes into account the above fairness constraints and extends the conventional notion of regret in a natural way. Our primary contribution is to obtain a complete characterization of a class of Fair-MAB algorithms via two parameters: the unfairness tolerance and the learning algorithm used as a black-box. For this class of algorithms, we provide a fairness guarantee that holds uniformly over time, irrespective of the choice of the learning algorithm. Further, when the learning algorithm is UCB1, we show that our algorithm achieves constant r-Regret for a large enough time horizon. Finally, we analyze the cost of fairness in terms of the conventional notion of regret. We conclude by experimentally validating our theoretical results.

Observe Before Play: Multi-Armed Bandit with Pre-Observations

We consider the stochastic multi-armed bandit (MAB) problem in a setting where a player can pay to pre-observe arm rewards before playing an arm in each round. Apart from the usual trade-off between exploring new arms to find the best one and exploiting the arm believed to offer the highest reward, we encounter an additional dilemma: pre-observing more arms gives a higher chance to play the best one, but incurs a larger cost. For the single-player setting, we design an Observe-Before-Play Upper Confidence Bound (OBP-UCB) algorithm for K arms with Bernoulli rewards, and prove a T-round regret upper bound O(K2log T). In the multi-player setting, collisions will occur when players select the same arm to play in the same round. We design a centralized algorithm, C-MP-OBP, and prove its T-round regret relative to an offline greedy strategy is upper bounded in O(K4/M2log T) for K arms and M players. We also propose distributed versions of the C-MP-OBP policy, called D-MP-OBP and D-MP-Adapt-OBP, achieving logarithmic regret with respect to collision-free target policies. Experiments on synthetic data and wireless channel traces show that C-MP-OBP and D-MP-OBP outperform random heuristics and offline optimal policies that do not allow pre-observations.

Fast mmwave Beam Alignment via Correlated Bandit Learning

Beam alignment (BA) is to ensure the transmitter and receiver beams are accurately aligned to establish a reliable communication link in millimeter-wave (mmwave) systems. Existing BA methods search the entire beam space to identify the optimal transmit-receive beam pair, which incurs significant BA latency on the order of seconds in the worst case. In this paper, we develop a learning algorithm to reduce BA latency, namely Hierarchical Beam Alignment (HBA) algorithm. We first formulate the BA problem as a stochastic multi-armed bandit problem with the objective to maximize the cumulative received signal strength within a certain period. The proposed algorithm takes advantage of the correlation structure among beams such that the information from nearby beams is extracted to identify the optimal beam, instead of searching the entire beam space. Furthermore, the prior knowledge on the channel fluctuation is incorporated in the proposed algorithm to further accelerate the BA process. Theoretical analysis indicates that the proposed algorithm is asymptotically optimal. Extensive simulation results demonstrate that the proposed algorithm can identify the optimal beam with a high probability and reduce the BA latency from hundreds of milliseconds to a few milliseconds in the multipath channel, as compared to the existing BA method in IEEE 802.11ad.

Phase Transitions and Cyclic Phenomena in Bandits with Switching Constraints

We consider the classical stochastic multi-armed bandit problem with a constraint that limits the total cost incurred by switching between actions to be no larger than a given switching budget. For this problem, we prove matching upper and lower bounds on the optimal (i.e., minimax) regret, and provide efficient rate-optimal algorithms. Surprisingly, the optimal regret of this problem exhibits a non-conventional growth rate in terms of the time horizon and the number of arms. Consequently, we discover surprising regarding how the optimal regret rate changes with respect to the switching budget: when the number of arms is fixed, there are equal-length phases, where the optimal regret rate remains (almost) the same within each phase and exhibits abrupt changes between phases; when the number of arms grows with the time horizon, such abrupt changes become subtler and may disappear, but a generalized notion of phase transitions involving certain new measurements still exists. The results enable us to fully characterize the trade-off between the regret rate and the incurred switching cost in the stochastic multi-armed bandit problem, contributing new insights to this fundamental problem. Under the general switching cost structure, the results reveal interesting connections between bandit problems and graph traversal problems, such as the shortest Hamiltonian path problem.