Finite-time Analysis of the Multiarmed Bandit Problem

ArXiv, 2011

In the classic Bayesian restless multi-armed bandit (RMAB) problem, there are $N$ arms, with rewards on all arms evolving at each time as Markov chains with known parameters. A player seeks to activate $K \geq 1$ arms at each time in order to maximize the expected total reward obtained over multiple plays. RMAB is a challenging problem that is known to be PSPACE-hard in general. We consider in this work the even harder non-Bayesian RMAB, in which the parameters of the Markov chain are assumed to be unknown \emph{a priori}. We develop an original approach to this problem that is applicable when the corresponding Bayesian problem has the structure that, depending on the known parameter values, the optimal solution is one of a prescribed finite set of policies. In such settings, we propose to learn the optimal policy for the non-Bayesian RMAB by employing a suitable meta-policy which treats each policy from this finite set as an arm in a different non-Bayesian multi-armed bandit proble...

We consider a stochastic bandit problem with infinitely many arms. In this setting, the learner has no chance of trying all the arms even once and has to dedicate its limited number of samples only to a certain number of arms. All previous algorithms for this setting were designed for minimizing the cumulative regret of the learner. In this paper, we propose an algorithm aiming at minimizing the simple regret. As in the cumulative regret setting of infinitely many armed bandits, the rate of the simple regret will depend on a parameter \$\backslash beta\$ characterizing the distribution of the near-optimal arms. We prove that depending on \$\backslash beta\$, our algorithm is minimax optimal either up to a multiplicative constant or up to a \$\backslash log(n)\$ factor. We also provide extensions to several important cases: when \$\backslash beta\$ is unknown, in a natural setting where the near-optimal arms have a small variance, and in the case of unknown time horizon.

2011

Abstract In the classic Bayesian restless multi-armed bandit (RMAB) problem, there are N arms, with rewards on all arms evolving at each time as Markov chains with known parameters. A player seeks to activate K��� 1 arms at each time in order to maximize the expected total reward obtained over multiple plays. RMAB is a challenging problem that is known to be PSPACE-hard in general.

We develop asymptotically optimal policies for the multi armed bandit (MAB), problem, under a cost constraint. This model is applicable in situations where each sample (or activation) from a population (bandit) incurs a known bandit dependent cost. Successive samples from each population are iid random variables with unknown distribution. The objective is to have a feasible policy for deciding from which population to sample from, so as to maximize the expected sum of outcomes of n total samples or equivalently to minimize the regret due to lack on information of sample distributions, For this problem we consider the class of feasible uniformly fast (f-UF) convergent policies, that satisfy sample path wise the cost constraint. We first establish a necessary asymptotic lower bound for the rate of increase of the regret function of f-UF policies. Then we construct a class of f-UF policies and provide conditions under which they are asymptotically optimal within the class of f-UF policies, achieving this asymptotic lower bound. At the end we provide the explicit form of such policies for the case in which the unknown distributions are Normal with unknown means and known variances.

2012

Online learning algorithms are designed to learn even when their input is generated by an adversary. The widely-accepted formal definition of an online algorithm's ability to learn is the game-theoretic notion of regret. We argue that the standard definition of regret becomes inadequate if the adversary is allowed to adapt to the online algorithm's actions. We define the alternative notion of policy regret, which attempts to provide a more meaningful way to measure an online algorithm's performance against adaptive adversaries. Focusing on the online bandit setting, we show that no bandit algorithm can guarantee a sublinear policy regret against an adaptive adversary with unbounded memory. On the other hand, if the adversary's memory is bounded, we present a general technique that converts any bandit algorithm with a sublinear regret bound into an algorithm with a sublinear policy regret bound. We extend this result to other variants of regret, such as switching regret, internal regret, and swap regret.

2020

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 learn...

2007

We provide a framework to exploit dependencies among arms in multi-armed bandit problems, when the dependencies are in the form of a generative model on clusters of arms. We find an optimal MDP-based policy for the discounted reward case, and also give an approximation of it with formal error guarantee. We discuss lower bounds on regret in the undiscounted reward scenario, and propose a general two-level bandit policy for it. We propose three different instantiations of our general policy and provide theoretical justifications of how the regret of the instantiated policies depend on the characteristics of the clusters. Finally, we empirically demonstrate the efficacy of our policies on large-scale realworld and synthetic data, and show that they significantly outperform classical policies designed for bandits with independent arms.

2019

In this paper, we study multi-armed bandit problems in an explore-then-commit setting. In our proposed explore-then-commit setting, the goal is to identify the best arm after a pure experimentation (exploration) phase and exploit it once or for a given finite number of times. We identify that although the arm with the highest expected reward is the most desirable objective for infinite exploitations, it is not necessarily the one that is most probable to have the highest reward in a single or finite-time exploitations. Alternatively, we advocate the idea of risk-aversion where the objective is to compete against the arm with the best risk-return trade-off. We propose two algorithms whose objectives are to select the arm that is most probable to reward the most. Using a new notion of finite-time exploitation regret, we find an upper bound of order ln 1 for the minimum number of experiments before commitment, to guarantee upper bound for regret. As compared to existing risk-averse bandit algorithms, our algorithms do not rely on hyper-parameters, resulting in a more robust behavior, which is verified by numerical evaluations.

IEEE Journal of Selected Topics in Signal Processing, 2000

In the Multi-Armed Bandit (MAB) problem, there are a given set of arms with unknown reward distributions. At each time, a player selects one arm to play, aiming to maximize the total expected reward over a horizon of length T . The essence of the problem is the tradeoff between exploration and exploitation: playing a less explored arm to learn its reward statistics for future benefit or playing the arm with the largest sample mean at the current time. An approach based on a Deterministic Sequencing of Exploration and Exploitation (DSEE) is developed for constructing sequential arm selection policies.

IEEE Transactions on Information Theory, 2013

We consider the restless multi-armed bandit (RMAB) problem with unknown dynamics in which a player chooses M out of N arms to play at each time. The reward state of each arm transits according to an unknown Markovian rule when it is played and evolves according to an arbitrary unknown random process when it is passive. The performance of an arm selection policy is measured by regret, defined as the reward loss with respect to the case where the player knows which M arms are the most rewarding and always plays the M best arms. We construct a policy with an interleaving exploration and exploitation epoch structure that achieves a regret with logarithmic order when arbitrary (but nontrivial) bounds on certain system parameters are known. When no knowledge about the system is available, we show that the proposed policy achieves a regret arbitrarily close to the logarithmic order. We further extend the problem to a decentralized setting where multiple distributed players share the arms without information exchange. Under both an exogenous restless model and an endogenous restless model, we show that a decentralized extension of the proposed policy preserves the logarithmic regret order as in the centralized setting. The results apply to adaptive learning in various dynamic systems and communication networks, as well as financial investment.