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Register a new userFurther Optimal Regret Bounds for Thompson Sampling
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Shipra Agrawal
Publication date
2012
fields
Informatics Engineering
and research's language is
English
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Thompson Sampling is one of the oldest heuristics for multi-armed bandit problems. It is a randomized algorithm based on Bayesian ideas, and has recently generated significant interest after several studies demonstrated it to have better empirical performance compared to the state of the art methods. In this paper, we provide a novel regret analysis for Thompson Sampling that simultaneously proves both the optimal problem-dependent bound of $(1+epsilon)sum_i frac{ln T}{Delta_i}+O(frac{N}{epsilon^2})$ and the first near-optimal problem-independent bound of $O(sqrt{NTln T})$ on the expected regret of this algorithm. Our near-optimal problem-independent bound solves a COLT 2012 open problem of Chapelle and Li. The optimal problem-dependent regret bound for this problem was first proven recently by Kaufmann et al. [ALT 2012]. Our novel martingale-based analysis techniques are conceptually simple, easily extend to distributions other than the Beta distribution, and also extend to the more general contextual bandits setting [Manuscript, Agrawal and Goyal, 2012].
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Thompson Sampling is one of the oldest heuristics for multi-armed bandit problems. It is a randomized algorithm based on Bayesian ideas, and has recently generated significant interest after several studies demonstrated it to have better empirical performance compared to the state-of-the-art methods. However, many questions regarding its theoretical performance remained open. In this paper, we design and analyze a generalization of Thompson Sampling algorithm for the stochastic contextual multi-armed bandit problem with linear payoff functions, when the contexts are provided by an adaptive adversary. This is among the most important and widely studi
In this paper, we consider the worst-case regret of Linear Thompson Sampling (LinTS) for the linear bandit problem. citet{russo2014learning} show that the Bayesian regret of LinTS is bounded above by $widetilde{mathcal{O}}(dsqrt{T})$ where $T$ is the time horizon and $d$ is the number of parameters. While this bound matches the minimax lower-bounds for this problem up to logarithmic factors, the existence of a similar worst-case regret bound is still unknown. The only known worst-case regret bound for LinTS, due to cite{agrawal2013thompson,abeille2017linear}, is $widetilde{mathcal{O}}(dsqrt{dT})$ which requires the posterior variance to be inflated by a factor of $widetilde{mathcal{O}}(sqrt{d})$. While this bound is far from the minimax optimal rate by a factor of $sqrt{d}$, in this paper we show that it is the best possible one can get, settling an open problem stated in cite{russo2018tutorial}. Specifically, we construct examples to show that, without the inflation, LinTS can incur linear regret up to time $exp(Omega(d))$. We then demonstrate that, under mild conditions, a slightly modified version of LinTS requires only an $widetilde{mathcal{O}}(1)$ inflation where the constant depends on the diversity of the optimal arm.
We study the combinatorial pure exploration problem Best-Set in stochastic multi-armed bandits. In a Best-Set instance, we are given $n$ arms with unknown reward distributions, as well as a family $mathcal{F}$ of feasible subsets over the arms. Our goal is to identify the feasible subset in $mathcal{F}$ with the maximum total mean using as few samples as possible. The problem generalizes the classical best arm identification problem and the top-$k$ arm identification problem, both of which have attracted significant attention in recent years. We provide a novel instance-wise lower bound for the sample complexity of the problem, as well as a nontrivial sampling algorithm, matching the lower bound up to a factor of $ln|mathcal{F}|$. For an important class of combinatorial families, we also provide polynomial time implementation of the sampling algorithm, using the equivalence of separation and optimization for convex program, and approximate Pareto curves in multi-objective optimization. We also show that the $ln|mathcal{F}|$ factor is inevitable in general through a nontrivial lower bound construction. Our results significantly improve several previous results for several important combinatorial constraints, and provide a tighter understanding of the general Best-Set problem. We further introduce an even more general problem, formulated in geometric terms. We are given $n$ Gaussian arms with unknown means and unit variance. Consider the $n$-dimensional Euclidean space $mathbb{R}^n$, and a collection $mathcal{O}$ of disjoint subsets. Our goal is to determine the subset in $mathcal{O}$ that contains the $n$-dimensional vector of the means. The problem generalizes most pure exploration bandit problems studied in the literature. We provide the first nearly optimal sample complexity upper and lower bounds for the problem.
In the classical best arm identification (Best-$1$-Arm) problem, we are given $n$ stochastic bandit arms, each associated with a reward distribution with an unknown mean. We would like to identify the arm with the largest mean with probability at least $1-delta$, using as few samples as possible. Understanding the sample complexity of Best-$1$-Arm has attracted significant attention since the last decade. However, the exact sample complexity of the problem is still unknown. Recently, Chen and Li made the gap-entropy conjecture concerning the instance sample complexity of Best-$1$-Arm. Given an instance $I$, let $mu_{[i]}$ be the $i$th largest mean and $Delta_{[i]}=mu_{[1]}-mu_{[i]}$ be the corresponding gap. $H(I)=sum_{i=2}^nDelta_{[i]}^{-2}$ is the complexity of the instance. The gap-entropy conjecture states that $Omegaleft(H(I)cdotleft(lndelta^{-1}+mathsf{Ent}(I)right)right)$ is an instance lower bound, where $mathsf{Ent}(I)$ is an entropy-like term determined by the gaps, and there is a $delta$-correct algorithm for Best-$1$-Arm with sample complexity $Oleft(H(I)cdotleft(lndelta^{-1}+mathsf{Ent}(I)right)+Delta_{[2]}^{-2}lnlnDelta_{[2]}^{-1}right)$. If the conjecture is true, we would have a complete understanding of the instance-wise sample complexity of Best-$1$-Arm. We make significant progress towards the resolution of the gap-entropy conjecture. For the upper bound, we provide a highly nontrivial algorithm which requires [Oleft(H(I)cdotleft(lndelta^{-1} +mathsf{Ent}(I)right)+Delta_{[2]}^{-2}lnlnDelta_{[2]}^{-1}mathrm{polylog}(n,delta^{-1})right)] samples in expectation. For the lower bound, we show that for any Gaussian Best-$1$-Arm instance with gaps of the form $2^{-k}$, any $delta$-correct monotone algorithm requires $Omegaleft(H(I)cdotleft(lndelta^{-1} + mathsf{Ent}(I)right)right)$ samples in expectation.
In mobile health (mHealth) smart devices deliver behavioral treatments repeatedly over time to a user with the goal of helping the user adopt and maintain healthy behaviors. Reinforcement learning appears ideal for learning how to optimally make these sequential treatment decisions. However, significant challenges must be overcome before reinforcement learning can be effectively deployed in a mobile healthcare setting. In this work we are concerned with the following challenges: 1) individuals who are in the same context can exhibit differential response to treatments 2) only a limited amount of data is available for learning on any one individual, and 3) non-stationary responses to treatment. To address these challenges we generalize Thompson-Sampling bandit algorithms to develop IntelligentPooling. IntelligentPooling learns personalized treatment policies thus addressing challenge one. To address the second challenge, IntelligentPooling updates each users degree of personalization while making use of available data on other users to speed up learning. Lastly, IntelligentPooling allows responsivity to vary as a function of a users time since beginning treatment, thus addressing challenge three. We show that IntelligentPooling achieves an average of 26% lower regret than state-of-the-art. We demonstrate the promise of this approach and its ability to learn from even a small group of users in a live clinical trial.
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