Thompson sampling (TS) is a class of algorithms for sequential decision-making, which requires maintaining a posterior distribution over a model. However, calculating exact posterior distributions is intractable for all but the simplest models. Consequently, efficient computation of an approximate posterior distribution is a crucial problem for scalable TS with complex models, such as neural networks. In this paper, we use distribution optimization techniques to approximate the posterior distribution, solved via Wasserstein gradient flows. Based on the framework, a principled particle-optimization algorithm is developed for TS to approximate the posterior efficiently. Our approach is scalable and does not make explicit distribution assumptions on posterior approximations. Extensive experiments on both synthetic data and real large-scale data demonstrate the superior performance of the proposed methods.
Thompson sampling, a Bayesian method for balancing exploration and exploitation in bandit problems, has theoretical guarantees and exhibits strong empirical performance in many domains. Traditional Thompson sampling, however, assumes perfect compliance, where an agent's chosen action is treated as the implemented action. This article introduces a stochastic noncompliance model that relaxes this assumption. We prove that any noncompliance in a 2-armed Bernoulli bandit increases existing regret bounds. With our noncompliance model, we derive Thompson sampling variants that explicitly handle both…
In this note, we present a version of the Thompson sampling algorithm for the problem of online linear generalization with full information (i.e., the experts setting), studied by Kalai and Vempala, 2005. The algorithm uses a Gaussian prior and time-varying Gaussian likelihoods, and we show that it essentially reduces to Kalai and Vempala's Follow-the-Perturbed-Leader strategy, with exponentially distributed noise replaced by Gaussian noise. This implies sqrt(T) regret bounds for Thompson sampling (with time-varying likelihood) for online learning with full information.
Thompson Sampling provides an efficient technique to introduce prior knowledge in the multi-armed bandit problem, along with providing remarkable empirical performance. In this paper, we revisit the Thompson Sampling algorithm under rewards drawn from symmetric $α$-stable distributions, which are a class of heavy-tailed probability distributions utilized in finance and economics, in problems such as modeling stock prices and human behavior. We present an efficient framework for posterior inference, which leads to two algorithms for Thompson Sampling in this setting. We prove finite-time regret…
This technical note presents a new approach to carrying out the kind of exploration achieved by Thompson sampling, but without explicitly maintaining or sampling from posterior distributions. The approach is based on a bootstrap technique that uses a combination of observed and artificially generated data. The latter serves to induce a prior distribution which, as we will demonstrate, is critical to effective exploration. We explain how the approach can be applied to multi-armed bandit and reinforcement learning problems and how it relates to Thompson sampling. The approach is particularly wel…
The University of Tokyo（东京大学）；RIKEN AIP（理化学研究所先进智能项目中心）
We investigate finite stochastic partial monitoring, which is a general model for sequential learning with limited feedback. While Thompson sampling is one of the most promising algorithms on a variety of online decision-making problems, its properties for stochastic partial monitoring have not been theoretically investigated, and the existing algorithm relies on a heuristic approximation of the posterior distribution. To mitigate these problems, we present a novel Thompson-sampling-based algorithm, which enables us to exactly sample the target parameter from the posterior distribution. Beside…
Recent advances in deep reinforcement learning have made significant strides in performance on applications such as Go and Atari games. However, developing practical methods to balance exploration and exploitation in complex domains remains largely unsolved. Thompson Sampling and its extension to reinforcement learning provide an elegant approach to exploration that only requires access to posterior samples of the model. At the same time, advances in approximate Bayesian methods have made posterior approximation for flexible neural network models practical. Thus, it is attractive to consider a…
Thompson sampling provides a solution to bandit problems in which new observations are allocated to arms with the posterior probability that an arm is optimal. While sometimes easy to implement and asymptotically optimal, Thompson sampling can be computationally demanding in large scale bandit problems, and its performance is dependent on the model fit to the observed data. We introduce bootstrap Thompson sampling (BTS), a heuristic method for solving bandit problems which modifies Thompson sampling by replacing the posterior distribution used in Thompson sampling by a bootstrap distribution. …
STOR-i Centre for Doctoral Training, Lancaster University（兰卡斯特大学STOR-i博士训练中心）；Department of Mathematics and Statistics, Lancaster University（兰卡斯特大学数学与统计系）
Thompson Sampling is a well established approach to bandit and reinforcement learning problems. However its use in continuum armed bandit problems has received relatively little attention. We provide the first bounds on the regret of Thompson Sampling for continuum armed bandits under weak conditions on the function class containing the true function and sub-exponential observation noise. Our bounds are realised by analysis of the eluder dimension, a recently proposed measure of the complexity of a function class, which has been demonstrated to be useful in bounding the Bayesian regret of Thom…
College of Information and Computer Science（信息与计算机科学学院）；University of Massachusetts（马萨诸塞大学）；VinAI（VinAI研究院）
We study the effects of approximate inference on the performance of Thompson sampling in the $k$-armed bandit problems. Thompson sampling is a successful algorithm for online decision-making but requires posterior inference, which often must be approximated in practice. We show that even small constant inference error (in $α$-divergence) can lead to poor performance (linear regret) due to under-exploration (for $α<1$) or over-exploration (for $α>0$) by the approximation. While for $α> 0$ this is unavoidable, for $α\leq 0$ the regret can be improved by adding a small amount of forced exploratio…
Google Research（谷歌研究院）；University of Alberta（阿尔伯塔大学）；DeepMind（深度思维）
Efficient exploration in bandits is a fundamental online learning problem. We propose a variant of Thompson sampling that learns to explore better as it interacts with bandit instances drawn from an unknown prior. The algorithm meta-learns the prior and thus we call it MetaTS. We propose several efficient implementations of MetaTS and analyze it in Gaussian bandits. Our analysis shows the benefit of meta-learning and is of a broader interest, because we derive a novel prior-dependent Bayes regret bound for Thompson sampling. Our theory is complemented by empirical evaluation, which shows that …
MediaTek Research（联发科研究中心）；Secondmind；Imperial College London（帝国理工学院）；University of Cambridge（剑桥大学）
Thompson Sampling (TS) from Gaussian Process (GP) models is a powerful tool for the optimization of black-box functions. Although TS enjoys strong theoretical guarantees and convincing empirical performance, it incurs a large computational overhead that scales polynomially with the optimization budget. Recently, scalable TS methods based on sparse GP models have been proposed to increase the scope of TS, enabling its application to problems that are sufficiently multi-modal, noisy or combinatorial to require more than a few hundred evaluations to be solved. However, the approximation error int…
We study the use of policy gradient algorithms to optimize over a class of generalized Thompson sampling policies. Our central insight is to view the posterior parameter sampled by Thompson sampling as a kind of pseudo-action. Policy gradient methods can then be tractably applied to search over a class of sampling policies, which determine a probability distribution over pseudo-actions (i.e., sampled parameters) as a function of observed data. We also propose and compare policy gradient estimators that are specialized to Bayesian bandit problems. Numerical experiments demonstrate that direct p…
In this paper, we propose a Double Thompson Sampling (D-TS) algorithm for dueling bandit problems. As indicated by its name, D-TS selects both the first and the second candidates according to Thompson Sampling. Specifically, D-TS maintains a posterior distribution for the preference matrix, and chooses the pair of arms for comparison by sampling twice from the posterior distribution. This simple algorithm applies to general Copeland dueling bandits, including Condorcet dueling bandits as its special case. For general Copeland dueling bandits, we show that D-TS achieves $O(K^2 \log T)$ regret. …
McCombs School of Business, The University of Texas at Austin（德克萨斯大学奥斯汀分校麦库姆斯商学院）
Thompson sampling is an efficient algorithm for sequential decision making, which exploits the posterior uncertainty to address the exploration-exploitation dilemma. There has been significant recent interest in integrating Bayesian neural networks into Thompson sampling. Most of these methods rely on global variable uncertainty for exploration. In this paper, we propose a new probabilistic modeling framework for Thompson sampling, where local latent variable uncertainty is used to sample the mean reward. Variational inference is used to approximate the posterior of the local variable, and sem…
We consider Thompson sampling for linear bandit problems with finitely many independent arms, where rewards are sampled from normal distributions that are linearly dependent on unknown parameter vectors and with unknown variance. Specifically, with a Bayesian formulation we consider multivariate normal-gamma priors to represent environment uncertainty for all involved parameters. We show that our chosen sampling prior is a conjugate prior to the reward model and derive a Bayesian regret bound for Thompson sampling under the condition that the 5/2-moment of the variance distribution exist.
Carnegie Mellon University（卡内基梅隆大学）；University of Massachusetts, Amherst（马萨诸塞大学阿默斯特分校）
We design and analyse variations of the classical Thompson sampling (TS) procedure for Bayesian optimisation (BO) in settings where function evaluations are expensive, but can be performed in parallel. Our theoretical analysis shows that a direct application of the sequential Thompson sampling algorithm in either synchronous or asynchronous parallel settings yields a surprisingly powerful result: making $n$ evaluations distributed among $M$ workers is essentially equivalent to performing $n$ evaluations in sequence. Further, by modeling the time taken to complete a function evaluation, we show…
Restless bandit problems are instances of non-stationary multi-armed bandits. These problems have been studied well from the optimization perspective, where the goal is to efficiently find a near-optimal policy when system parameters are known. However, very few papers adopt a learning perspective, where the parameters are unknown. In this paper, we analyze the performance of Thompson sampling in episodic restless bandits with unknown parameters. We consider a general policy map to define our competitor and prove an $\tilde{\mathcal{O}}(\sqrt{T})$ Bayesian regret bound. Our competitor is flexi…
We consider the stochastic multi-armed bandit problem with a prior distribution on the reward distributions. We are interested in studying prior-free and prior-dependent regret bounds, very much in the same spirit as the usual distribution-free and distribution-dependent bounds for the non-Bayesian stochastic bandit. Building on the techniques of Audibert and Bubeck [2009] and Russo and Roy [2013] we first show that Thompson Sampling attains an optimal prior-free bound in the sense that for any prior distribution its Bayesian regret is bounded from above by $14 \sqrt{n K}$. This result is unim…
We derive an alternative proof for the regret of Thompson sampling (\ts) in the stochastic linear bandit setting. While we obtain a regret bound of order $\widetilde{O}(d^{3/2}\sqrt{T})$ as in previous results, the proof sheds new light on the functioning of the \ts. We leverage on the structure of the problem to show how the regret is related to the sensitivity (i.e., the gradient) of the objective function and how selecting optimal arms associated to \textit{optimistic} parameters does control it. Thus we show that \ts can be seen as a generic randomized algorithm where the sampling distribu…
Tel Aviv University（特拉维夫大学）；Google Research（谷歌研究院）
We study the Thompson sampling algorithm in an adversarial setting, specifically, for adversarial bit prediction. We characterize the bit sequences with the smallest and largest expected regret. Among sequences of length $T$ with $k < \frac{T}{2}$ zeros, the sequences of largest regret consist of alternating zeros and ones followed by the remaining ones, and the sequence of smallest regret consists of ones followed by zeros. We also bound the regret of those sequences, the worse case sequences have regret $O(\sqrt{T})$ and the best case sequence have regret $O(1)$. We extend our results to a m…
Microsoft Research（微软研究院）；Stanford University（斯坦福大学）
We address online combinatorial optimization when the player has a prior over the adversary's sequence of losses. In this framework, Russo and Van Roy proposed an information-theoretic analysis of Thompson Sampling based on the information ratio, resulting in optimal worst-case regret bounds. In this paper we introduce three novel ideas to this line of work. First we propose a new quantity, the scale-sensitive information ratio, which allows us to obtain more refined first-order regret bounds (i.e., bounds of the form $\sqrt{L^*}$ where $L^*$ is the loss of the best combinatorial action). Seco…
We study the logistic bandit, in which rewards are binary with success probability $\exp(βa^\top θ) / (1 + \exp(βa^\top θ))$ and actions $a$ and coefficients $θ$ are within the $d$-dimensional unit ball. While prior regret bounds for algorithms that address the logistic bandit exhibit exponential dependence on the slope parameter $β$, we establish a regret bound for Thompson sampling that is independent of $β$. Specifically, we establish that, when the set of feasible actions is identical to the set of possible coefficient vectors, the Bayesian regret of Thompson sampling is $\tilde{O}(d\sqrt{…
Approximate Thompson sampling with Langevin Monte Carlo broadens its reach from Gaussian posterior sampling to encompass more general smooth posteriors. However, it still encounters scalability issues in high-dimensional problems when demanding high accuracy. To address this, we propose an approximate Thompson sampling strategy, utilizing underdamped Langevin Monte Carlo, where the latter is the go-to workhorse for simulations of high-dimensional posteriors. Based on the standard smoothness and log-concavity conditions, we study the accelerated posterior concentration and sampling using a spec…
Thompson sampling has been shown to be an effective policy across a variety of online learning tasks. Many works have analyzed the finite time performance of Thompson sampling, and proved that it achieves a sub-linear regret under a broad range of probabilistic settings. However its asymptotic behavior remains mostly underexplored. In this paper, we prove an asymptotic convergence result for Thompson sampling under the assumption of a sub-linear Bayesian regret, and show that the actions of a Thompson sampling agent provide a strongly consistent estimator of the optimal action. Our results rel…
We provide an information-theoretic analysis of Thompson sampling that applies across a broad range of online optimization problems in which a decision-maker must learn from partial feedback. This analysis inherits the simplicity and elegance of information theory and leads to regret bounds that scale with the entropy of the optimal-action distribution. This strengthens preexisting results and yields new insight into how information improves performance.
Microsoft Research NYC（微软研究院纽约中心）；Cornell University（康奈尔大学）
We study the stochastic multi-armed bandit problem with the graph-based feedback structure introduced by Mannor and Shamir. We analyze the performance of the two most prominent stochastic bandit algorithms, Thompson Sampling and Upper Confidence Bound (UCB), in the graph-based feedback setting. We show that these algorithms achieve regret guarantees that combine the graph structure and the gaps between the means of the arm distributions. Surprisingly this holds despite the fact that these algorithms do not explicitly use the graph structure to select arms; they observe the additional feedback …
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 $(1+ε)\sum_i \frac{\ln T}{Δ_i}+O(\frac{N}{ε^2})$ and the first near-optimal problem-independent bound of $O(\sqrt{NT\ln T})$ on the expected regret of this…
Information-theoretic Bayesian regret bounds of Russo and Van Roy capture the dependence of regret on prior uncertainty. However, this dependence is through entropy, which can become arbitrarily large as the number of actions increases. We establish new bounds that depend instead on a notion of rate-distortion. Among other things, this allows us to recover through information-theoretic arguments a near-optimal bound for the linear bandit. We also offer a bound for the logistic bandit that dramatically improves on the best previously available, though this bound depends on an information-theore…
Approximate Thompson Sampling for Learning Linear Quadratic Regulators with $O(\sqrt{T})$ Regret
Department of Applied Artificial Intelligence, Seoul National University of Science and Technology（应用人工智能系，首尔科学技术大学）；Department of Electrical and Computer Engineering and ASRI, Seoul National University（电气电子工程系和ASRI，首尔国立大学）
We consider the problem of global optimization of a function over a continuous domain. In our setup, we can evaluate the function sequentially at points of our choice and the evaluations are noisy. We frame it as a continuum-armed bandit problem with a Gaussian Process prior on the function. In this regime, most algorithms have been developed to minimize some form of regret. In this paper, we study the convergence of the sequential point $x^t$ to the global optimizer $x^*$ for the Thompson Sampling approach. Under some assumptions and regularity conditions, we prove concentration bounds for $x…
Chemical space is so large that brute force searches for new interesting molecules are infeasible. High-throughput virtual screening via computer cluster simulations can speed up the discovery process by collecting very large amounts of data in parallel, e.g., up to hundreds or thousands of parallel measurements. Bayesian optimization (BO) can produce additional acceleration by sequentially identifying the most useful simulations or experiments to be performed next. However, current BO methods cannot scale to the large numbers of parallel measurements and the massive libraries of molecules cur…
National University of Singapore（新加坡国立大学）；MIT（麻省理工学院）
Bayesian optimization (BO) is a prominent approach to optimizing expensive-to-evaluate black-box functions. The massive computational capability of edge devices such as mobile phones, coupled with privacy concerns, has led to a surging interest in federated learning (FL) which focuses on collaborative training of deep neural networks (DNNs) via first-order optimization techniques. However, some common machine learning tasks such as hyperparameter tuning of DNNs lack access to gradients and thus require zeroth-order/black-box optimization. This hints at the possibility of extending BO to the FL…
Stanford University（斯坦福大学）；Microsoft Research NYC（微软研究院纽约）
We consider incentivized exploration: a version of multi-armed bandits where the choice of arms is controlled by self-interested agents, and the algorithm can only issue recommendations. The algorithm controls the flow of information, and the information asymmetry can incentivize the agents to explore. Prior work achieves optimal regret rates up to multiplicative factors that become arbitrarily large depending on the Bayesian priors, and scale exponentially in the number of arms. A more basic problem of sampling each arm once runs into similar factors. We focus on the price of incentives: the …
In stochastic bandit problems, a Bayesian policy called Thompson sampling (TS) has recently attracted much attention for its excellent empirical performance. However, the theoretical analysis of this policy is difficult and its asymptotic optimality is only proved for one-parameter models. In this paper we discuss the optimality of TS for the model of normal distributions with unknown means and variances as one of the most fundamental example of multiparameter models. First we prove that the expected regret of TS with the uniform prior achieves the theoretical bound, which is the first result …
