Thompson Sampling: An Asymptotically Optimal Finite Time Analysis

The question of the optimality of Thompson Sampling for solving the stochastic multi-armed bandit problem had been open since 1933. In this paper we answer it positively for the case of Bernoulli rewards by providing the first finite-time analysis that matches the asymptotic rate given in the Lai and Robbins lower bound for the cumulative regret. The proof is accompanied by a numerical comparison with other optimal policies, experiments that have been lacking in the literature until now for the Bernoulli case.Comment: 15 pages, 2 figures, submitted to ALT (Algorithmic Learning Theory

A two-armed bandit based scheme for accelerated decentralized learning

The two-armed bandit problem is a classical optimization problem where a decision maker sequentially pulls one of two arms attached to a gambling machine, with each pull resulting in a random reward. The reward distributions are unknown, and thus, one must balance between exploiting existing knowledge about the arms, and obtaining new information. Bandit problems are particularly fascinating because a large class of real world problems, including routing, QoS control, game playing, and resource allocation, can be solved in a decentralized manner when modeled as a system of interacting gambling machines. Although computationally intractable in many cases, Bayesian methods provide a standard for optimal decision making. This paper proposes a novel scheme for decentralized decision making based on the Goore Game in which each decision maker is inherently Bayesian in nature, yet avoids computational intractability by relying simply on updating the hyper parameters of sibling conjugate priors, and on random sampling from these posteriors. We further report theoretical results on the variance of the random rewards experienced by each individual decision maker. Based on these theoretical results, each decision maker is able to accelerate its own learning by taking advantage of the increasingly more reliable feedback that is obtained as exploration gradually turns into exploitation in bandit problem based learning. Extensive experiments demonstrate that the accelerated learning allows us to combine the benefits of conservative learning, which is high accuracy, with the benefits of hurried learning, which is fast convergence. In this manner, our scheme outperforms recently proposed Goore Game solution schemes, where one has to trade off accuracy with speed. We thus believe that our methodology opens avenues for improved performance in a number of applications of bandit based decentralized decision making

Further Optimal Regret Bounds for Thompson Sampling

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 $O(\sqrt{NT\ln T})$ 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].Comment: arXiv admin note: substantial text overlap with arXiv:1111.179

Towards Thompson Sampling for Complex Bayesian Reasoning

Paper III, IV, and VI are not available as a part of the dissertation due to the copyright.Thompson Sampling (TS) is a state-of-art algorithm for bandit problems set in a Bayesian framework. Both the theoretical foundation and the empirical efficiency of TS is wellexplored for plain bandit problems. However, the Bayesian underpinning of TS means that TS could potentially be applied to other, more complex, problems as well, beyond the bandit problem, if suitable Bayesian structures can be found. The objective of this thesis is the development and analysis of TS-based schemes for more complex optimization problems, founded on Bayesian reasoning. We address several complex optimization problems where the previous state-of-art relies on a relatively myopic perspective on the problem. These includes stochastic searching on the line, the Goore game, the knapsack problem, travel time estimation, and equipartitioning. Instead of employing Bayesian reasoning to obtain a solution, they rely on carefully engineered rules. In all brevity, we recast each of these optimization problems in a Bayesian framework, introducing dedicated TS based solution schemes. For all of the addressed problems, the results show that besides being more effective, the TS based approaches we introduce are also capable of solving more adverse versions of the problems, such as dealing with stochastic liars.publishedVersio

Accelerated Bayesian learning for decentralized two-armed bandit based decision making with applications to the Goore Game

The two-armed bandit problem is a classical optimization problem where a decision maker sequentially pulls one of two arms attached to a gambling machine, with each pull resulting in a random reward. The reward distributions are unknown, and thus, one must balance between exploiting existing knowledge about the arms, and obtaining new information. Bandit problems are particularly fascinating because a large class of real world problems, including routing, Quality of Service (QoS) control, game playing, and resource allocation, can be solved in a decentralized manner when modeled as a system of interacting gambling machines. Although computationally intractable in many cases, Bayesian methods provide a standard for optimal decision making. This paper proposes a novel scheme for decentralized decision making based on the Goore Game in which each decision maker is inherently Bayesian in nature, yet avoids computational intractability by relying simply on updating the hyper parameters of sibling conjugate priors, and on random sampling from these posteriors. We further report theoretical results on the variance of the random rewards experienced by each individual decision maker. Based on these theoretical results, each decision maker is able to accelerate its own learning by taking advantage of the increasingly more reliable feedback that is obtained as exploration gradually turns into exploitation in bandit problem based learning. Extensive experiments, involving QoS control in simulated wireless sensor networks, demonstrate that the accelerated learning allows us to combine the benefits of conservative learning, which is high accuracy, with the benefits of hurried learning, which is fast convergence. In this manner, our scheme outperforms recently proposed Goore Game solution schemes, where one has to trade off accuracy with speed. As an additional benefit, performance also becomes more stable. We thus believe that our methodology opens avenues for improved performance in a number of applications of bandit based decentralized decision making

Solving dynamic bandit problems and decentralized games using the kalman bayesian learning automaton

Multi-armed bandit problems have been subject to a lot of research in computer science because it captures the fundamental dilemma of exploration versus exploitation in reinforcement learning. The goal of a bandit problem is to determine the optimal balance between the gain of new information (exploration) and immediate reward maximization (exploitation). Dynamic bandit problems are especially challenging because they involve changing environments. Combined with game theory, where one analyze the behavior of agents in multi-agent settings, bandit problems serves as a framework for benchmarking the applicability of learning algorithms in various situations. In this thesis, we investigate a novel approach to the multi-armed bandit problem, the Kalman Bayesian Learning Automaton, an algorithm which applies concepts from Kalman filtering, a powerful technique for probabilistic reasoning over time. To determine the effectiveness of such an approach we have conducted an empirical study of the Kalman Bayesian Learning Automaton in multi-armed dynamic bandit problems and selected games from game theory. Specifically, we evaluate the performance of the Kalman Bayesian Learning Automaton in randomly changing environments, switching environments, the Goore game, the Prisoners Dilemma and zero-sum games. The scalability and robustness of the algorithm are also examined. Indeed, we reveal that the strength of the Kalman Bayesian Learning Automatons lies in its excellent tracking abilities, and are among the top performers in all experiments. Unfortunately, it is dependent on tuning of parameters. We believe further work on the approach could solve the parameter problem, but even with the need to tune parameters we consider the Kalman Bayesian Learning Automaton a strong solution to dynamic multi-armed bandit problems and definitely has the potential to be applied in various applications and multi-agent settings