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].

Novel Exploration Techniques (NETs) for Malaria Policy Interventions

The task of decision-making under uncertainty is daunting, especially for problems which have significant complexity. Healthcare policy makers across the globe are facing problems under challenging constraints, with limited tools to help them make data driven decisions. In this work we frame the process of finding an optimal malaria policy as a stochastic multi-armed bandit problem, and implement three agent based strategies to explore the policy space. We apply a Gaussian Process regression to the findings of each agent, both for comparison and to account for stochastic results from simulating the spread of malaria in a fixed population. The generated policy spaces are compared with published results to give a direct reference with human expert decisions for the same simulated population. Our novel approach provides a powerful resource for policy makers, and a platform which can be readily extended to capture future more nuanced policy spaces.

Thompson Sampling for Noncompliant Bandits

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 observed and latent noncompliance. With extensive empirical analysis, we demonstrate that our algorithms either match or outperform traditional Thompson sampling in both compliant and noncompliant environments.

Optimal Testing in the Experiment-rich Regime

Motivated by the widespread adoption of large-scale A/B testing in industry, we propose a new experimentation framework for the setting where potential experiments are abundant (i.e., many hypotheses are available to test), and observations are costly; we refer to this as the experiment-rich regime. Such scenarios require the experimenter to internalize the opportunity cost of assigning a sample to a particular experiment. We fully characterize the optimal policy and give an algorithm to compute it. Furthermore, we develop a simple heuristic that also provides intuition for the optimal policy. We use simulations based on real data to compare both the optimal algorithm and the heuristic to other natural alternative experimental design frameworks. In particular, we discuss the paradox of power: high-powered classical tests can lead to highly inefficient sampling in the experiment-rich regime.

Confidence Intervals for Policy Evaluation in Adaptive Experiments

Adaptive experiments can result in considerable cost savings in multi-armed trials by enabling analysts to quickly focus on the most promising alternatives. Most existing work on adaptive experiments (which include multi-armed bandits) has focused maximizing the speed at which the analyst can identify the optimal arm and/or minimizing the number of draws from sub-optimal arms. In many scientific settings, however, it is not only of interest to identify the optimal arm, but also to perform a statistical analysis of the data collected from the experiment. Naive approaches to statistical inference with adaptive inference fail because many commonly used statistics (such as sample means or inverse propensity weighting) do not have an asymptotically Gaussian limiting distribution centered on the estimate, and so confidence intervals constructed from these statistics do not have correct coverage. But, as shown in this paper, carefully designed data-adaptive weighting schemes can be used to overcome this issue and restore a relevant central limit theorem, enabling hypothesis testing. We validate the accuracy of the resulting confidence intervals in numerical experiments.