Nonparametric Stochastic Contextual Bandits

We analyze the $K$-armed bandit problem where the reward for each arm is a noisy realization based on an observed context under mild nonparametric assumptions. We attain tight results for top-arm identification and a sublinear regret of $\widetilde{O}\Big(T^{\frac{1+D}{2+D}}\Big)$, where $D$ is the context dimension, for a modified UCB algorithm that is simple to implement ($k$NN-UCB). We then give global intrinsic dimension dependent and ambient dimension independent regret bounds. We also discuss recovering topological structures within the context space based on expected bandit performance and provide an extension to infinite-armed contextual bandits. Finally, we experimentally show the improvement of our algorithm over existing multi-armed bandit approaches for both simulated tasks and MNIST image classification.Comment: AAAI 201

The art of clustering bandits.

Multi-armed bandit problems are receiving a great deal of attention because they adequately formalize the exploration-exploitation trade-offs arising in several industrially relevant applications, such as online advertisement and, more generally, recommendation systems. In many cases, however, these applications have a strong social component, whose integration in the bandit algorithms could lead to a dramatic performance increase. For instance, we may want to serve content to a group of users by taking advantage of an underlying network of social relationships among them. The purpose of this thesis is to introduce novel and principled algorithmic approaches to the solution of such networked bandit problems. Starting from a global (Laplacian-based) strategy which allocates a bandit algorithm to each network node (user), and allows it to "share" signals (contexts and payoffs) with the neghboring nodes, our goal is to derive and experimentally test more scalable approaches based on different ways of clustering the graph nodes. More importantly, we shall investigate the case when the graph structure is not given ahead of time, and has to be inferred based on past user behavior. A general difficulty arising in such practical scenarios is that data sequences are typically nonstationary, implying that traditional statistical inference methods should be used cautiously, possibly replacing them with by more robust nonstochastic (e.g., game-theoretic) inference methods. In this thesis, we will firstly introduce the centralized clustering bandits. Then, we propose the corresponding solution in decentralized scenario. After that, we explain the generic collaborative clustering bandits. Finally, we extend and showcase the state-of-the-art clustering bandits that we developed in the quantification problem

Regret Analysis of Stochastic and Nonstochastic Multi-armed Bandit Problems

Multi-armed bandit problems are the most basic examples of sequential decision problems with an exploration-exploitation trade-off. This is the balance between staying with the option that gave highest payoffs in the past and exploring new options that might give higher payoffs in the future. Although the study of bandit problems dates back to the Thirties, exploration-exploitation trade-offs arise in several modern applications, such as ad placement, website optimization, and packet routing. Mathematically, a multi-armed bandit is defined by the payoff process associated with each option. In this survey, we focus on two extreme cases in which the analysis of regret is particularly simple and elegant: i.i.d. payoffs and adversarial payoffs. Besides the basic setting of finitely many actions, we also analyze some of the most important variants and extensions, such as the contextual bandit model.Comment: To appear in Foundations and Trends in Machine Learnin