Stochastic Low-rank Tensor Bandits for Multi-dimensional Online Decision Making

TL;DR

This paper introduces stochastic low-rank tensor bandits for multi-dimensional online decision making, proposing algorithms with theoretical regret bounds and demonstrating superior performance over existing methods in simulations and real data.

Contribution

It develops novel algorithms for tensor bandits, including tensor elimination, tensor epoch-greedy, and tensor ensemble sampling, with theoretical analysis and practical effectiveness.

Findings

Tensor elimination achieves the best regret bounds.

Tensor epoch-greedy has sharper dimension dependency.

Algorithms outperform state-of-the-art methods in experiments.

Abstract

Multi-dimensional online decision making plays a crucial role in many real applications such as online recommendation and digital marketing. In these problems, a decision at each time is a combination of choices from different types of entities. To solve it, we introduce stochastic low-rank tensor bandits, a class of bandits whose mean rewards can be represented as a low-rank tensor. We consider two settings, tensor bandits without context and tensor bandits with context. In the first setting, the platform aims to find the optimal decision with the highest expected reward, a.k.a, the largest entry of true reward tensor. In the second setting, some modes of the tensor are contexts and the rest modes are decisions, and the goal is to find the optimal decision given the contextual information. We propose two learning algorithms tensor elimination and tensor epoch-greedy for tensor bandits without context, and derive finite-time regret bounds for them. Comparing with existing competitive methods, tensor elimination has the best overall regret bound and tensor epoch-greedy has a sharper dependency on dimensions of the reward tensor. Furthermore, we develop a practically effective Bayesian algorithm called tensor ensemble sampling for tensor bandits with context. Extensive simulations and real analysis in online advertising data back up our theoretical findings and show that our algorithms outperform various state-of-the-art approaches that ignore the tensor low-rank structure.