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

TL;DR

This work introduces stochastic low-rank tensor bandits, where the mean reward is modeled as a Tucker-decomposed tensor $\mathcal X = \mathcal S \times_1 \mathbf U_1 \times_2 \cdots \times_d \mathbf U_d$, enabling efficient online decision making over multi-dimensional arms. It develops two non-contextual algorithms, tensor elimination and tensor epoch-greedy, with distinct regret scalings that leverage low-rank structure, and a Bayesian contextual approach, tensor ensemble sampling, to handle contextual tensor bandits. Theoretical results show that tensor elimination achieves a sharp regret bound and tensor epoch-greedy provides favorable dimension dependence, while contextual settings enjoy Bayes-regret guarantees through approximate Thompson sampling. Simulations and online advertising data demonstrate that exploiting tensor low-rank structure yields substantial performance gains over vectorized and matricized baselines, validating the practical impact for multi-dimensional online decision making.

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.

Figures (4)

• Figure 1: An example of interactive multi-dimensional online decision making. The rewards from all sequential multi-dimensional decisions formulate a tensor.
• Figure 2: Cumulative regrets with varying dimension $p$ and signal strength $w$. The shaded areas represent the confidence bands.
• Figure 3: The left plot illustrates the reward tensor formulation in our online advertising data. The right plot shows cumulative regrets of tensor epoch-greedy, tensor ensemble sampling and vectorized UCB in the contextual tensor bandit real data.
• Figure 4: Cumulative regrets in non-contextual tensor bandit real data.

Theorems & Definitions (2)

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