Sparse Optimistic Information Directed Sampling

Abstract: Many high-dimensional online decision-making problems can be modeled as stochastic sparse linear bandits. Most existing algorithms are designed to achieve optimal worst-case regret in either the data-rich regime, where polynomial depen- dence on the ambient dimension is unavoidable, or the data-poor regime, where dimension-independence is possible at the cost of worse dependence on the num- ber of rounds. In contrast, the sparse Information Directed Sampling (IDS) algo- rithm satisfies a Bayesian regret bound that has the optimal rate in both regimes simultaneously. In this work, we explore the use of Sparse Optimistic Informa- tion Directed Sampling (SOIDS) to achieve the same adaptivity in the worst-case setting, without Bayesian assumptions. Through a novel analysis that enables the use of a time-dependent learning rate, we show that SOIDS can optimally balance information and regret. Our results extend the theoretical guarantees of IDS, pro- viding the first algorithm that simultaneously achieves optimal worst-case regret in both the data-rich and data-poor regimes. We empirically demonstrate the good performance of SOIDS.