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Poster
The Hardness Analysis of Thompson Sampling for Combinatorial Semi-bandits with Greedy Oracle
Fang Kong · Yueran Yang · Wei Chen · Shuai Li

Wed Dec 08 12:30 AM -- 02:00 AM (PST) @
in Poster Session 3 »
Thompson sampling (TS) has attracted a lot of interest in the bandit area. It was introduced in the 1930s but has not been theoretically proven until recent years. All of its analysis in the combinatorial multi-armed bandit (CMAB) setting requires an exact oracle to provide optimal solutions with any input. However, such an oracle is usually not feasible since many combinatorial optimization problems are NP-hard and only approximation oracles are available. An example \cite{WangC18} has shown the failure of TS to learn with an approximation oracle. However, this oracle is uncommon and is designed only for a specific problem instance. It is still an open question whether the convergence analysis of TS can be extended beyond the exact oracle in CMAB. In this paper, we study this question under the greedy oracle, which is a common (approximation) oracle with theoretical guarantees to solve many (offline) combinatorial optimization problems. We provide a problem-dependent regret lower bound of order $\Omega(\log T/\Delta^2)$ to quantify the hardness of TS to solve CMAB problems with greedy oracle, where $T$ is the time horizon and $\Delta$ is some reward gap. We also provide an almost matching regret upper bound. These are the first theoretical results for TS to solve CMAB with a common approximation oracle and break the misconception that TS cannot work with approximation oracles.
Keywords: Optimization · Bandits

Author Information

Fang Kong (Shanghai Jiao Tong University)
Yueran Yang (Shanghai Jiao Tong University)
Wei Chen (Microsoft Research)
Shuai Li (Shanghai Jiao Tong University)

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