Improved Algorithms for Linear Stochastic Bandits

TLDR

The study aims to improve the theoretical analysis and empirical performance of algorithms for stochastic and linear stochastic multi‑armed bandit problems. The improvement is achieved by constructing smaller confidence sets using a novel tail inequality for vector‑valued martingales. The modified algorithm attains high‑probability constant regret, improves the regret bound by a logarithmic factor, and shows substantial empirical gains. References: Rusmevichientong & Tsitsiklis (2010), Li et al.

Abstract

We improve the theoretical analysis and empirical performance of algorithms for the stochastic multi-armed bandit problem and the linear stochastic multi-armed bandit problem. In particular, we show that a simple modification of Auer's UCB algorithm (Auer, 2002) achieves with high probability constant regret. More importantly, we modify and, consequently, improve the analysis of the algorithm for the for linear stochastic bandit problem studied by Auer (2002), Dani et al. (2008), Rusmevichientong and Tsitsiklis (2010), Li et al. (2010). Our modification improves the regret bound by a logarithmic factor, though experiments show a vast improvement. In both cases, the improvement stems from the construction of smaller confidence sets. For their construction we use a novel tail inequality for vector-valued martingales.