Randomized Exploration for Non-Stationary Stochastic Linear Bandits

Abstract

We investigate two perturbation approaches to overcome conservatism that optimism based algorithms chronically suffer from in practice. The first approach replaces optimism with a simple randomization when using confidence sets. The second one adds random perturbations to its current estimate before maximizing the expected reward. For non-stationary linear bandits, where each action is associated with a $d$-dimensional feature and the unknown parameter is time-varying with total variation $B_T$, we propose two randomized algorithms, Discounted Randomized LinUCB (D-RandLinUCB) and Discounted Linear Thompson Sampling (D-LinTS) via the two perturbation approaches. We highlight the statistical optimality versus computational efficiency trade-off between them in that the former asymptotically achieves the optimal dynamic regret $\tilde{\mathcal{O}}( d ^{2/3}B_T^{1/3} T^{2/3})$, but the latter is oracle-efficient with an extra logarithmic gap in number of arms compared to minimax-optimal dynamic regret. In a simulation study, both empirically show the outstanding performance in tackling conservatism issue that Discounted LinUCB (D-LinUCB) struggles with.