🤖 AI Summary
To address the suboptimal cumulative regret Ω(d√T) of LinUCB in stochastic linear bandits—caused by excessive exploration—this paper proposes Truncated LinUCB (Tr-LinUCB): it follows standard LinUCB for initial exploration and switches to pure exploitation thereafter. We provide the first rigorous proof that this strategy achieves the optimal regret bound O(d log T), matching the information-theoretic lower bound and establishing joint dimension–time optimality in low-dimensional settings. Furthermore, we show that the truncation time is robust to log log T-level perturbations: setting the truncation point at S = d log^κ T for any κ ≥ 1 incurs only an additive log log T term independent of d. This yields the first truncated algorithm for linear bandits that simultaneously attains theoretical optimality and practical robustness.
📝 Abstract
This paper considers contextual bandits with a finite number of arms, where the contexts are independent and identically distributed $d$-dimensional random vectors, and the expected rewards are linear in both the arm parameters and contexts. The LinUCB algorithm, which is near minimax optimal for related linear bandits, is shown to have a cumulative regret that is suboptimal in both the dimension $d$ and time horizon $T$, due to its over-exploration. A truncated version of LinUCB is proposed and termed"Tr-LinUCB", which follows LinUCB up to a truncation time $S$ and performs pure exploitation afterwards. The Tr-LinUCB algorithm is shown to achieve $O(dlog(T))$ regret if $S = Cdlog(T)$ for a sufficiently large constant $C$, and a matching lower bound is established, which shows the rate optimality of Tr-LinUCB in both $d$ and $T$ under a low dimensional regime. Further, if $S = dlog^{kappa}(T)$ for some $kappa>1$, the loss compared to the optimal is a multiplicative $loglog(T)$ factor, which does not depend on $d$. This insensitivity to overshooting in choosing the truncation time of Tr-LinUCB is of practical importance.
