Revisiting Weighted Strategy for Non-stationary Parametric Bandits and MDPs

TL;DR

This paper refines the analysis of weighted strategies for non-stationary parametric bandits, leading to simpler algorithms with improved regret bounds and extending these insights to non-stationary MDPs with function approximation.

Contribution

It introduces a refined analysis framework for weighted strategies, simplifying algorithm design and improving regret bounds for linear, generalized linear, and self-concordant bandits, and extends to non-stationary MDPs.

Findings

Simpler weight-based algorithms matching the efficiency of window/restart strategies.

Improved regret bounds for generalized linear bandits.

Extension of the framework to non-stationary MDPs with function approximation.

Abstract

Non-stationary parametric bandits have attracted much attention recently. There are three principled ways to deal with non-stationarity, including sliding-window, weighted, and restart strategies. As many non-stationary environments exhibit gradual drifting patterns, the weighted strategy is commonly adopted in real-world applications. However, previous theoretical studies show that its analysis is more involved and the algorithms are either computationally less efficient or statistically suboptimal. This paper revisits the weighted strategy for non-stationary parametric bandits. In linear bandits (LB), we discover that this undesirable feature is due to an inadequate regret analysis, which results in an overly complex algorithm design. We propose a \emph{refined analysis framework}, which simplifies the derivation and, importantly, produces a simpler weight-based algorithm that is as efficient as window/restart-based algorithms while retaining the same regret as previous studies. Furthermore, our new framework can be used to improve regret bounds of other parametric bandits, including Generalized Linear Bandits (GLB) and Self-Concordant Bandits (SCB). For example, we develop a simple weighted GLB algorithm with an $\tilde{O}(k_\mu^{5/4} c_\mu^{-3/4} d^{3/4} P_T^{1/4}T^{3/4})$ regret, improving the $\tilde{O}(k_\mu^{2} c_\mu^{-1}d^{9/10} P_T^{1/5}T^{4/5})$ bound in prior work, where $k_\mu$ and $c_\mu$ characterize the reward model's nonlinearity, $P_T$ measures the non-stationarity, $d$ and $T$ denote the dimension and time horizon. Moreover, we extend our framework to non-stationary Markov Decision Processes (MDPs) with function approximation, focusing on Linear Mixture MDP and Multinomial Logit (MNL) Mixture MDP. For both classes, we propose algorithms based on the weighted strategy and establish dynamic regret guarantees using our analysis framework.