Gap-Dependent Bounds for Federated $Q$-learning

TL;DR

This paper introduces a gap-dependent analysis for federated Q-learning in finite-horizon MDPs, achieving improved regret and communication bounds by exploiting MDP structures, unlike previous worst-case approaches.

Contribution

It provides the first gap-dependent regret and communication bounds for federated Q-learning, revealing faster convergence and reduced communication costs under benign MDP conditions.

Findings

Achieves $ ext{log } T$ regret bounds using gap-dependent analysis.

Refines communication cost bounds to remove dependence on $MSA$ in the $ ext{log } T$ term.

Shows multi-agent speedup pattern in regret bounds.

Abstract

We present the first gap-dependent analysis of regret and communication cost for on-policy federated $Q$-Learning in tabular episodic finite-horizon Markov decision processes (MDPs). Existing FRL methods focus on worst-case scenarios, leading to $\sqrt{T}$-type regret bounds and communication cost bounds with a $\log T$ term scaling with the number of agents $M$, states $S$, and actions $A$, where $T$ is the average total number of steps per agent. In contrast, our novel framework leverages the benign structures of MDPs, such as a strictly positive suboptimality gap, to achieve a $\log T$-type regret bound and a refined communication cost bound that disentangles exploration and exploitation. Our gap-dependent regret bound reveals a distinct multi-agent speedup pattern, and our gap-dependent communication cost bound removes the dependence on $MSA$ from the $\log T$ term. Notably, our gap-dependent communication cost bound also yields a better global switching cost when $M=1$, removing $SA$ from the $\log T$ term.