Finite-Time Error Bounds for Greedy-GQ
It provides theoretical guidance for step-size selection in non-convex two-timescale RL algorithms, though it is incremental as it builds on existing methods.
This paper develops tight finite-time error bounds for the Greedy-GQ algorithm in reinforcement learning, showing convergence rates of O(1/√T) under i.i.d. settings and O(log T/√T) under Markovian settings, and matching sample complexity for a variant.
Greedy-GQ with linear function approximation, originally proposed in \cite{maei2010toward}, is a value-based off-policy algorithm for optimal control in reinforcement learning, and it has a non-linear two timescale structure with the non-convex objective function. This paper develops its tightest finite-time error bounds. We show that the Greedy-GQ algorithm converges as fast as $\mathcal{O}({1}/{\sqrt{T}})$ under the i.i.d.\ setting and $\mathcal{O}({\log T}/{\sqrt{T}})$ under the Markovian setting. We further design a variant of the vanilla Greedy-GQ algorithm using the nested-loop approach, and show that its sample complexity is $\mathcal{O}({\log(1/ε)ε^{-2}})$, which matches with the one of the vanilla Greedy-GQ. Our finite-time error bounds match with one of the stochastic gradient descent algorithms for general smooth non-convex optimization problems, despite its additonal challenge in the two time-scale updates. Our finite-sample analysis provides theoretical guidance on choosing step-sizes for faster convergence in practice, and suggests the trade-off between the convergence rate and the quality of the obtained policy. Our techniques provide a general approach for finite-sample analysis of non-convex two timescale value-based reinforcement learning algorithms.