[2508.05984] Parameter-free Optimal Rates for Nonlinear Semi-Norm Contractions with Applications to $Q$-Learning

Computer Science > Machine Learning arXiv:2508.05984 (cs) [Submitted on 8 Aug 2025 (v1), last revised 21 Mar 2026 (this version, v2)] Title:Parameter-free Optimal Rates for Nonlinear Semi-Norm Contractions with Applications to $Q$-Learning Authors:Ankur Naskar, Gugan Thoppe, Vijay Gupta View a PDF of the paper titled Parameter-free Optimal Rates for Nonlinear Semi-Norm Contractions with Applications to $Q$-Learning, by Ankur Naskar and 2 other authors View PDF HTML (experimental) Abstract:Algorithms for solving \textit{nonlinear} fixed-point equations -- such as average-reward \textit{$Q$-learning} and \textit{TD-learning} -- often involve semi-norm contractions. Achieving parameter-free optimal convergence rates for these methods via Polyak--Ruppert averaging has remained elusive, largely due to the non-monotonicity of such semi-norms. We close this gap by (i.) recasting the averaged error as a linear recursion involving a nonlinear perturbation, and (ii.) taming the nonlinearity by coupling the semi-norm's contraction with the monotonicity of a suitably induced norm. Our main result yields the first parameter-free $\tilde{O}(1/\sqrt{t})$ optimal rates for $Q$-learning in both average-reward and exponentially discounted settings, where $t$ denotes the iteration index. The result applies within a broad framework that accommodates synchronous and asynchronous updates, single-agent and distributed deployments, and data streams obtained either from simulators or along Markovi...