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

Algorithms for solving $\mathit{\text{nonlinear}}$ fixed-point equations – such as average-reward $Q\mathit{\text{-learning}}$ and $\mathit{\text{TD-learning}}$ – 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 $\tilde{O}(1\mathrm{/}\sqrt{t})$ 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 Markovian trajectories.