Parameter-free Optimal Rates for Nonlinear Semi-Norm Contractions with Applications to $Q$-Learning
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| Format: | Preprint |
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2025
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| _version_ | 1866917355828805632 |
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| author | Naskar, Ankur Thoppe, Gugan Gupta, Vijay |
| author_facet | Naskar, Ankur Thoppe, Gugan Gupta, Vijay |
| contents | 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 Markovian trajectories. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_05984 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Parameter-free Optimal Rates for Nonlinear Semi-Norm Contractions with Applications to $Q$-Learning Naskar, Ankur Thoppe, Gugan Gupta, Vijay Machine Learning 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 Markovian trajectories. |
| title | Parameter-free Optimal Rates for Nonlinear Semi-Norm Contractions with Applications to $Q$-Learning |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2508.05984 |