analiza aiqbench
This paper is a theoretical analysis of almost sure convergence rates for stochastic approximation methods under Markovian noise, with direct applicability to algorithms like Q-learning and linear temporal difference learning. Its main contribution is a proof technique combining a Poisson-equation correction with Moreau-envelope smoothing, yielding rate bounds that approach the classical n^-1 benchmark under specific learning-rate conditions. For ML readers, the result matters as a sharpening of convergence guarantees for RL algorithms, though it is a mathematical result rather than an empirical one and its practical impact depends on how tight these bounds prove in real training settings.
Almost Sure Convergence Rates of Stochastic Approximation and Reinforcement Learning via a Poisson-Moreau Drift
arXiv:2605.07104v2 Announce Type: replace Abstract: Establishing almost sure convergence rates for stochastic approximation and reinforcement learning under Markovian noise is a fundamental theoretical challenge. We make progress towards this challenge for a class of stochastic approximation algorithms whose expected updates are contractive, a setting that arises in many reinforcement learning algorithms such as $Q$-learning and linear temporal difference learning. Specifically, for a power-law learning rate $O(n^{-\eta})$ with $\eta \in (1/2, 1)$, we obtain an almost sure convergence rate arbitrarily close to $o(n^{1 - 2\eta})$. For the harmonic learning rate $\alpha_n=\alpha/(n+1)$, the almost sure squared-error rate is $o(n^{-\zeta})$ for every $\zeta<\min\{1,2(1-\kappa)\alpha\}$, where $\kappa$ is the contraction factor. Thus exponents arbitrarily close to one are available when $2(1-\kappa)\alpha\geq1$. This approaches the $n^{-1}$ polynomial exponent of classical law-of-the-iterated-logarithm benchmarks. Key to our analysis is a novel Lyapunov drift construction that applies a Poisson-equation based correction for Markovian noise to the well-established Moreau-envelope smoothing for the contractive mapping.
Abstract:Establishing almost sure convergence rates for stochastic approximation and reinforcement learning under Markovian noise is a fundamental theoretical challenge. We make progress towards this challenge for a class of stochastic approximation algorithms whose expected updates are contractive, a setting that arises in many reinforcement learning algorithms such as $Q$-learning and linear temporal difference learning. Specifically, for a power-law learning rate $O(n^{-\eta})$ with $\eta \in (1/2, 1)$, we obtain an almost sure convergence rate arbitrarily close to $o(n^{1 - 2\eta})$. For the harmonic learning rate $\alpha_n=\alpha/(n+1)$, the almost sure squared-error rate is $o(n^{-\zeta})$ for every $\zeta<\min\{1,2(1-\kappa)\alpha\}$, where $\kappa$ is the contraction factor. Thus exponents arbitrarily close to one are available when $2(1-\kappa)\alpha\geq1$. This approaches the $n^{-1}$ polynomial exponent of classical law-of-the-iterated-logarithm benchmarks. Key to our analysis is a novel Lyapunov drift construction that applies a Poisson-equation based correction for Markovian noise to the well-established Moreau-envelope smoothing for the contractive mapping.
Machine Learning (cs.LG); Optimization and Control (math.OC); Machine Learning (stat.ML)
arXiv:2605.07104 [cs.LG]
(or arXiv:2605.07104v2 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2605.07104 Focus to learn more arXiv-issued DOI via DataCite
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.
Text integral sub licența deschisă a publicației · 248 cuvinte
Subiecte conexe
Provably Efficient Federated Reinforcement Learning with Linear Function Approximation and Logarithmic Communication Cost
arXiv · 2 sept. 2026
On the Complexity of Offline Reinforcement Learning with $Q^\star$-Approximation and Partial Coverage
arXiv · 9 iun. 2026
Reinforcement LearningtoHarness Approximation Errors for Long-Time QuantumSimulation
arXiv · 21 aug. 2026
Understanding Gap-Dependent Regret for Optimism-Based Reinforcement Learning with Linear Function Approximation
arXiv · 6 oct. 2026