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This research advances offline reinforcement learning by establishing finite-sample guarantees for fitted Q-iteration when actions are continuous functions, such as radiation therapy fluence maps or robotic trajectories. The authors introduce a critic-relative coverage condition to bypass the difficulty of defining probability densities for functional actions while accommodating both kernel ridge regression and neural network approximators. These theoretical results provide rigorous regret bounds that improve upon constant-action baselines, offering a robust framework for high-dimensional decision-making in physics-informed applications.
Finite-Sample Theory for Fitted Q-Iteration When Actions Are Functions
arXiv:2609.36390v1 Announce Type: cross Abstract: Offline reinforcement learning seeks optimal decision rules from previously collected data. In some applications, a decision can be an entire function, such as a fluence map in radiation therapy or a smooth movement trajectory in robotics. In this paper, we study the finite-sample theory for fitted Q-iteration (FQI) with functional actions in a discounted infinite-horizon setting. Three major difficulties arise in this setting: first, the absence of a Lebesgue probability density for functional actions complicates coverage descriptions; second, conventional coverage requirements can be restrictive; and third, the large functional action space makes greedy optimization in FQI challenging. To address these difficulties, we study smoothness-regularized policy search under a critic-relative coverage condition. This condition measures how well logged data distinguish relevant action-value differences without requiring an action density. Our main theorem gives finite-sample guarantees for learned-policy regret relative to the best value within a fixed smooth class of functional-action policies. The results allow trajectory lengths to be either bounded or growing and Q-functions to be fitted by either functional-input kernel ridge regression or adaptive functional neural networks. For a few examples, we can obtain polynomially decaying regret bounds in the number of logged transitions, up to logarithmic factors, with logarithmically many FQI iterations. Numerical experiments show gains of learned functional-action policies over constant-action policies and support our adoption of a critic-relative coverage condition.
Abstract:Offline reinforcement learning seeks optimal decision rules from previously collected data. In some applications, a decision can be an entire function, such as a fluence map in radiation therapy or a smooth movement trajectory in robotics. In this paper, we study the finite-sample theory for fitted Q-iteration (FQI) with functional actions in a discounted infinite-horizon setting. Three major difficulties arise in this setting: first, the absence of a Lebesgue probability density for functional actions complicates coverage descriptions; second, conventional coverage requirements can be restrictive; and third, the large functional action space makes greedy optimization in FQI challenging. To address these difficulties, we study smoothness-regularized policy search under a critic-relative coverage condition. This condition measures how well logged data distinguish relevant action-value differences without requiring an action density. Our main theorem gives finite-sample guarantees for learned-policy regret relative to the best value within a fixed smooth class of functional-action policies. The results allow trajectory lengths to be either bounded or growing and Q-functions to be fitted by either functional-input kernel ridge regression or adaptive functional neural networks. For a few examples, we can obtain polynomially decaying regret bounds in the number of logged transitions, up to logarithmic factors, with logarithmically many FQI iterations. Numerical experiments show gains of learned functional-action policies over constant-action policies and support our adoption of a critic-relative coverage condition.
Methodology (stat.ME); Machine Learning (cs.LG); Statistics Theory (math.ST); Machine Learning (stat.ML)
arXiv:2609.36390 [stat.ME]
(or arXiv:2609.36390v1 [stat.ME] for this version)
https://doi.org/10.48550/arXiv.2609.36390 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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