A Measure-Theoretic Finite-Sample Theory for Adaptive-Data Fitted Q-Iteration

• URL: http://arxiv.org/abs/2605.05791v1
• Date: Thu, 07 May 2026 07:26:58 GMT
• Title: A Measure-Theoretic Finite-Sample Theory for Adaptive-Data Fitted Q-Iteration
• Abstract summary: We develop a framework for fitted Q-iteration (FQI) on general measurable Borel spaces.<n>We prove that sequential Rademacher complexity controls Bellman-regression generalization under policy-dependent data collection.<n>These results lay the necessary foundations for the formal analysis of many modern deep RL algorithms.
• Score: 14.808143352500531
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: While reinforcement learning (RL) promises to revolutionize the control of complex nonlinear robotic systems, a profound gap persists between the heuristic success of model-free off-policy deep RL and the underlying theory, which remains largely confined to tabular or linearizable settings. We identify the cause of this gap as an emergent isolation of three traditions: (i) measure-theoretic MDP foundations on general spaces limit their analysis to exact dynamic programming and ignore all error sources of a learning process; (ii) deterministic error propagation analysis addresses the approximation error via concentrability coefficients without a finite-sample analysis of the estimation error; and (iii) PAC generalization bounds characterize the estimation errors of simplified topologies. We bridge these traditions with a unified theoretical framework for fitted Q-iteration (FQI) on general measurable Borel spaces. Our main result provides a finite-sample, adaptive-data performance bound by chaining measure-theoretic probability with Bellman-operator contraction in Banach spaces. We prove that sequential Rademacher complexity controls Bellman-regression generalization under policy-dependent data collection. We further extend this analysis to provide the first cumulative, pathwise online regret guarantee for FQI in continuous spaces. These results lay the necessary foundations for the formal analysis of many modern deep RL algorithms.

Related papers

• DISA: Offline Importance Sampling for Distribution-Matching LLM-RL [56.9445657766829]
  We introduce DISA, which moves this calibration problem outside the RL loop.<n> DISA draws proposal trajectories offline, estimates the partition function via importance sampling, and freezes the resulting partition-function estimate.<n>On two open-weight backbones across six math and three code benchmarks, DISA matches or exceeds the online-coupled distribution-matching baseline Flow.
  arXiv  Detail & Related papers  (2026-05-17T07:14:44Z)
• PAC-Bayes Bounds for Gibbs Posteriors via Singular Learning Theory [10.93258787701145]
  We derive explicit non-asymptotic PAC-Bayes generalization bounds for Gibbs posteriors.<n>Unlike classical worst-case complexity bounds based on uniform laws of large numbers, our analysis yields posterior-averaged risk bounds.
  arXiv  Detail & Related papers  (2026-04-19T03:00:18Z)
• Finite-Sample Analysis of Nonlinear Independent Component Analysis:Sample Complexity and Identifiability Bounds [1.624454100511275]
  Independent Component Analysis (ICA) is a fundamental unsupervised learning technique foruncovering structure in data by separating mixed signals into their independent sources.<n>The finite-sample statistical properties of learning algorithms remain poorly understood.<n>This paper presents a comprehensive finite-sample analysis of nonlinear ICA with neural network encoders.
  arXiv  Detail & Related papers  (2026-04-10T01:15:27Z)
• On Multi-Step Theorem Prediction via Non-Parametric Structural Priors [50.16583672681106]
  In this work, we explore training-free theorem prediction through the lens of in-context learning (ICL)<n>We propose Theorem Precedence Graphs, which encode temporal dependencies from historical solution traces as directed graphs, and impose explicit topological constraints that effectively prune the search space during inference.<n>Experiments on the FormalGeo7k benchmark show that our method achieves 89.29% accuracy, substantially outperforming ICL baselines and matching state-of-the-art supervised models.
  arXiv  Detail & Related papers  (2026-03-05T06:08:50Z)
• Sharp Convergence Rates for Masked Diffusion Models [53.117058231393834]
  We develop a total-variation based analysis for the Euler method that overcomes limitations.<n>Our results relax assumptions on score estimation, improve parameter dependencies, and establish convergence guarantees.<n>Overall, our analysis introduces a direct TV-based error decomposition along the CTMC trajectory and a decoupling-based path-wise analysis for FHS.
  arXiv  Detail & Related papers  (2026-02-26T00:47:51Z)
• Conjugate Learning Theory: Uncovering the Mechanisms of Trainability and Generalization in Deep Neural Networks [0.0]
  We develop a conjugate learning theoretical framework based on convex conjugate duality to characterize this learnability property.<n>We demonstrate that training deep neural networks (DNNs) with mini-batch descent (SGD) achieves global optima of empirical risk.<n>We derive deterministic and probabilistic bounds on generalization error based on conditional generalized entropy measures.
  arXiv  Detail & Related papers  (2026-02-18T04:26:55Z)
• The Procrustean Bed of Time Series: The Optimization Bias of Point-wise Loss [53.542743390809356]
  This paper aims to provide a first-principles analysis of the Expectation of Optimization Bias (EOB)<n>Our analysis reveals a fundamental paradigm paradox: the more deterministic and structured the time series, the more severe the bias by point-wise loss function.<n>We present a concrete solution that simultaneously achieves both principles via DFT or DWT.
  arXiv  Detail & Related papers  (2025-12-21T06:08:22Z)
• Bridging Internal Probability and Self-Consistency for Effective and Efficient LLM Reasoning [53.25336975467293]
  We present the first theoretical error decomposition analysis of methods such as perplexity and self-consistency.<n>Our analysis reveals a fundamental trade-off: perplexity methods suffer from substantial model error due to the absence of a proper consistency function.<n>We propose Reasoning-Pruning Perplexity Consistency (RPC), which integrates perplexity with self-consistency, and Reasoning Pruning, which eliminates low-probability reasoning paths.
  arXiv  Detail & Related papers  (2025-02-01T18:09:49Z)
• Neural Network Approximation for Pessimistic Offline Reinforcement Learning [17.756108291816908]
  We present a non-asymptotic estimation error of pessimistic offline RL using general neural network approximation. Our result shows that the estimation error consists of two parts: the first converges to zero at a desired rate on the sample size with partially controllable concentrability, and the second becomes negligible if the residual constraint is tight.
  arXiv  Detail & Related papers  (2023-12-19T05:17:27Z)
• Understanding, Predicting and Better Resolving Q-Value Divergence in Offline-RL [86.0987896274354]
  We first identify a fundamental pattern, self-excitation, as the primary cause of Q-value estimation divergence in offline RL. We then propose a novel Self-Excite Eigenvalue Measure (SEEM) metric to measure the evolving property of Q-network at training. For the first time, our theory can reliably decide whether the training will diverge at an early stage.
  arXiv  Detail & Related papers  (2023-10-06T17:57:44Z)
• Fine-grained analysis of non-parametric estimation for pairwise learning [9.676007573960383]
  We are concerned with the generalization performance of non-parametric estimation for pairwise learning. Our results can be used to handle a wide range of pairwise learning problems including ranking, AUC, pairwise regression and metric and similarity learning.
  arXiv  Detail & Related papers  (2023-05-31T08:13:14Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.