Asymptotic Analysis of Sample-averaged Q-learning

2024-10-14

AI Summary

This paper addresses the insufficient modeling of model uncertainty in reinforcement learning under complex, uncertain environments. We propose Sample-Averaged Q-Learning (SA-QL), a framework that explicitly captures data variability and estimation uncertainty via time-varying batch aggregation of rewards and next-state samples. Theoretically, we establish, for the first time under mild regularity conditions, the asymptotic normality of SA-QL estimators using the Functional Central Limit Theorem (FCLT), and introduce a hyperparameter-free stochastic scaling method to construct valid confidence intervals. Empirically, evaluations on challenging domains—including a wind-affected grid world and a slippery ice lake—demonstrate that batch scheduling critically influences both learning efficiency and uncertainty quantification accuracy. Our confidence intervals achieve coverage exceeding 93%, and the theoretically derived error bounds align closely with empirical observations.

Technology Category

Machine Learning: Calibration & Uncertainty QuantificationReasoning under Uncertainty: Stochastic OptimizationMultiagent Systems: Multiagent Systems under Uncertainty

Application Category

Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for rankingSemantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactionsEconomics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

Reinforcement learning (RL) has emerged as a key approach for training agents in complex and uncertain environments. Incorporating statistical inference in RL algorithms is essential for understanding and managing uncertainty in model performance. This paper introduces a generalized framework for time-varying batch-averaged Q-learning, termed sample-averaged Q-learning (SA-QL), which extends traditional single-sample Q-learning by aggregating samples of rewards and next states to better account for data variability and uncertainty. We leverage the functional central limit theorem (FCLT) to establish a novel framework that provides insights into the asymptotic normality of the sample-averaged algorithm under mild conditions. Additionally, we develop a random scaling method for interval estimation, enabling the construction of confidence intervals without requiring extra hyperparameters. Extensive numerical experiments across classic stochastic OpenAI Gym environments, including windy gridworld and slippery frozenlake, demonstrate how different batch scheduling strategies affect learning efficiency, coverage rates, and confidence interval widths. This work establishes a unified theoretical foundation for sample-averaged Q-learning, providing insights into effective batch scheduling and statistical inference for RL algorithms.

Problem

• Analyzes asymptotic behavior of sample-averaged Q-learning
• Develops confidence interval estimation without extra hyperparameters
• Explores batch scheduling impact on reinforcement learning efficiency

Innovation

• Generalized time-varying batch-averaged Q-learning
• Functional central limit theorem application
• Random scaling for confidence intervals