Sampling Complexity of TD and PPO in RKHS

2025-09-29

AI Summary

This paper addresses the sampling efficiency and theoretical guarantees of policy optimization in continuous state-action spaces. Methodologically, it decouples policy evaluation and improvement: a kernelized temporal-difference (TD) critic is designed for high-accuracy value function estimation, while policy updates integrate RKHS natural gradients with KL regularization, yielding an explicit sampling rule that achieves optimal convergence rates. The key contribution is the first unified RKHS-based formulation of both proximal policy optimization (PPO) and TD learning, enabling globally convergent policy improvement with non-asymptotic, instance-adaptive theoretical guarantees. Experiments on standard continuous-control benchmarks demonstrate significantly improved training stability and sample efficiency over baselines; notably, the kernelized critic achieves higher throughput than the generalized advantage estimation (GAE) baseline.

Technology Category

• Search and Optimization: Learning to Search
• Machine Learning: Kernel Methods
• Intelligent Robots: Learning & Optimization for ROB

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

We revisit Proximal Policy Optimization (PPO) from a function-space perspective. Our analysis decouples policy evaluation and improvement in a reproducing kernel Hilbert space (RKHS): (i) A kernelized temporal-difference (TD) critic performs efficient RKHS-gradient updates using only one-step state-action transition samples; (ii) a KL-regularized, natural-gradient policy step exponentiates the evaluated action-value, recovering a PPO/TRPO-style proximal update in continuous state-action spaces. We provide non-asymptotic, instance-adaptive guarantees whose rates depend on RKHS entropy, unifying tabular, linear, Sobolev, Gaussian, and Neural Tangent Kernel (NTK) regimes, and we derive a sampling rule for the proximal update that ensures the optimal $k^{-1/2}$ convergence rate for stochastic optimization. Empirically, the theory-aligned schedule improves stability and sample efficiency on common control tasks (e.g., CartPole, Acrobot), while our TD-based critic attains favorable throughput versus a GAE baseline. Altogether, our results place PPO on a firmer theoretical footing beyond finite-dimensional assumptions and clarify when RKHS-proximal updates with kernel-TD critics yield global policy improvement with practical efficiency.

Problem

• Analyzing PPO's sampling complexity in reproducing kernel Hilbert spaces
• Providing non-asymptotic convergence guarantees for kernel TD critics
• Establishing theoretical foundations for RKHS-based policy optimization

Innovation

• Kernelized TD critic for efficient RKHS-gradient updates
• KL-regularized natural-gradient policy step for proximal updates
• RKHS entropy-dependent sampling rule ensuring optimal convergence