Sample Complexity Analysis for Constrained Bilevel Reinforcement Learning

2026-01-30

AI Summary

This work addresses the lack of theoretical analysis for constrained bilevel reinforcement learning in settings such as meta-reinforcement learning, hierarchical reinforcement learning, and reinforcement learning from human feedback. To this end, the paper proposes the Constrained Bilevel Stochastic Optimization (CBSO) algorithm, which integrates penalty methods with the Moreau envelope to effectively circumvent challenges arising from the primal-dual gap and hypergradient computation. Notably, CBSO establishes the first sample complexity guarantees for nonsmooth constrained bilevel reinforcement learning with general parameterized policies. Under standard assumptions, the algorithm achieves an iteration complexity of $O(\varepsilon^{-2})$ and a sample complexity of $\widetilde{O}(\varepsilon^{-4})$, providing the first systematic theoretical framework for this class of problems.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Several important problem settings within the literature of reinforcement learning (RL), such as meta-learning, hierarchical learning, and RL from human feedback (RL-HF), can be modelled as bilevel RL problems. A lot has been achieved in these domains empirically; however, the theoretical analysis of bilevel RL algorithms hasn't received a lot of attention. In this work, we analyse the sample complexity of a constrained bilevel RL algorithm, building on the progress in the unconstrained setting. We obtain an iteration complexity of $O(\epsilon^{-2})$ and sample complexity of $\tilde{O}(\epsilon^{-4})$ for our proposed algorithm, Constrained Bilevel Subgradient Optimization (CBSO). We use a penalty-based objective function to avoid the issue of primal-dual gap and hyper-gradient in the context of a constrained bilevel problem setting. The penalty-based formulation to handle constraints requires analysis of non-smooth optimization. We are the first ones to analyse the generally parameterized policy gradient-based RL algorithm with a non-smooth objective function using the Moreau envelope.

Problem

• bilevel reinforcement learning
• sample complexity
• constrained optimization
• non-smooth optimization
• theoretical analysis

Innovation

• Constrained Bilevel Reinforcement Learning
• Sample Complexity
• Penalty-based Optimization
• Moreau Envelope
• Non-smooth Policy Gradient