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On The Sample Complexity Bounds In Bilevel Reinforcement Learning

Mudit Gaur, Utsav Singh, Amrit Singh Bedi, Raghu Pasupathu, Vaneet Aggarwal

TL;DR

This paper addresses the lack of theoretical guarantees for sample efficiency in bilevel reinforcement learning (BRL) with continuous state-action spaces. It introduces a penalty-based proxy objective $\Phi_{\sigma}$ and a fully first-order, Hessian-free algorithm, proving the first BRL sample complexity bound of $\tilde{Ω}(ε^{-3})$. The analysis hinges on the Polyak-Łojasiewicz (PL) condition to handle non-convex lower levels and extends to standard bilevel optimization with unbiased gradients, achieving the same rate and improving upon prior results. The work advances BRL theory and provides scalable, practical guidance for AI alignment and RLHF applications, avoiding costly second-order computations while delivering tight sample-efficiency guarantees.

Abstract

Bilevel reinforcement learning (BRL) has emerged as a powerful framework for aligning generative models, yet its theoretical foundations, especially sample complexity bounds, remain underexplored. In this work, we present the first sample complexity bound for BRL, establishing a rate of $\mathcal{O}(ε^{-3})$ in continuous state-action spaces. Traditional MDP analysis techniques do not extend to BRL due to its nested structure and non-convex lower-level problems. We overcome these challenges by leveraging the Polyak-Łojasiewicz (PL) condition and the MDP structure to obtain closed-form gradients, enabling tight sample complexity analysis. Our analysis also extends to general bi-level optimization settings with non-convex lower levels, where we achieve state-of-the-art sample complexity results of $\mathcal{O}(ε^{-3})$ improving upon existing bounds of $\mathcal{O}(ε^{-6})$. Additionally, we address the computational bottleneck of hypergradient estimation by proposing a fully first-order, Hessian-free algorithm suitable for large-scale problems.

On The Sample Complexity Bounds In Bilevel Reinforcement Learning

TL;DR

This paper addresses the lack of theoretical guarantees for sample efficiency in bilevel reinforcement learning (BRL) with continuous state-action spaces. It introduces a penalty-based proxy objective and a fully first-order, Hessian-free algorithm, proving the first BRL sample complexity bound of . The analysis hinges on the Polyak-Łojasiewicz (PL) condition to handle non-convex lower levels and extends to standard bilevel optimization with unbiased gradients, achieving the same rate and improving upon prior results. The work advances BRL theory and provides scalable, practical guidance for AI alignment and RLHF applications, avoiding costly second-order computations while delivering tight sample-efficiency guarantees.

Abstract

Bilevel reinforcement learning (BRL) has emerged as a powerful framework for aligning generative models, yet its theoretical foundations, especially sample complexity bounds, remain underexplored. In this work, we present the first sample complexity bound for BRL, establishing a rate of in continuous state-action spaces. Traditional MDP analysis techniques do not extend to BRL due to its nested structure and non-convex lower-level problems. We overcome these challenges by leveraging the Polyak-Łojasiewicz (PL) condition and the MDP structure to obtain closed-form gradients, enabling tight sample complexity analysis. Our analysis also extends to general bi-level optimization settings with non-convex lower levels, where we achieve state-of-the-art sample complexity results of improving upon existing bounds of . Additionally, we address the computational bottleneck of hypergradient estimation by proposing a fully first-order, Hessian-free algorithm suitable for large-scale problems.

Paper Structure

This paper contains 18 sections, 11 theorems, 117 equations, 1 figure, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Works
  3. Problem Formulation
  4. Proposed Approach
  5. Theoretical Analysis
  6. Proof sketch of Theorem \ref{['thm']}:
  7. Standard Bilevel Optimization: A Special Case
  8. Conclusion
  9. Acknowledgment
  10. Proof of Lemma \ref{['thm_supp']}
  11. Proof of Theorem \ref{['thm']}
  12. Supplementary Lemmas For Theorem \ref{['thm']}
  13. Proof of Theorem \ref{['thm_bi']}
  14. Supplementary Lemmas For Theorem \ref{['thm_bi']}
  15. Experiments
  16. ...and 3 more sections

Key Result

Theorem 1

Suppose Assumptions assump_1-assump_6 hold and we have $0 < \eta \le \frac{1}{2L}$, $0 \le \tau \le \frac{1}{L_{J}}$, $0 \le \tau^{'} \le \frac{1}{L_{h}}$ where $L,L_{J},L_{\sigma}$ are the smoothness constants of $\Phi_{\sigma}$,$J$ and $h_{\sigma}$ respectively. Then from Algorithm algo_1, we obt If we set $\sigma^{2} = \tilde{\Omega}(\epsilon)$, $B = \tilde{\Omega}(\epsilon^{-2})$, $n = \tild

Figures (1)

  • Figure 1: Training curves on Walker locomotion task (left) from the DeepMind Control Suite tassa2018deepmind and the Door Open manipulation task (right) from Meta-world mclean2025metaworld. The solid line and shaded regions respectively, denote mean and standard deviation of the success rate, across multiple seeds. Blue curve: PEBBLE, Red curve: OURS.

Theorems & Definitions (21)

  • Theorem 1
  • Lemma 1
  • Theorem 2
  • proof
  • Lemma 2: Uniform bound for a sample-based KL gradient estimator
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 11 more