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Bilevel reinforcement learning via the development of hyper-gradient without lower-level convexity

Yan Yang, Bin Gao, Ya-xiang Yuan

TL;DR

This work develops a fully first-order framework for bilevel reinforcement learning by deriving the hyper-gradient without assuming lower-level convexity, using the fixed-point structure of entropy-regularized RL. It introduces model-based (M-SoBiRL) and model-free (SoBiRL) algorithms, and extends to stochastic settings (Stoc-SoBiRL), with convergence guarantees: $\mathcal{O}(\epsilon^{-1})$ for deterministic methods and $\widetilde{\mathcal{O}}(\epsilon^{-1.5})$ outer iterations and $\widetilde{\mathcal{O}}(\epsilon^{-3.5})$ samples for the stochastic variant. The hyper-gradient combines exploitation and exploration, enabling joint optimization of reward shaping and RLHF-style objectives while requiring only first-order oracles. Empirical results on RLHF tasks (Atari, Mujoco) and a synthetic BiRL problem corroborate the effectiveness and scalability of the approach, highlighting the practical impact of first-order BiRL methods in complex hierarchical RL settings.

Abstract

Bilevel reinforcement learning (RL), which features intertwined two-level problems, has attracted growing interest recently. The inherent non-convexity of the lower-level RL problem is, however, to be an impediment to developing bilevel optimization methods. By employing the fixed point equation associated with the regularized RL, we characterize the hyper-gradient via fully first-order information, thus circumventing the assumption of lower-level convexity. This, remarkably, distinguishes our development of hyper-gradient from the general AID-based bilevel frameworks since we take advantage of the specific structure of RL problems. Moreover, we design both model-based and model-free bilevel reinforcement learning algorithms, facilitated by access to the fully first-order hyper-gradient. Both algorithms enjoy the convergence rate $O(ε^{-1})$. To extend the applicability, a stochastic version of the model-free algorithm is proposed, along with results on its iteration and sample complexity. In addition, numerical experiments demonstrate that the hyper-gradient indeed serves as an integration of exploitation and exploration.

Bilevel reinforcement learning via the development of hyper-gradient without lower-level convexity

TL;DR

This work develops a fully first-order framework for bilevel reinforcement learning by deriving the hyper-gradient without assuming lower-level convexity, using the fixed-point structure of entropy-regularized RL. It introduces model-based (M-SoBiRL) and model-free (SoBiRL) algorithms, and extends to stochastic settings (Stoc-SoBiRL), with convergence guarantees: for deterministic methods and outer iterations and samples for the stochastic variant. The hyper-gradient combines exploitation and exploration, enabling joint optimization of reward shaping and RLHF-style objectives while requiring only first-order oracles. Empirical results on RLHF tasks (Atari, Mujoco) and a synthetic BiRL problem corroborate the effectiveness and scalability of the approach, highlighting the practical impact of first-order BiRL methods in complex hierarchical RL settings.

Abstract

Bilevel reinforcement learning (RL), which features intertwined two-level problems, has attracted growing interest recently. The inherent non-convexity of the lower-level RL problem is, however, to be an impediment to developing bilevel optimization methods. By employing the fixed point equation associated with the regularized RL, we characterize the hyper-gradient via fully first-order information, thus circumventing the assumption of lower-level convexity. This, remarkably, distinguishes our development of hyper-gradient from the general AID-based bilevel frameworks since we take advantage of the specific structure of RL problems. Moreover, we design both model-based and model-free bilevel reinforcement learning algorithms, facilitated by access to the fully first-order hyper-gradient. Both algorithms enjoy the convergence rate . To extend the applicability, a stochastic version of the model-free algorithm is proposed, along with results on its iteration and sample complexity. In addition, numerical experiments demonstrate that the hyper-gradient indeed serves as an integration of exploitation and exploration.
Paper Structure (38 sections, 44 theorems, 271 equations, 4 figures, 2 tables, 5 algorithms)

This paper contains 38 sections, 44 theorems, 271 equations, 4 figures, 2 tables, 5 algorithms.

Table of Contents

  1. INTRODUCTION
  2. RELATED WORK
  3. PROBLEM FORMULATION
  4. Entropy-regularized MDPs
  5. Bilevel RL Formulation
  6. Applications of Bilevel RL
  7. MODEL-BASED SOFT BILEVEL REINFORCEMENT LEARNING
  8. MODEL-FREE SOFT BILEVEL REINFORCEMENT LEARNING
  9. THE STOCHASTIC EXTENSION
  10. THEORETICAL ANALYSIS
  11. EXPERIMENTS
  12. RELATED WORK IN BILEVEL OPTIMIZATION
  13. VECTOR AND MATRIX NOTATION
  14. PRELIMINARIES ON NUMERICAL LINEAR ALGEBRA
  15. PROOF IN SECTION \ref{['sec:MSoBiRL']}
  16. ...and 23 more sections

Key Result

Proposition 4.1

For any $x\in\mathbb{R}^n$, $\varphi(x,\cdot)$ is a contraction mapping, i.e., $\|\nabla_v \varphi(x,v)\|_{\infty}=\gamma<1$, and the matrix ${I-\nabla_{v}\varphi(x,v)}$ is invertible. Consequently, $V^*(x)$ is the unique fixed point of $\varphi(x,\cdot)$, with a well-defined derivative $\nabla V^*( Additionally, $\nabla_v \varphi\left( {x,V^*(x)} \right)$ coincides with the $\gamma$-scaled transi

Figures (4)

  • Figure 1: Comparison of algorithms on the Atari game, BeamRider, evaluated by the ground-truth reward. Each bilevel algorithm collects a total of $3000$ trajectory pairs. The running average over $15$ consecutive episodes is adopted for the presentation, and results are averaged over $5$ seeds.
  • Figure 2: Comparison on the Atari games---Seaquest and SpaceInvaders---evaluated by the ground-truth reward. Each bilevel algorithm collects a total of $3000$ trajectory pairs. The results are averaged over $5$ seeds.
  • Figure 3: Comparison of algorithms on the Mujoco simulations---HalfCheetah, Walker2d, and Hopper---evaluated by the ground-truth reward. Each bilevel algorithm collects a total of $3000$ trajectory pairs. The results are averaged over $5$ seeds.
  • Figure 4: A synthetic bilevel RL problem to verify the model-based algorithm, M-SoBiRL. Metrics are the hyper-gradient norm $\left\| {\nabla \phi(x)} \right\| _2$ and the upper-level loss $f(x,\pi)$.

Theorems & Definitions (83)

  • Proposition 4.1
  • Proposition 5.1
  • Proposition 5.2: Hyper-gradient
  • Theorem 7.4: Model-based
  • Proposition 7.5
  • Theorem 7.6: Model-free
  • Theorem 7.7: Stochastic
  • Proposition C.1
  • proof
  • Definition C.2
  • ...and 73 more