Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sample Complexity Analysis for Constrained Bilevel Reinforcement Learning

Naman Saxena, Vaneet Aggarwal

TL;DR

The paper addresses constrained bilevel reinforcement learning where the inner problem enforces inequality constraints. It proposes Constrained Bilevel Subgradient Optimization (CBSO), a penalty-based objective that sidesteps primal-dual gaps and leverages the Moreau envelope to handle non-smoothness. The authors prove non-asymptotic guarantees for CBSO, achieving an iteration complexity of $O(\epsilon^{-2})$ and a sample complexity of $\tilde{O}(\epsilon^{-4})$, under mild assumptions and without convexity of the inner problem. This work is the first to provide convergence guarantees for constrained bilevel RL and its optimization counterpart, and the techniques extend to general constrained bilevel optimization with nonconvex inner problems. The results have implications for safety-aware RL settings, RLHF, and hierarchical/RL-based systems requiring constraint satisfaction during learning.

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 $O(ε^{-2})$ and sample complexity of $\tilde{O}(ε^{-4})$ 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.

Sample Complexity Analysis for Constrained Bilevel Reinforcement Learning

TL;DR

The paper addresses constrained bilevel reinforcement learning where the inner problem enforces inequality constraints. It proposes Constrained Bilevel Subgradient Optimization (CBSO), a penalty-based objective that sidesteps primal-dual gaps and leverages the Moreau envelope to handle non-smoothness. The authors prove non-asymptotic guarantees for CBSO, achieving an iteration complexity of and a sample complexity of , under mild assumptions and without convexity of the inner problem. This work is the first to provide convergence guarantees for constrained bilevel RL and its optimization counterpart, and the techniques extend to general constrained bilevel optimization with nonconvex inner problems. The results have implications for safety-aware RL settings, RLHF, and hierarchical/RL-based systems requiring constraint satisfaction during learning.

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 and sample complexity of 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.
Paper Structure (25 sections, 23 theorems, 171 equations, 2 tables, 1 algorithm)

This paper contains 25 sections, 23 theorems, 171 equations, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Works
  3. Bilevel optimization
  4. Bilevel RL
  5. Constrained bilevel optimization
  6. Problem Formulation
  7. Constrained Markov Decision Process (CMDP)
  8. Constrained Bilevel Reinforcement Learning
  9. Proposed Approach
  10. Theoretical Analysis
  11. Proof Sketch
  12. Constrained Bilevel Optimization
  13. Conclusion
  14. Extended Related Work
  15. Bilevel optimization
  16. ...and 10 more sections

Key Result

Lemma 4.1

Let $\bar{x}$ be the $\epsilon$-optimal solution of the penalty-based objective (Eq. eq:app-biobj3). Then, $\bar{x}$ is also an $\epsilon'$-optimal solution of the relaxed objective (Eq. eq:biobj4) with at most $\epsilon_\lambda$ violation of the constraints. Here, $C_f=\max_{x,y}|f(x,y)|$, $C_g=\ma

Theorems & Definitions (49)

  • Lemma 4.1
  • Definition 4.2
  • Definition 5.2
  • Theorem 5.7
  • proof
  • Remark 5.8
  • Definition 5.9
  • Lemma 5.10
  • Definition 5.11
  • Lemma 5.12
  • ...and 39 more