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Minimax Optimal and Computationally Efficient Algorithms for Distributionally Robust Offline Reinforcement Learning

Zhishuai Liu, Pan Xu

TL;DR

The paper tackles distributionally robust offline reinforcement learning with function approximation by focusing on $d$-rectangular linear DRMDPs. It introduces two algorithms, DRPVI and VA-DRPVI, that achieve minimax-optimal instance-dependent suboptimality bounds by leveraging a novel variance-aware function-approximation mechanism and an uncertainty-decomposition framework. A range-shrinkage property of robust value functions is identified, enabling variance-based improvements and tighter bounds that scale favorably with the horizon. An information-theoretic lower bound demonstrates the intrinsic role of the uncertainty function, establishing near-optimality of the proposed methods and highlighting fundamental limits under distributional perturbations. The results collectively show that robust offline RL with linear function approximation is more challenging than standard offline RL, yet achieve computationally efficient, minimax-optimal learning in the $d$-rectangular DRMDP setting.

Abstract

Distributionally robust offline reinforcement learning (RL), which seeks robust policy training against environment perturbation by modeling dynamics uncertainty, calls for function approximations when facing large state-action spaces. However, the consideration of dynamics uncertainty introduces essential nonlinearity and computational burden, posing unique challenges for analyzing and practically employing function approximation. Focusing on a basic setting where the nominal model and perturbed models are linearly parameterized, we propose minimax optimal and computationally efficient algorithms realizing function approximation and initiate the study on instance-dependent suboptimality analysis in the context of robust offline RL. Our results uncover that function approximation in robust offline RL is essentially distinct from and probably harder than that in standard offline RL. Our algorithms and theoretical results crucially depend on a novel function approximation mechanism incorporating variance information, a new procedure of suboptimality and estimation uncertainty decomposition, a quantification of the robust value function shrinkage, and a meticulously designed family of hard instances, which might be of independent interest.

Minimax Optimal and Computationally Efficient Algorithms for Distributionally Robust Offline Reinforcement Learning

TL;DR

The paper tackles distributionally robust offline reinforcement learning with function approximation by focusing on -rectangular linear DRMDPs. It introduces two algorithms, DRPVI and VA-DRPVI, that achieve minimax-optimal instance-dependent suboptimality bounds by leveraging a novel variance-aware function-approximation mechanism and an uncertainty-decomposition framework. A range-shrinkage property of robust value functions is identified, enabling variance-based improvements and tighter bounds that scale favorably with the horizon. An information-theoretic lower bound demonstrates the intrinsic role of the uncertainty function, establishing near-optimality of the proposed methods and highlighting fundamental limits under distributional perturbations. The results collectively show that robust offline RL with linear function approximation is more challenging than standard offline RL, yet achieve computationally efficient, minimax-optimal learning in the -rectangular DRMDP setting.

Abstract

Distributionally robust offline reinforcement learning (RL), which seeks robust policy training against environment perturbation by modeling dynamics uncertainty, calls for function approximations when facing large state-action spaces. However, the consideration of dynamics uncertainty introduces essential nonlinearity and computational burden, posing unique challenges for analyzing and practically employing function approximation. Focusing on a basic setting where the nominal model and perturbed models are linearly parameterized, we propose minimax optimal and computationally efficient algorithms realizing function approximation and initiate the study on instance-dependent suboptimality analysis in the context of robust offline RL. Our results uncover that function approximation in robust offline RL is essentially distinct from and probably harder than that in standard offline RL. Our algorithms and theoretical results crucially depend on a novel function approximation mechanism incorporating variance information, a new procedure of suboptimality and estimation uncertainty decomposition, a quantification of the robust value function shrinkage, and a meticulously designed family of hard instances, which might be of independent interest.
Paper Structure (71 sections, 28 theorems, 182 equations, 3 figures, 1 table, 3 algorithms)

This paper contains 71 sections, 28 theorems, 182 equations, 3 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Notations:
  3. Most Related Work
  4. DRMDPs.
  5. DRMDPs with linear function approximation.
  6. Problem Formulation
  7. Standard MDPs.
  8. Distributionally robust MDPs.
  9. $d$-rectangular linear DRMDPs.
  10. Robust Bellman equation.
  11. Offline Dataset and the Learning Goal.
  12. Warmup: Robust Pessimistic Value Iteration
  13. Distributionally Robust Variance-Aware Pessimistic Value Iteration
  14. Intuition
  15. Variance estimation
  16. ...and 56 more sections

Key Result

Theorem 4.4

Under assumption:linear MDPassumption:feature coverage, $\forall K>\max\{512\log(2dH^2/\delta)/\kappa^2, 20449d^2H^2/\kappa\}$ and $\delta\in(0,1)$, if we set $\lambda=1$ and $\beta_1=\tilde{O}(\sqrt{d}H)$ in alg:DR-PVI, then with probability at least $1-\delta$, $\forall s\in{\mathcal{S}}$, the sub where $\bm{\Lambda}_h$ is the empirical covariance matrix defined in eq:covariance matrix.

Figures (3)

  • Figure 1: The source and the target linear MDP environments. The value on each arrow represents the transition probability. For the source MDP, there are five states and three steps, with the initial state being $x_1$, the fail state being $x_4$, and $x_5$ being an absorbing state with reward 1. The target MDP on the right is obtained by perturbing the transition probability at the first step of the source MDP, with others remaining the same.
  • Figure 2: Simulation results under different source domains. The $x$-axis represents the perturbation level corresponding to different target environments. $\rho_{1,4}$ is the input uncertainty level for our VA-DRPVI algorithm. $\Vert\xi\Vert_1$ is the hyperparameter of the linear DRMDP environment.
  • Figure 3: The nominal environment and the worst case environment. The value on each arrow represents the transition probability. The MDP has two states and $H$ steps. For the nominal environment, both $x_1$ and $x_2$ are absorbing states, which means that the state will always stay at the initial state in the nominal environment. The worst case environment on the right is obtained by perturbing the transition probability at the first step of the nominal environment, with others remain the same.

Theorems & Definitions (33)

  • Remark 4.1
  • Remark 4.2
  • Theorem 4.4
  • Corollary 4.5
  • Remark 4.6
  • Lemma 5.1: Range Shrinkage
  • Theorem 5.2
  • Corollary 5.3
  • Remark 5.4
  • Theorem 6.1
  • ...and 23 more