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On the Theory of Policy Gradient Methods: Optimality, Approximation, and Distribution Shift

Alekh Agarwal, Sham M. Kakade, Jason D. Lee, Gaurav Mahajan

TL;DR

This work provides a rigorous global-convergence and average-case analysis of policy gradient methods in discounted MDPs, covering both tabular and function-approximation settings. It introduces a distribution-shift perspective and a distribution-mismatch coefficient to quantify exploration challenges, and it yields dimension-free rates for natural policy gradient in the tabular regime. For function approximation, it develops an estimation/transfer-error decomposition, enabling meaningful guarantees without worst-case state-space dependence and handling both log-linear and neural policies via Q-NPG and NPG frameworks. The results bridge policy-gradient optimization with supervised learning under distribution shift, offering insights into sample complexity, regularization, and the role of conditioning in practical reinforcement learning algorithms.

Abstract

Policy gradient methods are among the most effective methods in challenging reinforcement learning problems with large state and/or action spaces. However, little is known about even their most basic theoretical convergence properties, including: if and how fast they converge to a globally optimal solution or how they cope with approximation error due to using a restricted class of parametric policies. This work provides provable characterizations of the computational, approximation, and sample size properties of policy gradient methods in the context of discounted Markov Decision Processes (MDPs). We focus on both: "tabular" policy parameterizations, where the optimal policy is contained in the class and where we show global convergence to the optimal policy; and parametric policy classes (considering both log-linear and neural policy classes), which may not contain the optimal policy and where we provide agnostic learning results. One central contribution of this work is in providing approximation guarantees that are average case -- which avoid explicit worst-case dependencies on the size of state space -- by making a formal connection to supervised learning under distribution shift. This characterization shows an important interplay between estimation error, approximation error, and exploration (as characterized through a precisely defined condition number).

On the Theory of Policy Gradient Methods: Optimality, Approximation, and Distribution Shift

TL;DR

This work provides a rigorous global-convergence and average-case analysis of policy gradient methods in discounted MDPs, covering both tabular and function-approximation settings. It introduces a distribution-shift perspective and a distribution-mismatch coefficient to quantify exploration challenges, and it yields dimension-free rates for natural policy gradient in the tabular regime. For function approximation, it develops an estimation/transfer-error decomposition, enabling meaningful guarantees without worst-case state-space dependence and handling both log-linear and neural policies via Q-NPG and NPG frameworks. The results bridge policy-gradient optimization with supervised learning under distribution shift, offering insights into sample complexity, regularization, and the role of conditioning in practical reinforcement learning algorithms.

Abstract

Policy gradient methods are among the most effective methods in challenging reinforcement learning problems with large state and/or action spaces. However, little is known about even their most basic theoretical convergence properties, including: if and how fast they converge to a globally optimal solution or how they cope with approximation error due to using a restricted class of parametric policies. This work provides provable characterizations of the computational, approximation, and sample size properties of policy gradient methods in the context of discounted Markov Decision Processes (MDPs). We focus on both: "tabular" policy parameterizations, where the optimal policy is contained in the class and where we show global convergence to the optimal policy; and parametric policy classes (considering both log-linear and neural policy classes), which may not contain the optimal policy and where we provide agnostic learning results. One central contribution of this work is in providing approximation guarantees that are average case -- which avoid explicit worst-case dependencies on the size of state space -- by making a formal connection to supervised learning under distribution shift. This characterization shows an important interplay between estimation error, approximation error, and exploration (as characterized through a precisely defined condition number).

Paper Structure

This paper contains 41 sections, 40 theorems, 291 equations, 2 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Tabular case:
  4. Function Approximation:
  5. Related Work
  6. Setting
  7. Policy Parameterizations.
  8. Policy gradients.
  9. The performance difference lemma.
  10. The distribution mismatch coefficient.
  11. Notation.
  12. Warmup: Constrained Tabular Parameterization
  13. Gradient Domination
  14. Convergence Rates for Projected Gradient Ascent
  15. A Lower Bound: Vanishing Gradients and Saddle Points
  16. ...and 26 more sections

Key Result

Lemma 3.1

There is an MDP $M$ (described in Figure fig:noncon) such that the optimization problem $V^{\pi_\theta} (s)$ is not concave for both the direct and softmax parameterizations.

Figures (2)

  • Figure 1: (Non-concavity example) A deterministic MDP corresponding to Lemma \ref{['lemma:softmax-noncon']} where $V^{\pi_\theta} (s)$ is not concave. Numbers on arrows represent the rewards for each action.
  • Figure 2: (Vanishing gradient example) A deterministic, chain MDP of length $H+2$. We consider a policy where $\pi(a | s_i) = \theta_{s_i,a}$ for $i=1,2,\ldots,H$. Rewards are $0$ everywhere other than $r(s_{H+1}, a_1) = 1$. See Proposition \ref{['proposition:small_grad']}.

Theorems & Definitions (58)

  • Lemma 3.1
  • Lemma 3.2
  • Definition 3.1: Distribution mismatch coefficient
  • Lemma 4.1: Gradient domination
  • Theorem 4.1
  • Proposition 4.1: Vanishing gradients at suboptimal parameters
  • Remark 4.1
  • Remark 4.2
  • Remark 4.3
  • Theorem 5.1: Global convergence for softmax parameterization
  • ...and 48 more