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Statistical Analysis of Policy Space Compression Problem

Majid Molaei, Marcello Restelli, Alberto Maria Metelli, Matteo Papini

TL;DR

This research focuses on determining the necessary sample size to learn this compressed set accurately, using R\'enyi divergence to measure the similarity between true and estimated policy distributions and the $l_1$ norm, determining sample size requirements for both model-based and model-free settings.

Abstract

Policy search methods are crucial in reinforcement learning, offering a framework to address continuous state-action and partially observable problems. However, the complexity of exploring vast policy spaces can lead to significant inefficiencies. Reducing the policy space through policy compression emerges as a powerful, reward-free approach to accelerate the learning process. This technique condenses the policy space into a smaller, representative set while maintaining most of the original effectiveness. Our research focuses on determining the necessary sample size to learn this compressed set accurately. We employ Rényi divergence to measure the similarity between true and estimated policy distributions, establishing error bounds for good approximations. To simplify the analysis, we employ the $l_1$ norm, determining sample size requirements for both model-based and model-free settings. Finally, we correlate the error bounds from the $l_1$ norm with those from Rényi divergence, distinguishing between policies near the vertices and those in the middle of the policy space, to determine the lower and upper bounds for the required sample sizes.

Statistical Analysis of Policy Space Compression Problem

TL;DR

This research focuses on determining the necessary sample size to learn this compressed set accurately, using R\'enyi divergence to measure the similarity between true and estimated policy distributions and the norm, determining sample size requirements for both model-based and model-free settings.

Abstract

Policy search methods are crucial in reinforcement learning, offering a framework to address continuous state-action and partially observable problems. However, the complexity of exploring vast policy spaces can lead to significant inefficiencies. Reducing the policy space through policy compression emerges as a powerful, reward-free approach to accelerate the learning process. This technique condenses the policy space into a smaller, representative set while maintaining most of the original effectiveness. Our research focuses on determining the necessary sample size to learn this compressed set accurately. We employ Rényi divergence to measure the similarity between true and estimated policy distributions, establishing error bounds for good approximations. To simplify the analysis, we employ the norm, determining sample size requirements for both model-based and model-free settings. Finally, we correlate the error bounds from the norm with those from Rényi divergence, distinguishing between policies near the vertices and those in the middle of the policy space, to determine the lower and upper bounds for the required sample sizes.

Paper Structure

This paper contains 20 sections, 10 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Controlled Markov Processes
  4. Policy Optimization
  5. Importance Sampling and Rényi Divergence
  6. Concentration Bounds for Empirical Estimates
  7. Problem Formulation
  8. Estimation Error Range
  9. Geometry of the Problem
  10. Upper Bound to the Error Threshold
  11. Analysis using Total Variation
  12. Number of Samples with Known Model
  13. Number of Samples with Unknown Model
  14. Analysis using Rényi Divergence
  15. The Relationship between sigma and sigmaTV
  16. ...and 5 more sections

Key Result

lemma 2.1

Let $\{ X_i \}_{i = 1}^N$ be i.i.d. random values over $[a]$ such that $\mathbb{P} (X_i = m) = p_m$, and let $\hat{p}_m = \frac{1}{N} \sum_{i = 1}^N \mathds{1} (X_i = m)$ be the empirical estimate of $p_m$. For every confidence $\delta \in (0, 1)$, it holds:

Theorems & Definitions (10)

  • lemma 2.1
  • lemma 2.2
  • Proposition 1
  • lemma 5.1
  • lemma 6.1
  • theorem 6.2
  • lemma 6.3
  • theorem 6.4
  • lemma 6.5
  • theorem 6.6