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Operator World Models for Reinforcement Learning

Pietro Novelli, Marco Pratticò, Massimiliano Pontil, Carlo Ciliberto

TL;DR

This work introduces a novel approach based on learning a world model of the environment using conditional mean embeddings that leads to POWR, a new RL algorithm for which it is proved convergence rates to the global optimum.

Abstract

Policy Mirror Descent (PMD) is a powerful and theoretically sound methodology for sequential decision-making. However, it is not directly applicable to Reinforcement Learning (RL) due to the inaccessibility of explicit action-value functions. We address this challenge by introducing a novel approach based on learning a world model of the environment using conditional mean embeddings. Leveraging tools from operator theory we derive a closed-form expression of the action-value function in terms of the world model via simple matrix operations. Combining these estimators with PMD leads to POWR, a new RL algorithm for which we prove convergence rates to the global optimum. Preliminary experiments in finite and infinite state settings support the effectiveness of our method

Operator World Models for Reinforcement Learning

TL;DR

This work introduces a novel approach based on learning a world model of the environment using conditional mean embeddings that leads to POWR, a new RL algorithm for which it is proved convergence rates to the global optimum.

Abstract

Policy Mirror Descent (PMD) is a powerful and theoretically sound methodology for sequential decision-making. However, it is not directly applicable to Reinforcement Learning (RL) due to the inaccessibility of explicit action-value functions. We address this challenge by introducing a novel approach based on learning a world model of the environment using conditional mean embeddings. Leveraging tools from operator theory we derive a closed-form expression of the action-value function in terms of the world model via simple matrix operations. Combining these estimators with PMD leads to POWR, a new RL algorithm for which we prove convergence rates to the global optimum. Preliminary experiments in finite and infinite state settings support the effectiveness of our method
Paper Structure (40 sections, 29 theorems, 86 equations, 2 figures, 1 algorithm)

This paper contains 40 sections, 29 theorems, 86 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions
  3. Problem Formulation and Policy Mirror Descent
  4. Policy Mirror Descent (PMD)
  5. PMD in Reinforcement Learning
  6. Operator World Models
  7. Conditional Expectation Operators
  8. Operator Formulation of RL
  9. Learning the World Model via Conditional Mean Embeddings
  10. Estimating the Action-value Function $q_\pi$
  11. Proposed Algorithm: POWR
  12. Choice of the Feature Maps
  13. POWR
  14. Theoretical Analysis
  15. POWR is Well-defined
  16. ...and 25 more sections

Key Result

Theorem 1

In the tabular setting, let $(\pi_t)_{t\in{\mathbb{N}}}$ be a sequence of policies obtained by applying the PMD update in eq:point-wise-PMD-classic with functions $\hat{q}_{\pi_{t}}:\Omega\to{\mathbb{R}}$ in place of $q_{\pi_t}$ and $D$ a suitable Bregman divergence. For any $T\in{\mathbb{N}}$ and $

Figures (2)

  • Figure 1: The plots show the average cumulative reward in different environments with respect to the timesteps (i.e. number of interactions with MDP). The dark lines represent the mean of the cumulative reward and the shaded area is the minimum and maximum values reached across $7$ independent runs. The horizontal dashed lines represent the reward threshold proposed by the Gym library brockman2016openai.
  • Figure 2: Mean timestep at which various algorithms attain a specified reward threshold during their training. The reward targets are set at $0.8$ for FrozenLake-v1, $6$ for Taxi-v3, and $-110$ for MountainCar-v0. The absence of a box indicates that the corresponding algorithm was unable to meet the reward threshold within the training process.

Theorems & Definitions (52)

  • Theorem 1: Inexact PMD (Sec. 5 in xiao2022) -- Informal
  • Proposition 1: Well-specified CME
  • Definition 1: $({\mathcal{G}},{\mathcal{F}})$-compatibility
  • Proposition 1
  • Proposition 1: Separable Spaces
  • Theorem 2
  • Corollary 2
  • Theorem 3: Convergenge of Inexact PMD
  • Lemma 3: Implications of the Simulation Lemma
  • Theorem 4
  • ...and 42 more