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Rethinking Model-based, Policy-based, and Value-based Reinforcement Learning via the Lens of Representation Complexity (2312.17248v2)

Published 28 Dec 2023 in cs.LG, cs.AI, cs.CC, stat.ML, and cs.DS

Abstract: Reinforcement Learning (RL) encompasses diverse paradigms, including model-based RL, policy-based RL, and value-based RL, each tailored to approximate the model, optimal policy, and optimal value function, respectively. This work investigates the potential hierarchy of representation complexity -- the complexity of functions to be represented -- among these RL paradigms. We first demonstrate that, for a broad class of Markov decision processes (MDPs), the model can be represented by constant-depth circuits with polynomial size or Multi-Layer Perceptrons (MLPs) with constant layers and polynomial hidden dimension. However, the representation of the optimal policy and optimal value proves to be $\mathsf{NP}$-complete and unattainable by constant-layer MLPs with polynomial size. This demonstrates a significant representation complexity gap between model-based RL and model-free RL, which includes policy-based RL and value-based RL. To further explore the representation complexity hierarchy between policy-based RL and value-based RL, we introduce another general class of MDPs where both the model and optimal policy can be represented by constant-depth circuits with polynomial size or constant-layer MLPs with polynomial size. In contrast, representing the optimal value is $\mathsf{P}$-complete and intractable via a constant-layer MLP with polynomial hidden dimension. This accentuates the intricate representation complexity associated with value-based RL compared to policy-based RL. In summary, we unveil a potential representation complexity hierarchy within RL -- representing the model emerges as the easiest task, followed by the optimal policy, while representing the optimal value function presents the most intricate challenge.

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Citations (1)
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Summary

  • The paper demonstrates that representing the environment model in RL is computationally simpler than approximating optimal policies and value functions.
  • It presents a framework quantifying representation complexity using constant-depth circuits and MLPs, highlighting a gap in computational demands.
  • The findings imply that RL algorithms tailored to sample efficiency must account for the increasing complexity from models to value functions.

Model Complexity in Reinforcement Learning

Introduction

In the field of Reinforcement Learning (RL), researchers typically focus on algorithms that fall into one of three categories: model-based RL, policy-based RL, and value-based RL. These methods involve approximating different components: the environment model, the optimal policy, and the optimal value function, respectively. Although analysis on statistical and optimization errors within these methods is extensive, the aspect of approximation errors — particularly the complexity of representing the crucial functions — has been less explored. This paper seeks to understand whether a hierarchy exists in the complexity required to represent functions within these RL paradigms.

Representation Complexity Framework

The concept of representation complexity serves as a pivotal cornerstone in understanding the computational demands of functions in RL frameworks. It provides a structured perspective on the function classes necessary to capture key elements of RL paradigms. This research investigates representation complexity by leveraging metrics from the realms of computational complexity and the expressiveness of Multi-Layer Perceptrons (MLPs).

Representation Complexity Results

The investigation yields several important findings:

  • For a wide range of Markov decision processes (MDPs), the models can be represented using constant-depth circuits with polynomial size or constant-layer MLPs with polynomial hidden dimensions. This signifies a relatively lower complexity for representing models.
  • Representing the optimal policy and value function is harder. Specifically, such representations for certain MDP classes prove to be NP-complete and unattainable via constant-layer MLPs of polynomial size. This highlights a clear complexity gap between model-based and model-free RL frameworks, i.e., policy-based and value-based RL.
  • Delving further into the intricacies of model-free RL paradigms, it is discovered that while both the model and optimal policy can be captured using constant-depth circuits or constant-layer MLPs of polynomial size, the optimal value functions are proven to be P-complete suggesting a more complex representation that is intricate within value-based RL compared to policy-based RL.

Practical Implications

The paper establishes a nuanced hierarchy in representation complexity across different RL categories: the underlying model is the easiest to represent, followed by difficulty in representing the optimal policy, with the most significant challenge emerging in representing the optimal value function. This hierarchy presents insights into the design of RL algorithms, especially from a sample efficiency perspective. Theoretical insights are provided on the significant role representation complexity may play in determining the disparate sample efficiency observed in different RL algorithms.

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