Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Limits of Pure Exploration in POMDPs: When the Observation Entropy is Enough

Riccardo Zamboni, Duilio Cirino, Marcello Restelli, Mirco Mutti

TL;DR

The paper addresses pure exploration in partially observable environments by proposing Maximum Observation Entropy (MOE), a tractable surrogate for maximizing latent-state entropy $H(S|\pi)$ through the observable $H(X|\pi)$. It develops a theoretical framework linking the MOE–MSE gap to the observation matrix $\mathbb{O}$ via spectral bounds $\log(1/\sigma_{\max}(\mathbb{O}^{\circ -1}))$ and $\log(\sigma_{\max}(\mathbb{O}))$, along with an information-based bound involving $H(X|S,\pi)$ and $H(S|X,\pi)$. The authors propose a trajectory-based policy-gradient approach for MOE, including Reg-MOE when $\mathbb{O}$ is known, and validate the methods on gridworlds, showing MOE can closely approximate MSE in well-behaved observation regimes and that Reg-MOE can mitigate gaps when observation noise is structured. Overall, the work provides a scalable blueprint for state-entropy maximization under partial observability and highlights intrinsic limits, motivating future belief-based extensions and more nuanced objectives.

Abstract

The problem of pure exploration in Markov decision processes has been cast as maximizing the entropy over the state distribution induced by the agent's policy, an objective that has been extensively studied. However, little attention has been dedicated to state entropy maximization under partial observability, despite the latter being ubiquitous in applications, e.g., finance and robotics, in which the agent only receives noisy observations of the true state governing the system's dynamics. How can we address state entropy maximization in those domains? In this paper, we study the simple approach of maximizing the entropy over observations in place of true latent states. First, we provide lower and upper bounds to the approximation of the true state entropy that only depends on some properties of the observation function. Then, we show how knowledge of the latter can be exploited to compute a principled regularization of the observation entropy to improve performance. With this work, we provide both a flexible approach to bring advances in state entropy maximization to the POMDP setting and a theoretical characterization of its intrinsic limits.

The Limits of Pure Exploration in POMDPs: When the Observation Entropy is Enough

TL;DR

The paper addresses pure exploration in partially observable environments by proposing Maximum Observation Entropy (MOE), a tractable surrogate for maximizing latent-state entropy through the observable . It develops a theoretical framework linking the MOE–MSE gap to the observation matrix via spectral bounds and , along with an information-based bound involving and . The authors propose a trajectory-based policy-gradient approach for MOE, including Reg-MOE when is known, and validate the methods on gridworlds, showing MOE can closely approximate MSE in well-behaved observation regimes and that Reg-MOE can mitigate gaps when observation noise is structured. Overall, the work provides a scalable blueprint for state-entropy maximization under partial observability and highlights intrinsic limits, motivating future belief-based extensions and more nuanced objectives.

Abstract

The problem of pure exploration in Markov decision processes has been cast as maximizing the entropy over the state distribution induced by the agent's policy, an objective that has been extensively studied. However, little attention has been dedicated to state entropy maximization under partial observability, despite the latter being ubiquitous in applications, e.g., finance and robotics, in which the agent only receives noisy observations of the true state governing the system's dynamics. How can we address state entropy maximization in those domains? In this paper, we study the simple approach of maximizing the entropy over observations in place of true latent states. First, we provide lower and upper bounds to the approximation of the true state entropy that only depends on some properties of the observation function. Then, we show how knowledge of the latter can be exploited to compute a principled regularization of the observation entropy to improve performance. With this work, we provide both a flexible approach to bring advances in state entropy maximization to the POMDP setting and a theoretical characterization of its intrinsic limits.
Paper Structure (19 sections, 4 theorems, 17 equations, 5 figures, 1 algorithm)

This paper contains 19 sections, 4 theorems, 17 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Preliminaries
  4. Notation.
  5. Problem Formulation
  6. A Formal Characterization of Maximum Observation Entropy
  7. Towards Principled Policy Gradients for MOE
  8. Known Observation Matrix.
  9. Numerical Validation
  10. Related Work
  11. Conclusions
  12. Missing Proofs
  13. Additional Details on the Experiments
  14. Domains
  15. Observations.
  16. ...and 4 more sections

Key Result

Theorem 4.1

Let $\mathbb{M}$ a POMDP and let $\pi \in \Pi \subseteq \Delta_{\mathcal{X}}^{\mathcal{A}}$ a policy. Then, it holds

Figures (5)

  • Figure 1: Entropy on latent states (MSE) achieved by PG for MSE, PG for MOE, and PG for Reg-MOE in gridworlds with various $\mathbb{O}$. We report the average and 95% c.i. over 16 runs.
  • Figure 2: Heatmaps representing the observation matrix $\mathbb{O}$ employed in the experiments of Section \ref{['sec:experiments']}. Note that in Figure \ref{['subfig:obs_image3']} the colormap has logarithmic scale.
  • Figure 3: Comparison of the performance with different values of the learning rate for various algorithms and domains.
  • Figure 4: A comparison of different values of regularization for varying emission matrices' quality and settings with and without glasses. For the low value of regularization, the performances of Reg-MOE are equivalent to the MOE performances.
  • Figure 5: Comparison of the policies learned by PG for MOE and PG for Reg-MOE over 2000 episodes. The magnitude of each arrow is proportional to the probability of the policy to choose that action, after marginalizing over all the possible observations emitted in that state.

Theorems & Definitions (7)

  • Theorem 4.1: Spectral Approximation Bounds
  • proof
  • Theorem 4.2: Information Approximation Bound
  • proof
  • Proposition 5.1: Policy Gradient for MOE
  • Corollary 5.2: Actionable Lower Bound
  • proof