Spectral Alignment in Forward-Backward Representations via Temporal Abstraction

TL;DR

This paper investigates how temporal abstraction can reduce spectral mismatch in forward-backward representations, improving low-rank approximation and stability in continuous control tasks.

Contribution

It provides a spectral analysis showing temporal abstraction acts as a low-pass filter, enabling better low-rank successor representations in continuous environments.

Findings

Temporal abstraction suppresses high-frequency spectral components.

Alignment via temporal abstraction improves stability at high discount factors.

Spectral analysis guides effective long-horizon representation learning.

Abstract

Forward-backward (FB) representations provide a powerful framework for learning the successor representation (SR) in continuous spaces by enforcing a low-rank factorization. However, a fundamental spectral mismatch often exists between the high-rank transition dynamics of continuous environments and the low-rank bottleneck of the FB architecture, making accurate low-rank representation learning difficult. In this work, we analyze temporal abstraction as a mechanism to mitigate this mismatch. By characterizing the spectral properties of the transition operator, we show that temporal abstraction acts analogously to a low-pass filter that suppresses high-frequency spectral components. This suppression reduces the effective rank of the induced SR while preserving a formal bound on the resulting value function error. Empirically, we show that this alignment is a key factor for stable FB learning, particularly at high discount factors where bootstrapping becomes error-prone. Our results identify temporal abstraction as a principled mechanism for shaping the spectral structure of the underlying MDP and enabling effective long-horizon representations in continuous control.