Spectral Alignment in Forward-Backward Representations via Temporal Abstraction
This addresses a fundamental challenge in learning successor representations for continuous control, though it appears incremental as it builds on existing forward-backward frameworks.
The paper tackled the spectral mismatch between high-rank transition dynamics and low-rank forward-backward representations in continuous spaces, showing that temporal abstraction reduces this mismatch and enables stable learning, particularly at high discount factors.
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 as 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.