Temporal abstraction is shown to align the spectrum of forward-backward representations, stabilizing learning at high discount factors in continuous control
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Synopsis
The work analyzes temporal abstraction as a mechanism to mitigate the spectral mismatch in forward-backward (FB) representations, showing by characterizing the spectral properties of the transition operator that temporal abstraction acts analogously to a low-pass filter suppressing high-frequency spectral components, thereby reducing the effective rank of the induced successor representation while preserving a formal bound on the value function error, and empirically showing that this alignment is a key factor for stable FB learning at high discount factors.
Figure 1 : Q-function via Successor Representation (SR). The SR enables rapid value inference for arbitrary goals (e.g., star marker ). Low-rank structure in SR is desirable for navigation, as it preserves topological features (e.g., rooms) while suppressing transient dynamics. Top: In discrete MDPs, the SR can be computed from the transition matrix. Bottom: In continuous domains, FB ( FB ) learning approximates the SR, where the embedding dimension controls the rank of the approximation. Low-rank structure can arise through (1) explicit constraints ( Low-Rank column; e.g., SVD or small embeddings), (2) long horizons ( High γ \gamma column), or (3) temporal abstraction ( Temp. Abs. column; e.g., action repetition). In continuous settings, temporal abstraction provides the spectral alignment needed for effective bootstrapping, whereas high γ \gamma or overly restrictive bottlenecks can impair representation learning, leading to Q-functions with many erroneous local maxima.
arXivInterpretation
The paper identifies a fundamental spectral mismatch between the high-rank transition dynamics of continuous environments and the low-rank bottleneck of the FB architecture, which makes accurate low-rank representation learning difficult. It attributes the difficulty in FB representation learning explicitly to a mismatch in spectral structure rather than solely to optimization or function approximation issues. Based on a characterization of the spectral properties of the transition operator, constituting a theoretical analysis.
The paper shows that temporal abstraction is spectrally analogous to a low-pass filter that suppresses high-frequency spectral components, reducing the effective rank of the induced successor representation while preserving a formal bound on the value function error. It provides a principled spectral account of temporal abstraction, elevating its role from an empirical trick to a mechanism that shapes the spectral structure of the underlying MDP. A formal bound is given as a theoretical guarantee, making this an analytical result.
Experiments indicate that this spectral alignment is a key factor for stable FB learning, particularly at high discount factors where bootstrapping becomes error-prone. It links the theoretical notion of spectral alignment to learning stability at high discount factors in practice. Based on empirical observations in continuous control tasks; the abstract does not report specific task counts or metric values.
Perspective
The result targets successor representation learning based on FB representations in continuous control, in settings that require long-horizon representations and involve high discount factors; in such settings, temporal abstraction can serve as a principled mechanism for shaping the spectral structure of the underlying MDP, reducing effective rank while preserving a value function error bound. For researchers and engineers working on low-rank representation learning, temporal abstraction design, and long-horizon reinforcement learning, this perspective can guide the choice of representation structure and abstraction scale.
The abstract does not report the experimental environments, number of tasks, discount factor values, or specific performance metrics, so the quantitative relationship between spectral alignment and stability still needs to be checked in the main text; the exact form of the formal bound, its assumptions, and its tightness also need confirmation in the main text. In addition, how the specific implementation of temporal abstraction (such as abstraction scale and temporal span selection) affects the results is not expanded in the abstract and remains an open question worth attention.