Spectral Alignment in Forward-Backward Representations via Temporal Abstraction
| Authors | Seyed Mahdi B. Azad et al. |
| Year | 2026 |
| Field | Machine Learning |
| arXiv | 2603.20103 |
| Download | |
| Categories | cs.LG, cs.AI, cs.RO |
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 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.
Engineering Breakdown
Plain English
This paper addresses a fundamental problem in learning successor representations (SR) in continuous spaces: the forward-backward architecture enforces a low-rank bottleneck that conflicts with the high-rank spectral properties of real environment dynamics. The authors analyze temporal abstraction—grouping multiple timesteps together—as a solution and show mathematically that it acts as a low-pass filter to suppress high-frequency spectral components, effectively reducing the mismatch between the environment's complexity and the model's capacity constraints. Their analysis provides formal bounds on value function approximation error, showing this trade-off can preserve performance while enabling accurate low-rank representation learning.
Core Technical Contribution
The core innovation is formalizing temporal abstraction as a spectral filtering mechanism that bridges the rank mismatch in forward-backward SR learning. Prior work treated temporal abstraction informally or used it ad-hoc; this paper provides rigorous spectral analysis showing precisely how aggregating timesteps suppresses high-frequency components of the transition operator's spectrum. The authors prove that this suppression reduces effective rank while maintaining formal bounds on downstream value function error—moving from empirical tricks to principled theory. This theoretical characterization enables principled selection of abstraction levels rather than manual tuning.
How It Works
The method starts with the transition dynamics in continuous spaces, which have a full or high-rank spectrum that the low-rank factorization in forward-backward architecture cannot capture. Temporal abstraction groups τ consecutive timesteps into single macro-actions, which corresponds to raising the original transition matrix to the τ-th power (M^τ). Spectral analysis shows that this operation naturally attenuates eigenvalues corresponding to high-frequency oscillations while preserving low-frequency structure, effectively filtering the spectrum. The authors then prove a bound on how well the resulting low-rank SR approximates the true SR, quantifying the error introduced by this filtering. The output is a practical recipe: choose τ large enough to suppress problematic high-frequency components while small enough that the bound remains acceptable for your task.
Production Impact
In production RL systems operating in continuous spaces (robotics, autonomous systems, complex simulations), this provides a principled way to reduce model capacity requirements without manually tuning abstraction levels. Instead of experimenting with different frame-skip rates or action repeat counts empirically, engineers can analyze the spectrum of their environment's dynamics (via eigendecomposition of empirical transition matrices) and select τ to target specific spectral modes. This directly reduces memory footprint and computation per step—critical for resource-constrained edge deployment or high-frequency control. The trade-off is slightly stale value estimates (error bounded by the spectral filtering), but the paper quantifies this formally, allowing engineers to make explicit accuracy-vs-efficiency decisions rather than guessing.
Limitations and When Not to Use This
The paper assumes access to or ability to estimate the spectral properties of the transition operator, which is non-trivial in truly high-dimensional or unknown environments—you may need to collect substantial data to estimate eigenvalues reliably. The temporal abstraction approach fundamentally trades off temporal resolution for representation quality, which fails for tasks requiring precise short-timescale decisions or high-frequency feedback (e.g., stabilization of fast oscillations). The analysis applies specifically to the forward-backward SR architecture and may not transfer directly to other representation learning schemes (e.g., contrastive methods or world models). Additionally, the paper appears incomplete (abstract cuts off mid-sentence on 'value func[tion]'), suggesting key results or experimental validation may not be fully presented.
Research Context
This work builds directly on the successor representation framework (Dayan, 1993) and recent advances in SR for continuous control, particularly the forward-backward factorization approach that enables low-rank learning. It contributes to the broader trend of using spectral theory to understand and design deep RL architectures, similar to recent work on the spectrum of value function approximators and neural network kernels. The paper opens a research direction on principled spectral mismatch mitigation—moving beyond frame-skipping tricks to formal analysis of why and when temporal abstraction helps. This connects to ongoing work on abstraction in RL and state representation learning, where managing dimensionality and frequency content is a core challenge.
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