Universal Sequence Preconditioning (2502.06545v2)

Abstract: We study the problem of preconditioning in the setting of sequential prediction. From the theoretical lens of linear dynamical systems, we show that applying a convolution to the input sequence translates to applying a polynomial to the unknown transition matrix in the hidden space. With this insight, we develop a novel preconditioning method that convolves the input sequence with the coefficients of the Chebyshev or Legendre polynomials. We formally prove that this improves the regret of two distinct prediction methods. Moreover, using this preconditioning technique on either method gives the first sublinear regret bounds that are also hidden dimension independent (up to logarithmic factors) even when the hidden transition matrix is asymmetric. From rigorous experiments on synthetic data we show that our simple preconditioning method generalizes to both 1) settings where the data is not from a linear dynamical system and 2) a broad range of learning algorithms, including recurrent neural networks.

• The paper extends spectral filtering to asymmetric LDS by constructing a Chebyshev polynomial-based spectral basis for handling complex eigenvalues.
• It achieves dimension-free sublinear regret bounds (˜O(T^(9/10))), overcoming traditional scalability issues in high-dimensional systems.
• An efficient online convex optimization algorithm is introduced, enhancing practical applications in control engineering and real-time forecasting.

Dimension-Free Regret for Learning Asymmetric Linear Dynamical Systems

Linear Dynamical Systems (LDS) are a central topic in control theory, time-series analysis, and machine learning due to their foundational role in modeling dynamical processes. These systems are typically characterized by their state-space representations, where future outputs depend linearly on the current state and inputs. The paper under review tackles a long-standing challenge in learning partially observable LDS. Specifically, it addresses the problem of learning LDS governed by asymmetric and marginally stable transition matrices through an innovative framework that achieves dimension-free regret bounds.

Traditionally, methods to learn marginally stable systems with non-symmetric matrices have faced computational hurdles, often involving regret bounds that scale unfavorably with the hidden dimensions. Earlier approaches such as Kalman filtering and other spectral methods have demonstrated satisfactory results for symmetric matrices but struggled with asymmetries. This paper proposes a hybrid method combining spectral filtering with Chebyshev polynomials, effectively allowing learning with dimension-free guarantees.

Key Contributions

1. Generalization of Spectral Filtering: The paper extends existing spectral filtering techniques to asymmetric LDS. By constructing a novel spectral basis using Chebyshev polynomials, the authors demonstrate that it is feasible to handle systems with complex eigenvalues, thus broadening the applicability of previous methodologies focused only on symmetric matrices. The approach notably requires no assumptions about the perturbation model or matrix symmetries.
2. Novel Use of Chebyshev Polynomials: The utilization of Chebyshev polynomials in the complex plane is a key technical innovation. This choice ensures robustness and efficiency of computations, providing a balanced trade-off between computational complexity and theoretical approximation guarantees.
3. Sublinear Regret Bound Achievement: Through rigorous theoretical development, the paper establishes sublinear regret bounds, specifically $\tilde{O}(T^{9/10})$, which do not depend on the dimensions of the state space. This result is significant due to the pervasive presence of high-dimensional space in real-world systems.
4. Efficient Algorithm for Online Learning: An algorithm leveraging the proposed spectral filtering basis in an online convex optimization framework is presented. Sublinear regret is demonstrated against the best possible linear predictor, showing the method's efficacy in practical settings.

Implications and Future Directions

The theoretical implications of this paper offer a significant advancement in the landscape of learning linear dynamical systems. The proposed method not only extends the capability to more general classes of LDS but also opens avenues for future research into similarly challenging scenarios like non-linear dynamical systems. Moreover, the absence of dependence on the hidden dimension marks a crucial step toward scalability in large-scale applications.

Practically, this research offers potential enhancements for adaptive systems in control engineering and real-time forecasting systems that require robust handling of uncertainties and noise—factors often prevalent in non-generative real-world scenarios.

Future work could explore refining these methods for broader types of nonlinear dynamical systems, potentially leveraging these insights to bridge gaps between linear assumptions and real-world complexities. Moreover, extending these frameworks to other optimization objectives or under varied model uncertainties could lead to more generalized applications across disciplines integrating control, learning, and signal processing.

In conclusion, this paper introduces a versatile enhancement to spectral filtering methodologies with its application to asymmetric, marginally stable systems, free from hidden dimension constraints, thereby positioning itself as a pivotal contribution to both the theoretical and practical domains of dynamical system learning.