Learning Without Mixing: Towards A Sharp Analysis of Linear System Identification (1802.08334v4)

Abstract: We prove that the ordinary least-squares (OLS) estimator attains nearly minimax optimal performance for the identification of linear dynamical systems from a single observed trajectory. Our upper bound relies on a generalization of Mendelson's small-ball method to dependent data, eschewing the use of standard mixing-time arguments. Our lower bounds reveal that these upper bounds match up to logarithmic factors. In particular, we capture the correct signal-to-noise behavior of the problem, showing that more unstable linear systems are easier to estimate. This behavior is qualitatively different from arguments which rely on mixing-time calculations that suggest that unstable systems are more difficult to estimate. We generalize our technique to provide bounds for a more general class of linear response time-series.

• The paper extends the small-ball method to dependent data, establishing new performance bounds for OLS.
• The paper finds that unstable systems yield a higher signal-to-noise ratio, simplifying parameter estimation.
• The paper generalizes the OLS analysis to broader linear time-series, promising wider applicability in dynamical systems.

Toward a Sharp Analysis of Linear System Identification Without Mixing

The paper "Learning Without Mixing: Towards A Sharp Analysis of Linear System Identification" presents an in-depth exploration of estimating the parameters of linear dynamical systems from single trajectory observations. The authors focus on the ordinary least-squares (OLS) estimator, demonstrating its capability to achieve nearly minimax optimal performance in the given context.

Core Contributions

1. Establishment of New Performance Bounds: The research extends Mendelson's small-ball method to dependent data, avoiding traditional mixing-time arguments. This adaptation allows for a more accurate representation of OLS performance, particularly in estimating the parameters of linear systems.
2. Reevaluation of System Stability: It is traditionally believed that unstable systems are harder to estimate due to their slower mixing rates. However, the results reveal a counterintuitive conclusion that more unstable systems present a higher signal-to-noise ratio, thus making parameter estimation easier.
3. Generalization to Linear Time-Series: The methodology developed for linear dynamical systems is extended to a broader class of linear response time-series, highlighting its adaptability and broader applicability.

Technical Highlights

• The paper circumvents mixing-time limitations by providing a concentration-based framework for analyzing linear systems. This approach directly tackles the statistical identifiability without assuming fast mixing to a stationary distribution, thus accommodating systems that exhibit marginal stability (e.g., where the spectral radius $\rho(A) \leq 1$).
• Theoretical results include both upper and lower bounds for the OLS estimator. For $\rho(A) < 1$, OLS performance ties closely to the minimum eigenvalue of the controllability Gramian, $\Gamma_T$, an intrinsic measure of the system's ability to respond to noise.
• When addressing $k$-block conditions in block martingale settings, the authors leverage a block martingale small-ball condition, extending the applicability of Mendelson's original small-ball technique. This enables estimations under conditions of high interdependencies among system trajectories.

Implications and Future Directions

Practically, the insights found in this analysis suggest more efficient resource allocation when measuring and estimating the parameters of dynamical systems in fields like control theory, robotics, and reinforcement learning. With these bounds, practitioners can better anticipate the required sample complexity for identifying system parameters effectively.

Theoretically, the analysis opens doors for further exploration in non-mixing time series data, suggesting potential application in other areas like finance or economics where data does not conform to mixing assumptions.

Future research could explore sharper bounds by minimizing logarithmic factors in the presented analysis or generalizing these techniques to multi-dimensional systems with more complex interactions than those modeled with the current linear assumptions. Additionally, expanding this research to adaptive or reinforcement learning frameworks where inputs are dynamically altered based on estimation feedback would be intriguing.

Overall, this paper provides a critical reassessment of commonly held beliefs about system stability and identifiability, advancing our understanding of parameter estimation in dynamically complex systems.