[논문 리뷰] Near optimal finite time identification of arbitrary linear dynamical systems

Tuhin Sarkar, Alexander Rakhlin|arXiv (Cornell University)|2018. 12. 04.

Control Systems and Identification인용 수 70

한 줄 요약

이 논문은 단일 궤적을 이용해 일반 LTI 시스템의 추정을 위한 비점근적 유한시간 오차 경계를 OLS로 도출하며, 안정적, 경계적으로 안정적, 그리고 폭발적(me) 구간을 다루고, OLS가 불일치할 수 있는 조건을 보여준다.

ABSTRACT

We derive finite time error bounds for estimating general linear time-invariant (LTI) systems from a single observed trajectory using the method of least squares. We provide the first analysis of the general case when eigenvalues of the LTI system are arbitrarily distributed in three regimes: stable, marginally stable, and explosive. Our analysis yields sharp upper bounds for each of these cases separately. We observe that although the underlying process behaves quite differently in each of these three regimes, the systematic analysis of a self--normalized martingale difference term helps bound identification error up to logarithmic factors of the lower bound. On the other hand, we demonstrate that the least squares solution may be statistically inconsistent under certain conditions even when the signal-to-noise ratio is high.

연구 동기 및 목표

• Provide sharp non-asymptotic identification error bounds for the OLS estimator of X_{t+1}=AX_t+η_{t+1} without input
• Develop bounds valid for arbitrary eigenvalue distributions across stable, marginally stable, and explosive regimes
• Show that regularity of A is essential for consistency of OLS in explosive settings
• Highlight the role of the sample covariance and self-normalized martingale terms in error control
• Demonstrate potential inconsistency of OLS when regularity conditions fail

제안 방법

• Model the system X_{t+1}=AX_t+η_{t+1} with Λ as Jordan form of A and analyze Y_T=∑_{t=0}^T X_tX_t' and S_T=∑_{t=0}^T X_tη_{t+1}'
• Develop non-asymptotic bounds for the self-normalized martingale term Propositions leveraging subgaussian noise and self-normalized concentration (Proposition 3.1)
• Characterize Y_T behavior in three regimes: S0 (stable), S1 (marginally stable), S2 (explosive), and derive deterministic upper/lower bounds (V_up, V_dn)
• Handle explosive case by transforming to z_t=A^{-t}x_t and analyzing U_T and F_T with invertibility conditions under regularity
• Use anti-concentration and subgaussian tail inequalities to sharpen bounds and obtain near-optimal rates (up to log factors)
• Provide lower bounds showing dependence on δ and demonstrate that OLS may be inconsistent when A is irregular

실험 결과

연구 질문

• RQ1 What finite-time non-asymptotic error bounds can be derived for the OLS estimator of A in X_{t+1}=AX_t+η_{t+1} across stable, marginally stable, and explosive regimes?
• RQ2 Under what regularity conditions on A does OLS remain consistent for arbitrary eigenvalue distributions?
• RQ3 How do sample covariance properties and cross-terms between covariates and noise influence identification error in each regime?
• RQ4 Are the derived bounds tight, and do lower bounds match the upper bounds in different regimes?
• RQ5 Can these results extend to the presence of inputs U_t and heavy-tailed noise distributions?

주요 결과

• For stable and marginally stable A, non-asymptotic error scales as O(sqrt(log(1/δ))/√T) or similar near-optimal rates (up to logarithmic factors)
• For explosive A, error decays exponentially with time T when regularity holds, with δ-dependent bounds that scale as 1/δ
• The analysis provides sharp, regime-specific bounds by coupling the sample covariance with a self-normalized martingale term
• Regularity of A (geometric multiplicity of eigenvalues >1 equal to 1) is necessary for OLS consistency in explosive settings, otherwise OLS is inconsistent
• When A is irregular, even with high signal-to-noise ratio, OLS can be inconsistent due to ill-conditioned sample covariance
• The results cover the general case of eigenvalues arbitrarily distributed across stable, marginally stable, and explosive regimes, with matching lower bounds in some cases