Variance Estimation For Dynamic Regression via Spectrum Thresholding

We consider the dynamic linear regression problem, where the predictor vector may vary with time. This problem can be modeled as a linear dynamical system, where the parameters that need to be learned are the variance of both the process noise and the observation noise. While variance estimation for dynamic regression is a natural problem, with a variety of applications, existing approaches to this problem either lack guarantees or only have asymptotic guarantees without explicit rates. In addition, all existing approaches rely strongly on Guassianity of the noises. In this paper we study the global system operator: the operator that maps the noise vectors to the output. In particular, we obtain estimates on its spectrum, and as a result derive the first known variance estimators with finite sample complexity guarantees. Moreover, our results hold for arbitrary sub Gaussian distributions of noise terms. We evaluate the approach on synthetic and real-world benchmarks.