Learning structure from unstructured data

Abstract

This thesis develops statistical tools and algorithmic techniques for non-asymptotic system identification of dynamical systems from noisy input-output data. Specifically, we address the question: "For a fixed length of noisy data generated by an unknown model, what is the best approximation that can be estimated?"; this is in contrast to traditional system identification which answers the question of estimating the unknown model when data length tends to infinity. The importance of such analyses and tools cannot be overstated in applications such as reinforcement learning where a popular design principle is system identification for control. Typically, in such settings we are presented with two problems: first, we are given access only to a finite noisy data set; and second, the hidden state dimension or model order is unknown. The first problem limits our ability to comment on the finite time performance of estimation algorithms; and the second problem prevents appropriate parametrizations for model identification. The goal of this thesis is to address these issues for a large class of dynamical systems. The premise of our approach relies on the existence of suitable low order approximations of the true model that can be constructed from finite, albeit noisy, data. Since the true model order is apriori unknown, we simply estimate low order approximations of this model from data. The order of these approximations grow as we accumulate more data. By such a method, we construct consistent finite time estimators of the underlying data generating model. This principle of constructing low order estimates directly from data is different from the status quo of constructing the largest possible model and then performing a reduction procedure to obtain estimates. We show that in many cases our method outperforms existing algorithms in finite time.