  There is a growing interest in using robust control theory to analyze and
design optimization and machine learning algorithms. This paper studies a class
of nonconvex optimization problems whose cost functions satisfy the so-called
Regularity Condition (RC). Empirical studies show that accelerated gradient
descent (AGD) algorithms (e.g. Nesterov's acceleration and Heavy-ball) with
proper initializations often work well in practice. However, the convergence of
such AGD algorithms is largely unknown in the literature. The main contribution
of this paper is the analytical characterization of the convergence regions of
AGD under RC via robust control tools. Since such optimization problems arise
frequently in many applications such as phase retrieval, training of neural
networks and matrix sensing, our result shows promise of robust control theory
in these areas.
