Gradient descent (GD) converges linearly to the global optimum for even nonconvex problems when the loss function satisfies certain benign geometric properties that are strictly weaker than strong convexity. One important property studied in the literature is the so-called Regularity Condition (RC). The RC property has been proven valid for many problems such as deep linear neural networks, shallow neural networks with nonlinear activations, phase retrieval, to name a few. Moreover, accelerated first-order methods (e.g. Nesterov's accelerated gradient and Heavy-ball) achieve great empirical success when the parameters are tuned properly but lack theoretical understandings in the nonconvex setting. In this paper, we use tools from robust control to analytically characterize the region of hyperparameters that ensure linear convergence of accelerated first-order methods under RC. Our results apply to all functions satisfying RC and therefore are more general than results derived for speci 查看全文>>