Escaping Saddle Points Efficiently with Occupation-Time-Adapted Perturbations
This addresses the challenge of avoiding saddle points in optimization for machine learning practitioners, though it is incremental as it builds on perturbed gradient descent frameworks.
The paper tackles the problem of escaping saddle points in non-convex optimization by introducing a new perturbation mechanism based on occupation time, leading to algorithms that converge to second-order stationary points at least as fast as existing methods and outperform them empirically.
Motivated by the super-diffusivity of self-repelling random walk, which has roots in statistical physics, this paper develops a new perturbation mechanism for optimization algorithms. In this mechanism, perturbations are adapted to the history of states via the notion of occupation time. After integrating this mechanism into the framework of perturbed gradient descent (PGD) and perturbed accelerated gradient descent (PAGD), two new algorithms are proposed: perturbed gradient descent adapted to occupation time (PGDOT) and its accelerated version (PAGDOT). PGDOT and PAGDOT are shown to converge to second-order stationary points at least as fast as PGD and PAGD, respectively, and thus they are guaranteed to avoid getting stuck at non-degenerate saddle points. The theoretical analysis is corroborated by empirical studies in which the new algorithms consistently escape saddle points and outperform not only their counterparts, PGD and PAGD, but also other popular alternatives including stochastic gradient descent, Adam, AMSGrad, and RMSProp.