A Generic Approach for Escaping Saddle points
This addresses a central challenge for practitioners using first-order methods in large-scale nonconvex optimization, though it is incremental as it builds on existing second-order methods.
The paper tackles the problem of saddle points in nonconvex optimization by introducing a generic framework that minimizes expensive Hessian computations while provably converging to second-order critical points, achieving competitive convergence results with empirical performance.
A central challenge to using first-order methods for optimizing nonconvex problems is the presence of saddle points. First-order methods often get stuck at saddle points, greatly deteriorating their performance. Typically, to escape from saddles one has to use second-order methods. However, most works on second-order methods rely extensively on expensive Hessian-based computations, making them impractical in large-scale settings. To tackle this challenge, we introduce a generic framework that minimizes Hessian based computations while at the same time provably converging to second-order critical points. Our framework carefully alternates between a first-order and a second-order subroutine, using the latter only close to saddle points, and yields convergence results competitive to the state-of-the-art. Empirical results suggest that our strategy also enjoys a good practical performance.