Proximal methods avoid active strict saddles of weakly convex functions
This work addresses convergence issues in optimization for machine learning and related fields, providing a theoretical guarantee that is applicable to generic semi-algebraic problems.
The paper tackles the problem of ensuring convergence to local minimizers in nonsmooth optimization by introducing a strict saddle property for weakly convex functions, proving that proximal algorithms with random initialization avoid strict saddles under this property.
We introduce a geometrically transparent strict saddle property for nonsmooth functions. This property guarantees that simple proximal algorithms on weakly convex problems converge only to local minimizers, when randomly initialized. We argue that the strict saddle property may be a realistic assumption in applications, since it provably holds for generic semi-algebraic optimization problems.