A nearly linearly convergent first-order method for nonsmooth functions with quadratic growth

Classical results show that gradient descent converges linearly to minimizers of smooth strongly convex functions. A natural question is whether there exists a locally nearly linearly convergent method for nonsmooth functions with quadratic growth. This work designs such a method for a wide class of nonsmooth and nonconvex locally Lipschitz functions, including max-of-smooth, Shapiro’s decomposable … Read more

Subgradient methods near active manifolds: saddle point avoidance, local convergence, and asymptotic normality

Nonsmooth optimization problems arising in practice, whether in signal processing, statistical estimation, or modern machine learning, tend to exhibit beneficial smooth substructure: their domains stratify into “active manifolds” of smooth variation, which common proximal algorithms “identify” in finite time. Identification then entails a transition to smooth dynamics, and permits the use of second-order information for … Read more