Linearly Convergent Algorithms for Nonsmooth Problems with Unknown Smooth Pieces
This addresses optimization of complex functions with unknown structure, providing theoretical guarantees and practical termination criteria for researchers and practitioners in mathematical optimization.
The authors developed algorithms for optimizing piecewise smooth functions where the smooth pieces are unknown, achieving global linear convergence for functions with quadratic growth and matching state-of-the-art complexity for weakly-convex problems.
We develop efficient algorithms for optimizing piecewise smooth (PWS) functions where the underlying partition of the domain into smooth pieces is \emph{unknown}. For PWS functions satisfying a quadratic growth (QG) condition, we propose a bundle-level (BL) type method that achieves global linear convergence -- to our knowledge, the first such result for any algorithm for this problem class. We extend this method to handle approximately PWS functions and to solve weakly-convex PWS problems, improving the state-of-the-art complexity to match the benchmark for smooth non-convex optimization. Furthermore, we introduce the first verifiable and accurate termination criterion for PWS optimization. Similar to the gradient norm in smooth optimization, this certificate tightly characterizes the optimality gap under the QG condition, and can moreover be evaluated without knowledge of any problem parameters. We develop a search subroutine for this certificate and embed it within a guess-and-check framework, resulting in an almost parameter-free algorithm for both the convex QG and weakly-convex settings.