Nonsmooth Convex Optimization using the Specular Gradient Method with Root-Linear Convergence
This work provides a theoretical convergence guarantee for a simple first-order method on nonsmooth convex problems, which is incremental but addresses a known bottleneck in optimization theory.
The paper identifies a special case of the subgradient method, called the specular gradient method, that achieves root-linear convergence for minimizing one-dimensional convex functions without additional assumptions, and provides an implementation that avoids explicit specular derivative calculation.
In this paper, we find the special case of the subgradient method minimizing a one-dimensional real-valued function, which we term the specular gradient method, that converges root-linearly without any additional assumptions except the convexity. Furthermore, we suggest a way to implement the specular gradient method without explicitly calculating specular derivatives.