Fast Nonsmooth Regularized Risk Minimization with Continuation
This addresses a bottleneck in machine learning optimization for researchers and practitioners dealing with nonsmooth functions, offering a flexible and efficient solution with proven convergence.
The paper tackles the difficulty of optimizing nonsmooth regularized risk minimization problems by proposing a continuation algorithm that achieves fast convergence rates of O(1/T^2) for strongly convex problems and O(1/T) for general convex problems, and outperforms state-of-the-art methods in experiments on classification and regression tasks.
In regularized risk minimization, the associated optimization problem becomes particularly difficult when both the loss and regularizer are nonsmooth. Existing approaches either have slow or unclear convergence properties, are restricted to limited problem subclasses, or require careful setting of a smoothing parameter. In this paper, we propose a continuation algorithm that is applicable to a large class of nonsmooth regularized risk minimization problems, can be flexibly used with a number of existing solvers for the underlying smoothed subproblem, and with convergence results on the whole algorithm rather than just one of its subproblems. In particular, when accelerated solvers are used, the proposed algorithm achieves the fastest known rates of $O(1/T^2)$ on strongly convex problems, and $O(1/T)$ on general convex problems. Experiments on nonsmooth classification and regression tasks demonstrate that the proposed algorithm outperforms the state-of-the-art.