Smooth Minimization of Nonsmooth Functions with Parallel Coordinate Descent Methods

Abstract

We study the performance of a family of randomized parallel coordinate descent methods for minimizing the sum of a nonsmooth and separable convex functions. The problem class includes as a special case L1-regularized L1 regression and the minimization of the exponential loss ("AdaBoost problem"). We assume the input data defining the loss function is contained in a sparse $m\times n$ matrix $A$ with at most $\omega$ nonzeros in each row. Our methods need $O(n \beta/\tau)$ iterations to find an approximate solution with high probability, where $\tau$ is the number of processors and $\beta = 1 + (\omega-1)(\tau-1)/(n-1)$ for the fastest variant. The notation hides dependence on quantities such as the required accuracy and confidence levels and the distance of the starting iterate from an optimal point. Since $\beta/\tau$ is a decreasing function of $\tau$, the method needs fewer iterations when more processors are used. Certain variants of our algorithms perform on average only $O(\nnz(A)/n)$ arithmetic operations during a single iteration per processor and, because $\beta$ decreases when $\omega$ does, fewer iterations are needed for sparser problems.