Smooth minimization of nonsmooth functions with parallel coordinate descent methods
This work provides an incremental improvement in optimization algorithms for machine learning, addressing efficiency in parallel computing for specific sparse data problems.
The paper tackles the problem of minimizing nonsmooth convex functions, including L1-regularized L1 regression and AdaBoost, using randomized parallel coordinate descent methods, achieving an iteration complexity of O(n β/τ) that decreases with more processors and sparser data.
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 $ω$ nonzeros in each row. Our methods need $O(n β/τ)$ iterations to find an approximate solution with high probability, where $τ$ is the number of processors and $β= 1 + (ω-1)(τ-1)/(n-1)$ 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 $β/τ$ is a decreasing function of $τ$, 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 $β$ decreases when $ω$ does, fewer iterations are needed for sparser problems.