Reference EPFL-REPORT-229241 URL: http: arxiv.org abs 1512.04011 Record created on 2017-06-21, modified on 2017-07-12
Coordinate descent on @math -regularized problems with @math can be interpreted as the iterative minimization of a quadratic approximation of the smooth part of the objective (as in a one-dimensional Newton step), followed by a shrinkage step resulting from the @math part. In the single-coordinate update case, this is at the core of @cite_15 @cite_17 , and widely used in, e.g., solvers based on the primal formulation of @math -regularized objectives @cite_23 @cite_13 @cite_24 @cite_25 @cite_19 . When changing more than one coordinate at a time, again employing a quadratic upper bound on the smooth part, this results in a two-loop method as in @cite_15 for the special case of logistic regression. This idea is crucial for the distributed setting.
@cite_24 @cite_19 @cite_23 @cite_15 @cite_13 @cite_25 @cite_17
1673797905 38875623 2099119387 2050968963
One of the key steps at each iteration of a randomized block coordinate descent method consists in determining the update to a block of variables. Existing algorithms assume that in order to compute the update, a particular subproblem is solved exactly. In this work, we relax this requirement and allow for the subproblem to be solved inexactly, leading to an inexact block coordinate descent method. Our approach incorporates the best known results for exact updates as a special case. Moreover, these theoretical guarantees are complemented by practical considerations: the use of iterative techniques to determine the update and the use of preconditioning for further acceleration. We propose a new stochastic dual coordinate ascent technique that can be applied to a wide range of regularized learning problems. Our method is based on alternating direction method of multipliers (ADMM) to deal with complex regularization functions such as structured regularizations. Although the original ADMM is a batch method, the proposed method offers a stochastic update rule where each iteration requires only one or few sample observations. Moreover, our method can naturally afford mini-batch update and it gives speed up of convergence. We show that, under mild assumptions, our method converges exponentially. The numerical experiments show that our method actually performs efficiently. 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 @math matrix @math with at most @math nonzeros in each row. Our methods need @math iterations to find an approximate solution with high probability, where @math is the number of processors and @math 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 @math is a decreasing function of @math , the method needs fewer iterations when more processors are used. Certain variants of our algorithms perform on average only @math arithmetic operations during a single iteration per processor and, because @math decreases when @math does, fewer iterations are needed for sparser problems. The block coordinate descent (BCD) method is widely used for minimizing a continuous function @math of several block variables. At each iteration of this method, a single block of variables is optimized, while the remaining variables are held fixed. To ensure the convergence of the BCD method, the subproblem of each block variable needs to be solved to its unique global optimal. Unfortunately, this requirement is often too restrictive for many practical scenarios. In this paper, we study an alternative inexact BCD approach which updates the variable blocks by successively minimizing a sequence of approximations of @math which are either locally tight upper bounds of @math or strictly convex local approximations of @math . The main contributions of this work include the characterizations of the convergence conditions for a fairly wide class of such methods, especially for the cases where the objective functions are either nondifferentiable or nonconvex. Our results unify and extend the existing convergence results ...