An Accelerated Proximal Coordinate Gradient Method and its Application to Regularized Empirical Risk Minimization

We consider the problem of minimizing the sum of two convex functions: one is smooth and given by a gradient oracle, and the other is separable over blocks of coordinates and has a simple known structure over each block. We develop an accelerated randomized proximal coordinate gradient (APCG) method for minimizing such convex composite functions. For strongly convex functions, our method achieves faster linear convergence rates than existing randomized proximal coordinate gradient methods. Without strong convexity, our method enjoys accelerated sublinear convergence rates. We show how to apply the APCG method to solve the regularized empirical risk minimization (ERM) problem, and devise efficient implementations that avoid full-dimensional vector operations. For ill-conditioned ERM problems, our method obtains improved convergence rates than the state-of-the-art stochastic dual coordinate ascent (SDCA) method

Alternating Randomized Block Coordinate Descent

Block-coordinate descent algorithms and alternating minimization methods are fundamental optimization algorithms and an important primitive in large-scale optimization and machine learning. While various block-coordinate-descent-type methods have been studied extensively, only alternating minimization -- which applies to the setting of only two blocks -- is known to have convergence time that scales independently of the least smooth block. A natural question is then: is the setting of two blocks special? We show that the answer is "no" as long as the least smooth block can be optimized exactly -- an assumption that is also needed in the setting of alternating minimization. We do so by introducing a novel algorithm AR-BCD, whose convergence time scales independently of the least smooth (possibly non-smooth) block. The basic algorithm generalizes both alternating minimization and randomized block coordinate (gradient) descent, and we also provide its accelerated version -- AAR-BCD. As a special case of AAR-BCD, we obtain the first nontrivial accelerated alternating minimization algorithm.Comment: Version 1 appeared Proc. ICML'18. v1 -> v2: added remarks about how accelerated alternating minimization follows directly from the results that appeared in ICML'18; no new technical results were needed for thi

Optimization with Sparsity-Inducing Penalties

Sparse estimation methods are aimed at using or obtaining parsimonious representations of data or models. They were first dedicated to linear variable selection but numerous extensions have now emerged such as structured sparsity or kernel selection. It turns out that many of the related estimation problems can be cast as convex optimization problems by regularizing the empirical risk with appropriate non-smooth norms. The goal of this paper is to present from a general perspective optimization tools and techniques dedicated to such sparsity-inducing penalties. We cover proximal methods, block-coordinate descent, reweighted $\ell_2$-penalized techniques, working-set and homotopy methods, as well as non-convex formulations and extensions, and provide an extensive set of experiments to compare various algorithms from a computational point of view