Efficiency of Coordinate Descent Methods on Huge-Scale Optimization Problems

Abstract

In this paper we propose new methods for solving huge-scale optimization problems. For problems of this size, even the simplest full-dimensional vector operations are very expensive. Hence, we propose to apply an optimization technique based on random partial update of decision variables. For these methods, we prove the global estimates for the rate of convergence. Surprisingly enough, for certain classes of objective functions, our results are better than the standard worst-case bounds for deterministic algorithms. We present constrained and unconstrained versions of the method, and its accelerated variant. Our numerical test confirms a high efficiency of this technique on problems of very big size.

Randomized Sparse Block Kaczmarz as Randomized Dual Block-Coordinate Descent

“…For this particular choice (non-asymptotic) convergence rates were only recently derived in [2], although the convergence of the method was extensively studied in the literature under various assumptions [13,3]. Instead of using a deterministic cyclic order, randomized strategies were proposed in [14,12,16] for choosing a block to update at each iteration of the BCGD method. At iteration k, an index i k is generated randomly according to the probability distribution vector p ∈ ∆ c .…”

“…At iteration k, an index i k is generated randomly according to the probability distribution vector p ∈ ∆ c . In [14] the distribution vector was chosen as…”

“…The above stochastic results from [14] for minimizing convex differentiable functions were generalized in [16] for minimizing the sum of a smooth convex function and a block-separable convex function. …”

“…Despite these observations, the strong convexity of the dual function ψ ε opens the way for various accelerations along the lines of [14,16,18,17]. We omit this here, however.…”

Asynchronous incremental block-coordinate descent

“…Randomized coordinate descent has been shown to be competitive with the classical gradient descent method, in the sense that it requires less work per iteration, but a comparable number of iterations to converge [8]. In this section, we demonstrate that a similar property holds for asynchronous incremental block-coordinate descent: if the amount of work required to evaluate a partial gradient is proportional to its block size, then incremental block-coordinate descent can always be expected to be more efficient than a corresponding incremental gradient descent algorithm.…”