Abstract
In this work we study the parallel coordinate descent method (PCDM) proposed by Richtárik and Takáč [Parallel coordinate descent methods for big data optimization, Math. Program. Ser. A (2015), pp. 1–52] for minimizing a regularized convex function. We adopt elements from the work of Lu and Xiao [On the complexity analysis of randomized block-coordinate descent methods, Math. Program. Ser. A 152(1–2) (2015), pp. 615–642], and combine them with several new insights, to obtain sharper iteration complexity results for PCDM than those presented in [Richtárik and Takáč, Parallel coordinate descent methods for big data optimization, Math. Program. Ser. A (2015), pp. 1–52]. Moreover, we show that PCDM is monotonic in expectation, which was not confirmed in [Richtárik and Takáč, Parallel coordinate descent methods for big data optimization, Math. Program. Ser. A (2015), pp. 1–52], and we also derive the first high probability iteration complexity result where the initial levelset is unbounded.
Highlights
Block coordinate descent methods are being thrust into the optimization spotlight because of a dramatic increase in the size of real world problems, and because of the ‘Big data’ phenomenon
In this work we study the parallel coordinate descent method (PCDM) proposed by Richtárik and Takác [Parallel coordinate descent methods for big data optimization, Math
PCDM in [24] was analysed (for a general convex composite function of the form (1)) under a monotonicity assumption; if, at any iteration of PCDM, an update was computed that would lead to a higher objective value than the objective value at the current point, that update is rejected
Summary
Block coordinate descent methods are being thrust into the optimization spotlight because of a dramatic increase in the size of real world problems, and because of the ‘Big data’ phenomenon. It is little wonder, when these seemingly simple methods, with low iteration costs and low memory requirements, can solve problems where the dimension is more than one billion, in a matter of hours [24]. State-of-the-art block coordinate descent method is the Parallel (block) Coordinate Descent Method (PCDM) of Richtárik and Takác [24] This method selects the coordinates to update randomly and the update is determined by minimizing an overapproximation of the objective function at the current point
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