In this talk, we first show that coordinate descent methods can be efficiently applied to the optimization of nonsmooth convex functions with explicit max-structure. We derive a quadratic separable overapproximation of the smoothed function obtained by Nesterov's smoothing and from it we derive the iteration complexity of the Parallel Coordinate Descent Method applied to this class of functions.
In a second part, we compare the Parallel Coordinate Descent Method with the Monte Carlo approach, where we launch several times the serial randomized coordinate descent method in order to reduce the variance of the random process describing the value of the function. Using the theoretical iteration complexity, we show that, depending on the problem at stake, both approaches may be competitive, or that a nested parallelism may be better.
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