Optimization of InfConvolution Regularized Nonconvex Composite Problems
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Proceedings of Machine Learning Research, PMLR 89:547556, 2019.
Abstract
In this work, we consider nonconvex composite problems that involve infconvolution with a Legendre function, which gives rise to an anisotropic generalization of the proximal mapping and Moreauenvelope. In a convex setting such problems can be solved via alternating minimization of a splitting formulation, where the consensus constraint is penalized with a Legendre function. In contrast, for nonconvex models it is in general unclear that this approach yields stationary points to the infimal convolution problem. To this end we analytically investigate local regularity properties of the Moreauenvelope function under proxregularity, which allows us to establish the equivalence between stationary points of the splitting model and the original infconvolution model. We apply our theory to characterize stationary points of the penalty objective, which is minimized by the elastic averaging SGD (EASGD) method for distributed training, showing that perfect consensus between the workers is attainable via a finite penalty parameter. Numerically, we demonstrate the practical relevance of the proposed approach on the important task of distributed training of deep neural networks.
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