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NESTT: A Nonconvex Primal-Dual Splitting Method for Distributed and Stochastic Optimization

Abstract

We study a stochastic and distributed algorithm for nonconvex problems whose objective consists of a sum of NN nonconvex Li/NL_i/N-smooth functions, plus a nonsmooth regularizer. The proposed NonconvEx primal-dual SpliTTing (NESTT) algorithm splits the problem into NN subproblems, and utilizes an augmented Lagrangian based primal-dual scheme to solve it in a distributed and stochastic manner. With a special non-uniform sampling, a version of NESTT achieves ϵ\epsilon-stationary solution using O((i=1NLi/N)2/ϵ)\mathcal{O}((\sum_{i=1}^N\sqrt{L_i/N})^2/\epsilon) gradient evaluations, which can be up to O(N)\mathcal{O}(N) times better than the (proximal) gradient descent methods. It also achieves Q-linear convergence rate for nonconvex 1\ell_1 penalized quadratic problems with polyhedral constraints. Further, we reveal a fundamental connection between primal-dual based methods and a few primal only methods such as IAG/SAG/SAGA.Comment: 35 pages, 2 figure

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