NESTT: A Nonconvex Primal-Dual Splitting Method for Distributed and Stochastic Optimization
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We study a stochastic and distributed algorithm for nonconvex problems whose objective consists of a sum of N nonconvex L_i/N-smooth functions, plus a nonsmooth regularizer. The proposed NonconvEx primal-dual SpliTTing (NESTT) algorithm splits the problem into N 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-stationary solution using O((∑_i=1^N√(L_i/N))^2/ϵ) gradient evaluations, which can be up to O(N) times better than the (proximal) gradient descent methods. It also achieves Q-linear convergence rate for nonconvex ℓ_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.
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