Part of Advances in Neural Information Processing Systems 29 (NIPS 2016)
Davood Hajinezhad, Mingyi Hong, Tuo Zhao, Zhaoran Wang
We study a stochastic and distributed algorithm for nonconvex problems whose objective consists a sum N nonconvex Li/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((∑Ni=1√Li/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 ℓ1 penalized quadratic problems with polyhedral constraints. Further, we reveal a fundamental connection between {\it primal-dual} based methods and a few {\it primal only} methods such as IAG/SAG/SAGA.