  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
$\epsilon$-stationary solution using
$\mathcal{O}((\sum_{i=1}^N\sqrt{L_i/N})^2/\epsilon)$ gradient evaluations,
which can be up to $\mathcal{O}(N)$ times better than the (proximal) gradient
descent methods. It also achieves Q-linear convergence rate for nonconvex
$\ell_1$ 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.
