  In this paper, we propose a faster stochastic alternating direction method of
multipliers (ADMM) for nonconvex optimization by using a new stochastic
path-integrated differential estimator (SPIDER), called as SPIDER-ADMM.
Moreover, we prove that the SPIDER-ADMM achieves a record-breaking incremental
first-order oracle (IFO) complexity of $\mathcal{O}(n+n^{1/2}\epsilon^{-1})$
for finding an $\epsilon$-approximate stationary point, which improves the
deterministic ADMM by a factor $\mathcal{O}(n^{1/2})$, where $n$ denotes the
sample size. As one of major contribution of this paper, we provide a new
theoretical analysis framework for nonconvex stochastic ADMM methods with
providing the optimal IFO complexity. Based on this new analysis framework, we
study the unsolved optimal IFO complexity of the existing non-convex SVRG-ADMM
and SAGA-ADMM methods, and prove they have the optimal IFO complexity of
$\mathcal{O}(n+n^{2/3}\epsilon^{-1})$. Thus, the SPIDER-ADMM improves the
existing stochastic ADMM methods by a factor of $\mathcal{O}(n^{1/6})$.
Moreover, we extend SPIDER-ADMM to the online setting, and propose a faster
online SPIDER-ADMM. Our theoretical analysis shows that the online SPIDER-ADMM
has the IFO complexity of $\mathcal{O}(\epsilon^{-\frac{3}{2}})$, which
improves the existing best results by a factor of
$\mathcal{O}(\epsilon^{-\frac{1}{2}})$. Finally, the experimental results on
benchmark datasets validate that the proposed algorithms have faster
convergence rate than the existing ADMM algorithms for nonconvex optimization.
