  Many real-world problems not only have complicated nonconvex functional
constraints but also use a large number of data points. This motivates the
design of efficient stochastic methods on finite-sum or expectation constrained
problems. In this paper, we design and analyze stochastic inexact augmented
Lagrangian methods (Stoc-iALM) to solve problems involving a nonconvex
composite (i.e. smooth+nonsmooth) objective and nonconvex smooth functional
constraints. We adopt the standard iALM framework and design a subroutine by
using the momentum-based variance-reduced proximal stochastic gradient method
(PStorm) and a postprocessing step. Under certain regularity conditions
(assumed also in existing works), to reach an $\varepsilon$-KKT point in
expectation, we establish an oracle complexity result of $O(\varepsilon^{-5})$,
which is better than the best-known $O(\varepsilon^{-6})$ result. Numerical
experiments on the fairness constrained problem and the Neyman-Pearson
classification problem with real data demonstrate that our proposed method
outperforms an existing method with the previously best-known complexity
result.
