Stochastic inexact augmented Lagrangian method for nonconvex expectation constrained optimization

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 $$\varepsilon $$ ε -KKT point in expectation, we establish an oracle complexity result of $$O(\varepsilon ^{-5})$$ O ( ε - 5 ) , which is better than the best-known $$O(\varepsilon ^{-6})$$ O ( ε - 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.