First-order methods for Stochastic Variational Inequality problems with Function Constraints

The monotone Variational Inequality (VI) is an important problem in machine learning. In numerous instances, the VI problems are accompanied by function constraints which can possibly be data-driven, making the projection operator challenging to compute. In this paper, we present novel first-order methods for function constrained VI (FCVI) problem under various settings, including smooth or nonsmooth problems with a stochastic operator and/or stochastic constraints. First, we introduce the~{\texttt{OpConEx}} method and its stochastic variants, which employ extrapolation of the operator and constraint evaluations to update the variables and the Lagrangian multipliers. These methods achieve optimal operator or sample complexities when the FCVI problem is either (i) deterministic nonsmooth, or (ii) stochastic, including smooth or nonsmooth stochastic constraints. Notably, our algorithms are simple single-loop procedures and do not require the knowledge of Lagrange multipliers to attain these complexities. Second, to obtain the optimal operator complexity for smooth deterministic problems, we present a novel single-loop Adaptive Lagrangian Extrapolation~(\texttt{AdLagEx}) method that can adaptively search for and explicitly bound the Lagrange multipliers. Furthermore, we show that all of our algorithms can be easily extended to saddle point problems with coupled function constraints, hence achieving similar complexity results for the aforementioned cases. To our best knowledge, many of these complexities are obtained for the first time in the literature.