Constrained Nonconvex Stochastic Optimization with One Projection

Constrained nonconvex optimization has seen increasing application in modern machine learning, such as safe LLM alignment/finetuning, transfer learning and rank-constrained continual learning. Projection onto a functional constraint can cost substantially more than a stochastic first-order update. We study whether this operation can be deferred until the end of nonconvex stochastic optimization. For weakly convex, possibly nonsmooth objectives and regular convex constraints, we give a penalized proximal method that uses $\tilde{O}({ϵ}^{-4})$ stochastic subgradients and one projection onto the constraint set. The returned point is exactly feasible and has a small Moreau-envelope stationarity measure; for smooth objectives, the same algorithm controls the projected-gradient mapping. For smooth nonconvex constraints, we assume a global lower bound on the infeasible constraint slope which permits arbitrary, possibly infeasible initialization. An exact-penalty variant attains the same stochastic-oracle order and returns an exactly feasible point within $O(ϵ)$ of an $ϵ$-KKT point. All intermediate updates use only projections onto a simple Euclidean ball. These are oracle-complexity guarantees: the cost of the single terminal projection is separate, and no efficient projection algorithm for a general nonconvex set is assumed to follow from regularity.