Matching lower bound for one-projection stochastic optimization

Establish a matching lower bound for the stochastic-oracle complexity of smooth nonconvex optimization with convex constraints under the assumptions and oracle access used by the Penalized Proximal Subgradient Method with One Projection.

Background

The paper proves a stochastic-oracle complexity rate of O~(ϵ−4)\widetilde O(\epsilon^{-4}) for smooth nonconvex objectives with convex functional constraints while using exactly one projection onto the feasible set, and notes that this exponent matches known unconstrained benchmarks. However, the comparison is only at the level of complexity exponents. A corresponding lower bound has not been established for the paper’s more specific constrained setting, including its regularity assumptions and oracle model. Establishing such a bound would determine whether the reported rate is optimal for this one-projection framework rather than merely comparable to the unconstrained lower-bound exponent.

References

The \widetilde O(\epsilon{-4}) rate matches the optimal unconstrained smooth nonconvex exponent under the oracle models of \citet{arjevani2023lower,jin2026bounded}, while using one final projection. This is a comparison of complexity exponents; a matching lower bound under all assumptions and oracle access of Theorem~\ref{thm:convex-smooth} is not established here.

— Constrained Nonconvex Stochastic Optimization with One Projection  (2609.34099 - Deng et al., 28 Sep 2026) in Remark following Theorem 3.1, Section 3.2 (Smooth objectives)