High-probability stationarity guarantees

Establish high-probability convergence guarantees for the constrained nonsmooth nonconvex optimization algorithms, analogous to the expectation bounds proved for their stationarity measures.

Background

The paper's main guarantees control the expectation of a constrained Goldstein residual or Goldstein Frank-Wolfe gap. The authors identify analogous high-probability guarantees as an unresolved direction, while noting that standard amplification techniques may be applicable. Such results would strengthen the probabilistic form of the convergence guarantees beyond expected stationarity.

References

Finally, similarly to \citet{liu2024}, our guarantees control the expectation of a stationarity measure (in our case, a stronger one, as discussed in Remark~\ref{rem: stronger stat}), whereas obtaining analogous high-probability guarantees should be possible using standard amplification techniques, which we do not pursue here.

— Optimal Algorithms for Nonsmooth Nonconvex Stochastic Optimization with Constraints  (2610.13787 - Buzaglo et al., 9 Oct 2026) in Discussion section