Optimizing the Preconditioner: A Black-box Online-to-Nonconvex Conversion with Static Regret Minimization Oracles

Stochastic nonconvex optimization is central to training deep networks and LLMs in modern machine learning. We give a black-box reduction from stochastic nonconvex optimization to ordinary static regret minimization in online convex optimization (OCO), thereby resolving the open problem posed by Chen and Hazan (2024). Our reduction maintains a predictable gradient tracker, while a black-box online learner $\mathcal{A}$ selects a preconditioner that transforms this tracker into the update direction. Given a $β$-smooth function with a range bounded by $M$ and an unbiased gradient oracle with variance bounded by $σ^2$, we bound the expected average squared gradient norm by $O(σ\sqrt{Mβ/T}+\sqrt{Mβ}\mathrm{Reg}_T(\mathcal{A})/T+\frac{Mβ}{T})$, where $\mathrm{Reg}_T(\mathcal{A})$ is the static regret of $\mathcal{A}$. Thus, any OCO oracle with $O(\sqrt{T})$ regret recovers the classical $O(T^{-1/2})$ convergence rate. We further extend the framework to nonsmooth nonconvex objectives, still relying only on ordinary static regret, and attain the optimal convergence rate for Goldstein-type stationarity. Finally, we conduct numerical experiments on nonconvex objectives to illustrate how the reduction exploits online-selected preconditioners while using the same stochastic-oracle budget as stochastic gradient descent.