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

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arXiv:2607.17607v2 Announce Type: replace Abstract: We study whether stochastic nonconvex optimization can be reduced to ordinary static regret minimization in online convex optimization in a black-box manner. For smooth nonconvex objectives, our reduction maintains a predictable gradient tracker, while a black-box online learner selects a preconditioner that determines how this tracker is transformed into the update direction. The learner receives linear convex losses and is evaluated against a single fixed comparator over one undiscounted online game. For a $\beta$-smooth objective with range bounded by $M$ and an unbiased stochastic-gradient oracle with variance bounded by \(\sigma^2\), we establish $$\frac{1}{T}\sum_{t=1}^T \mathbb E\!\left[\|\nabla f(x_t)\|_2^2\right] \lesssim \frac{\sigma\sqrt{M\beta}}{\sqrt T} + \frac{\sqrt{M\beta}\, \mathscr R_T(\mathcal A,I_d)}{T} + \frac{M\beta}{T}.$$ Consequently, any black-box OCO algorithm with $\mathscr R_T(\mathcal A,I_d)=O(\sqrt T)$ recovers the classical $O(\frac{1}{\sqrt{T}})$ convergence rate. We further show that the same black-box framework extends beyond the smooth setting to Lipschitz nonconvex objectives without Lipschitz continuous gradients. Importantly, this extension continues to rely only on an ordinary static-regret guarantee and requires no stronger notion of online regret. When the OCO oracle admits square-root static regret, the resulting conversion achieves the optimal $O(T^{-2/7})$ convergence rate for the corresponding Goldstein stationary point. These results resolve the open problem posed by …

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