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

2026-07-20

AI Summary

This work proposes a black-box framework that unifies stochastic non-convex optimization by reducing it to static regret minimization in online convex optimization. By introducing a predictable gradient tracker and leveraging a black-box online learner to adaptively select a preconditioner for generating update directions, the method relies solely on standard static regret guarantees to handle both smooth and non-smooth non-convex objectives. This approach provides a theoretical justification for adaptive algorithms such as AdaGrad, resolving an open problem posed by Chen and Hazan (2024). In terms of convergence, the method achieves the classical $O(1/\sqrt{T})$ rate for smooth objectives and attains the optimal $O(T^{-2/7})$ rate for converging to a Goldstein stationary point under non-smooth Lipschitz non-convex settings.

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 $β$-smooth objective with range bounded by $M$ and an unbiased stochastic-gradient oracle with variance bounded by \(σ^2\), we establish $$\frac{1}{T}\sum_{t=1}^T \mathbb E\!\left[\|\nabla f(x_t)\|_2^2\right] \lesssim \frac{σ\sqrt{Mβ}}{\sqrt T} + \frac{\sqrt{Mβ}\, \mathscr R_T(\mathcal A,I_d)}{T} + \frac{Mβ}{T}.$$ 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 Chen and Hazan (2024). More broadly, our framework separates optimizer design into gradient prediction and online preconditioner selection, providing a principled perspective on how adaptive optimization methods such as AdaGrad and Shampoo may be understood through static regret and applied in nonconvex optimization.

Problem

• nonconvex optimization
• online convex optimization
• static regret
• black-box reduction
• preconditioner

Innovation

• black-box reduction
• static regret minimization
• preconditioner optimization
• nonconvex optimization
• Goldstein stationary point