Improving Online-to-Nonconvex Conversion for Smooth Optimization via Double Optimism
Francisco Patitucci, Ruichen Jiang, Aryan Mokhtari
TL;DR
The paper addresses smooth nonconvex minimization by unifying deterministic and stochastic guarantees through online-to-nonconvex conversion. It introduces Online Doubly Optimistic Gradient (ODOG), which uses an extrapolated-gradient hint and a doubly optimistic update to remove the inner fixed-point loop and achieve a unified rate that interpolates between deterministic and stochastic regimes. The key contributions are a simple, two-gradient-query algorithm with a novel hint construction and a rigorous complexity analysis showing $\mathcal{O}(\varepsilon^{-1.75} + σ^2 \varepsilon^{-3.5})$ performance, plus an adaptive step-size scheme that preserves this rate without tuning. This approach improves practicality while maintaining theoretical tightness, and it suggests a path toward fully parameter-free, adaptive nonconvex optimizers. Overall, the work advances the efficiency and applicability of online-to-nonconvex conversion for smooth nonconvex optimization by combining simplicity, modularity, and optimal rates across settings.
Abstract
A recent breakthrough in nonconvex optimization is the online-to-nonconvex conversion framework of [Cutkosky et al., 2023], which reformulates the task of finding an $\varepsilon$-first-order stationary point as an online learning problem. When both the gradient and the Hessian are Lipschitz continuous, instantiating this framework with two different online learners achieves a complexity of $O(\varepsilon^{-1.75}\log(1/\varepsilon))$ in the deterministic case and a complexity of $O(\varepsilon^{-3.5})$ in the stochastic case. However, this approach suffers from several limitations: (i) the deterministic method relies on a complex double-loop scheme that solves a fixed-point equation to construct hint vectors for an optimistic online learner, introducing an extra logarithmic factor; (ii) the stochastic method assumes a bounded second-order moment of the stochastic gradient, which is stronger than standard variance bounds; and (iii) different online learning algorithms are used in the two settings. In this paper, we address these issues by introducing an online optimistic gradient method based on a novel doubly optimistic hint function. Specifically, we use the gradient at an extrapolated point as the hint, motivated by two optimistic assumptions: that the difference between the hint and the target gradient remains near constant, and that consecutive update directions change slowly due to smoothness. Our method eliminates the need for a double loop and removes the logarithmic factor. Furthermore, by simply replacing full gradients with stochastic gradients and under the standard assumption that their variance is bounded by $σ^2$, we obtain a unified algorithm with complexity $O(\varepsilon^{-1.75} + σ^2 \varepsilon^{-3.5})$, smoothly interpolating between the best-known deterministic rate and the optimal stochastic rate.