Improving Online-to-Nonconvex Conversion for Smooth Optimization via Double Optimism

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 $\mathcal{O}(\varepsilon^{-1.75}\log(1/\varepsilon))$ in the deterministic case and a complexity of $\mathcal{O}(\varepsilon^{-3.5})$ 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 $\sigma^2$, we obtain a unified algorithm with complexity $\mathcal{O}(\varepsilon^{-1.75} + \sigma^2 \varepsilon^{-3.5})$, smoothly interpolating between the best-known deterministic rate and the optimal stochastic rate.