Improving Online-to-Nonconvex Conversion for Smooth Optimization via Double Optimism
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| Format: | Preprint |
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2025
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| _version_ | 1866912889397313536 |
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| author | Patitucci, Francisco Jiang, Ruichen Mokhtari, Aryan |
| author_facet | Patitucci, Francisco Jiang, Ruichen Mokhtari, Aryan |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_03167 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Improving Online-to-Nonconvex Conversion for Smooth Optimization via Double Optimism Patitucci, Francisco Jiang, Ruichen Mokhtari, Aryan Optimization and Control Machine Learning 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. |
| title | Improving Online-to-Nonconvex Conversion for Smooth Optimization via Double Optimism |
| topic | Optimization and Control Machine Learning |
| url | https://arxiv.org/abs/2510.03167 |