This work investigates stepsize-based acceleration of gradient descent with {\em anytime} convergence guarantees. For smooth (non-strongly) convex optimization, we propose a stepsize schedule that allows gradient descent to achieve convergence guarantees of $O(T^{-1.03})$ for any stopping time $T$, where the stepsize schedule is predetermined without prior knowledge of the stopping time. This result provides an affirmative answer to a COLT open problem \citep{kornowski2024open} regarding whether stepsize-based acceleration can yield anytime convergence rates of $o(T^{-1})$. We further extend our theory to yield anytime convergence guarantees of $\exp(-\Omega(T/\kappa^{0.97}))$ for smooth and strongly convex optimization, with $\kappa$ being the condition number.

翻译：本研究探讨了基于步长的梯度下降加速方法，并提供了具有任意时间收敛保证的理论分析。针对光滑（非强凸）凸优化问题，我们提出了一种步长调度策略，使得梯度下降能够在任意停止时间$T$处实现$O(T^{-1.03})$的收敛保证，且该步长调度无需预先知道停止时间即可预先确定。这一结果为COLT开放问题\citep{kornowski2024open}——即基于步长的加速方法能否产生$o(T^{-1})$的任意时间收敛速率——提供了肯定答案。我们进一步将理论拓展至光滑强凸优化问题，获得了$\exp(-\Omega(T/\kappa^{0.97}))$的任意时间收敛保证，其中$\kappa$为条件数。