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 $O(T^{-1.119})$ 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 $\exp(-\Omega(T/\kappa^{0.893}))$ for smooth and strongly convex optimization, with $\kappa$ being the condition number.

翻译：本研究探讨了具有任意时间收敛保证的基于步长的梯度下降加速方法。针对光滑（非强）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$O(T^{-1.119})$的收敛保证，且该步长调度策略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}——即基于步长的加速能否产生$o(T^{-1})$的任意时间收敛率——提供了肯定答案。我们进一步扩展理论，针对光滑强凸优化问题，在条件数为$\kappa$的情况下，实现了$\exp(-\Omega(T/\kappa^{0.893}))$的任意时间收敛保证。