Anytime Acceleration of Gradient Descent

This work investigates stepsize-based acceleration of gradient descent with anytime convergence guarantees. For smooth (non-strongly) convex optimization, we propose a stepsize schedule that allows gradient descent to achieve convergence guarantees of $O\big(T^{-\frac{2\log_2\rho}{1+\log_2\rho}}\big) \approx O(T^{-1.119})$ for any stopping time $T$, where $\rho=\sqrt{2}+1$ is the silver ratio and the stepsize schedule is predetermined without prior knowledge of the stopping time. This result provides an affirmative answer to a COLT open problem 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.