Gradient Methods with Online Scaling Part I. Theoretical Foundations

This paper establishes the theoretical foundations of the online scaled gradient methods (OSGM), a framework that utilizes online learning to adapt stepsizes and provably accelerate first-order methods. OSGM quantifies the effectiveness of a stepsize by a feedback function motivated from a convergence measure and uses the feedback to adjust the stepsize through an online learning algorithm. Consequently, instantiations of OSGM achieve convergence rates that are asymptotically no worse than the optimal stepsize. OSGM yields desirable convergence guarantees on smooth convex problems, including 1) trajectory-dependent global convergence on smooth convex objectives; 2) an improved complexity result on smooth strongly convex problems, and 3) local superlinear convergence. Notably, OSGM constitutes a new family of first-order methods with non-asymptotic superlinear convergence, joining the celebrated quasi-Newton methods. Finally, OSGM explains the empirical success of the popular hypergradient-descent heuristic in optimization for machine learning.

Gradient Methods with Online Scaling

We introduce a framework to accelerate the convergence of gradient-based methods with online learning. The framework learns to scale the gradient at each iteration through an online learning algorithm and provably accelerates gradient-based methods asymptotically. In contrast with previous literature, where convergence is established based on worst-case analysis, our framework provides a strong convergence guarantee with respect to the optimal scaling matrix for the iteration trajectory. For smooth strongly convex optimization, our results provide an $O(\kappa^\star \log(1/\varepsilon)$) complexity result, where $\kappa^\star$ is the condition number achievable by the optimal preconditioner, improving on the previous $O(\sqrt{n}\kappa^\star \log(1/\varepsilon))$ result. In particular, a variant of our method achieves superlinear convergence on convex quadratics. For smooth convex optimization, we show for the first time that the widely-used hypergradient descent heuristic improves on the convergence of gradient descent.