Towards Unified Acceleration of High-Order Algorithms under Hölder Continuity and Uniform Convexity

In this paper, through a very intuitive {\em vanilla proximal method} perspective, we derive accelerated high-order optimization algorithms for minimizing a convex function that has H\"{o}lder continuous derivatives. In this general convex setting, we propose a {\em unified acceleration algorithm} with an iteration complexity that matches the lower iteration complexity bound given in \cite{grapiglia2019tensor}. If the function is further uniformly convex, we propose a {\em general restart scheme}. The iteration complexity of the algorithm matches existing lower bounds in most important cases. For practical implementation, we introduce a new and effective heuristic that significantly simplifies the binary search procedure required by the algorithm, which makes the algorithm in general settings as efficient as the special case \cite{grapiglia2019tensor}. On large-scale classification datasets, our algorithm demonstrates clear and consistent advantages of high-order acceleration methods over first-order ones, in terms of run-time complexity. Our formulation considers the more general composite setting in which the objective function may contain a second possibly non-smooth convex term. Our analysis and proofs are also applicable to the general case in which the high-order smoothness conditions are with respect to non-Euclidean norms.