Unified Acceleration of High-Order Algorithms under H\"{o}lder Continuity and Uniform Convexity

Understand

In this paper, through a very intuitive 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 concise unified acceleration framework (UAF), which reconciles the two different high-order acceleration approaches, one by Nesterov and Baes [29, 3, 33] and one by Monteiro and Svaiter [25].
• As result, the UAF unifies the high-order acceleration instances [29, 3, 33, 15, 16, 25, 19, 6, 14] of the two approaches by only two problem-related parameters and two additional parameters for framework design.
• Furthermore, the UAF (and its analysis) is the first approach to make high-order methods applicable for high-order smoothness conditions with respect to non-Euclidean norms.