  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. If the
function is further uniformly convex, we propose a general restart scheme for
the UAF. The iteration complexities of instances of both the UAF and the
restarted UAF match existing lower bounds in most important cases [2, 16]. For
practical implementation, we introduce a new and effective heuristic that
significantly simplifies the binary search procedure required by the framework.
We use experiments to verify the effectiveness of the heuristic and demonstrate
clear and consistent advantages of high-order acceleration methods over
first-order ones, in terms of run-time complexity. Finally, the UAF is proposed
directly in the general composite convex setting, thus show that the existing
high-order algorithms [29, 3, 33, 16, 6, 14] can be naturally extended to the
general composite convex setting.
