  Since Nesterov's seminal 1983 work, many accelerated first-order optimization
methods have been proposed, but their analyses lacks a common unifying
structure. In this work, we identify a geometric structure satisfied by a wide
range of first-order accelerated methods. Using this geometric insight, we
present several novel generalizations of accelerated methods. Most interesting
among them is a method that reduces the squared gradient norm with
$\mathcal{O}(1/K^4)$ rate in the prox-grad setup, faster than the
$\mathcal{O}(1/K^3)$ rates of Nesterov's FGM or Kim and Fessler's FPGM-m.
