  Quasar convexity is a condition that allows some first-order methods to
efficiently minimize a function even when the optimization landscape is
non-convex. Previous works develop near-optimal accelerated algorithms for
minimizing this class of functions, however, they require a subroutine of
binary search which results in multiple calls to gradient evaluations in each
iteration, and consequently the total number of gradient evaluations does not
match a known lower bound. In this work, we show that a recently proposed
continuized Nesterov acceleration can be applied to minimizing quasar convex
functions and achieves the optimal bound with a high probability. Furthermore,
we find that the objective functions of training generalized linear models
(GLMs) satisfy quasar convexity, which broadens the applicability of the
relevant algorithms, while known practical examples of quasar convexity in
non-convex learning are sparse in the literature. We also show that if a smooth
and one-point strongly convex, Polyak-Lojasiewicz, or quadratic-growth function
satisfies quasar convexity, then attaining an accelerated linear rate for
minimizing the function is possible under certain conditions, while
acceleration is not known in general for these classes of functions.
