  In this paper, we provide near-optimal accelerated first-order methods for
minimizing a broad class of smooth nonconvex functions that are strictly
unimodal on all lines through a minimizer. This function class, which we call
the class of smooth quasar-convex functions, is parameterized by a constant
$\gamma \in (0,1]$, where $\gamma = 1$ encompasses the classes of smooth convex
and star-convex functions, and smaller values of $\gamma$ indicate that the
function can be "more nonconvex." We develop a variant of accelerated gradient
descent that computes an $\epsilon$-approximate minimizer of a smooth
$\gamma$-quasar-convex function with at most $O(\gamma^{-1} \epsilon^{-1/2}
\log(\gamma^{-1} \epsilon^{-1}))$ total function and gradient evaluations. We
also derive a lower bound of $\Omega(\gamma^{-1} \epsilon^{-1/2})$ on the
worst-case number of gradient evaluations required by any deterministic
first-order method, showing that, up to a logarithmic factor, no deterministic
first-order method can improve upon ours.
