Near-Optimal Methods for Minimizing Star-Convex Functions and Beyond

In this paper, we provide near-optimal accelerated first-order methods for minimizing a broad class of smooth nonconvex functions that are 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 $$\gamma \in (0,1]$$: $$\gamma = 1$$ 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.