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.

翻译：本文针对一类光滑非凸函数的极小化问题，提出了近优的加速一阶方法。这类函数在通过最小化点的所有直线上严格单峰，我们将其称为光滑类拟凸函数类，由常数$\gamma \in (0,1]$参数化。当$\gamma=1$时，该函数类包含光滑凸函数和星凸函数；$\gamma$值越小表示函数可能"更非凸"。我们开发了一种加速梯度下降的变体算法，对于光滑$\gamma$-类拟凸函数，仅需至多$O(\gamma^{-1} \epsilon^{-1/2} \log(\gamma^{-1} \epsilon^{-1}))$次函数与梯度评估即可计算$\epsilon$-近似最小点。同时，我们推导出任意确定性一阶方法在最坏情况下所需梯度评估次数的下界为$\Omega(\gamma^{-1} \epsilon^{-1/2})$，这表明除了对数因子外，任何确定性一阶方法均无法优于我们的算法。