论文标题
Nesterov在非convex优化中的加速方法的注释：弱估计序列方法
A Note on Nesterov's Accelerated Method in Nonconvex Optimization: a Weak Estimate Sequence Approach
论文作者
论文摘要
我们提出了一种适用于Nesterov的最佳一阶方法的加速梯度下降算法的变体，用于虚弱的Quasi-Convex和弱Quasi-trong-trong-trong-convex函数。我们表明，通过调整所谓的估计序列方法，衍生算法可实现弱Quasi-Convex和弱Quasi-trong-trong-rong-trong-Convex的最佳收敛速率，以甲骨文的复杂性。特别是，对于具有Lipschitz连续梯度的弱Quasi-Convex函数，我们需要$ O（\ frac {1} {\ sqrt {\ sqrt {\ varepsilon}}）$ iterations $ iterations $ iterations才能获取$ \ varepsilon $ -sondolution $ -SONOLUTION;对于虚弱的Quasi-Strongly-convex函数，迭代复杂度为$ o \ left（\ ln \ left（\ frac {1} {\ varepsilon} \ right）\ right）$。此外，我们讨论了这些算法对线性二次最佳控制问题的含义。
We present a variant of accelerated gradient descent algorithms, adapted from Nesterov's optimal first-order methods, for weakly-quasi-convex and weakly-quasi-strongly-convex functions. We show that by tweaking the so-called estimate sequence method, the derived algorithm achieves optimal convergence rate for weakly-quasi-convex and weakly-quasi-strongly-convex in terms of oracle complexity. In particular, for a weakly-quasi-convex function with Lipschitz continuous gradient, we require $O(\frac{1}{\sqrt{\varepsilon}})$ iterations to acquire an $\varepsilon$-solution; for weakly-quasi-strongly-convex functions, the iteration complexity is $O\left( \ln\left(\frac{1}{\varepsilon}\right) \right)$. Furthermore, we discuss the implications of these algorithms for linear quadratic optimal control problem.
