论文标题
自适应正则化，用于使用不精确函数值和随机扰动的衍生物进行非convex优化
Adaptive Regularization for Nonconvex Optimization Using Inexact Function Values and Randomly Perturbed Derivatives
论文作者
论文摘要
提出了一种允许衍生物中随机噪声和不精确函数值随机噪声的正则化算法，以计算任何顺序不受约束的优化问题的任何顺序的局部临界点。对于Lipschitz连续$ p $ -th导数并给定任意最佳订单$ q \ leq p $的目标函数，这表明该算法最多将计算出这种观点$ o \ left（\ left（\ min_ {j \ in \ {1，\ ldots，q \}}ε_jj\ right） $ε_j$是$ j $ th订单准确性的公差。此绑定最多会变为$ o \ left（\ left（\ min_ {j \ in \ {1，\ ldots，q \}}ε_j\ right）^{ - \ frac {q（p+1）} {p+1）} {p}}}} {p}}} \ right）此外，这些边界在准确性公差的顺序上是锋利的。还概述了对凸的约束问题的扩展。
A regularization algorithm allowing random noise in derivatives and inexact function values is proposed for computing approximate local critical points of any order for smooth unconstrained optimization problems. For an objective function with Lipschitz continuous $p$-th derivative and given an arbitrary optimality order $q \leq p$, it is shown that this algorithm will, in expectation, compute such a point in at most $O\left(\left(\min_{j\in\{1,\ldots,q\}}ε_j\right)^{-\frac{p+1}{p-q+1}}\right)$ inexact evaluations of $f$ and its derivatives whenever $q\in\{1,2\}$, where $ε_j$ is the tolerance for $j$th order accuracy. This bound becomes at most $O\left(\left(\min_{j\in\{1,\ldots,q\}}ε_j\right)^{-\frac{q(p+1)}{p}}\right)$ inexact evaluations if $q>2$ and all derivatives are Lipschitz continuous. Moreover these bounds are sharp in the order of the accuracy tolerances. An extension to convexly constrained problems is also outlined.
