The subdifferential of measurable composite max integrands and smoothing approximation
The subdifferential of measurable composite max integrands and smoothing approximation
复制标题
可测复合最大被积函数的次微分与平滑近似
DOI：
10.1007/s10107-019-01441-9
复制
发表时间：
2020
影响因子：
2.7
通讯作者：
Hailin Sun
中科院分区：
文献类型：
--
作者：
James V. Burke;Xiaojun Chen;Hailin Sun
非光滑随机被积函数期望的次微分涉及随机优化中的许多基本问题和挑战性问题。已知对于Clarke正则被积函数，期望的Clarke次微分等于其Clarke次微分的期望。特别地，这适用于凸被积函数。然而，很少有人知道的计算Clarke次梯度的期望的非正则被积。这一贡献的重点是近似克拉克次梯度的期望随机被积函数的平滑方法应用于被积函数。一个框架，如何进行沿着这条道路的发展，然后应用到一类可测的复合最大积分。这类包含非正规的被积函数从随机互补问题以及随机优化问题中产生的统计学习。
The subdifferential calculus for the expectation of nonsmooth random integrands involves many fundamental and challenging problems in stochastic optimization. It is known that for Clarke regular integrands, the Clarke subdifferential of the expectation equals the expectation of their Clarke subdifferential. In particular, this holds for convex integrands. However, little is known about the calculation of Clarke subgradients for the expectation of non-regular integrands. The focus of this contribution is to approximate Clarke subgradients for the expectation of random integrands by smoothing methods applied to the integrand. A framework for how to proceed along this path is developed and then applied to a class ofmeasurable composite max integrands. This class contains non-regular integrands from stochastic complementarity problems as well as stochastic optimization problems arising in statistical learning.