论文标题
希尔伯特空间中非凸问题的随机近端梯度方法
Stochastic Proximal Gradient Methods for Nonconvex Problems in Hilbert Spaces
论文作者
论文摘要
对于有限维问题，随机近似方法长期以来一直用于解决随机优化问题。他们在无限维度问题上的应用不了解，尤其是对于非洞穴目标。本文介绍了应用于希尔伯特空间的随机近端梯度方法的收敛结果，这是由具有随机输入和系数的偏微分方程（PDE）约束的优化问题所激发的。我们研究了非凸和非平滑问题的随机算法，其中非平滑部分是凸的，而非凸的部分是期望，假定它具有Lipschitz的连续梯度。优化变量是希尔伯特空间的元素。我们显示了算法与固定点产生的随机序列的强极点的几乎确定收敛性。我们演示了在跟踪型功能上的随机近端梯度算法，其功能具有$ l^1 $ - 苯甲酸术语，该项受半连续PDE和盒子约束约束，其中输入项和系数均受到不确定性。我们验证确保算法收敛并显示模拟的条件。
For finite-dimensional problems, stochastic approximation methods have long been used to solve stochastic optimization problems. Their application to infinite-dimensional problems is less understood, particularly for nonconvex objectives. This paper presents convergence results for the stochastic proximal gradient method applied to Hilbert spaces, motivated by optimization problems with partial differential equation (PDE) constraints with random inputs and coefficients. We study stochastic algorithms for nonconvex and nonsmooth problems, where the nonsmooth part is convex and the nonconvex part is the expectation, which is assumed to have a Lipschitz continuous gradient. The optimization variable is an element of a Hilbert space. We show almost sure convergence of strong limit points of the random sequence generated by the algorithm to stationary points. We demonstrate the stochastic proximal gradient algorithm on a tracking-type functional with a $L^1$-penalty term constrained by a semilinear PDE and box constraints, where input terms and coefficients are subject to uncertainty. We verify conditions for ensuring convergence of the algorithm and show a simulation.
