随机不动点问题的随机函数迭代
Random Function Iterations for Stochastic Fixed Point Problems
摘要：
研究了寻找相应Markov算子的不变测度的随机函数迭代的收敛性。我们把寻找这样一个不变测度的问题称为随机不动点问题。这推广了早期研究随机可行性问题的工作，即找到概率为1的点，即随机函数的不动点[Hermer，Luke，Sturm，2019]。当不存在这样的点时，随机可行性问题被称为不相容的，但仍然在某些假设下，更一般的随机不动点问题有解，并且随机函数迭代收敛到相应的马尔可夫算子的不变测度。有两种主要的收敛类型：在随机可行的情况下，迭代几乎必然收敛到一个不动点，以及更一般的分布依分布收敛。我们展示了如何在确定性不动点理论的共同结构可以利用建立存在的不变测度和收敛的马尔可夫链。我们表明，较弱的假设比通常遇到的分析马尔可夫链保证线性/几何收敛。这个框架专门针对当前感兴趣的许多应用，包括，例如，大规模分布式计算的随机算法，以及具有计算误差的确定性迭代过程。在这项研究中开发的理论提供了一个坚实的基础，描述简单的计算方法的收敛性，而无需假设无限精度的算术或消失的计算误差。
英文摘要：
We study the convergence of random function iterations for finding an invariant measure of the corresponding Markov operator. We call the problem of finding such an invariant measure the stochastic fixed point problem. This generalizes earlier work studying the stochastic feasibility problem}, namely, to find points that are, with probability 1, fixed points of the random functions [Hermer, Luke, Sturm, 2019]. When no such points exist, the stochastic feasibility problem is called inconsistent, but still under certain assumptions, the more general stochastic fixed point problem has a solution and the random function iterations converge to an invariant measure for the corresponding Markov operator. There are two major types of convergence: almost sure convergence of the iterates to a fixed point in the case of stochastic feasibility, and convergence in distribution more generally. We show how common structures in deterministic fixed point theory can be exploited to establish existence of invariant measures and convergence of the Markov chain. We show that weaker assumptions than are usually encountered in the analysis of Markov chains guarantee linear/geometric convergence. This framework specializes to many applications of current interest including, for instance, stochastic algorithms for large-scale distributed computation, and deterministic iterative procedures with computational error. The theory developed in this study provides a solid basis for describing the convergence of simple computational methods without the assumption of infinite precision arithmetic or vanishing computational errors.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
