具有非加性迭代依赖马尔可夫噪声的随机递归包含
Stochastic Recursive Inclusions with Non-Additive Iterate-Dependent Markov Noise
AI总结：
本文研究了具有多值漂移函数和非加性迭代依赖马尔可夫噪声的随机逼近方案的渐进行为，通过线性插值轨迹证明其为极限微分包含的渐近伪轨迹，并利用极限集定理描述递归的极限集。
摘要：
在本文中，我们研究了具有多值漂移函数和非加性迭代依赖马尔可夫噪声的随机逼近方案的渐进行为。我们证明，此类递归的线性插值轨迹是通过平均递归的多值漂移函数相对于马尔可夫噪声的平稳分布所得到的极限微分包含的渐近伪轨迹。然后利用[1]中的极限集定理，用极限微分包含的动力学来表征递归的极限集。接着，我们陈述了两种马尔可夫噪声假设的变体，使得递归的分析类似于本文所展示的分析。这些包括在非加性迭代依赖马尔可夫噪声存在下的受控随机逼近、次梯度下降、近似漂移问题和不连续动力学的分析等场景。
英文摘要：
In this paper we study the asymptotic behavior of stochastic approximation schemes with set-valued drift function and non-additive iterate-dependent Markov noise. We show that a linearly interpolated trajectory of such a recursion is an asymptotic pseudotrajectory for the flow of a limiting differential inclusion obtained by averaging the set-valued drift function of the recursion w.r.t. the stationary distributions of the Markov noise. The limit set theorem in [1] is then used to characterize the limit sets of the recursion in terms of the dynamics of the limiting differential inclusion. We then state two variants of the Markov noise assumption under which the analysis of the recursion is similar to the one presented in this paper. Scenarios where our recursion naturally appears are presented as applications. These include controlled stochastic approximation, subgradient descent, approximate drift problem and analysis of discontinuous dynamics all in the presence of non-additive iterate-dependent Markov noise.
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