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首页> 外文期刊>IEEE Transactions on Automatic Control >Self-Tuned Stochastic Approximation Schemes for Non-Lipschitzian Stochastic Multi-User Optimization and Nash Games
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Self-Tuned Stochastic Approximation Schemes for Non-Lipschitzian Stochastic Multi-User Optimization and Nash Games

机译：非Lipschitzian随机多用户优化和Nash游戏的自调整随机逼近方案

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摘要

We consider multi-user optimization problems and Nash games with stochastic convex objectives, instances of which arise in decentralized control problems. The associated equilibrium conditions of both problems can be cast as Cartesian stochastic variational inequality problems with mappings that are strongly monotone but not necessarily Lipschitz continuous. Consequently, most of the currently available stochastic approximation schemes cannot address such problems. First, through a user-specific local smoothing, we derive an approximate map that is shown to be Lipschitz continuous with a prescribed constant. Second, motivated by the need for a robust scheme that can be implemented in a distributed fashion, we develop a distributed self-tuned stochastic approximation scheme (DSSA) that adapts to problem parameters. Importantly, this scheme is provably convergent in an almost-sure sense and displays the optimal rate of convergence in mean squared error, i.e., . A locally randomized variant is also provided to ensure that the scheme can contend with stochastic non-Lipschitzian multi-user problems. We conclude with numerics derived from a stochastic Nash-Cournot game.
机译：我们考虑具有随机凸目标的多用户优化问题和Nash游戏，其实例出现在分散控制问题中。可以将这两个问题的相关均衡条件转换为具有很强单调但不一定是Lipschitz连续的映射的笛卡尔随机变分不等式问题。因此，大多数当前可用的随机逼近方案不能解决这些问题。首先，通过用户特定的局部平滑，我们得出一个近似图，该图显示为具有规定常数的Lipschitz连续。其次，出于对可以以分布式方式实施的健壮方案的需求的激励，我们开发了一种适用于问题参数的分布式自调整随机近似方案（DSSA）。重要的是，该方案在几乎确定的意义上证明是收敛的，并且显示出均方误差的最佳收敛速率，即。还提供了本地随机变量，以确保该方案可以应对随机的非Lipschitzian多用户问题。我们以随机Nash-Cournot游戏得出的数字作为结论。

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