随机环境下的自交互和学习

The project will focus on the study of self-interacting processes in random structures, and on the learning features that arise from the interaction. These processes model phenomena observed in physics, economics, population genetics and ethology. They are generally governed by a simple local rule, which creates a global structure as a result of its repeated iteration. ..For instance, self-interacting random walks can be self-repelling or self-attracting, in other words more likely to stay away from or to come back to the places already visited before. These random walks are not Markov processes and eventually either localize on particular subsets or scatter on the graph. ..Our goal is to develop generic theoretical tools for the study of these processes. ..A first aim is to develop the link between self-interacting random walks and statistical physics. This equivalence was recently shown for the Edge-Reinforced Random Walk (Sabot-Tarrès, 2011) with the supersymmetric hyperbolic sigma model, a random Schrödinger operator and Dynkin’s isomorphism, but other interacting walks arising from Bayesian statistics have similar structure and are likely to display a similar link. ..A second objective is to describe, under general assumptions, the behavior of the distribution of the local time of these self-interacting random walks. This approach was carried out by Tóth in the 90’s on the integer graph, and could not be extended to general graphs, but recent analysis introduced similar techniques in the cases of Brownian polymers, self-avoiding random walks and strongly reinforced random walks, which suggests that results could be obtained in a more general framework...A third aim is to apply our techniques to the study of self-interacting processes proposed in biology, physics and economics. We will analyse multiparticle self-interacting processes, which provide useful models in biology (ants, bacteria), and start by a description of the large particle limit. We will also study related models arising in game theory and statistical learning. We have already conducted promising research in that area, in particular on reinforcement learning models language or network formation, online learning and multi-armed bandit algorithms. Our techniques allow us to approach the topic from a different angle.

本项目研究随机环境中的自交互过程（self-interacting processes）以及在交互过程中产生的学习功能。这些随机过程给物理、经济学、群体遗传学、行为学中所观测到的不同现象提供很好的数学模型。它们一般由简单的局部规则支配，由于这些局部规则的长期重复作用会产生全局的结构变化。..举例来说，自交互游走可以是自排斥也可以是自吸引，也就是说它们可以倾向于远离已访问过的地方也可以倾向于重回已访问过的地方。这些游走并不是马尔科夫过程，它们最终要么只在一个小区域活动要么在整个图上都有可能出现。..我们的目标就是建立一般的理论工具来研究上面提到的过程。..我们的首要目标是建立自交互随机游走和统计物理之间的联系。Sabot和Tarrès于2011年找到了一个这样的联系：他们建立了边增强随机游走（Edge-Reinforced Random Walk）和超对称双曲sigma模型、随机薛定谔算子、

本课题的目标是对随机结构中的自交互和学习过程进行严谨的研究，并着重于发展用于理解自交互随机游走的新的通用技术。我们主要关注了以下问题：.（a）.自交互随机游走和统计物理之间的联系.（b）.自交互随机游走的局部时过程的动态演化.（c）.树上的一次强化随机游走..问题（a）源于边强化随机游走，点强化跳过程和统计物理中的各种对象（超对称双曲西格玛模型，随机薛定谔算子，Ray-Knight/Dynkin同构）之间的联系。..最近和Bacallado以及Sabot（两篇预印本可见于Arxiv，2021）一起，我们将那些性质推广到一类随机游走。这种随机游走源于对可变次序可逆马尔科夫链的贝叶斯分析，被称为*-边强化随机游走(*-ERRW, Bacallado, 2006)。我们定义与之相关联的对应的连续时间*-点强化跳过程并引入对应的数个统计场和恒等式。..和Lupu以及Sabot一起，我们在2020年引入一种被称为线性强化运动的新的自交互过程。它可以被视为点强化跳过程在一维线上的细网格极限，并能够用Bass-Burdzy流和扩散的混合来表示。..和Lupu以及Sabot一起，我们在2019年通过一种自交互随机游走描述了符号高斯自由场和圈系综之间耦合的逆转。我们也定义了一类自排斥一维扩散过程。它是上面提到的线性强化运动的共轭，逆转了Ray-Knight恒等式。..和Merkl以及Rolles一起，我们在2019年对在点强化跳过程的极限中的超对称双曲西格玛变量提供了一种诠释，并在2021年为一般图上的暂留点强化跳过程引入了随机交织。..对问题（b），和Huang, Kious以及Sidoravicius一起，我们证明了连续时间马尔科夫链的局部时，末离树以及循环数的联合密度的一个简单公式。在和Collevecchio的共同工作中（正在进行中），我们通过一个涉及局部时的有趣公式描述一次强化随机游走。..对问题（c），Collevecchio, Kious和Sidoravicius 在2019年得到了树上的一次强化随机游走的常返暂留判断准则。