Reinforced random walks (RRWs), including vertex-reinforced random walks (VRRWs) and edge-reinforced random walks (ERRWs), model random walks where the transition probabilities evolve based on prior visitation history~\cite{mgr, fmk, tarres, volkov}. These models have found applications in various areas, such as network representation learning~\cite{xzzs}, reinforced PageRank~\cite{gly}, and modeling animal behaviors~\cite{smouse}, among others. However, statistical estimation of the parameters governing RRWs remains underexplored. This work focuses on estimating the initial edge weights of ERRWs using observed trajectory data. Leveraging the connections between an ERRW and a random walk in a random environment (RWRE)~\cite{mr, mr2}, as given by the so-called ``magic formula", we propose an estimator based on the generalized method of moments. To analyze the sample complexity of our estimator, we exploit the hyperbolic Gaussian structure embedded in the random environment to bound the fluctuations of the underlying random edge conductances.

翻译：增强随机游走（RRWs），包括顶点增强随机游走（VRRWs）和边增强随机游走（ERRWs），建模了转移概率基于先前访问历史演化的随机游走模型~\cite{mgr, fmk, tarres, volkov}。这些模型已应用于多个领域，例如网络表示学习~\cite{xzzs}、增强PageRank~\cite{gly}以及动物行为建模~\cite{smouse}等。然而，控制RRWs参数的统计估计问题仍未被充分探索。本文聚焦于利用观测到的轨迹数据估计ERRWs的初始边权重。通过利用ERRW与随机环境中的随机游走（RWRE）之间的关联~\cite{mr, mr2}（即所谓的“魔法公式”），我们提出基于广义矩方法的估计量。为分析该估计量的样本复杂度，我们利用随机环境中嵌入的双曲高斯结构来约束底层随机边电导的波动。