On Statistical Estimation of Edge-Reinforced Random Walks

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~citemgr, fmk, tarres, volkov. These models have found applications in various areas, such as network representation learning~citexzzs, reinforced PageRank~citegly, and modeling animal behaviors~citesmouse, 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)~citemr, 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.