đ¤ AI Summary
This work addresses the fundamental limitation of traditional Markov chain Monte Carlo (MCMC) integration, whose convergence rate is typically restricted to $ t^{-1/2} $, hindering efficient high-precision estimation. For the first time, we rigorously incorporate the true self-avoiding walk (TSAW) mechanism into the MCMC framework by dynamically adjusting transition probabilities through penalties on empirical visitation frequencies in a finite state space, thereby discouraging redundant sampling. Leveraging adaptive dynamics based on irreducible Markov kernels and a careful analysis of occupation measures, we establish that, for any bounded integrand, the integration error converges almost surely at a rate of $ O(\sqrt{\log t}/t) $. This rate substantially improves upon classical MCMC methods, yielding significantly enhanced sample efficiency and estimation accuracy.
đ Abstract
We study true self-avoiding walk (TSAW) as a mechanism for improving empirical integral estimation via Markov chain Monte Carlo (MCMC). We consider finite-state adaptive sampling dynamics associated with an irreducible Markov kernel $P$ on a finite set, with stationary distribution $Ī$, in which the transition probabilities are penalized according to empirical overuse.
Our main result is that the empirical occupation counts $L_t(i)$ and transition counts $N_t(i,j)$ of the resulting TSAW-based walk satisfy \[ L_t(i)-tĪ_i = O(\sqrt{\log t}) \quad\text{and}\quad N_t(i,j)-tĪ_iP_{ij}=O(\sqrt{\log t}) \qquad\text{almost surely} \] for every state $i$ and every edge $(i,j)$ with $P_{ij}>0$. Consequently, for every bounded function $f:V\to\mathbb R$, the error of our integral estimator converges as \[ \left|\frac1t\sum_{s=0}^{t-1} f(X_s)-\sum_{i\in V}Ī_i f(i)\right| = O\left(\frac{\sqrt{\log t}}{t}\right) \qquad\text{almost surely}. \]
These results show that, in contrast with the usual $t^{-1/2}$ error scaling for empirical averages under standard random-walk-based methods, TSAW-based estimator yields empirical integral errors of order $O(\sqrt{\log t}/t)$ almost surely, thereby achieving a substantially sharper dependence on the sample size $t$.