True Self-Avoiding Walk for Accelerating Markov-Chain Monte Carlo Integration
This work provides a significantly faster convergence rate for MCMC integral estimation, which is crucial for researchers and practitioners who rely on MCMC for complex simulations and data analysis, especially in fields like physics, statistics, and machine learning.
This paper introduces a true self-avoiding walk (TSAW) mechanism to accelerate Markov Chain Monte Carlo (MCMC) integration. The authors demonstrate that their TSAW-based estimator achieves an error scaling of O(sqrt(log t)/t) almost surely, which is a substantial improvement over the standard O(t^-1/2) error scaling of traditional random-walk-based MCMC methods.
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 $π$, 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 \[ L_t(i)-tπ_i = O(\sqrt{\log t}) \quad\text{and}\quad N_t(i,j)-tπ_iP_{ij}=O(\sqrt{\log t}) \qquad\text{almost surely} \] 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 \[ \left|\frac1t\sum_{s=0}^{t-1} f(X_s)-\sum_{i\in V}π_i f(i)\right| = O\left(\frac{\sqrt{\log t}}{t}\right) \qquad\text{almost surely}. \] 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 $t$.