🤖 AI Summary
This paper addresses the finite-sample error and computational inefficiency of sequential Monte Carlo (SMC) methods in static spaces. We propose a theoretical analysis framework grounded in interpolation distribution design. First, we establish finite-sample convergence guarantees for SMC without assuming bounded importance weights—an unprecedented result. Second, via $L_2$-error analysis and Markov chain mixing time theory, we derive explicit theoretical bounds quantifying how interpolation distribution choice affects estimation error. Third, we provide rigorous theoretical justification for adaptive path selection based on relative effective sample size (REES), overcoming the bias and degeneracy inherent in conventional data-tempering approaches. Experiments demonstrate that our adaptive method automatically approximates the optimal interpolation sequence, achieving both significantly reduced computational complexity and improved approximation accuracy.
📝 Abstract
We prove a bound on the finite sample error of sequential Monte Carlo (SMC) on static spaces using the $L_2$ distance between interpolating distributions and the mixing times of Markov kernels. This result is unique in that it is the first finite sample convergence result for SMC that does not require an upper bound on the importance weights. Using this bound we show that careful selection of the interpolating distributions can lead to substantial improvements in the computational complexity of the algorithm. This result also justifies the adaptive selection of SMC distributions using the relative effective sample size commonly used in the literature and we establish conditions guaranteeing the approximation accuracy of the adaptive SMC approach. We then demonstrate empirically that this procedure provides nearly-optimal sequences of distributions in an automatic fashion for realistic examples.