Workshop Details

Particle Filtering (PF) methods are an established class of procedures for performing inference in non-linear state-space models. Resampling is a key ingredient of PF, necessary to obtain low variance likelihood and states estimates. However, traditional resampling methods result in PF-based loss functions being non-differentiable with respect to model and PF parameters. In a variational inference context, resampling also yields high variance gradient estimates of the PF-based evidence lower bound. By leveraging optimal transport ideas, we introduce a principled differentiable particle filter and provide convergence results. We demonstrate this novel method on a variety of applications including robot localization.

Joint work with Adrien Corenflos, James Thornton and George Deligiannidis

Sequential Monte Carlo algorithms have been widely accepted as a powerful computational tool for making inference with dynamical systems. A key step in sequential Monte Carlo is resampling, which plays a role of steering the algorithm towards the future dynamics. Several strategies have been used in practice, including multinomial resampling, residual resampling, optimal resampling, stratified resampling, and optimal transport resampling. We will review some of these approaches and show that in one-dimensional cases the optimal transport resampling is equivalent to stratified resampling on the sorted particles, and they both minimize the resampling variance as well as the expected squared energy distance between the original and resampled empirical distributions. In general 𝑑-dimensional cases, if the particles are first sorted using the Hilbert curve, we show that the variance of stratified resampling is O (m -(1+2d-1)) with 𝑚 being the number of resampled particles and that this is optimal for ordered stratified resampling , as conjectured in Gerber et al. (2019). In light of these results, we show that, for dimension 𝑑 > 1 , the mean square error of sequential quasi-Monte Carlo with 𝑛 particles can be  𝑂 (𝑛−1− 4 𝑑(𝑑+4) if Hilbert curve resampling is used and a specific low-discrepancy set is chosen. To our knowledge, this is the first known convergence rate lower than 𝑜(𝑛 -1). The presentation is based on the joint work with Wenshuo Wang, Yichao Li, and Ke Deng.