Streaming Variational Monte Carlo

Abstract

Nonlinear state-space models are powerful tools to describe dynamical structures in complex time series. In a streaming setting where data are processed one sample at a time, simultaneously inferring the state and their nonlinear dynamics has posed significant challenges in practice. We develop a novel online learning framework, leveraging variational inference and sequential Monte Carlo, which enables flexible and accurate Bayesian joint filtering. Our method provides a filtering posterior arbitrarily close to the true filtering distribution for a wide class of dynamics models and observation models. Specifically, the proposed framework can efficiently infer a posterior over the dynamics using sparse Gaussian processes. Constant time complexity per sample makes our approach amenable to online learning scenarios and suitable for real-time applications. * equal contribution Preprint. Under review.

Online Variational Filtering and Parameter Learning

“…The approach developed here can be thought of as a way to correct the approximate ELBOs computed in [16,44] in a principled manner, which takes into account the discrepancy between the filtering and approximate filtering distributions, and maintains the correct gradient dependencies in the computation graph. Finally [43] relies on the PF to do online variational inference. However the variational approximation of the filtering distribution is only implicit at its expression includes an intractable expectation and, like any other PF technique, its performance is expected to degrade significantly with the state dimension [6].…”

“…We next evaluate the performance of our algorithm for state estimation in non-linear, high-dimensional SSMs. We reproduce the Chaotic Recurrent Neural Network (CRNN) example in [43], but with state dimension d x = 5, 20, and 100. This non-linear model is an Euler approximation of the continuous-time recurrent neural network dynamics…”

“…where we use a 1-layer Multi-Layer Perceptron (MLP) with 100 neurons for each q φt t (x t−1 |x t ). We generate a dataset of length 100 using the same settings as [43], and each algorithm is run 10 times to report the mean and standard deviation. We also match approximately the computational complexity for all methods.…”

“…We follow closely [43] using the same parameter settings for data generation. We use 1 million particles for EnKF and BPF for dimension d x = 5 and 20, and 250000 particles for dimension d x = 100 in order to match the computation cost.…”

Ensemble Kalman Variational Objectives: Nonlinear Latent Trajectory Inference with A Hybrid of Variational Inference and Ensemble Kalman Filter

“…In recent years, Variational Inference (VI) combined with Bayesian nonlinear filtering has been actively researched. In particular, VI based on Sequential Monte Carlo (SMC) methods [42,38,48,25,61,47,37,40] achieved tighter bound of the log marginal likelihood theoretically and experimentally. These methods, however, suffer from two drawbacks: particle degeneracy and biased gradient estimates.…”