Enhanced SMC²: Leveraging Gradient Information from Differentiable Particle Filters Within Langevin Proposals

Sequential Monte Carlo Squared <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(\mathbf{SMC}^{2})$</tex> is a Bayesian method which can infer the states and parameters of non-linear, non-Gaussian state-space models. The standard random-walk proposal in SMC<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> faces challenges, particularly with high-dimensional parameter spaces. This study outlines a novel approach by harnessing first-order gradients derived from a Common Random Numbers - Particle Filter (CRN-PF) using PyTorch. The resulting gradients can be leveraged within a Langevin proposal without accept/reject. Including Langevin dynamics within the proposal can result in a higher effective sample size and more accurate parameter estimates when compared with the random-walk. The resulting algorithm is parallelized on distributed memory using Message Passing Interface (MPI) and runs in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathcal{O}(\log_{2}N)$</tex> time complexity. Utilizing 64 computational cores we obtain a 51x speed-up when compared to a single core. A GitHub link is given which provides access to the code.