  State space models (SSMs) provide a flexible framework for modeling complex
time series via a latent stochastic process. Inference for nonlinear,
non-Gaussian SSMs is often tackled with particle methods that do not scale well
to long time series. The challenge is two-fold: not only do computations scale
linearly with time, as in the linear case, but particle filters additionally
suffer from increasing particle degeneracy with longer series. Stochastic
gradient MCMC methods have been developed to scale Bayesian inference for
finite-state hidden Markov models and linear SSMs using buffered stochastic
gradient estimates to account for temporal dependencies. We extend these
stochastic gradient estimators to nonlinear SSMs using particle methods. We
present error bounds that account for both buffering error and particle error
in the case of nonlinear SSMs that are log-concave in the latent process. We
evaluate our proposed particle buffered stochastic gradient using stochastic
gradient MCMC for inference on both long sequential synthetic and
minute-resolution financial returns data, demonstrating the importance of this
class of methods.
