In this paper, we establish the almost sure convergence of two-timescale stochastic gradient descent algorithms in continuous time under general noise and stability conditions, extending well known results in discrete time. We analyse algorithms with additive noise and those with non-additive noise. In the non-additive case, our analysis is carried out under the assumption that the noise is a continuous-time Markov process, controlled by the algorithm states. The algorithms we consider can be applied to a broad class of bilevel optimisation problems. We study one such problem in detail, namely, the problem of joint online parameter estimation and optimal sensor placement for a partially observed diffusion process. We demonstrate how this can be formulated as a bilevel optimisation problem, and propose a solution in the form of a continuous-time, two-timescale, stochastic gradient descent algorithm. Furthermore, under suitable conditions on the latent signal, the filter, and the filter derivatives, we establish almost sure convergence of the online parameter estimates and optimal sensor placements to the stationary points of the asymptotic log-likelihood and asymptotic filter covariance, respectively. We also provide numerical examples, illustrating the application of the proposed methodology to a partially observed Beneš equation, and a partially observed stochastic advection-diffusion equation.
The first author was funded by the EPSRC CDT in the Mathematics of Planet Earth (grant number EP/L016613/1) and the National Physical Laboratory. The second author was partially funded by JPMorgan Chase & Co. under a J.P. Morgan A.I. Research Award (2019, 2021).
We are very grateful to D. Crisan and A. Forbes for many helpful discussions and suggestions. We are also thankful to the two anonymous referees and the associate editor, whose constructive comments greatly improved the paper.
"Two-timescale stochastic gradient descent in continuous time with applications to joint online parameter estimation and optimal sensor placement." Bernoulli 29 (2) 1137 - 1165, May 2023. https://doi.org/10.3150/22-BEJ1493