Two-Timescale Stochastic Gradient Descent in Continuous Time with Applications to Joint Online Parameter Estimation and Optimal Sensor Placement

TL;DR

This paper proves the almost sure convergence of two-timescale stochastic gradient descent algorithms in continuous time, applicable to complex bilevel optimization problems like joint parameter estimation and sensor placement in stochastic systems.

Contribution

It extends convergence results to continuous time with general noise, including non-additive Markov noise, and applies these to joint estimation and sensor placement problems.

Findings

Proved convergence of continuous-time two-timescale SGD under broad conditions.

Applied the method to joint parameter estimation and sensor placement in stochastic PDEs.

Demonstrated effectiveness through numerical examples on complex systems.

Abstract

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 Bene\v{s} equation, and a partially observed stochastic advection-diffusion equation.