Adrian N. Bishop

Ajay Jasra, Jeremy Heng, Yaxian Xu et al.

We consider a class of finite time horizon nonlinear stochastic optimal control problem, where the control acts additively on the dynamics and the control cost is quadratic. This framework is flexible and has found applications in many domains. Although the optimal control admits a path integral representation for this class of control problems, efficient computation of the associated path integrals remains a challenging Monte Carlo task. The focus of this article is to propose a new Monte Carlo approach that significantly improves upon existing methodology. Our proposed methodology first tackles the issue of exponential growth in variance with the time horizon by casting optimal control estimation as a smoothing problem for a state space model associated with the control problem, and applying smoothing algorithms based on particle Markov chain Monte Carlo. To further reduce computational cost, we then develop a multilevel Monte Carlo method which allows us to obtain an estimator of the optimal control with $\mathcal{O}(ε^2)$ mean squared error with a computational cost of $\mathcal{O}(ε^{-2}\log(ε)^2)$. In contrast, a computational cost of $\mathcal{O}(ε^{-3})$ is required for existing methodology to achieve the same mean squared error. Our approach is illustrated on two numerical examples, which validate our theory.

Adrian N. Bishop, Edwin V. Bonilla

We consider the Bayesian optimal filtering problem: i.e. estimating some conditional statistics of a latent time-series signal from an observation sequence. Classical approaches often rely on the use of assumed or estimated transition and observation models. Instead, we formulate a generic recurrent neural network framework and seek to learn directly a recursive mapping from observational inputs to the desired estimator statistics. The main focus of this article is the approximation capabilities of this framework. We provide approximation error bounds for filtering in general non-compact domains. We also consider strong time-uniform approximation error bounds that guarantee good long-time performance. We discuss and illustrate a number of practical concerns and implications of these results.

Adrian N. Bishop

The capability of recurrent neural networks to approximate trajectories of a random dynamical system, with random inputs, on non-compact domains, and over an indefinite or infinite time horizon is considered. The main result states that certain random trajectories over an infinite time horizon may be approximated to any desired accuracy, uniformly in time, by a certain class of deep recurrent neural networks, with simple feedback structures. The formulation here contrasts with related literature on this topic, much of which is restricted to compact state spaces and finite time intervals. The model conditions required here are natural, mild, and easy to test, and the proof is very simple.