The generator gradient estimator is an adjoint state method for stochastic differential equations

TL;DR

This paper reveals that the generator gradient estimator for SDEs is an adjoint state method, connecting it to existing techniques and highlighting its scalability with the number of states rather than parameters.

Contribution

It demonstrates that the generator gradient estimator is an adjoint state method and relates it to the exact Integral Path Algorithm for stochastic chemical reaction networks.

Findings

The estimator scales with the number of states, not parameters.

It is an analogue to the exact Integral Path Algorithm.

The method applies to overparameterized SDEs like Neural SDEs.

Abstract

Motivated by the increasing popularity of overparameterized Stochastic Differential Equations (SDEs) like Neural SDEs, Wang, Blanchet and Glynn recently introduced the generator gradient estimator, a novel unbiased stochastic gradient estimator for SDEs whose computation time remains stable in the number of parameters. In this note, we demonstrate that this estimator is in fact an adjoint state method, an approach which is known to scale with the number of states and not the number of parameters in the case of Ordinary Differential Equations (ODEs). In addition, we show that the generator gradient estimator is a close analogue to the exact Integral Path Algorithm (eIPA) estimator which was introduced by Gupta, Rathinam and Khammash for a class of Continuous-Time Markov Chains (CTMCs) known as stochastic chemical reactions networks (CRNs).