Geometric Methods for Sampling, Optimisation, Inference and Adaptive Agents

TL;DR

This chapter explores how geometric structures underpin sampling, optimisation, inference, and adaptive decision-making, leading to algorithms that leverage symplectic geometry, information geometry, and Poisson geometry for improved efficiency and robustness.

Contribution

It introduces a unifying geometric framework that connects various fields and derives algorithms exploiting these structures for better sampling, optimisation, and adaptive agents.

Findings

Symplectic geometry enables accelerated sampling and optimisation.

Information geometry guides robust estimation methods.

Preserving information geometry improves adaptive decision-making.

Abstract

In this chapter, we identify fundamental geometric structures that underlie the problems of sampling, optimisation, inference and adaptive decision-making. Based on this identification, we derive algorithms that exploit these geometric structures to solve these problems efficiently. We show that a wide range of geometric theories emerge naturally in these fields, ranging from measure-preserving processes, information divergences, Poisson geometry, and geometric integration. Specifically, we explain how (i) leveraging the symplectic geometry of Hamiltonian systems enable us to construct (accelerated) sampling and optimisation methods, (ii) the theory of Hilbertian subspaces and Stein operators provides a general methodology to obtain robust estimators, (iii) preserving the information geometry of decision-making yields adaptive agents that perform active inference. Throughout, we emphasise the rich connections between these fields; e.g., inference draws on sampling and optimisation, and adaptive decision-making assesses decisions by inferring their counterfactual consequences. Our exposition provides a conceptual overview of underlying ideas, rather than a technical discussion, which can be found in the references herein.