Geometric Methods for Sampling, Optimisation, Inference and Adaptive Agents (2203.10592v3)

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.

• The paper introduces novel geometric frameworks that integrate differential and symplectic methods to improve optimisation stability and convergence.
• The paper demonstrates enhanced sampling techniques via Hamiltonian Monte Carlo with improved convergence and robustness in high-dimensional spaces.
• The paper develops adaptive agent models using active inference and kernel-based methodologies to refine decision-making processes in AI systems.

Geometric Methods for Sampling, Optimisation, Inference and Adaptive Agents: A Summary

The chapter titled "Geometric Methods for Sampling, Optimisation, Inference, and Adaptive Agents" presents a nuanced exploration of the interconnections between geometric structures and several core areas of mathematics and statistics—namely, optimisation, sampling, inference, and adaptive decision-making in the context of dynamic agents. This synthesis is achieved by examining the underlying geometric frameworks that these areas share. The authors propose novel algorithms that leverage such geometric insights to address complex problems efficiently.

Geometric Frameworks and Their Implications

The core premise of the chapter is that many mathematical disciplines fundamentally rely on differential geometry, specifically through the lens of information geometry. Notably, the authors highlight the application of symplectic geometry in constructing enhanced optimisation and sampling methods. The chapter further argues for the preservation of information geometry in decision-making processes, which leads to the development of adaptive agents capable of sophisticated inference.

Key Contributions and Results

1. Optimisation and Symplectic Integrators: The text delves deeply into accelerated optimisation techniques by applying geometric integration principles. Through the use of symplectic methods, it addresses the issues of stability and convergence rates, especially within the constraints of manifold optimisation.
2. Hamiltonian Methods for Sampling: Leveraging the deterministic structure afforded by Hamiltonian systems, the chapter discusses the strength of Hamiltonian Monte Carlo (HMC) methods. The integration of non-reversibility and hypocoercivity into sampling methods is explicitly linked to improved convergence rates and robustness.
3. Inference Using Kernel Methods: For statistical inference, the authors employ kernel-based discrepancies, such as Maximum Mean Discrepancy (MMD), to better align model assumptions with data-driven needs. This section also elucidates the use of Stein’s method for designing inference schemes centered around kernel Stein discrepancies.
4. Development of Adaptive Agents with Active Inference: Returning to the roots of decision theory and control processes, adaptive agents are modeled using active inference frameworks. This allows agents to incorporate feedback and adaptively refine their decision-making hypothesis—a concept critical for understanding cognitive processes in neuroscience and progressing AI systems.

Implications and Future Directions

This work primarily impacts the domains of algorithmic development and artificial intelligence, offering a robust theoretical groundwork for future advancements. The blending of geometric theory with statistical mechanics promulgates methodologies that hold promise in efficiently solving high-dimensional problems, such as those encountered in machine learning and robotics.

From a theoretical perspective, the authors suggest extending the current methodologies to broader classes of dynamical systems, which implies that future explorations might focus on manifold sampling or further optimising consensus mechanisms in machine learning contexts. Practically, adapting geometric methods may significantly enhance the performance of AI systems, suggesting a natural trajectory towards developing better, more responsive, and context-aware AI agents.

In conclusion, by drawing on a rich array of mathematical theories and providing actionable insights, this chapter significantly contributes to advancing the intersection of computational mathematics and AI. It elucidates how geometric perspectives lend themselves to efficient, scalable solutions in optimisation and sampling, reinforcing that such rigorous frameworks are pivotal to the next generation of intelligent systems. This interdisciplinary approach paves the way for further research into leveraging geometry to address increasingly complex AI challenges.