Efficient Sampling on Riemannian Manifolds via Langevin MCMC (2402.10357v1)

Abstract: We study the task of efficiently sampling from a Gibbs distribution $d \pi^* = e^{-h} d {vol}_g$ over a Riemannian manifold $M$ via (geometric) Langevin MCMC; this algorithm involves computing exponential maps in random Gaussian directions and is efficiently implementable in practice. The key to our analysis of Langevin MCMC is a bound on the discretization error of the geometric Euler-Murayama scheme, assuming $\nabla h$ is Lipschitz and $M$ has bounded sectional curvature. Our error bound matches the error of Euclidean Euler-Murayama in terms of its stepsize dependence. Combined with a contraction guarantee for the geometric Langevin Diffusion under Kendall-Cranston coupling, we prove that the Langevin MCMC iterates lie within $\epsilon$-Wasserstein distance of $\pi^*$ after $\tilde{O}(\epsilon^{-2})$ steps, which matches the iteration complexity for Euclidean Langevin MCMC. Our results apply in general settings where $h$ can be nonconvex and $M$ can have negative Ricci curvature. Under additional assumptions that the Riemannian curvature tensor has bounded derivatives, and that $\pi^*$ satisfies a $CD(\cdot,\infty)$ condition, we analyze the stochastic gradient version of Langevin MCMC, and bound its iteration complexity by $\tilde{O}(\epsilon^{-2})$ as well.

• The paper establishes that the geometric Euler-Murayama discretization achieves error bounds equivalent to the Euclidean case for properly chosen step sizes.
• It extends stochastic gradient Langevin dynamics to manifold settings by leveraging curvature conditions and bounded gradient assumptions.
• The findings improve sampling efficiency in non-Euclidean spaces, enabling more precise Bayesian inference in complex, manifold-structured models.

Exploring Efficient Sampling on Riemannian Manifolds via Langevin MCMC

The field of Machine Learning and Bayesian inference has long benefited from advancements in sampling techniques, particularly those that navigate high-dimensional spaces efficiently. The paper under review introduces significant improvements in Langevin Markov Chain Monte Carlo (MCMC) algorithms specifically tailored for Riemannian manifolds, exploring their theoretical underpinnings and practical implementations.

Theoretical Foundation and Innovative Contributions

At the heart of this research is the adaptation of Langevin MCMC methods for efficient sampling over Riemannian manifolds. Traditional MCMC methods have seen wide usage in Euclidean spaces, but their adaptation to manifolds introduces both challenges and opportunities. The paper establishes a quantitative comparison of discretization errors between geometric Langevin MCMC implementations and continuous-time dynamics on manifolds. Remarkably, it proves that for carefully chosen step sizes, the geometric Euler-Murayama discretization scheme achieves a bound on error that matches the Euclidean case, effectively bridging the gap in error analysis between Euclidean spaces and Riemannian manifolds.

The researchers extend their analysis to stochastic gradient Langevin dynamics (SGLD) on manifolds under certain curvature conditions and bounded gradients. The implications are profound, suggesting a pathway to efficiently sample from distributions over manifolds with potentially complex geometry.

Practical Implications and Speculations on Future Developments

This leap in theoretical understanding presents several practical implications. First and foremost, it enables the application of Langevin MCMC methods in areas where the underlying structure is inherently non-Euclidean, such as in the paper of shapes, graphs, and various data structures modeled on manifolds. It opens the door for more precise Bayesian inference procedures in these domains, potentially improving the performance of algorithms in computer vision, natural language processing, and beyond.

Another avenue touched upon involves the computational efficiency of these methods. The established error bounds and contraction rates provide a solid foundation for the development of more computationally efficient sampling algorithms, which could significantly reduce the time and resources required for Bayesian computations in complex models.

Looking towards the future, this paper sets a clear direction for expanding the repertoire of tools available for sampling and optimization on manifolds. The focus on Riemannian manifolds in particular is apt, considering the manifold hypothesis in learning—the idea that high-dimensional data in fact lies on low-dimensional manifolds. As machine learning models grow increasingly complex, efficiently navigating these underlying spaces will become paramount. This research not only contributes to this goal but also questions our understanding of the dynamics of Langevin MCMC on curved spaces, with implications that stretch beyond the immediate field of computational statistics to the very fabric of geometric learning and optimization.

In conclusion, the exploration of Langevin MCMC on Riemannian manifolds as presented in this paper is a substantial contribution to the field of computational mathematics and statistical learning. It not only extends theoretical models to more complex spaces but also opens up new practical possibilities and efficiencies in computational methods. As the community builds upon these foundations, we can expect a significant broadening of the scope and scalability of statistical inference methods applied to the manifold-structured data that pervades machine learning applications.