Ai-Sampler: Adversarial Learning of Markov kernels with involutive maps (2406.02490v1)

Abstract: Markov chain Monte Carlo methods have become popular in statistics as versatile techniques to sample from complicated probability distributions. In this work, we propose a method to parameterize and train transition kernels of Markov chains to achieve efficient sampling and good mixing. This training procedure minimizes the total variation distance between the stationary distribution of the chain and the empirical distribution of the data. Our approach leverages involutive Metropolis-Hastings kernels constructed from reversible neural networks that ensure detailed balance by construction. We find that reversibility also implies $C_2$-equivariance of the discriminator function which can be used to restrict its function space.

• The paper introduces an adversarial MCMC framework using reversible neural networks to parameterize involutive maps for efficient sampling.
• The methodology leverages reversible kernels and a GAN-inspired objective to ensure detailed balance and enhance mixing in high-dimensional spaces.
• Results show significant improvements in sample quality and effective sample sizes compared to traditional approaches like HMC in complex distributions.

Overview of "Ai-Sampler: Adversarial Learning of Markov kernels with involutive maps"

The paper "Ai-Sampler: Adversarial Learning of Markov kernels with involutive maps" introduces a novel approach to improving Markov Chain Monte Carlo (MCMC) methods. These methods are a cornerstone in statistical sampling for unnormalized complex probability distributions. Despite their widespread utility, MCMC algorithms have not fully capitalized on recent advances in deep neural network technologies due to difficulties in measuring sample quality and establishing convergence. The authors propose an innovative solution by employing neural networks to parameterize reversible transition kernels, creating an adversarial learning framework for efficient sampling.

Methodology

The central premise of the paper is leveraging involutive Metropolis-Hastings kernels constructed from reversible neural networks. Reversibility ensures detailed balance, a critical property for convergence to the desired stationary distribution. Traditional MCMC methods face challenges in defining suitable objective functions that balance sample quality with exploration. The proposed approach circumvents these challenges by framing the problem as an adversarial game between a parameterized mapping and a discriminator.

The technique utilizes time-reversible neural networks to parameterize involutive maps. These maps inherently satisfy the necessary detailed balance conditions through their construction, obviating the need for additional auxiliary variables to enforce reversibility. This intrinsic reversibility is achieved by exploiting the relationship between time-reversible dynamical systems and detailed balance conditions.

The authors also design an adversarial training objective inspired by Generative Adversarial Networks (GANs) to refine the transition kernel. The discriminator in this setting serves to distinguish between samples from the target distribution and those generated by the transition kernel, driving the learning process towards minimizing the total variation distance between the two distributions.

Results

The method exhibits superior proficiency in traversing complex distribution landscapes compared to conventional techniques like Hamiltonian Monte Carlo (HMC) and deterministic proposals using neural networks (e.g., A-NICE-MC). Numerical results on synthetic 2D densities demonstrate enhanced mixing and effective sampling from multimodal distributions. The method shows robust performance in maintaining high effective sample sizes, particularly in high-dimensional datasets typical in Bayesian logistic regression problems, such as the heart, german, and australian datasets.

Moreover, the adversarial objective facilitates a gradual bootstrap process where sample quality progressively improves as it guides the proposal towards the desired density.

Implications

The implications of this paper are manifold. Practically, it provides a scalable MCMC framework that can efficiently sample from complex, high-dimensional distributions. This ability is crucial for applications in Bayesian inference where posterior distributions from large and intricate models must be effectively sampled. Theoretically, the framework enriches our understanding of how neural network architectures can be integrated within classical statistical methods to enhance performance while maintaining theoretical guarantees of convergence and unbiasedness.

Speculation on Future Developments

Future work could delve into extending the Ai-Sampler framework to other probabilistic models that benefit from efficient sampling. Furthermore, exploring different architectures and parameterizations that adhere to the involutivity constraint could lead to broader applications in normalizing flows and variational inference contexts. Given the modular nature of the proposed framework, integrating it with more advanced adversarial strategies or building upon the symplectic geometry of transition kernels could yield further gains in sampling efficiency.

In conclusion, the paper presents a sophisticated treatment of MCMC methodologies, innovatively merging adversarial learning and reversible neural network architectures. This fusion tackles some of the longstanding challenges in the field, paving the way for more robust solutions in probabilistic inference and beyond.