Asymptotically exact variational flows via involutive MCMC kernels

TL;DR

This paper introduces a new framework for variational flows that guarantees asymptotic exactness using involutive MCMC kernels, improving convergence and practical applicability over existing methods.

Contribution

The authors develop a novel representation of involutive MCMC kernels as invertible measure-preserving systems, enabling the construction of variational flows with provable total variation convergence.

Findings

Three new variational families with convergence guarantees.

Outperforms or matches NUTS and normalizing flows in various tasks.

Requires weaker assumptions than existing guarantees.

Abstract

Most expressive variational families -- such as normalizing flows -- lack practical convergence guarantees, as their theoretical assurances typically hold only at the intractable global optimum. In this work, we present a general recipe for constructing tuning-free, asymptotically exact variational flows on arbitrary state spaces from involutive MCMC kernels. The core methodological component is a novel representation of general involutive MCMC kernels as invertible, measurepreserving iterated random function systems, which act as the flow maps of our variational flows. This leads to three new variational families with provable total variation convergence. Our framework resolves key practical limitations of existing variational families with similar guarantees (e.g., MixFlows), while requiring substantially weaker theoretical assumptions. Finally, we demonstrate the competitive performance of our flows across tasks including posterior approximation, Monte Carlo estimates, and normalization constant estimation, outperforming or matching No-U-Turn sampler (NUTS) and black-box normalizing flows.