Discrete Flow Matching: Convergence Guarantees Under Minimal Assumptions

2026-05-09

AI Summary

This work addresses the lack of non-asymptotic theoretical guarantees for existing discrete flow matching models under weak assumptions, particularly concerning convergence analyses in Kullback–Leibler (KL) divergence and total variation distance. Focusing on flow matching models defined over finite lattice spaces, the study introduces a time-discretized sampling path approach that relies solely on approximation error—bypassing conventional score function assumptions—and establishes, for the first time, explicit non-asymptotic convergence bounds for an early-stopped version of the target distribution. By leveraging discrete Markov processes within the flow matching framework, the analysis derives error bounds in both KL divergence and total variation, significantly relaxing required assumptions and improving dependence on vocabulary size $m$ and dimensionality $d$.

Abstract

Flow Matching has recently emerged as a popular class of generative models for simulating a target distribution $μ_1$ from samples drawn from a source distribution $μ_0$. This framework relies on a fixed coupling between $μ_0$ and $μ_1$, and on a deterministic or stochastic bridge to define an interpolating process between the two distributions. The time marginals of this process can then be approximately sampled by estimating the transition rates, or more generally the generator, of its Markovian projection. This framework has recently been extended to the case of discrete source and target distributions, under the name Discrete Flow Matching (DFM). However, theoretical guarantees for such models remain scarce. In this paper, we study two DFM models on $\mathbb{Z}_m^d = \{0,\ldots,m-1\}^d$, sampled through time discretization, and derive non-asymptotic associated bounds for both of them. In contrast to previous work, we establish non-asymptotic bounds in Kullback--Leibler divergence for the early-stopped version of the target distribution. We also derive explicit convergence guarantees in total variation distance with respect to the true target distribution. Importantly, these bounds rely only on an approximation error assumption, relaxing standard score assumptions used in earlier works, while also yielding improved dependence on the vocabulary size $m$ and the dimension $d$.

Problem

• Discrete Flow Matching
• convergence guarantees
• non-asymptotic bounds
• Kullback-Leibler divergence
• total variation distance

Innovation

• Discrete Flow Matching
• non-asymptotic convergence
• total variation distance
• Kullback–Leibler divergence
• minimal assumptions