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Theoretical guarantees in KL for Diffusion Flow Matching

NeurIPS 2024 · Probabilistic and scientific modeling › Variational and generative sampling · approach: Flow matching and flow sampling · 2 citations (OpenAlex)

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Provides non-asymptotic KL guarantees for diffusion flow matching (Brownian-bridge interpolant, Markovian projection, Euler-Maruyama sampling) under moment and score-integrability assumptions plus an L2 drift-error assumption, with and without early stopping.

Problem

Theoretical convergence guarantees for flow matching / stochastic interpolant generative models that use a stochastic (Brownian bridge) interpolant, accounting for both drift-estimation and time-discretization errors, under mild assumptions.

Approach

Analyze the Markovian projection of the Brownian-bridge stochastic interpolant and bound KL between data and the Euler-Maruyama-simulated DFM output using only moment conditions and integrability of scores of the target, base and coupling, with no Lipschitz or manifold assumptions; an early-stopping variant handles targets without densities.

Key results

• KL(nu* | nu_1^theta) bounded by epsilon^2 plus h(h^{1/8}+1) times a term of dimension and 8th-order moment/score quantities, for uniform step size h without early stopping.
• Choosing N_h proportional to the bound constants over epsilon^2 gives approximation error O(epsilon^2) with complexity O(epsilon^-2).
• With early stopping at 1-delta and independent coupling, the KL bound holds for the smoothed target at time 1-delta, with complexity scaling as max(d^4 eps^-4, eps^-6).

Limitations

• Requires 8th-order moments and L^8 integrability of score of target, base and coupling, which is strong.
• Dimension dependence is poor (d^4) in the early-stopping regime.
• Only Brownian bridge interpolant analyzed; no statistical (estimation) analysis of the drift learning.

Experimental setup

Metrics: KL divergence

Baselines: Lipschitz-based flow matching bounds probability flow ODE bounds SGM KL bounds

Builds on: flow matching stochastic interpolants score-based generative models Markovian projection

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