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Markovian Flow Matching: Accelerating MCMC with Continuous Normalizing Flows

NeurIPS 2024 · Probabilistic and scientific modeling › Variational and generative sampling · approach: Stochastic gradient MCMC · 3 citations (OpenAlex)

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Proposes Markovian Flow Matching (MFM), an adaptive MCMC method that trains a continuous normalizing flow on-the-fly with flow matching using MCMC samples, combining MALA with a flow-informed random-walk kernel and adaptive tempering.

Problem

Sampling from unnormalized multimodal distributions with MCMC, where local samplers mix slowly and global proposals need careful design; prior flow-based MCMC uses discrete normalizing flows and cannot exploit simulation-free CNF training.

Approach

Use the conditional flow matching objective with samples drawn from an adaptive MCMC chain (MALA local kernel plus flow-informed random-walk MH kernel, with ESS-based adaptive tempering) to train a CNF on the fly, which then furnishes non-local proposals.

Key results

• On the 4-mode Gaussian mixture (K=5e3), MFM with k_Q=100 reached MMD 1.05e-3 in 39.2 s versus AT-SMC MMD 3.95e-2 and DDS MMD 1.76e-4 at 114 s, with DDS being the closest approximation.
• On the Field system, MFM achieved KSD U-stat ~2.6 at about 52 s while DDS took about 2400 s, i.e. roughly 10-25x faster than competing methods as claimed.
• On the 16-mode mixture, FAB marginally beat MFM on MMD (1.51e-3 vs 1.81e-2 for MFM k_Q=100 at K=1e3) but at ~100 s versus ~11 s for MFM.

Measured effects reported in the paper

Surprising findings

• DDS gave a closer approximation to the 4-mode target than MFM and even than FM trained on true target samples, suggesting a limitation of the FM objective with the chosen architecture.
• NF-MCMC performed poorly on the Field system without an informed base measure, despite tuning attempts.

Limitations

• Convergence result only guarantees convergence to a local minimum of the flow matching objective, not generalization to the target.
• No non-asymptotic convergence rates are established.
• Architecture choice of the flow network was not explored in depth and may strongly affect performance.

Experimental setup

Datasets: 4-mode Gaussian mixture 16-mode Gaussian mixture Field system (stochastic Allen-Cahn) Log-Gaussian Cox process (Finnish Scots pine saplings)

Benchmarks: 4-mode Gaussian mixture 16-mode Gaussian mixture Field system Log-Gaussian Cox process

Metrics: MMD KSD U-statistic KSD V-statistic expected log target under learned flow wall-clock seconds

Baselines: Denoising Diffusion Sampler (DDS) NF-MCMC Flow Annealed Importance Sampling Bootstrap (FAB) Adaptive Tempered SMC (AT-SMC) FM with target samples

Builds on: flow matching neutraMCMC Markovian score climbing Denoising Diffusion Samplers MALA

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