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Chain of Log-Concave Markov Chains
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Proposes a non-Markovian sampler that decomposes sampling from an unnormalized density into a chain of log-concave conditional sampling problems via accumulating noisy measurements with a single fixed noise scale, generalizing walk-jump sampling.
Problem
Sampling from unnormalized densities in high dimensions with multimodal, ill-conditioned structure, where standard Langevin MCMC mixes slowly between modes.
Approach
Smooth the target with m isotropic Gaussian measurements at a fixed noise level, sample each measurement sequentially via Markov chains conditioned on the running empirical mean (which yields log-concave conditionals, "once log-concave, always log-concave"), then denoise with an empirical Bayes jump; generalizes walk-jump sampling.
Key results
- On the elliptical Gaussian, OAT with sigma=1 or 2 outperforms Langevin MCMC without smoothing (sigma=0) in 2-Wasserstein distance for all tested dimensions d.
- On the Gaussian mixture (alpha=1/5, mu=3), OAT achieves consistently low 2-Wasserstein distance as d increases while AAO and other schemes deteriorate.
- In a two-mode tunneling experiment, OAT walkers initialized at the dominant mode (80% mass) tunnel to the smaller mode quickly, whereas unsmoothed Langevin walkers remain stuck.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use one-at-a-time walk-jump sampling (OAT) with Gaussian smoothing instead of Langevin MCMC without smoothing | Langevin MCMC · Elliptical Gaussian | 2-Wasserstein distance | Helped |
| Use OAT with $\sigma=1$ or $\sigma=2$ instead of Langevin MCMC without smoothing | Langevin MCMC · Elliptical Gaussian | 2-Wasserstein distance | Helped |
| Use OAT instead of other tested sampling schemes | Walk-jump sampling and Langevin MCMC · Mixture of Gaussians | 2-Wasserstein distance | Helped |
| Use OAT with a suitable noise level instead of the best-performing unsmoothed underdamped Langevin MCMC | Underdamped Langevin MCMC · Mixture of Gaussians | 2-Wasserstein distance | Helped |
| Initialize OAT at the dominant mode and use smoothing to enable mode transitions | OAT sampling · Two-mode Gaussian mixture, initialized at $(3, 3)$ | Mode tunneling | Helped |
| Use unsmoothed Langevin MCMC instead of OAT for mode transitions | Langevin MCMC · Two-mode Gaussian mixture, initialized at $(3, 3)$ | Mode tunneling | Hurt |
| Sample all measurements at once (AAO) instead of sequentially with OAT (ablation) | Walk-jump sampling · Elliptical Gaussian | 2-Wasserstein distance | Hurt |
| Use single-measurement walk-jump sampling ($m=1$) instead of OAT as dimension increases (ablation) | Walk-jump sampling · Mixture of Gaussians | 2-Wasserstein distance | Hurt |
| Accumulate measurements sequentially instead of sampling all measurements in parallel | Sampling from $(\sigma,m)$-densities for anisotropic Gaussians · Anisotropic Gaussian | Condition number | Helped |
| Increase the number of accumulated measurements in the sequential scheme | Conditional sampling densities · General density | Log-concavity | Helped |
Surprising findings
- All-at-once sampling of the m smoothed measurements has a condition number 1+m*sigma^-2*tau_max^2 that grows with m, so it is worse than single-measurement sampling.
- The authors show analytically that the conditional densities' condition numbers decrease monotonically with t, but the stronger monotonicity of the Hessian bound does not hold in general (the remark on monotonicity).
Limitations
- Does not establish that the log-concave sampling strategy is optimal.
- Gradients of the smoothed log-density must be estimated for general targets; the plug-in score estimator is complex and not studied in depth.
- Experiments are restricted to synthetic low-dimensional-marginal test densities with Wasserstein computed on 1D projections.
Experimental setup
Benchmarks: elliptical Gaussian (synthetic) Gaussian mixture (synthetic)
Metrics: 2-Wasserstein distance (1D projection approximation)
Baselines: Langevin MCMC without smoothing (sigma=0) all-at-once walk-jump sampling (AAO) single-measurement walk-jump sampling (m=1)
Builds on: walk-jump sampling underdamped Langevin MCMC annealed importance sampling
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