Research map / Probabilistic and scientific modeling / Variational and generative sampling
Stochastic Localization via Iterative Posterior Sampling
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Proposes SLIPS, a learning-free stochastic localization sampler that estimates the SL denoiser with MALA posterior sampling and uses a flexible noise schedule plus a "duality of log-concavity" initialization to sample unnormalized densities.
Problem
Sampling from unnormalized multi-modal densities in high dimension, where MCMC, SMC, AIS and learned score/VI methods struggle; stochastic localization has been explored for sampling only sparsely and without a practical general strategy.
Approach
A general SL observation process Y_t = alpha(t) X + sigma W_t with flexible denoising schedule g (Geom schemes), integrated via Euler-Maruyama with SNR-adapted time steps, where the denoiser is estimated by MALA posterior samples (no learning), and initialization is done by Langevin-within-Langevin at a time t0 chosen so both the posterior and the marginal are log-concave ("duality of log-concavit
Key results
- On Bayesian logistic regression, SLIPS Geom(1,1) reaches -86.32 ± 0.10 average predictive log-likelihood on Ionosphere vs -109.84 for RDMC and -87.92 for AIS.
- On 8-Gaussians, SLIPS Geom(1,1) achieves the lowest entropic 2-Wasserstein distance (0.74 ± 0.12) versus 0.91 for OAT and 1.10 for AIS.
- On the Gaussian mixture weight-estimation task, SLIPS recovers the relative weight at about 1% accuracy even in high dimensions, while other methods fail in most dimensions.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use SLIPS with Geom(1,1) instead of the strongest baseline | 8-Gaussians · 8-Gaussians | entropic regularized 2-Wasserstein distance: 0.91 ± 0.09 -> 0.74 ± 0.12 | Helped |
| Use SLIPS instead of OAT | Rings · Rings | entropic regularized 2-Wasserstein distance: 0.18 ± 0.02 -> 0.19 ± 0.01 | No clear effect |
| Use SLIPS Standard instead of the strongest baseline | Funnel · Funnel | sliced Kolmogorov-Smirnov distance: 0.035 ± 0.005 -> 0.024 ± 0.003 | Helped |
| Use SLIPS Geom(1,1) instead of the strongest baseline | Bayesian logistic regression · Sonar | average predictive posterior log-likelihood: -111.02 ± 0.13 -> -109.14 ± 0.09 | Helped |
| Use SLIPS Geom(1,1) instead of the strongest baseline | Bayesian logistic regression · Ionosphere | average predictive posterior log-likelihood: -87.82 ± 0.16 -> -86.32 ± 0.10 | Helped |
| Vary the denoising schedule among Standard, Geom(1,1), and Geom(2,1) (ablation) | SLIPS · 8-Gaussians, Rings, Funnel, Sonar, Ionosphere | reported benchmark metric | Mixed |
Surprising findings
- The paper reports that the score-based SDE formulation using posterior expectation of grad log pi was less stable than the denoiser-based formulation it adopted.
Limitations
- The only hyper-parameter requiring careful tuning is the initial time t0, with a 'duality of log-concavity' condition that is restrictive (dR^2/tau^2 < 2) for the theory.
- Theoretical log-concavity results rely on the assumption that the target is log-concave outside a compact or a convolution of a bounded-variance measure with a Gaussian.
- Experiments use oracle distance tuning of hyper-parameters, and baselines were tuned under similar budgets.
Experimental setup
Datasets: 8-Gaussians Rings Funnel Ionosphere Sonar Gaussian mixtures (high-dimensional) 1D phi^4 field model
Benchmarks: 8-Gaussians Rings Funnel Bayesian logistic regression (Ionosphere, Sonar) high-dimensional Gaussian mixtures phi^4 field system
Metrics: entropic regularized 2-Wasserstein distance sliced Kolmogorov-Smirnov distance average predictive log-likelihood relative weight estimation error sliced Wasserstein distance
Baselines: AIS SMC OAT RDMC
Builds on: stochastic localization Reverse Diffusion Monte Carlo Once-At-a-Time sampling Metropolis-adjusted Langevin algorithm Tweedie's formula
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