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Multilevel Generative Samplers for Investigating Critical Phenomena
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Proposes RiGCS, a multilevel sampler that replaces the nearest-neighbor heat-bath conditionals of MLMC-HB with autoregressive (PixelCNN) generative models to capture residual long-range and higher-order interactions, plus a sequential warm-start training scheme for 2D Ising near criticality.
Problem
Efficient Monte Carlo sampling of 2D lattice systems at criticality, where critical slowing down hurts MCMC and long receptive fields hurt generative samplers; MLMC-HB's nearest-neighbor heat-bath conditionals become inaccurate for D>=2 with slight scale-invariance violation.
Approach
Keep the multilevel ancestral sampling of MLMC-HB but replace the independent nearest-neighbor heat-bath conditionals at intermediate levels with small-receptive-field autoregressive conditional generative models that absorb long-range and higher-order renormalized interactions; train sequentially on growing lattices, warm-starting parameters from smaller systems via scale invariance.
Key results
- For N=128 Ising at beta=0.44, RiGCS free energy estimates are compatible with ground truth and comparable to Cluster-AIS, whereas HAN is highly biased.
- RiGCS's ESS is reported as a few orders of magnitude higher than HAN at N=128, the only generative model with non-vanishing ESS.
- RiGCS shows no mode-dropping (EMDM near 1) at N=128, while HAN's EMDM drops significantly.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| replace part of the independent heat-bath samplers with generative conditional models | MLMC-HB, yielding RiGCS · 2D Ising, N=128 | ESS: several orders of magnitude | Helped |
| replace part of the independent heat-bath samplers with generative conditional models | MLMC-HB, yielding RiGCS · 2D Ising, large lattices | ESS | Mixed |
| use sequential training with model transfer instead of random initialization (ablation) | RiGCS · 2D Ising, N=64 | training efficiency | Helped |
| use generative multilevel sampling instead of nearest-neighbor heat-bath approximations | MLMC-HB · 2D Ising, N≥32 | free energy estimation bias | Helped |
| use RiGCS instead of HAN | HAN · 2D Ising, N=128 | EMDM | Helped |
| scale VAN to N≥64 | VAN · 2D Ising, N≥64 | trainability: exceeds several weeks | Hurt |
Surprising findings
- MLMC-HB performs poorly already for N>=32 with strong bias, despite its efficiency under exact scale invariance.
Limitations
- Wolff cluster method remains more accurate and efficient than all generative samplers for large lattices.
- Only 2D Ising is evaluated; extension to Potts models and gauge theories is future work.
- Multilevel approximation error accumulates through ancestral sampling, so MLMC-HB alone has low acceptance for D>=2.
Experimental setup
Datasets: generated samples from the model (data-free reverse-KL training)
Benchmarks: 2D Ising model free energy estimation 2D Ising ESS and EMDM
Metrics: relative free energy error effective sample size effective mode-dropping measure
Baselines: MLMC-HB VAN HAN Wolff cluster method (Cluster-AIS)
Builds on: MLMC-HB Hierarchical Autoregressive Network PixelCNN Wolff cluster algorithm
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