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LEAPS: A discrete neural sampler via locally equivariant networks
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Proposes LEAPS, a discrete neural sampler that learns a CTMC rate matrix via a PINN-style objective with importance weights, using locally equivariant networks to make the proactive IS update cheap; evaluated on Ising and Potts models on a 15x15 lattice.
Problem
Sampling from discrete distributions known only up to normalization, where MCMC mixes slowly and AIS/SMC importance weights can have high variance; standard neural rate-matrix parameterizations make the importance-weight computation prohibitively expensive.
Approach
Learn a CTMC rate matrix that transports a base distribution to the target, with log-importance weights accumulated along the trajectory (Radon-Nikodym derivative of reverse-time CTMC). A "locally equivariant" parameterization of rates (flux from x to neighbor is negative of flux back) makes the weight update and PINN loss computable in one network forward pass, and a family of locally equivariant
Key results
- At the critical temperature of the 15x15 Ising model, LEAPS achieves an ESS of about 68% with 100 integration steps, whereas AIS needs around 10^6 steps to reach similar ESS.
- On the Potts model, LEAPS reaches an ESS of about 20% in 100 integration steps, while AIS has lower ESS even at 10^6 steps.
- On the DISCS benchmark, LEAPS shows significantly higher ESS than MCMC samplers even after accounting for function evaluations (as stated by the authors).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use LEAPS instead of AIS | Ising model at critical temperature · Ising model, critical temperature | ESS at a similar ESS target: ~10^6 integration steps (AIS) vs. 100 integration steps (LEAPS) | Helped |
| Use LEAPS instead of AIS | Potts model at critical temperature · Potts model, critical temperature | ESS: ∼ 20% ESS in 100 integration steps; AIS has a lower ESS even for 10^6 integration steps | Helped |
| Use LEAPS instead of MCMC samplers | Ising model · Ising model, DISCS benchmark | ESS | Helped |
| Use LEAPS instead of MCMC samplers | Potts model · Potts model, DISCS benchmark | ESS | Helped |
| Increase LEC network depth (ablation) | LEAPS on the simple Ising model · Ising model, L=15, J = 0.4, β = 0.7 | ESS | Helped |
| Remove transport and use only annealed dynamics with marginal-preserving MCMC updates (ablation) | LEAPS on the simple Ising model · Ising model, L=15, J = 0.4, β = 0.7 | magnetization histogram agreement | Hurt |
| Use deep LEC instead of LEA (ablation) | LEAPS on the simple Ising model · Ising model, L=15, J = 0.4, β = 0.7 | performance | Helped |
Limitations
- Comparison with MCMC samplers is not like-for-like: LEAPS counts neural network evaluations while MCMC counts energy calls, and ESS is measured differently.
- Local equivariance is not preserved under composition, so deep architectures require a specific hierarchical design.
- Experiments are limited to lattice spin models (Ising, Potts) on a 15x15 grid.
Experimental setup
Datasets: Ising model samples from target distribution (simulated) Potts model (DISCS benchmark setup)
Benchmarks: Ising model 15x15 lattice Potts model 15x15 lattice DISCS benchmark
Metrics: effective sample size magnetization histogram two-point correlation function
Baselines: annealed importance sampling Glauber dynamics DISCS MCMC samplers
Builds on: annealed importance sampling sequential Monte Carlo discrete diffusion models log-variance divergence
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