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Energy-based generator matching: A neural sampler for general state space
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Proposes energy-based generator matching (EGM), training neural samplers for arbitrary CTMPs (diffusion, flow, jump) from energy functions alone, using self-normalized importance sampling with a bootstrapping trick for variance reduction.
Problem
Sampling from Boltzmann distributions given only energy-function access, across continuous, discrete, and mixed state spaces, where data-driven generative training is unavailable.
Approach
Estimate the generator matching loss from energies via self-normalized importance sampling, and reduce importance-weight variance by bootstrapping: estimate the marginal generator through an intermediate time r>t using a learned intermediate energy surrogate.
Key results
- On 10x10 Ising at beta=0.2, EGM with bootstrapping reaches energy W1 0.39 vs 0.53 for Gibbs.
- On JointMoG (d=20), EGM with bootstrapping achieves energy W1 1.20 vs 7.76 for Gibbs.
- On GB-RBM, bootstrapping lowers energy W1 from 0.77 (Gibbs) to 0.20.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| add bootstrapping (ablation) | EGM · 5 x 5 Ising, β=0.2, J=1.0 | Energy W1: 0.20±0.06 -> 0.10±0.06 | Helped |
| add bootstrapping (ablation) | EGM · 5 x 5 Ising, β=0.2, J=1.0 | Magnetization W1: 0.02±0.01 -> 0.02±0.01 | No clear effect |
| add bootstrapping (ablation) | EGM · 5 x 5 Ising, β=0.4, J=1.0 | Energy W1: 3.73±0.39 -> 0.60±0.12 | Helped |
| add bootstrapping (ablation) | EGM · 5 x 5 Ising, β=0.4, J=1.0 | Magnetization W1: 0.24±0.02 -> 0.04±0.01 | Helped |
| add bootstrapping (ablation) | EGM · 10 x 10 Ising, β=0.2, J=1.0 | Energy W1: 0.84±0.07 -> 0.39±0.34 | Helped |
| add bootstrapping (ablation) | EGM · 10 x 10 Ising, β=0.2, J=1.0 | Magnetization W1: 0.02±0.00 -> 0.06±0.07 | Mixed |
| add bootstrapping (ablation) | EGM · 10 x 10 Ising, β=0.4, J=1.0 | Energy W1: 19.94±0.69 -> 2.51±0.16 | Helped |
| add bootstrapping (ablation) | EGM · 10 x 10 Ising, β=0.4, J=1.0 | Magnetization W1: 0.36±0.01 -> 0.24±0.01 | Helped |
| add bootstrapping (ablation) | EGM · GB-RBM (d=5) | Energy W1: 0.33±0.02 -> 0.20±0.15 | Helped |
| add bootstrapping (ablation) | EGM · GB-RBM (d=5) | x-W2: 0.63±0.10 -> 0.61±0.09 | Helped |
| add bootstrapping (ablation) | EGM · JointDW4 (d=12) | Energy W1: 2.45±0.07 -> 1.65±0.96 | Helped |
| add bootstrapping (ablation) | EGM · JointMoG (d=20) | Energy W1: 8.83±0.12 -> 1.20±0.32 | Helped |
Surprising findings
- Plain EGM without bootstrapping underperforms Gibbs on several settings, e.g., 10x10 Ising at beta=0.4 (19.94 vs 5.29 energy W1).
Limitations
- Importance weights remain high-variance; the estimator is biased (SNIS) and relies on a learned intermediate energy surrogate.
- No neural-sampler baselines for jump or hybrid samplers, so comparison is only against Gibbs sampling.
- Target score identity for general state spaces is not derived.
Experimental setup
Datasets: 2D Ising model (5x5, 10x10) GB-RBM JointDW4 JointMoG
Benchmarks: 2D Ising model GB-RBM JointDW4 JointMoG
Metrics: energy W1 magnetization W1 x-W2
Baselines: Gibbs sampler
Builds on: generator matching iDEM noise energy matching masked diffusion conditional optimal transport path
Closest papers by meaning
- Generator Matching: Generative modeling with arbitrary Markov processes (ICLR 2025 · similarity 0.90)
- Energy Matching: Unifying Flow Matching and Energy-Based Models for Generative Modeling (NeurIPS 2025 · similarity 0.88)
- Energy-Weighted Flow Matching for Offline Reinforcement Learning (ICLR 2025 · similarity 0.85)
- Improving Adversarial Energy-Based Model via Diffusion Process (ICML 2024 · similarity 0.84)
- Equilibrium Matching: Generative Modeling with Implicit Energy-Based Models (NeurIPS 2026 · similarity 0.84)
- Noise-conditioned Energy-based Annealed Rewards (NEAR): A Generative Framework for Imitation Learning from Observation (ICLR 2025 · similarity 0.83)
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