# Learning Graph Energies for Finite-Time Generation

arXiv: https://arxiv.org/abs/2610.19946 · HTML: https://mlsanity.com/paper/2610.19946/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 23 of its 28 references that are in the map.
- First posted on arXiv 14 Oct 2026.

## Builds on (most cited first, 23 of 23)

- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [DiGress: Discrete Denoising diffusion for graph generation](https://mlsanity.com/paper/2209.14734/) (2022)
- [A Continuous Time Framework for Discrete Denoising Models](https://mlsanity.com/paper/2205.14987/) (2022)
- [Guiding a Diffusion Model with a Bad Version of Itself](https://mlsanity.com/paper/2406.02507/) (2024)
- [Molecular Sets (MOSES): A Benchmarking Platform for Molecular Generation Models](https://mlsanity.com/paper/1811.12823/) (2018)
- [Fr'echet ChemNet Distance: A metric for generative models for molecules in drug discovery](https://mlsanity.com/paper/1803.09518/) (2018)
- [Learning Non-Convergent Non-Persistent Short-Run MCMC Toward Energy-Based Model](https://mlsanity.com/paper/1904.09770/) (2019)
- [Generative Flow Networks for Discrete Probabilistic Modeling](https://mlsanity.com/paper/2202.01361/) (2022)
- [DeFoG: Discrete Flow Matching for Graph Generation](https://mlsanity.com/paper/2410.04263/) (2024)
- [Fine-Tuning Discrete Diffusion Models via Reward Optimization with Applications to DNA and Protein Design](https://mlsanity.com/paper/2410.13643/) (2024)
- [A Langevin-like Sampler for Discrete Distributions](https://mlsanity.com/paper/2206.09914/) (2022)
- [A-NICE-MC: Adversarial Training for MCMC](https://mlsanity.com/paper/1706.07561/) (2017)
- [Cometh: A continuous-time discrete-state graph diffusion model](https://mlsanity.com/paper/2406.06449/) (2024)
- [Learning Discrete Energy-based Models via Auxiliary-variable Local Exploration](https://mlsanity.com/paper/2011.05363/) (2020)
- [Automatic Differentiation of Programs with Discrete Randomness](https://mlsanity.com/paper/2210.08572/) (2022)
- [Graph Diffusion Policy Optimization](https://mlsanity.com/paper/2402.16302/) (2024)
- [Explaining the effects of non-convergent sampling in the training of Energy-Based Models](https://mlsanity.com/paper/2301.09428/) (2023)
- [Differentiating Metropolis-Hastings to Optimize Intractable Densities](https://mlsanity.com/paper/2306.07961/) (2023)
- [Reinforcement Learning for Adaptive MCMC](https://mlsanity.com/paper/2405.13574/) (2024)
- [Reward-Guided Discrete Diffusion via Clean-Sample Markov Chain for Molecule and Biological Sequence Design](https://mlsanity.com/paper/2602.09424/) (2026)
- [TFG-Flow: Training-free Guidance in Multimodal Generative Flow](https://mlsanity.com/paper/2501.14216/) (2025)
- [Graph-GRPO: Training Graph Flow Models with Reinforcement Learning](https://mlsanity.com/paper/2603.10395/) (2026)
- [Graph Energy Matching: Transport-Aligned Energy-Based Modeling for Graph Generation](https://mlsanity.com/paper/2603.23398/) (2026)

## Abstract (arXiv)

Energy-based models can represent distributions over discrete structures such as molecular graphs, but an energy that is accurate at equilibrium need not yield accurate samples when the sampler stops after a fixed number of steps. We introduce finite-horizon energy learning, which adapts the acceptance energy of a Metropolis--Hastings sampler to the distribution it returns after a prescribed number of steps, with initialization, proposals, and annealing schedule held fixed. For graph generation, a frozen pretrained model supplies initialization and proposals, and a learned energy governs acceptance. A training-only discriminator scores completed samples, and coupled continuations of the accepted and rejected branches of a proposal give an acceptance-gradient estimator that is unbiased for a fixed discriminator under stated conditions, without equilibrated negative samples or differentiation through discrete states. Generation uses the learned energy in the original single-trajectory sampler, without discriminator evaluations or reranking. On twelve pairwise binary targets with exactly evaluated terminal laws, learned energies have lower median terminal total variation than the equilibrium-correct energy in every tested short-horizon setting, and than its numerically tuned scalar multiple in most settings. On MOSES, a learned two-coefficient combination of pretrained Graph Energy Matching (GEM) potentials lowers FCD by 3.7--10.5% relative to scale-calibrated GEM at matched sampling steps, with validity gains at the shortest budget and a sub-point validity trade-off at higher budgets.