Paper
One-Step Sampler for Boltzmann Distributions via Drifting
Authors
Wenhan Cao, Keyu Yan, Lin Zhao
Abstract
We present a drifting-based framework for amortized sampling of Boltzmann distributions defined by energy functions. The method trains a one-step neural generator by projecting samples along a Gaussian-smoothed score field from the current model distribution toward the target Boltzmann distribution. For targets specified only up to an unknown normalization constant, we derive a practical target-side drift from a smoothed energy and use two estimators: a local importance-sampling mean-shift estimator and a second-order curvature-corrected approximation. Combined with a mini-batch Gaussian mean-shift estimate of the sampler-side smoothed score, this yields a simple stop-gradient objective for stable one-step training. On a four-mode Gaussian-mixture Boltzmann target, our sampler achieves mean error $0.0754$, covariance error $0.0425$, and RBF MMD $0.0020$. Additional double-well and banana targets show that the same formulation also handles nonconvex and curved low-energy geometries. Overall, the results support drifting as an effective way to amortize iterative sampling from Boltzmann distributions into a single forward pass at test time.
Metadata
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"raw_xml": "<entry>\n <id>http://arxiv.org/abs/2603.17579v1</id>\n <title>One-Step Sampler for Boltzmann Distributions via Drifting</title>\n <updated>2026-03-18T10:35:16Z</updated>\n <link href='https://arxiv.org/abs/2603.17579v1' rel='alternate' type='text/html'/>\n <link href='https://arxiv.org/pdf/2603.17579v1' rel='related' title='pdf' type='application/pdf'/>\n <summary>We present a drifting-based framework for amortized sampling of Boltzmann distributions defined by energy functions. The method trains a one-step neural generator by projecting samples along a Gaussian-smoothed score field from the current model distribution toward the target Boltzmann distribution. For targets specified only up to an unknown normalization constant, we derive a practical target-side drift from a smoothed energy and use two estimators: a local importance-sampling mean-shift estimator and a second-order curvature-corrected approximation. Combined with a mini-batch Gaussian mean-shift estimate of the sampler-side smoothed score, this yields a simple stop-gradient objective for stable one-step training. On a four-mode Gaussian-mixture Boltzmann target, our sampler achieves mean error $0.0754$, covariance error $0.0425$, and RBF MMD $0.0020$. Additional double-well and banana targets show that the same formulation also handles nonconvex and curved low-energy geometries. Overall, the results support drifting as an effective way to amortize iterative sampling from Boltzmann distributions into a single forward pass at test time.</summary>\n <category scheme='http://arxiv.org/schemas/atom' term='cs.LG'/>\n <published>2026-03-18T10:35:16Z</published>\n <arxiv:primary_category term='cs.LG'/>\n <author>\n <name>Wenhan Cao</name>\n </author>\n <author>\n <name>Keyu Yan</name>\n </author>\n <author>\n <name>Lin Zhao</name>\n </author>\n </entry>"
}