Gaussian Processes for Query-Efficient Boltzmann Sampling

Neural samplers have achieved remarkable success in recent years, as efficient learning-based substitutes for traditional MCMC methods that significantly reduce the sampling cost after learning to sample. However, both neural samplers and MCMC methods remain sample-inefficient and therefore unsuitable for sampling problems where energy or likelihood evaluations are expensive. We develop a model-based sampling approach using a Gaussian process (GP) framework that trades computation for fewer evaluations by learning an energy surrogate and using its uncertainty to guide exploration. We distinguish two objectives: (1) producing accurate samples throughout exploration and (2) learning an accurate distribution for subsequent sampling. For (1), we analyze posterior-mean, optimistic and posterior-sampling policies using a KL-based notion of sampling regret and establish temperature-dependent bounds. For (2), we derive a Bayes-optimal one-step acquisition that selects evaluations to maximize the expected reduction in reverse KL divergence, together with practical approximations. Our experiments on low-dimensional sampling problems show the significant advantage of our model-based sampling method over neural samplers, MCMC methods, and existing GP-based sampling techniques.