Sampling with Mollified Interaction Energy Descent

• URL: http://arxiv.org/abs/2210.13400v1
• Date: Mon, 24 Oct 2022 16:54:18 GMT
• Title: Sampling with Mollified Interaction Energy Descent
• Authors: Lingxiao Li, Qiang Liu, Anna Korba, Mikhail Yurochkin, Justin Solomon
• Abstract summary: We present a new optimization-based method for sampling called mollified interaction energy descent (MIED) MIED minimizes a new class of energies on probability measures called mollified interaction energies (MIEs) We show experimentally that for unconstrained sampling problems our algorithm performs on par with existing particle-based algorithms like SVGD.
• Score: 57.00583139477843
• License: http://creativecommons.org/publicdomain/zero/1.0/
• Abstract: Sampling from a target measure whose density is only known up to a normalization constant is a fundamental problem in computational statistics and machine learning. In this paper, we present a new optimization-based method for sampling called mollified interaction energy descent (MIED). MIED minimizes a new class of energies on probability measures called mollified interaction energies (MIEs). These energies rely on mollifier functions -- smooth approximations of the Dirac delta originated from PDE theory. We show that as the mollifier approaches the Dirac delta, the MIE converges to the chi-square divergence with respect to the target measure and the gradient flow of the MIE agrees with that of the chi-square divergence. Optimizing this energy with proper discretization yields a practical first-order particle-based algorithm for sampling in both unconstrained and constrained domains. We show experimentally that for unconstrained sampling problems our algorithm performs on par with existing particle-based algorithms like SVGD, while for constrained sampling problems our method readily incorporates constrained optimization techniques to handle more flexible constraints with strong performance compared to alternatives.

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