Logarithmic integral optimization via adaptive importance sampling based surrogation methods

• 成果类型: Article; Early Access
• 署名作者: He, Ziyu; Liu, Junyi; Pang, Jong-Shi
• 署名单位: University of Southern California; University of Southern California; Tsinghua University
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-025-02254-9
• 发表日期: 2025
• 关键词: carlo maximum-likelihood; average approximation

摘要

This paper explores Logarithmic Integral Optimization (LIO) problems, providing a unified computational framework for various tasks in computational statistics. Key among these are Maximum Likelihood Estimation (MLE) and Maximum a Posteriori (MAP) inference for probabilistic models. Specifically, we investigate scenarios where the model consists of conditional density functions with intractable normalizers. This feature can pose substantial computational challenges for the associated LIO, especially when coupled with the growing prevalence of nonconvex and nondifferentiable modelings in contemporary applications. To address these challenges, we propose an efficient algorithm for LIO, termed Adaptive Importance Sampling-based Surrogation. This method is designed to simultaneously handle nonconvexity and nondifferentiability, while also improving the sampling approximation of the intractable integral term in LIO through variance reduction. The justification of this algorithm is supported by our analysis, which establishes an almost sure subsequential convergence to a necessary candidate for a local minimizer, referred to as a surrogation stationary point. Furthermore, we demonstrate the effectiveness of our algorithm through extensive numerical experiments, confirming its efficiency and stability in facilitating more advanced probabilistic models with intractable normalizers.