优化器作为检测器：用于潜在混合模型的随机梯度下降

中文摘要 AI 辅助

合并潜在子群体可能会掩盖变量间的关系，并导致误导性的回归结论，包括辛普森悖论（SP）。我们提出了一种基于恒定步长随机梯度下降（SGD）稳态动力学的检测器。与基于似然的混合检验和混杂因素搜索方法不同，它既不需要正态混合设定，也不需要观测到的候选混杂因素。两个线性片段在赢家通吃的平方误差损失下竞争，其归一化的终端分离度构成检验统计量。利用扩散近似，我们推导了在一般中心化标量协变量和对称噪声下高斯多元协变量情况下该统计量的渐近零分布。当协变量和噪声均为高斯分布时，依赖于分布的零中心简化为与维度无关的常数$4/\pi$。我们在正则条件下建立了渐近尺寸控制和针对固定混合备择假设的一致性。我们将该方法扩展到截距和部分混合异质性，并研究了内生性、异方差性和非线性误设定。模拟实验检验了校准、功效和稳健性。最后，一个三阶段的“检测-筛选-验证”工具包将异质性的证据与其实质性解释分开。在八个公开数据集中，它恢复了四个已确立的SP基准案例，并识别了据我们所知此前未被记录的四个案例。该检测器既不需要潜在组标签，也不需要预先指定混合成分的数量。

英文摘要

Pooling latent subpopulations can obscure relationships and yield misleading regression conclusions, including Simpson's paradox (SP). We propose a detector based on the steady-state dynamics of constant-step stochastic gradient descent (SGD). Unlike likelihood-based mixture tests and confounder-search methods, it requires neither a normal-mixture specification nor observed candidate confounders. Two linear pieces compete under a winner-take-all squared-error loss, and their normalized terminal separation forms the test statistic. Using diffusion approximations, we derive its asymptotic null distribution for general centered scalar covariates and for Gaussian multivariate covariates under symmetric noise. The distribution-dependent null center reduces to the dimension-free constant $4/π$ when both covariates and noise are Gaussian. We establish asymptotic size control and consistency against fixed mixture alternatives under regularity conditions. We extend the method to intercept and partial-mixture heterogeneity and study endogeneity, heteroskedasticity, and nonlinear misspecification. Simulations examine calibration, power, and robustness. Finally, a three-stage Detect--Screen--Verify toolkit separates evidence of heterogeneity from its substantive explanation. Across eight public datasets, it recovers four established SP benchmarks and identifies four cases that, to our knowledge, have not been documented previously. The detector requires neither latent-group labels nor a prespecified number of mixture components.

发表机构

• Institute of Operations Research and Analytics, National University of Singapore（新加坡国立大学运筹学与分析研究所）
• NUS Business School, National University of Singapore（新加坡国立大学商学院）
• School of Management, University of Science and Technology of China（中国科学技术大学管理学院）

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