相关设计下模型选择的统一描述复杂度框架
A Unified Descriptive-Complexity Framework for Model Selection under Correlated Designs
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中文总结 AI 辅助
针对强预测变量依赖与模型类不确定性下的模型选择难题，提出描述复杂度信息准则（DCIC），建立选择一致性与风险界，开发复杂度引导搜索路径，经实验验证其支持恢复与估计性能良好。
中文摘要 AI 辅助
在强预测变量依赖与模型类不确定性并存，尤其是存在指数级多模型时，模型选择变得极具挑战性。我们提出一种描述复杂度信息准则（DCIC），通过Kraft容许码长对大量候选模型集合进行正则化。在次Weibull噪声条件下，我们不依赖RIP型条件，而是通过近似误差分离建立选择一致性，同时得到在模型误设下仍有效的非渐近oracle风险界。该编码原则以极小的额外类识别成本将异构类置于统一复杂度尺度，此扩展在适当可识别性条件下实现类-模型恢复，并完成跨类风险适配。我们进一步开发复杂度引导的搜索路径，明确计算-统计权衡：大惩罚以高概率保留多项式大小的搜索区域，而小惩罚则优化oracle风险基准。数值实验表明，该方法在强依赖与模型类不确定性下，可实现稳定的支持恢复与良好的估计性能。
英文摘要
Model selection becomes particularly challenging under strong predictor dependence and model-class uncertainty, especially when there are exponentially many models. We propose a Descriptive-Complexity Information Criterion (DCIC) that regularizes large candidate model collections through Kraft-admissible code lengths. Under sub-Weibull noise, we establish selection consistency through approximation-error separation without relying on RIP-type conditions, together with nonasymptotic oracle risk bounds that remain valid under model misspecification. The same coding principle places heterogeneous classes on a common complexity scale at a small additional class-identification cost. This extension yields class--model recovery under suitable identifiability conditions and risk adaptation across classes. We further develop a complexity-guided search path that makes the computation--statistics trade-off explicit. Large penalties yield polynomial-size retained search regions with high probability, whereas smaller penalties sharpen the oracle risk benchmark. Numerical experiments illustrate stable support recovery and favorable estimation performance under strong dependence and model-class uncertainty.
发表机构
- Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）
- Citigroup（花旗集团）
- Beijing Institute of Mathematical Sciences and Applications（北京数学科学与应用研究院）
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