提出紧致的超参调优理论边界，解决多维调优泛化性难题。
Tight Bounds for Data-driven Multiple Hyper-parameter Tuning with Structured Loss Function
- 基于代数几何分析连通符号域，避免拓扑过计数，提升样本复杂度上界精度。
- 构建多阶段下界框架，证明上界在特定条件下可被严格逼近。
- 适用于广义双层验证与半代数问题，扩展性强，适合理论研究者。
数据驱动的算法设计将超参数调优视为统计学习问题，但模型性能对超参数的隐式非光滑依赖使得泛化保证难以建立。现有基于分段多项式的多维边界理论松弛，且缺乏完整的下界分析。本文通过建立多维数据驱动调优的紧致伪维数界来解决该问题。首先，利用实代数几何改进学习理论上界：通过分析块消去过程中的不变连通符号胞腔，而非孤立符号向量，避免了拓扑过计数，得到更严格的样本复杂度。其次，提出多范式下界框架，解耦组合与代数容量；通过构造不同范式下的打散实例，证明上界可被严格饱和。最后，将拓扑框架拓展至通用双层验证损失调优及更广泛的半代数应用场景。
原文摘要 · Abstract (English)
Data-driven algorithm design frames hyperparameter tuning as a statistical learning problem, but establishing generalization guarantees remains challenging due to the implicit, non-smooth dependence of model performance on hyperparameters. Existing multi-dimensional bounds under piecewise-polynomial assumptions remain theoretically loose and lack comprehensive lower bounds. We resolve this by establishing tight pseudo-dimension bounds for multi-dimensional data-driven tuning. First, we refine the learning-theoretic upper bound using real algebraic geometry; by analyzing invariant connected sign cells during block elimination rather than isolated sign vectors, we avoid topological over-counting to derive strictly sharper sample complexities. Second, we present a multi-regime lower-bound framework that disentangles combinatorial and algebraic capacities. By constructing shattered problem instances across distinct regimes, we prove our upper bounds are tightly saturated. Finally, we extend our topological framework to accommodate general bi-level validation-loss tuning and broader semi-algebraic applications.
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