用等周不等式改进二分类线性模型的泛化误差分析，更精准且适用范围广。
Improved generalization bounds for binary linear classification via isoperimetry
- 通过等周论证建立输出标签与加权输入向量的庞卡雷与对数-索博列夫不等式
- 在高维比例设定下，均匀泛化误差几乎必然收敛于其期望值
- 适用于无维度限制的统一大数定律，优于传统对逻辑回归的特定分析
我们通过等周论证研究二分类线性问题中均匀泛化误差围绕其期望的集中性。特别地，建立了输出标签与标签加权输入向量联合分布的庞卡雷不等式和对数-索博列夫不等式，并用于推导集中界。所得结果优于现有针对一般无界经验过程的界限，也优于专为逻辑回归设计的界限。在渐近分析中，我们还证明了在广泛设定（如比例高维情形）下，均匀泛化误差几乎必然收敛至其期望。利用此收敛性，我们在无维度限制条件下建立了统一的大数定律。
原文摘要 · Abstract (English)
We examine the concentration of uniform generalization errors around their expectation in binary linear classification problems via an isoperimetric argument. In particular, we establish Poincaré and log-Sobolev inequalities for the joint distribution of the output labels and the label-weighted input vectors, which we apply to derive concentration bounds. The derived results improve upon existing bounds obtained from general unbounded empirical processes, as well as that tailored specifically to logistic regression. In asymptotic analysis, we also show that almost sure convergence of uniform generalization errors to their expectation occurs in very broad settings, such as proportionally high-dimensional regimes. Using this convergence, we establish uniform laws of large numbers under dimension-free conditions.
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