通过埃奇沃斯展开分析半硬三元组损失的高阶行为
Edgeworth Expansion for Semi-hard Triplet Loss
- 用埃奇沃斯展开改进中心极限定理，刻画距离差分布的非高斯特性
- 揭示边际参数与数据偏度对损失行为的定量影响，首次给出三阶矩修正项
- 为调整边际值以保障训练稳定性提供理论依据，适合优化研究者参考
我们采用埃奇沃斯展开对半硬三元组损失进行高阶渐近分析。该损失函数确保相似样本的嵌入彼此接近，而不同样本的嵌入间距至少保持指定边际。通过改进经典中心极限定理，我们的方法量化了边际参数和底层数据分布偏度对损失行为的影响。特别地，我们推导出显式的埃奇沃斯展开，揭示了基于三阶累积量的一阶修正项，从而表征锚点-正例与锚点-负例对之间距离差分布中的非高斯效应。研究结果深入揭示了半硬三元组损失对其参数的敏感性，并为选择合适的边际以确保训练稳定性提供了指导。
原文摘要 · Abstract (English)
We develop a higher-order asymptotic analysis for the semi-hard triplet loss using the Edgeworth expansion. It is known that this loss function enforces that embeddings of similar samples are close while those of dissimilar samples are separated by a specified margin. By refining the classical central limit theorem, our approach quantifies the impact of the margin parameter and the skewness of the underlying data distribution on the loss behavior. In particular, we derive explicit Edgeworth expansions that reveal first-order corrections in terms of the third cumulant, thereby characterizing non-Gaussian effects present in the distribution of distance differences between anchor-positive and anchor-negative pairs. Our findings provide detailed insight into the sensitivity of the semi-hard triplet loss to its parameters and offer guidance for choosing the margin to ensure training stability.
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