Edgeworth Expansion for Semi-hard Triplet Loss

2025-03-17

AI Summary

Semi-hard triplet loss suffers from training instability due to unmodeled non-Gaussian behavior in the anchor-positive versus anchor-negative distance difference distribution. Method: We propose the first higher-order asymptotic analysis framework for triplet loss based on the Edgeworth expansion, explicitly characterizing how margin parameters and data distribution skewness jointly govern loss dynamics. By deriving third-order cumulant correction terms, we quantify non-Gaussian effects—particularly skewness—in the distance difference distribution. Contribution/Results: Our bias-corrected analytical approximation of the loss distribution provides a theoretical foundation for principled margin selection. Experiments demonstrate significantly improved training convergence stability and generalization consistency across benchmarks. This work bridges a critical theoretical gap in metric learning by establishing the first rigorous higher-order asymptotic analysis of triplet loss.

Technology Category

• Machine Learning: Learning with Manifolds
• Game Theory and Economic Paradigms: Adversarial Learning
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Social Networks and Social Media: Fairness and bias in social network and social media analysis
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

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.

Problem

• Analyzes semi-hard triplet loss using Edgeworth expansion
• Quantifies margin parameter impact on loss behavior
• Characterizes non-Gaussian effects in distance differences

Innovation

• Edgeworth expansion analyzes semi-hard triplet loss.
• Quantifies margin impact and data skewness effects.
• Derives corrections for non-Gaussian distance distributions.