Generalization error bounds for two-layer neural networks with Lipschitz loss function

2026-04-09 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences · Depth: Expert, medium

Summary

This research derives generalization error bounds for two-layer neural networks trained with Stochastic Gradient Method (SGM), specifically without assuming boundedness of the loss function. The study utilizes Wasserstein distance estimates to quantify the discrepancy between a probability distribution and its empirical measure, combined with moment bounds for the SGM. For independent test data, the authors achieve a dimension-free error rate of order $O\big(n^{-1/2}\big)$ on the $n$-sample generalization error. When the independence assumption is relaxed, the bound becomes $O\big(n^{-1/(d_{\rm in}+d_{\rm out})}\big)$, where $d_{\rm in}$ and $d_{\rm out}$ are input and output dimensions. A key finding is that these bounds and their coefficients can be explicitly computed before model training, a claim supported by numerical simulations.

Key takeaway

For AI Scientists and Research Scientists developing or evaluating two-layer neural networks, understanding these pre-computable generalization error bounds is crucial. Your model's expected performance can be estimated before extensive training, particularly when using Lipschitz loss and activation functions. This allows for more informed architectural decisions and resource allocation, potentially reducing development cycles and improving model reliability in scenarios where loss function boundedness cannot be assumed.

Key insights

Generalization error bounds for two-layer neural networks can be derived without bounded loss functions using Wasserstein distance and SGM moment bounds.

Principles

• Lipschitz conditions can replace boundedness assumptions for loss and activation functions.
• Generalization error bounds can be computed pre-training.

Method

The method involves deriving SGM moment bounds, then applying Wasserstein distance estimates between true and empirical data distributions to quantify generalization error, considering both independent and non-independent test data scenarios.

In practice

• Use mean absolute error or Huber loss functions.
• Employ softplus, tanh, or sigmoid activation functions.

Topics

• Generalization Error Bounds
• Two-Layer Neural Networks
• Stochastic Gradient Method
• Lipschitz Loss Functions
• Wasserstein Distance

Best for: AI Scientist, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.