Leveraging tails for adaptation

2026-06-19 · Source: stat.ML updates on arXiv.org · Field: Science & Research — Mathematics & Computational Sciences, Artificial Intelligence & Machine Learning · Depth: Expert, extended

Summary

This research investigates Bayesian posterior distribution contraction rates in nonparametric settings using p-exponential tailed priors, which generalize Laplace (p=1) and Gaussian (p=2) distributions. The study demonstrates that contraction rates improve as the tail parameter "p" decreases, achieving full adaptation to smoothness (up to logarithmic factors) in an appropriate p→0 regime. Applications include series priors in white noise regression and overparameterized shallow ReLU neural networks in random design regression. Specifically, overparameterized shallow ReLU networks are shown to adapt to any regularity from 0≤β≤2. A simulation study empirically validates the theoretical predictions, highlighting the benefits of heavier-tailed priors for improved adaptation and performance.

Key takeaway

For Research Scientists developing nonparametric regression models, this work suggests a shift towards heavier-tailed p-exponential priors, particularly in overparameterized neural networks. Adopting priors with a decreasing "p" (e.g., p_n = 2/log n) can significantly enhance adaptation to unknown function smoothness, reducing the need for complex hyperparameter tuning. This approach offers improved contraction rates and robust performance across diverse regularity settings, making your models more efficient and generalizable.

Key insights

Heavier p-exponential prior tails (p→0) significantly improve Bayesian posterior contraction rates, enabling full adaptation to unknown function smoothness.

Principles

• Bayesian posterior contraction rates improve with heavier p-exponential prior tails.
• Overparameterized networks, with p→0 tails, achieve full smoothness adaptation.
• Heavier-tailed priors can reduce hyperparameter estimation requirements.

Method

The study applies p-exponential priors to coefficients in series expansions for white noise regression and to weights in overparameterized shallow ReLU neural networks for random design regression, analyzing posterior contraction.

In practice

• Use p-exponential priors with p<1 for faster contraction.
• Employ p_n = 2/log n for SNN weights for adaptive performance.
• Consider overparameterized SNNs for unknown function regularity.

Topics

• Bayesian Nonparametrics
• p-Exponential Priors
• Posterior Contraction Rates
• Neural Network Adaptation
• Overparameterization
• White Noise Regression
• Shallow ReLU Networks

Best for: AI Scientist, Research Scientist, Data Scientist

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