Adaptive deep nonparametric regression from dependent data under covariate shift

2026-07-23 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Data Science & Analytics, Machine Learning Theory · Depth: Expert, extended

Summary

William Kengne and Ehud Mossa Ockegna introduce Sparse-Penalized Deep Neural Network (SPDNN) estimators for nonparametric quantile and Huber regression, specifically addressing covariate shift in dependent observations. This framework unifies theoretical guarantees for various classical models, including i.i.d., π-mixing, strong mixing, and ℼ-mixing processes, through a generalized Bernstein-type inequality. The proposed SPDNN estimators establish non-asymptotic error bounds for Hölder smooth functions. Notably, they adaptively attain (up to a logarithmic factor) minimax optimal convergence rates. The paper details unweighted and reweighted estimators, and a two-step pre-training procedure for cases where the density ratio between source and target distributions is unknown or unbounded, utilizing a truncated density ratio approach.

Key takeaway

For machine learning engineers and AI scientists developing robust regression models with time-series or other dependent data, this research provides a critical framework for handling covariate shift. You should implement Sparse-Penalized Deep Neural Networks (SPDNN) with Huber or quantile loss to achieve minimax optimal rates, even with distribution mismatches between training and testing sets. When the density ratio is unknown, leverage the two-step pre-training procedure, potentially with a truncated density ratio, to maintain performance and adaptivity.

Key insights

SPDNN estimators achieve minimax optimal rates for nonparametric regression under covariate shift with dependent data.

Principles

• A generalized Bernstein-type inequality unifies theory for various dependent data structures.
• Sparse-penalized regularization leads to adaptive estimators for optimal convergence rates.
• Huber regression offers robustness against outliers and heavy-tailed errors.

Method

A two-step pre-training procedure estimates the density ratio, then uses it to reweight the SPDNN estimator for the regression function, handling unknown or unbounded ratios.

In practice

• Apply SPDNN for robust regression with heavy-tailed data or outliers using Huber loss.
• Utilize truncated density ratio estimation for covariate shift when the ratio is unbounded.
• Extend deep learning guarantees to π-mixing, strong mixing, and ℼ-mixing time series.

Topics

• Deep Neural Networks
• Covariate Shift
• Nonparametric Regression
• Huber Regression
• Quantile Regression
• Sparse Regularization
• Dependent Data

Best for: Research Scientist, AI Scientist, Machine Learning Engineer

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