Distributed Prediction under Heterogeneity with Unidentifiable Parameter

2026-06-30

AI Summary

This work addresses the challenges of severe non-convexity and communication bottlenecks in distributed semi-parametric prediction arising from data heterogeneity and unidentifiable low-dimensional structures. The authors propose a novel framework that, for the first time, integrates a trace similarity penalty to adaptively handle heterogeneity and introduces invex relaxation combined with multi-step local updates to reduce communication costs while ensuring convergence to the global optimum. Rigorous theoretical analysis establishes model-free, non-asymptotic bounds on prediction error and proves that the method achieves the two-stage minimax optimal rate. Extensive experiments on both synthetic and real-world multi-center medical data demonstrate substantial improvements in prediction accuracy and communication efficiency over existing approaches.

Abstract

Predicting a response based on covariates is a fundamental problem in statistics and machine learning. However, profound difficulties arise when the underlying low-dimensional structural parameters are unidentifiable, as typified in dimension reduction contexts. Specifically,estimating these non-identifiable parameters inherently introduces severe nonconvexity. In distributed settings, this difficulty is further compounded by the challenges of data heterogeneity and communication cost. To overcome these intertwined barriers, we propose a novel distributed semiparametric framework. We formulate an adaptive homogeneity pursuit utilizing a trace-similarity penalty to effectively address data heterogeneity. To resolve the ensuing severe nonconvexity and communication bottlenecks, we introduce an invex relaxation technique coupled with a multi-step local update algorithm, ensuring stable convergence to global optimality with significantly reduced communication overhead. Theoretically, we establish a non-asymptotic model-free prediction error bound and prove that our estimator achieves a two-phase minimax optimal convergence rate and an sharper model-free prediction error bound. Furthermore, we provide theoretical guarantees for algorithmic convergence and communication efficiency. Extensive simulations and a real-world multi-center medical application validate the superiority of our method.

Problem

• distributed prediction
• parameter unidentifiability
• data heterogeneity
• nonconvexity
• communication cost

Innovation

• distributed prediction
• unidentifiable parameters
• data heterogeneity
• invex relaxation
• trace-similarity penalty