The Interplay Between Interpolation and Aggregation in Regression: Optimal Sample Complexity

2026-05-28 · Source: Machine Learning · Field: Technology & Digital — Artificial Intelligence & Machine Learning · Depth: Expert, quick

Summary

This theoretical work investigates the intricate relationship between interpolation and aggregation within the context of regression, specifically addressing optimal sample complexity. The research establishes that the γ-graph dimension serves as a crucial characteristic for defining learnability across a broad spectrum of natural aggregation procedures. A significant finding highlights that an exceptionally simple aggregation method, which combines three distinct interpolating hypotheses through the median, proves to be optimal among these procedures. Furthermore, this median-based approach is demonstrated to be strictly more powerful than traditional proper learning. The study also reveals that certain complex hypothesis classes are only learnable by aggregating an infinitely large number of hypotheses or by utilizing non-interpolating aggregation rules, where any finite interpolating aggregation fails to achieve even trivial performance.

Key takeaway

For AI Scientists designing regression models, understanding the interplay between interpolation and aggregation is crucial. You should consider that simple aggregation methods, like combining three interpolating hypotheses via the median, can achieve optimal learnability and surpass proper learning. Be aware that for certain complex hypothesis classes, finite interpolating aggregation may be insufficient, necessitating more advanced or infinite aggregation strategies to ensure effective learning.

Key insights

The γ-graph dimension characterizes learnability, and a median-based aggregation of three interpolating hypotheses is optimal.

Principles

• γ-graph dimension defines learnability for aggregation.
• Simple aggregation can outperform proper learning.
• Some hypothesis classes require infinite aggregation.

Method

Combine three interpolating hypotheses using the median to achieve optimal aggregation performance in regression.

Topics

• Regression Analysis
• Interpolation
• Aggregation Procedures
• Sample Complexity
• Learnability Theory
• γ-graph Dimension

Best for: Research Scientist, AI Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by Machine Learning.