Kernel Regression in Structured Non-IID Settings: Theory and Implications for Denoising Score Learning

2025-10-17

AI Summary

This work addresses the lack of generalization theory for kernel ridge regression (KRR) under non-i.i.d. data, specifically focusing on dependent data exhibiting signal–noise causal structure—e.g., multiple noisy observations sharing a common latent signal, as arises in denoising score learning. We propose a blockwise decomposition framework that, for the first time, systematically characterizes the joint influence of kernel spectrum, causal strength, and sampling mechanism on generalization error. By integrating causal modeling with spectral analysis, we derive an explicit upper bound on the excess risk, revealing how structural dependencies either exacerbate or mitigate overfitting. Our results yield interpretable, theoretically grounded sampling strategies for denoising score learning and provide rigorous generalization guarantees—thereby filling a critical gap in KRR theory for structured dependent data.

Technology Category

• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Causality
• Knowledge Representation and Reasoning: Knowledge Acquisition

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

Kernel ridge regression (KRR) is a foundational tool in machine learning, with recent work emphasizing its connections to neural networks. However, existing theory primarily addresses the i.i.d. setting, while real-world data often exhibits structured dependencies - particularly in applications like denoising score learning where multiple noisy observations derive from shared underlying signals. We present the first systematic study of KRR generalization for non-i.i.d. data with signal-noise causal structure, where observations represent different noisy views of common signals. By developing a novel blockwise decomposition method that enables precise concentration analysis for dependent data, we derive excess risk bounds for KRR that explicitly depend on: (1) the kernel spectrum, (2) causal structure parameters, and (3) sampling mechanisms (including relative sample sizes for signals and noises). We further apply our results to denoising score learning, establishing generalization guarantees and providing principled guidance for sampling noisy data points. This work advances KRR theory while providing practical tools for analyzing dependent data in modern machine learning applications.

Problem

• Extends kernel ridge regression theory to non-i.i.d. data settings
• Analyzes generalization bounds for signal-noise causal structures
• Establishes guarantees for denoising score learning applications

Innovation

• Novel blockwise decomposition for dependent data analysis
• Excess risk bounds incorporating kernel spectrum parameters
• Generalization guarantees for denoising score learning applications