A Unified Generalization Framework for Model Merging: Trade-offs, Non-Linearity, and Scaling Laws

TL;DR

This paper develops a unified theoretical framework for model merging, explaining its effectiveness and trade-offs, especially under heterogeneous finetuning conditions, and provides scaling laws to guide practical model merging strategies.

Contribution

It introduces a comprehensive theory integrating $L_2$-Stability to explain model merging, covering both linear and non-linear algorithms, and derives scaling laws for hyperparameter optimization.

Findings

Theoretical explanation of the optimization-generalization trade-off.

Unified framework for linear and non-linear merging algorithms.

Empirical validation of scaling laws across multiple architectures and tasks.

Abstract

Model merging efficiently aggregates capabilities from multiple fine-tuned models into a single one, operating purely in parameter space without original data or expensive re-computation. Despite empirical successes, a unified theory for its effectiveness under heterogeneous finetuning hyperparameters (e.g., varying learning rates, batch sizes) remains missing. Existing federated learning theories focus purely on optimization, which fails to explain model merging and inherently leads to theoretical paradoxes. To address this challenge, we pioneer the integration of $L_2$-Stability theory into heterogeneous environments to rigorously decouple the excess risk of the merged model $\boldsymbol{x}_{avg}$ into optimization and generalization errors. This comprehensive analysis yields three main contributions: (i) We mathematically establish the fundamental \textit{Optimization-Generalization Trade-off}, explicitly resolving the paradox of why over-trained experts lead to catastrophic merging collapse. (ii) \textit{A unified theoretical framework} is provided to explain not only linear merging algorithms (e.g., TA, AdaMerging) but also state-of-the-art \textit{non-linear} merging algorithms (e.g., TIES, DARE), proving how sparsification operators strictly tighten the generalization bound by suppressing task heterogeneity. (iii) Rather than heuristic guidelines, we derive \textit{Quantitative Scaling Laws} that theoretically predict the precise impact of hyperparameter choices, enabling practitioners to strategically construct ``merge-friendly'' experts. Extensive experiments on the ResNet and ViT architectures across 20 visual classification tasks, involving thousands of finetuning models, robustly confirm that our theoretical scaling laws accurately predict the empirical generalization behaviors of $\boldsymbol{x}_{avg}$.