On Task Vectors and Gradients

2025-08-22

AI Summary

Task arithmetic demonstrates strong empirical performance in model merging but lacks theoretical grounding. Method: This paper establishes, for the first time, an exact equivalence between task vectors and loss gradients, proving that a single fine-tuning step captures gradient information aligned with dominant fusion directions—revealing task arithmetic as gradient-driven approximate multitask learning. Leveraging second-order error analysis in feedforward networks and classical gradient descent theory, we rigorously justify the critical role of early-training dynamics. Contribution/Results: Experiments across seven vision benchmarks show that task vectors derived from just one epoch of fine-tuning achieve fusion performance comparable to those from fully converged models. This work bridges the theoretical gap in task arithmetic, elucidates its underlying mechanism, and offers a principled perspective for efficient model merging.

Technology Category

• Machine Learning: Transfer, Domain Adaptation, Multi-Task Learning
• Computer Vision: Learning & Optimization for CV
• Search and Optimization: Learning to Search

Application Category

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

Abstract

Task arithmetic has emerged as a simple yet powerful technique for model merging, enabling the combination of multiple finetuned models into one. Despite its empirical success, a clear theoretical explanation of why and when it works is lacking. This paper provides a rigorous theoretical foundation for task arithmetic by establishing a connection between task vectors and gradients of the task losses. We show that under standard gradient descent, a task vector generated from one epoch of finetuning is exactly equivalent to the negative gradient of the loss, scaled by the learning rate. For the practical multi-epoch setting, we prove that this equivalence holds approximately, with a second-order error term that we explicitly bound for feed-forward networks. Our empirical analysis across seven vision benchmarks corroborates our theory, demonstrating that the first-epoch gradient dominates the finetuning trajectory in both norm and direction. A key implication is that merging models finetuned for only a single epoch often yields performance comparable to merging fully converged models. These findings reframe task arithmetic as a form of approximate multitask learning, providing a clear rationale for its effectiveness and highlighting the critical role of early training dynamics in model merging.

Problem

• Lack of theoretical explanation for task arithmetic effectiveness
• Establishing connection between task vectors and gradient losses
• Analyzing early training dynamics impact on model merging

Innovation

• Task vectors equivalent to negative gradients
• Second-order error bounded for multi-epoch training
• First-epoch gradient dominates finetuning trajectory