On Task Vectors and Gradients
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arXiv
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| Main Authors: | , , , , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915562653745152 |
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| author | Zhou, Luca Solombrino, Daniele Crisostomi, Donato Bucarelli, Maria Sofia D'Inverno, Giuseppe Alessio Silvestri, Fabrizio Rodolà, Emanuele |
| author_facet | Zhou, Luca Solombrino, Daniele Crisostomi, Donato Bucarelli, Maria Sofia D'Inverno, Giuseppe Alessio Silvestri, Fabrizio Rodolà, Emanuele |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_16082 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Task Vectors and Gradients Zhou, Luca Solombrino, Daniele Crisostomi, Donato Bucarelli, Maria Sofia D'Inverno, Giuseppe Alessio Silvestri, Fabrizio Rodolà, Emanuele Machine Learning Artificial Intelligence 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. |
| title | On Task Vectors and Gradients |
| topic | Machine Learning Artificial Intelligence |
| url | https://arxiv.org/abs/2508.16082 |