On Task Vectors and Gradients

• URL: http://arxiv.org/abs/2508.16082v6
• Date: Mon, 20 Oct 2025 04:13:40 GMT
• Title: On Task Vectors and Gradients
• Authors: Luca Zhou, Daniele Solombrino, Donato Crisostomi, Maria Sofia Bucarelli, Giuseppe Alessio D'Inverno, Fabrizio Silvestri, Emanuele Rodolà,
• Abstract summary: We provide a rigorous theoretical foundation for task arithmetic by establishing a connection between task vectors and gradients of the task losses.<n>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.<n>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.
• Score: 24.021393654093103
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

Related papers

• Provable Benefit of Curriculum in Transformer Tree-Reasoning Post-Training [76.12556589212666]
  We show that curriculum post-training avoids the exponential complexity bottleneck.<n>Under outcome-only reward signals, reinforcement learning finetuning achieves high accuracy with sample complexity.<n>We establish guarantees for test-time scaling, where curriculum-aware querying reduces both reward oracle calls and sampling cost from exponential to order.
  arXiv  Detail & Related papers  (2025-11-10T18:29:54Z)
• Train with Perturbation, Infer after Merging: A Two-Stage Framework for Continual Learning [57.514786046966265]
  We propose textbfPerturb-and-Merge (P&M), a novel continual learning framework that integrates model merging into the CL paradigm to mitigate forgetting.<n>Our proposed approach achieves state-of-the-art performance on several continual learning benchmark datasets.
  arXiv  Detail & Related papers  (2025-05-28T14:14:19Z)
• When is Task Vector Provably Effective for Model Editing? A Generalization Analysis of Nonlinear Transformers [64.1656365676171]
  Task arithmetic refers to editing the pre-trained model by adding a weighted sum of task vectors.<n>This paper theoretically prove the effectiveness of task addition in simultaneously learning a set of irrelevant or irrelevant tasks.<n>We prove the proper selection for task arithmetic to achieve negation to out-of-domain tasks.
  arXiv  Detail & Related papers  (2025-04-15T08:04:39Z)
• ATM: Improving Model Merging by Alternating Tuning and Merging [16.12778778313037]
  We provide a theoretical motivation for task vectors by highlighting that, under single-epoch full-batch gradient descent, they are equivalent to multitask.<n>This insight leads us to reinterpret model merging as a single step in an iterative procedure that Alternates between Tuning and Merging.<n>We propose two applications of ATM: (1) as an alternative to multitask learning in scenarios where data sharing is restricted, and (2) as a lightweight refinement step to improve existing model merging methods using a small validation set.
  arXiv  Detail & Related papers  (2024-11-05T12:42:42Z)
• AdaMerging: Adaptive Model Merging for Multi-Task Learning [68.75885518081357]
  This paper introduces an innovative technique called Adaptive Model Merging (AdaMerging) It aims to autonomously learn the coefficients for model merging, either in a task-wise or layer-wise manner, without relying on the original training data. Compared to the current state-of-the-art task arithmetic merging scheme, AdaMerging showcases a remarkable 11% improvement in performance.
  arXiv  Detail & Related papers  (2023-10-04T04:26:33Z)
• Synergies between Disentanglement and Sparsity: Generalization and Identifiability in Multi-Task Learning [79.83792914684985]
  We prove a new identifiability result that provides conditions under which maximally sparse base-predictors yield disentangled representations. Motivated by this theoretical result, we propose a practical approach to learn disentangled representations based on a sparsity-promoting bi-level optimization problem.
  arXiv  Detail & Related papers  (2022-11-26T21:02:09Z)
• Analysis of Catastrophic Forgetting for Random Orthogonal Transformation Tasks in the Overparameterized Regime [9.184987303791292]
  We show that in permuted MNIST image classification tasks, the performance of multilayer perceptrons trained by vanilla gradient descent can be improved. We provide a theoretical explanation of this effect by studying a qualitatively similar two-task linear regression problem. We show that when a model is trained on the two tasks in sequence without any additional regularization, the risk gain on the first task is small.
  arXiv  Detail & Related papers  (2022-06-01T18:04:33Z)
• Instance-Level Relative Saliency Ranking with Graph Reasoning [126.09138829920627]
  We present a novel unified model to segment salient instances and infer relative saliency rank order. A novel loss function is also proposed to effectively train the saliency ranking branch. experimental results demonstrate that our proposed model is more effective than previous methods.
  arXiv  Detail & Related papers  (2021-07-08T13:10:42Z)
• A Theoretical Analysis of Fine-tuning with Linear Teachers [31.849269592822296]
  Fine-tuning is a common practice in deep learning, achieving excellent results on downstream tasks using relatively little training data. We show that the success of fine-tuning depends on the similarity between the source tasks and the target task, however measuring it is non trivial.
  arXiv  Detail & Related papers  (2021-07-04T14:15:50Z)
• Efficient Iterative Amortized Inference for Learning Symmetric and Disentangled Multi-Object Representations [8.163697683448811]
  We introduce EfficientMORL, an efficient framework for the unsupervised learning of object-centric representations. We show that optimization challenges caused by requiring both symmetry and disentanglement can be addressed by high-cost iterative amortized inference. We demonstrate strong object decomposition and disentanglement on the standard multi-object benchmark while achieving nearly an order of magnitude faster training and test time inference.
  arXiv  Detail & Related papers  (2021-06-07T14:02:49Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.