MOON: Multi-Objective OrthoNormalized Updates for Multitask Learning

• URL: http://arxiv.org/abs/2608.11749v1
• Date: Wed, 12 Aug 2026 07:41:31 GMT
• Title: MOON: Multi-Objective OrthoNormalized Updates for Multitask Learning
• Abstract summary: Most existing methods flatten model parameters into vectors and perform manipulation under Euclidean geometry.<n>In this paper, we show that in Euclidean space does not generally yield steepest descent under norm geometry.<n>We propose MOON, which performs manipulation under spectral gradient and norm geometry.
• Score: 11.574264042827558
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Multi-objective optimization (MOO) has demonstrated significant success in multi-task learning by mitigating task conflicts through gradient manipulation. However, most existing methods flatten model parameters into vectors and perform gradient manipulation under Euclidean geometry, thereby overlooking the matrix structure prevalent in modern architectures such as Transformers. In this paper, we show that gradient manipulation in Euclidean space does not generally yield the steepest descent direction under matrix geometry, potentially limiting optimization efficiency. Drawing from the theory of steepest descent for matrix-valued parameters, we propose MOON (Multi-Objective OrthoNormalized Updates), which performs gradient manipulation under spectral--nuclear norm geometry and uses the orthonormalized manipulated gradient for parameter updates. Theoretically, for smooth non-convex objectives, we establish convergence of the averaged Pareto-stationarity measure at rates of $\mathcal{O}(T^{-1/2})$ in the deterministic setting and $\mathcal{O}(T^{-1/4})$ under stochastic gradients. Empirical results across various benchmarks show that MOON consistently improves both optimization efficiency and final multi-task performance. Our code is available at https://github.com/KunlinLyu/MOON.

Related papers

• AdaGrad Meets Muon: Adaptive Stepsizes for Orthogonal Updates [5.049533819651459]
  We propose a new adaptive update, AdaGO, which combines a norm-based update with aGrad-type step.<n>AdaGO preserves the orthogonality of the update, which can be interpreted as a spectral descent, while adapting the stepsizes to the optimization landscape by scaling the direction with accumulated past gradients.
  arXiv  Detail & Related papers  (2025-09-03T03:42:22Z)
• FFT-based Dynamic Subspace Selection for Low-Rank Adaptive Optimization of Large Language Models [49.397861654088636]
  We propose a two-step procedure to approximate SVD/QR-based gradient projections into lower-dimensional spaces.<n>We show that our strategy achieves faster runtime and reduced memory usage by up to $25%$ across different model sizes.
  arXiv  Detail & Related papers  (2025-05-23T14:37:00Z)
• Improving Adaptive Moment Optimization via Preconditioner Diagonalization [11.01832755213396]
  We show that our approach can substantially enhance the convergence speed of modern adaptives.<n>For large language models like LLaMA, we can achieve a speedup of 2x compared to the baseline Adam.
  arXiv  Detail & Related papers  (2025-02-11T11:48:04Z)
• Neural Gradient Learning and Optimization for Oriented Point Normal Estimation [53.611206368815125]
  We propose a deep learning approach to learn gradient vectors with consistent orientation from 3D point clouds for normal estimation. We learn an angular distance field based on local plane geometry to refine the coarse gradient vectors. Our method efficiently conducts global gradient approximation while achieving better accuracy and ability generalization of local feature description.
  arXiv  Detail & Related papers  (2023-09-17T08:35:11Z)
• ELRA: Exponential learning rate adaption gradient descent optimization method [83.88591755871734]
  We present a novel, fast (exponential rate), ab initio (hyper-free) gradient based adaption. The main idea of the method is to adapt the $alpha by situational awareness. It can be applied to problems of any dimensions n and scales only linearly.
  arXiv  Detail & Related papers  (2023-09-12T14:36:13Z)
• Global optimization of MPS in quantum-inspired numerical analysis [0.0]
  The study focuses on the search for the lowest eigenstates of a Hamiltonian equation. Five algorithms are introduced: imaginary-time evolution, steepest gradient descent, an improved descent, an implicitly restarted Arnoldi method, and density matrix renormalization group (DMRG) optimization.
  arXiv  Detail & Related papers  (2023-03-16T16:03:51Z)
• Optimization using Parallel Gradient Evaluations on Multiple Parameters [51.64614793990665]
  We propose a first-order method for convex optimization, where gradients from multiple parameters can be used during each step of gradient descent. Our method uses gradients from multiple parameters in synergy to update these parameters together towards the optima.
  arXiv  Detail & Related papers  (2023-02-06T23:39:13Z)
• Leveraging Non-uniformity in First-order Non-convex Optimization [93.6817946818977]
  Non-uniform refinement of objective functions leads to emphNon-uniform Smoothness (NS) and emphNon-uniform Lojasiewicz inequality (NL) New definitions inspire new geometry-aware first-order methods that converge to global optimality faster than the classical $Omega (1/t2)$ lower bounds.
  arXiv  Detail & Related papers  (2021-05-13T04:23:07Z)
• Channel-Directed Gradients for Optimization of Convolutional Neural Networks [50.34913837546743]
  We introduce optimization methods for convolutional neural networks that can be used to improve existing gradient-based optimization in terms of generalization error. We show that defining the gradients along the output channel direction leads to a performance boost, while other directions can be detrimental.
  arXiv  Detail & Related papers  (2020-08-25T00:44:09Z)

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.