New Symmetrized Loss Functions Improve Neural Network Training with Noisy Labels
Researchers have published a theoretical analysis exploring whether intrinsic symmetries in training data generate conserved quantities during gradient-flow training of neural networks. The study proves that, for analytic non-polynomial loss functions, data symmetries generically do not produce additional integrals of motion, though exceptions exist for mean squared error loss under data augmentation. The findings introduce a framework based on 'tensorizable networks' to characterize when such conservation laws can arise.
A preprint posted to arXiv on June 9, 2026 investigates the relationship between data symmetries and conserved quantities in neural network training under gradient flow. The authors prove that when the loss function is analytic and non-polynomial, data symmetries do not generically induce extra integrals of motion—a result that constrains expectations about symmetry-driven conservation in most modern deep learning settings. However, for mean squared error loss, the paper identifies specific conditions under which data augmentation can yield additional conserved quantities. To systematically describe this phenomenon, the authors develop a framework centered on 'tensorizable networks,' a family of architectures in which parameter and input dependence can be separated via an intermediate representation. This family includes linear networks, polynomial networks, and Lightning Attention. The work contributes to the theoretical understanding of training dynamics and the role of symmetry in shaping the geometry of optimization.
What's missing
As a preprint, this work has not yet undergone peer review. The paper's scope is primarily theoretical; empirical validation of the conservation law framework on large-scale practical architectures beyond those explicitly analyzed (linear, polynomial networks, Lightning Attention) is not addressed. The generality of the 'tensorizable networks' framework to other modern architectures remains an open question.
What different sources said
- arXiv cs.LGCenter
Conservation Laws from Data Symmetry in Neural Networks
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