# On the Koopman-Based Generalization Bounds for Multi-Task Deep Learning

arXiv: https://arxiv.org/abs/2512.19199 · HTML: https://mlsanity.com/paper/2512.19199/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 32 references that are in the map.
- First posted on arXiv 22 Dec 2025.

## Builds on (most cited first, 17 of 17)

- [Spectrally-normalized margin bounds for neural networks](https://mlsanity.com/paper/1706.08498/) (2017)
- [Norm-Based Capacity Control in Neural Networks](https://mlsanity.com/paper/1503.00036/) (2015)
- [Size-Independent Sample Complexity of Neural Networks](https://mlsanity.com/paper/1712.06541/) (2017)
- [Sensitivity and Generalization in Neural Networks: an Empirical Study](https://mlsanity.com/paper/1802.08760/) (2018)
- [Orthogonal Deep Neural Networks](https://mlsanity.com/paper/1905.05929/) (2019)
- [Data-dependent Sample Complexity of Deep Neural Networks via Lipschitz Augmentation](https://mlsanity.com/paper/1905.03684/) (2019)
- [Improved Sample Complexities for Deep Networks and Robust Classification via an All-Layer Margin](https://mlsanity.com/paper/1910.04284/) (2019)
- [Robust Fine-Tuning of Deep Neural Networks with Hessian-based Generalization Guarantees](https://mlsanity.com/paper/2206.02659/) (2022)
- [Towards Optimal Sobolev Norm Rates for the Vector-Valued Regularized Least-Squares Algorithm](https://mlsanity.com/paper/2312.07186/) (2023)
- [Variation Spaces for Multi-Output Neural Networks: Insights on Multi-Task Learning and Network Compression](https://mlsanity.com/paper/2305.16534/) (2023)
- [Local Rademacher Complexity-based Learning Guarantees for Multi-Task Learning](https://mlsanity.com/paper/1602.05916/) (2016)
- [Provable Multi-Task Representation Learning by Two-Layer ReLU Neural Networks](https://mlsanity.com/paper/2307.06887/) (2023)
- [Koopman-based generalization bound: New aspect for full-rank weights](https://mlsanity.com/paper/2302.05825/) (2023)
- [Learning with Norm Constrained, Over-parameterized, Two-layer Neural Networks](https://mlsanity.com/paper/2404.18769/) (2024)
- [Fast Kernel Methods for Generic Lipschitz Losses via p-Sparsified Sketches](https://mlsanity.com/paper/2206.03827/) (2022)
- [Enhanced Feature Learning via Regularisation: Integrating Neural Networks and Kernel Methods](https://mlsanity.com/paper/2407.17280/) (2024)
- [Learning Analysis of Kernel Ridgeless Regression with Asymmetric Kernel Learning](https://mlsanity.com/paper/2406.01435/) (2024)

## Built on by (most cited first, 1 of 1)

- [Sample-Weighted End-to-End Trace-Norm Geometry for Multitask Learning](https://mlsanity.com/paper/2609.29520/) (2026)

## Abstract (arXiv)

The paper establishes generalization bounds for multitask deep neural networks using operator-theoretic techniques. The authors propose a tighter bound than those derived from conventional norm based methods by leveraging small condition numbers in the weight matrices and introducing a tailored Sobolev space as an expanded hypothesis space. This enhanced bound remains valid even in single output settings, outperforming existing Koopman based bounds. The resulting framework maintains key advantages such as flexibility and independence from network width, offering a more precise theoretical understanding of multitask deep learning in the context of kernel methods.