# Operator-Based Generalization Bound for Deep Learning: Insights on Multi-Task Learning

arXiv: https://arxiv.org/abs/2512.19184 · HTML: https://mlsanity.com/paper/2512.19184/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 40 references that are in the map.
- First posted on arXiv 22 Dec 2025.

## Builds on (most cited first, 25 of 25)

- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Spectrally-normalized margin bounds for neural networks](https://mlsanity.com/paper/1706.08498/) (2017)
- [Benign Overfitting in Linear Regression](https://mlsanity.com/paper/1906.11300/) (2019)
- [Multi-Task Learning with Deep Neural Networks: A Survey](https://mlsanity.com/paper/2009.09796/) (2020)
- [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)
- [Sketching as a Tool for Numerical Linear Algebra](https://mlsanity.com/paper/1411.4357/) (2014)
- [Generalization Properties of Learning with Random Features](https://mlsanity.com/paper/1602.04474/) (2016)
- [The Promises and Pitfalls of Deep Kernel Learning](https://mlsanity.com/paper/2102.12108/) (2021)
- [Deep Neural Tangent Kernel and Laplace Kernel Have the Same RKHS](https://mlsanity.com/paper/2009.10683/) (2020)
- [Convolutional Kernel Networks](https://mlsanity.com/paper/1406.3332/) (2014)
- [Compression based bound for non-compressed network: unified generalization error analysis of large compressible deep neural network](https://mlsanity.com/paper/1909.11274/) (2019)
- [Robust Fine-Tuning of Deep Neural Networks with Hessian-based Generalization Guarantees](https://mlsanity.com/paper/2206.02659/) (2022)
- [Randomized sketches for kernels: Fast and optimal non-parametric regression](https://mlsanity.com/paper/1501.06195/) (2015)
- [A representer theorem for deep kernel learning](https://mlsanity.com/paper/1709.10441/) (2017)
- [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)
- [Scalable Matrix-valued Kernel Learning for High-dimensional Nonlinear Multivariate Regression and Granger Causality](https://mlsanity.com/paper/1408.2066/) (2014)
- [Autoencoding any Data through Kernel Autoencoders](https://mlsanity.com/paper/1805.11028/) (2018)
- [Koopman-based generalization bound: New aspect for full-rank weights](https://mlsanity.com/paper/2302.05825/) (2023)
- [Entangled Kernels -- Beyond Separability](https://mlsanity.com/paper/2101.05514/) (2021)
- [Deep Learning with Kernels through RKHM and the Perron-Frobenius Operator](https://mlsanity.com/paper/2305.13588/) (2023)
- [Fast Kernel Methods for Generic Lipschitz Losses via p-Sparsified Sketches](https://mlsanity.com/paper/2206.03827/) (2022)

## 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)

This paper presents novel generalization bounds for vector-valued neural networks and deep kernel methods, focusing on multi-task learning through an operator-theoretic framework. Our key development lies in strategically combining a Koopman based approach with existing techniques, achieving tighter generalization guarantees compared to traditional norm-based bounds. To mitigate computational challenges associated with Koopman-based methods, we introduce sketching techniques applicable to vector valued neural networks. These techniques yield excess risk bounds under generic Lipschitz losses, providing performance guarantees for applications including robust and multiple quantile regression. Furthermore, we propose a novel deep learning framework, deep vector-valued reproducing kernel Hilbert spaces (vvRKHS), leveraging Perron Frobenius (PF) operators to enhance deep kernel methods. We derive a new Rademacher generalization bound for this framework, explicitly addressing underfitting and overfitting through kernel refinement strategies. This work offers novel insights into the generalization properties of multitask learning with deep learning architectures, an area that has been relatively unexplored until recent developments.