# Operator-Theoretic Generalization Bounds for Multitask Deep Learning

arXiv: https://arxiv.org/abs/2608.15982 · HTML: https://mlsanity.com/paper/2608.15982/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 27 references that are in the map.
- First posted on arXiv 17 Aug 2026.

## Builds on (most cited first, 15 of 15)

- [Exact solutions to the nonlinear dynamics of learning in deep linear neural networks](https://mlsanity.com/paper/1312.6120/) (2013)
- [Spectrally-normalized margin bounds for neural networks](https://mlsanity.com/paper/1706.08498/) (2017)
- [Stronger generalization bounds for deep nets via a compression approach](https://mlsanity.com/paper/1802.05296/) (2018)
- [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)
- [The Benefit of Multitask Representation Learning](https://mlsanity.com/paper/1505.06279/) (2015)
- [A vector-contraction inequality for Rademacher complexities](https://mlsanity.com/paper/1605.00251/) (2016)
- [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)
- [Local Rademacher Complexity-based Learning Guarantees for Multi-Task Learning](https://mlsanity.com/paper/1602.05916/) (2016)
- [Representer Theorem for Learning Koopman Operators](https://mlsanity.com/paper/2208.01681/) (2022)
- [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)

## 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)

We develop operator-theoretic generalization bounds for deep multi-output function classes by representing network layers as Koopman composition operators on vector-valued reproducing kernel Hilbert spaces. In vector-valued Sobolev RKHSs, we derive Rademacher complexity bounds for invertible and width-expanding injective architectures. The estimates separate the output-coupling contribution, represented by the trace of the task matrix, from the layerwise operator norms, Sobolev symbol ratios, determinant factors, and restriction constants generated by the linear maps. We then analyze a distinct one-dimensional Brownian/Cameron--Martin regime. Using the exact anchored derivative-norm characterization of the vector-valued Brownian RKHS, we obtain layerwise bounds for domain-preserving scalar linear maps and anchored diffeomorphic activations; the corresponding factors scale as |Wl|1/2 and |σl'|_∞1/2, respectively, and do not involve Sobolev smoothness exponents. Because the Sobolev and Brownian results concern different hypothesis spaces, neither is asserted to dominate the other uniformly. We additionally formulate shared operator learning across tasks, prove a finite-rank representer theorem, derive the exact finite-dimensional problem for squared loss, and establish a target-transfer bound when the learned operator is obtained independently of the target sample. Synthetic and MNIST studies examine stabilized Sobolev-inspired and Brownian-inspired complexity proxies; these empirical proxies are not evaluations of the proved bounds for rank-deficient architectures.