# Sample-Weighted End-to-End Trace-Norm Geometry for Multitask Learning

arXiv: https://arxiv.org/abs/2609.29520 · HTML: https://mlsanity.com/paper/2609.29520/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 23 of its 42 references that are in the map.
- First posted on arXiv 25 Aug 2026.

## Builds on (most cited first, 23 of 23)

- [Spectrally-normalized margin bounds for neural networks](https://mlsanity.com/paper/1706.08498/) (2017)
- [Sharp Minima Can Generalize For Deep Nets](https://mlsanity.com/paper/1703.04933/) (2017)
- [Implicit Regularization in Deep Matrix Factorization](https://mlsanity.com/paper/1905.13655/) (2019)
- [Implicit Regularization in Matrix Factorization](https://mlsanity.com/paper/1705.09280/) (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)
- [Path-SGD: Path-Normalized Optimization in Deep Neural Networks](https://mlsanity.com/paper/1506.02617/) (2015)
- [How do infinite width bounded norm networks look in function space?](https://mlsanity.com/paper/1902.05040/) (2019)
- [On Linear Identifiability of Learned Representations](https://mlsanity.com/paper/2007.00810/) (2020)
- [Fisher-Rao Metric, Geometry, and Complexity of Neural Networks](https://mlsanity.com/paper/1711.01530/) (2017)
- [Learning Dynamical Systems via Koopman Operator Regression in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2205.14027/) (2022)
- [The Role of Linear Layers in Nonlinear Interpolating Networks](https://mlsanity.com/paper/2202.00856/) (2022)
- [On Dropout and Nuclear Norm Regularization](https://mlsanity.com/paper/1905.11887/) (2019)
- [Trace norm regularization for multi-task learning with scarce data](https://mlsanity.com/paper/2202.06742/) (2022)
- [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)
- [An Inequality with Applications to Structured Sparsity and Multitask Dictionary Learning](https://mlsanity.com/paper/1402.1864/) (2014)
- [A Rescaling-Invariant Lipschitz Bound Based on Path-Metrics for Modern ReLU Network Parameterizations](https://mlsanity.com/paper/2405.15006/) (2024)
- [Operator-Based Generalization Bound for Deep Learning: Insights on Multi-Task Learning](https://mlsanity.com/paper/2512.19184/) (2025)
- [On the Koopman-Based Generalization Bounds for Multi-Task Deep Learning](https://mlsanity.com/paper/2512.19199/) (2025)
- [Operator-Theoretic Generalization Bounds for Multitask Deep Learning](https://mlsanity.com/paper/2608.15982/) (2026)

## Abstract (arXiv)

Multitask models combine a shared representation with task-specific outputs, but generalization bounds often control the two components separately. Such products can discard relative orientation and cancellation and can change under equivalent transformations of intermediate coordinates even when the represented predictors are unchanged. We study instead the sample-size-weighted trace norm of the end-to-end map from task coefficients to input-space predictors. For its fixed-radius class, we derive the exact empirical Rademacher complexity. The same quantity is characterized by eliminating a positive-definite task covariance after the representation acts and, in finite-dimensional intermediate spaces, by optimizing the separated product over all equivalent invertible refactorizations. Explicit constructions show unbounded orientation and factorization gaps and an exponential depth gap for cancelling linear layers. As a geometric application, finite-to-one Lipschitz shared maps yield an exact Sobolev task Gram matrix determined by multiplicity and local directional distortion. We evaluate the corresponding convex regularizer in two protocol-locked unseen suites. Across 252 paired held-out comparisons, weighted joint nuclear regularization improves average population excess over unweighted nuclear regularization by 0.00764, with a stratified-bootstrap 95% interval [0.00465, 0.01110]. Correct task counts improve average and least-sampled-quartile excess over shifted counts by 0.01072 and 0.02847; all 15 imbalanced rank-suite cells are positive and the balanced effect is zero. Weighted joint nuclear also outperforms weighted Frobenius and independent ridge. The least-sampled-quartile comparison with unweighted nuclear remains unresolved, delimiting rather than contradicting the average advantage. All seven predeclared gates pass.