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

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.