Trace-Norm Geometry of Composed Multitask Predictors

Abstract

Multitask predictors apply task-specific functions to a shared representation. Penalizing the representation and task functions separately can assign different costs to equivalent predictors and charge for directions the tasks do not use. We study the trace norm of the complete task predictors in a fixed input reproducing kernel Hilbert space, weighted by task sample sizes. For a finite-dimensional intermediate function space, this norm is the infimum of the product of the representation operator norm and task trace norm over invertible coordinate changes. Task inner products determine this norm; standard duality gives the exact empirical Rademacher complexity of its norm ball. We estimate the feature Gram matrix used in fitting. Under bounded leverage, one estimate controls the penalty for every task head and shared linear combination of fixed features, even when the Gram is singular and the coefficients are chosen after estimation. This yields comparisons of penalized empirical objectives and a generalization bound under an estimated norm constraint, with the stated loss and sampling assumptions. Controlled experiments show that signal alignment can favor either penalty. A SARCOS comparison finds close agreement between exact and estimated penalties on overlapping training and evaluation observations.

arXiv commentSubstantially revised and expanded. Added uniform guarantees for estimated feature Gram matrices, regularized fitting and generalization results within a fixed feature span, and additional experiments. Clarified assumptions and experimental protocols. Previously titled Sample-Weighted End-to-End Trace-Norm Geometry for Multitask Learning