Operator-Based Generalization Bounds for Multitask Deep Learning

TMLR Paper8863 Authors

Abstract: We study generalization bounds for compositions of functions through an operator-theoretic (Koopman) framework. Existing analyses in this direction are primarily restricted to scalar-valued settings and to Sobolev-type reproducing kernel Hilbert spaces (RKHSs), where the resulting bounds depend on smoothness parameters. We extend this framework to vector-valued RKHSs, enabling the analysis of multi-output function classes and making explicit how task-coupling kernels enter the resulting Rademacher complexity bounds. Within this setting, we derive bounds that depend on operator norms, singular values, and determinant-based geometric quantities associated with the underlying linear maps. We further introduce a vector-valued Brownian RKHS formulation, which replaces Sobolev smoothness assumptions by a first-order Cameron--Martin-type structure. In this regime, the resulting bounds no longer depend on Sobolev smoothness exponents and instead exhibit a milder spectral dependence involving only operator norms and determinant factors. This highlights a qualitative difference between Sobolev- and Brownian-based analyses at the level of function spaces. We additionally study a shared operator-learning formulation for multitask transfer in vector-valued RKHSs deriving an exact representer theorem, a finite-dimensional reduction of the corresponding operator-learning problem, and transfer bounds for the induced operator class. We illustrate these effects empirically on synthetic data and MNIST, comparing the behavior of Sobolev and Brownian bounds during training.

Submission Type: Regular submission (no more than 12 pages of main content)

Assigned Action Editor: ~changjian_shui1

Submission Number: 8863