Duality in RKHSs with Infinite Dimensional Outputs: Application to Robust Losses

TL;DR

This paper introduces a duality framework for operator-valued kernel methods with infinite-dimensional outputs, enabling robust loss functions like Huber and epsilon-insensitive, with theoretical and empirical robustness benefits.

Contribution

It develops a duality approach for OVK machines that supports a wide range of loss functions, extending beyond the traditional square norm loss.

Findings

The duality approach enables solving OVK problems with robust losses.

Theoretical stability analysis shows robustness benefits.

Empirical results demonstrate improved structured data performance.

Abstract

Operator-Valued Kernels (OVKs) and associated vector-valued Reproducing Kernel Hilbert Spaces provide an elegant way to extend scalar kernel methods when the output space is a Hilbert space. Although primarily used in finite dimension for problems like multi-task regression, the ability of this framework to deal with infinite dimensional output spaces unlocks many more applications, such as functional regression, structured output prediction, and structured data representation. However, these sophisticated schemes crucially rely on the kernel trick in the output space, so that most of previous works have focused on the square norm loss function, completely neglecting robustness issues that may arise in such surrogate problems. To overcome this limitation, this paper develops a duality approach that allows to solve OVK machines for a wide range of loss functions. The infinite dimensional Lagrange multipliers are handled through a Double Representer Theorem, and algorithms for $\epsilon$-insensitive losses and the Huber loss are thoroughly detailed. Robustness benefits are emphasized by a theoretical stability analysis, as well as empirical improvements on structured data applications.