  This paper advances the theory and practice of Domain Generalization (DG) in
machine learning. We consider the typical DG setting where the hypothesis is
composed of a representation mapping followed by a labeling function. Within
this setting, the majority of popular DG methods aim to jointly learn the
representation and the labeling functions by minimizing a well-known upper
bound for the classification risk in the unseen domain. In practice, however,
methods based on this theoretical upper bound ignore a term that cannot be
directly optimized due to its dual dependence on both the representation
mapping and the unknown optimal labeling function in the unseen domain. To
bridge this gap between theory and practice, we introduce a new upper bound
that is free of terms having such dual dependence, resulting in a fully
optimizable risk upper bound for the unseen domain. Our derivation leverages
classical and recent transport inequalities that link optimal transport metrics
with information-theoretic measures. Compared to previous bounds, our bound
introduces two new terms: (i) the Wasserstein-2 barycenter term that aligns
distributions between domains, and (ii) the reconstruction loss term that
assesses the quality of representation in reconstructing the original data.
Based on this new upper bound, we propose a novel DG algorithm named
Wasserstein Barycenter Auto-Encoder (WBAE) that simultaneously minimizes the
classification loss, the barycenter loss, and the reconstruction loss.
Numerical results demonstrate that the proposed method outperforms current
state-of-the-art DG algorithms on several datasets.
