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Domain adaptation arises as an important problem in statistical learningtheory when the data-generating processes differ between training and testsamples, respectively called source and target domains. Recent theoreticaladvances show that the success of domain adaptation algorithms heavily relieson their ability to minimize the divergence between the probabilitydistributions of the source and target domains. However, minimizing thisdivergence cannot be done independently of the minimization of other keyingredients such as the source risk or the combined error of the ideal jointhypothesis. The trade-off between these terms is often ensured by algorithmicsolutions that remain implicit and not directly reflected by the theoreticalguarantees. To get to the bottom of this issue, we propose in this paper a newtheoretical framework for domain adaptation through hierarchical optimaltransport. This framework provides more explicit generalization bounds andallows us to consider the natural hierarchical organization of samples in bothdomains into classes or clusters. Additionally, we provide a new divergencemeasure between the source and target domains called Hierarchical Wassersteindistance that indicates under mild assumptions, which structures have to bealigned to lead to a successful adaptation.

Theoretical Guarantees for Domain Adaptation with Hierarchical Optimal Transport