Theoretical Guarantees for Domain Adaptation with Hierarchical Optimal Transport

TL;DR

This paper introduces a new theoretical framework for domain adaptation using hierarchical optimal transport, providing explicit bounds and a novel divergence measure called Hierarchical Wasserstein distance, which considers class or cluster structures.

Contribution

It proposes a hierarchical optimal transport framework that explicitly links divergence minimization with source risk, offering new theoretical insights and a novel divergence measure for domain adaptation.

Findings

Hierarchical Wasserstein distance effectively measures domain divergence.

The framework provides explicit generalization bounds.

Aligning hierarchical structures improves adaptation success.

Abstract

Domain adaptation arises as an important problem in statistical learning theory when the data-generating processes differ between training and test samples, respectively called source and target domains. Recent theoretical advances show that the success of domain adaptation algorithms heavily relies on their ability to minimize the divergence between the probability distributions of the source and target domains. However, minimizing this divergence cannot be done independently of the minimization of other key ingredients such as the source risk or the combined error of the ideal joint hypothesis. The trade-off between these terms is often ensured by algorithmic solutions that remain implicit and not directly reflected by the theoretical guarantees. To get to the bottom of this issue, we propose in this paper a new theoretical framework for domain adaptation through hierarchical optimal transport. This framework provides more explicit generalization bounds and allows us to consider the natural hierarchical organization of samples in both domains into classes or clusters. Additionally, we provide a new divergence measure between the source and target domains called Hierarchical Wasserstein distance that indicates under mild assumptions, which structures have to be aligned to lead to a successful adaptation.