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\title{Domain Adaptation with Cauchy-Schwarz Divergence}

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\author[1]{Wenzhe~Yin}
\author[2,4]{Shujian~Yu\thanks{Corresponding author (yusj9011@gmail.com).}}
\author[2]{Yicong~Lin}
\author[1]{Jie~Liu}
\author[3]{Jan-Jakob~Sonke}
\author[1]{Efstratios~Gavves}
% % Add affiliations after the authors
\affil[1]{%
    %Institute for Informatics\\
    University of Amsterdam\\
    Amsterdam, The Netherlands
}

\affil[2]{%
    Vrije Universiteit Amsterdam\\
     Amsterdam, The Netherlands
}
\affil[3]{%
    Netherlands Cancer Institute\\
    Amsterdam, The Netherlands
  }
\affil[4]{%
    UiT - The Arctic University of Norway\\
    Troms\o, Norway
  }
  
  \begin{document}
\maketitle

\begin{abstract}
Domain adaptation aims to use training data from one or multiple source domains to learn a hypothesis that can be generalized to a different, but related, target domain. As such, having a reliable measure for evaluating the discrepancy of both marginal and conditional distributions is crucial. 
We introduce Cauchy-Schwarz (CS) divergence to the problem of unsupervised domain adaptation (UDA). The CS divergence offers a theoretically tighter generalization error bound {than the popular Kullback-Leibler divergence}. This holds for the general case of supervised learning, including multi-class classification and regression. Furthermore, we illustrate that the CS divergence enables a simple estimator on the discrepancy of both marginal and conditional distributions between source and target domains in the representation space, without requiring any distributional assumptions. 
We provide multiple examples to illustrate how the CS divergence can be conveniently used in both distance metric- or adversarial training-based UDA frameworks, resulting in compelling performance. The code of our paper is available at \url{https://github.com/ywzcode/CS-adv}.

\end{abstract}

\section{Introduction}
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\section{Related Work}
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\section{Method}
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\section{Experiments}
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\section{Conclusion}
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We introduce CS divergence to the problem of UDA, leading to an elegant estimation of the mismatch for both marginal (i.e., $D(p_s(\mathbf{z});p_t(\mathbf{z}))$) and conditional distributions (i.e., $D(p_s(y|\mathbf{z});p_t(y|\mathbf{z}))$). 
Compared to the MMD, it is more powerful and computationally efficient to distinguish two conditional distributions. Compared to the KL divergence, it is more stable and ensures a tighter generalization error bound. Integrating these favorable properties into a bi-classifier adversarial training framework, our method achieves SOTA performance in three UDA datasets. 

Finally, our result in Eq.~(\ref{eq:general_bound}), which combines CS divergence with the fundamental Pinsker's inequality, holds the potential to tighten bounds in various other applications. Further exploration of these possibilities is left for future work.

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\title{Domain Adaptation with Cauchy-Schwarz Divergence: \\
Supplementary Materials}

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\maketitle


\appendix

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\end{document}
