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Adaptive Learning of Density Ratios in RKHS

Werner Zellinger, Stefan Kindermann, Sergei V. Pereverzyev; 24(395):1−28, 2023.


Estimating the ratio of two probability densities from finitely many observations of the densities is a central problem in machine learning and statistics with applications in two-sample testing, divergence estimation, generative modeling, covariate shift adaptation, conditional density estimation, and novelty detection. In this work, we analyze a large class of density ratio estimation methods that minimize a regularized Bregman divergence between the true density ratio and a model in a reproducing kernel Hilbert space (RKHS). We derive new finite-sample error bounds, and we propose a Lepskii type parameter choice principle that minimizes the bounds without knowledge of the regularity of the density ratio. In the special case of square loss, our method adaptively achieves a minimax optimal error rate. A numerical illustration is provided.

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