Siming Zheng, Guohao Shen, Yuanyuan Lin, Jian Huang; 27(15):1−60, 2026.
Abstract
We consider the problem of density-ratio estimation using Bregman Divergence with Deep ReLU feedforward neural networks (BDD). We establish non-asymptotic error bounds for BDD density-ratio estimators, which are minimax optimal up to a logarithmic factor when the data distribution has finite support. As an application of our theoretical findings, we propose an estimator for the KL-divergence that is asymptotically normal, leveraging our convergence results for the deep density-ratio estimator and a data-splitting method. We also extend our results to cases with unbounded support and unbounded density ratios. Furthermore, we show that the BDD density-ratio estimator can mitigate the curse of dimensionality when data distributions are supported on an approximately low-dimensional manifold. Our results are applied to investigate the convergence properties of the telescoping density-ratio estimator proposed by Rhodes (2020). We provide sufficient conditions under which it achieves a lower error bound than a single-ratio estimator. Moreover, we conduct simulation studies to validate our main theoretical results and assess the performance of the BDD density-ratio estimator.