On a synergistic learning phenomenon in nonparametric domain adaptation

Abstract: Consider nonparametric domain adaptation for regression, which assumes the same conditional distribution of the response given the covariates but different marginal distributions of the covariates. An important goal is to understand how the source data may improve the minimax convergence rate of learning the regression function when the likelihood ratio of the covariate marginal distributions of the target data and the source data are unbounded. A previous work of Pathak et al. (2022) show that the minimax transfer learning rate is simply determined by the faster rate of using either the source or the target data alone. In this paper, we present a new synergistic learning phenomenon (SLP) that the minimax convergence rate based on both data may sometimes be faster (even much faster) than the better rate of convergence based on the source or target data only. The SLP occurs when and only when the target sample size is smaller (in order) than but not too much smaller than the source sample size in relation to the smoothness of the regression function and the nature of the covariate densities of the source and target distributions. Interestingly, the SLP happens in two different ways according to the relationship between the two sample sizes. One is that the target data help alleviate the difficulty in estimating the regression function at points where the density of the source data is close to zero and the other is that the source data (with its larger sample size than that of the target data) help the estimation at points where the density of the source data is not small. Extensions to handle unknown source and target parameters and smoothness of the regression function are also obtained.