Posterior drift refers to changes in the relationship between responses and covariates while the distributions of the covariates remain unchanged. In this work, we explore functional linear regression under posterior drift with transfer learning. Specifically, we investigate when and how auxiliary data can be leveraged to improve the estimation accuracy of the slope function in the target model when posterior drift occurs. We employ the approximated least square method together with a lasso penalty to construct an estimator that transfers beneficial knowledge from source data. Theoretical analysis indicates that our method avoids negative transfer under posterior drift, even when the contrast between slope functions is quite large. Specifically, the estimator is shown to perform at least as well as the classical estimator using only target data, and it enhances the learning of the target model when the source and target models are sufficiently similar. Furthermore, to address scenarios where covariate distributions may change, we propose an adaptive algorithm using aggregation techniques. This algorithm is robust against non-informative source samples and effectively prevents negative transfer. Simulation and real data examples are provided to demonstrate the effectiveness of the proposed algorithm.

This paper focuses on the design of spatial experiments to optimize the amount of information derived from the experimental data and enhance the accuracy of the resulting causal effect estimator. We propose a surrogate function for the mean squared error (MSE) of the estimator, which facilitates the use of classical graph cut algorithms to learn the optimal design. Our proposal offers three key advances: (1) it accommodates moderate to large spatial interference effects; (2) it adapts to different spatial covariance functions; (3) it is computationally efficient. Theoretical results and numerical experiments based on synthetic environments and a dispatch simulator that models a city-scale ridesharing market, further validate the effectiveness of our design. A python implementation of our method is available at https://github.com/Mamba413/CausalGraphCut.

Algorithmic fairness has become a central topic in machine learning, and mitigating disparities across different subpopulations has emerged as a rapidly growing research area. In this paper, we systematically study the classification of functional data under fairness constraints, ensuring the disparity level of the classifier is controlled below a pre-specified threshold. We propose a unified framework for fairness-aware functional classification, tackling an infinite-dimensional functional space, addressing key challenges from the absence of density ratios and intractability of posterior probabilities, and discussing unique phenomena in functional classification. We further design a post-processing algorithm Fair Functional Linear Discriminant Analysis classifier (Fair-FLDA), which targets at homoscedastic Gaussian processes and achieves fairness via group-wise thresholding. Under weak structural assumptions on eigenspace, theoretical guarantees on fairness and excess risk controls are established. As a byproduct, our results cover the excess risk control of the standard FLDA as a special case, which, to the best of our knowledge, is first time seen. Our theoretical findings are complemented by extensive numerical experiments on synthetic and real datasets, highlighting the practicality of our designed algorithm.