Scalable and Adaptive Distribution Regression via Random Features

Distribution regression seeks to learn mappings from probability distributions to response variables and has emerged as a unifying framework for kernel methods, multi-instance learning, and set-valued learning. Despite its success, kernel-based distribution regression remains computationally expensive and sensitive to kernel choice, motivating scalable and adaptive representations. Building on random feature approximations, we develop a finite-sample analysis of Random Feature Distribution Regression (RaDR) under two-stage sampling. Our perturbation analysis separates the effects of distribution-level sampling, finite within-bag sampling, and random feature approximation. We establish sufficient feature-budget conditions under which RaDR preserves the order of our excess-risk upper bound for exact two-stage kernel distribution regression while maintaining computational efficiency. We further propose Adaptive Random Feature Distribution Regression (A-RaDR), a unified score-guided framework that adaptively allocates random features according to downstream learning objectives. Experiments on synthetic benchmarks and 41 real-world datasets demonstrate that RaDR delivers substantial computational savings while maintaining competitive predictive performance, whereas A-RaDR further improves learning accuracy in distribution regression while retaining the computational speed up. Together, these results establish a practical and theoretically grounded framework for adaptive, scalable distribution learning