Safe Distributionally Robust Feature Selection under Covariate Shift

Feature selection is fundamental to robust data-centric AI, but most existing methods optimize predictive performance under a single data distribution. This often selects spurious features that fail under distribution shifts. Motivated by principles from causal invariance, we study feature selection from a stability perspective and introduce Causally-Guided Diffusion for Stable Feature Selection (CGDFS). In CGDFS, we formalized feature selection as approximate posterior inference over feature subsets, whose posterior mass favors low prediction error and low cross-environment variance. Our framework combines three key insights: First, we formulate feature selection as stability-aware posterior sampling. Here, causal invariance serves as a soft inductive bias rather than explicit causal discovery. Second, we train a diffusion model as a learned prior over plausible continuous selection masks, combined with a stability-aware likelihood that rewards invariance across environments. This diffusion prior captures structural dependencies among features and enables scalable exploration of the combinatorially large selection space. Third, we perform guided annealed Langevin sampling that combines the diffusion prior with the stability objective, which yields a tractable, uncertainty-aware posterior inference that avoids discrete optimization and produces robust feature selections. We evaluate CGDFS on open-source real-world datasets exhibiting distribution shifts. Across both classification and regression tasks, CGDFS consistently selects more stable and transferable feature subsets, which leads to improved out-of-distribution performance and greater selection robustness compared to sparsity-based, tree-based, and stability-selection baselines.

Distributional shifts arise when the target deployment environment differs from the source environment that generated the training data. Robust learning frameworks such as Distributionally Robust Optimization (DRO) and Robust Satisficing (RS) aim to address this challenge, yet their finite-sample guarantees under such shifts, and their systematic comparison, remain underexplored: existing analyses typically establish guarantees either in the source environment or for adversarial worst-case performance over an ambiguity set. This paper instead studies generalization error in the target environment---the excess loss under the shifted target distribution. Our contributions are threefold. First, we derive finite-sample generalization error bounds in the shifted target environment for both DRO and RS. These bounds explicitly characterize the trade-off between reduced sensitivity to shift and the regularization penalty induced by each method's robustness hyperparameter, and they avoid the curse of dimensionality associated with Wasserstein empirical concentration. Second, when partial shift information such as shift magnitude or direction is available, we propose information-directed hyperparameter calibrations and compare the two methods given the same information. Under these calibrations, and in the partial-information regimes we study, DRO and RS exhibit complementary theoretical and empirical behavior. Finally, we apply the framework to a network lot-sizing problem, using it to interpret how robust policies respond to positive shifts in the demand distribution. Together, these results fill a gap in understanding the statistical properties of robust learning methods under distributional shifts and provide a principled basis for comparing DRO and RS.

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