Support-Conditioned Risk Profiles for Learning under Nonuniform Design

Abstract

Prediction error is often summarized by aggregate validation or test metrics.
Under nonuniform design, such global averages can hide conditional failures in regions where the training inputs provide weak local support, and they can select hyperparameters that favor well-supported regions. This paper develops a post-training diagnostic framework for learning under heterogeneous empirical support. Given a fitted predictor, an empirical support field, and an independent test point, we define the support-conditioned risk profile as the conditional prediction error indexed by local support level. The profile asks where error occurs along the empirical support scale, rather than replacing the predictor or prescribing a universal support-dependent correction. We show that the binned profile is statistically estimable, with finite-sample concentration controlled by the effective bin size. We then prove that local support gaps impose an unavoidable error floor over H{\""o}lder smoothness classes: if a ball of radius $h$ around a target point contains no training sample, the local squared error has an order-$h^{2s}$ lower bound for (s)-smooth targets. A matching local polynomial upper bound identifies the corresponding rate $h^{2s}+\sigma^2/k$. For linear smoothers, including kernel ridge regression, we derive an exact support-conditioned bias--variance identity that explains how regularization reshapes the error profile across support levels. We also prove that support alone cannot determine the profile: identical support fields can produce opposite dense--weak error orderings when target variation changes location. These results yield a profile-aware diagnostic protocol for auditing, constrained hyperparameter re-ranking, and data-acquisition diagnosis. Experiments on synthetic and benchmark regression tasks support the estimability of the profile, validate the main support-gap and bias--variance mechanisms, and show that profile-aware decisions reveal trade-offs hidden by global-MSE validation.

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