Leave-One-Out Neighborhood Smoothing for Graphons: Berry-Esseen Bounds, Confidence Intervals, and Honest Tuning

TL;DR

This paper introduces a leave-one-out neighborhood smoothing method for graphon models that enables valid statistical inference, including confidence intervals and honest tuning, while maintaining optimal estimation rates.

Contribution

It proposes a novel leave-one-out modification to neighborhood smoothing that decouples dependencies, allowing for classical inference tools in graphon estimation.

Findings

Restores conditional independence for inference.

Derives Berry-Esseen bounds with explicit rates.

Provides finite-sample confidence intervals and honest tuning.

Abstract

Neighborhood smoothing methods achieve minimax-optimal rates for estimating edge probabilities under graphon models, but their use for statistical inference has remained limited. The main obstacle is that classical neighborhood smoothers select data-driven neighborhoods and average edges using the same adjacency matrix, inducing complex dependencies that invalidate standard concentration and normal approximation arguments. We introduce a leave-one-out modification of neighborhood smoothing for undirected simple graphs. When estimating a single entry P_ij, the neighborhood of node i is constructed from an adjacency matrix in which the jth row and column are set to zero, thereby decoupling neighborhood selection from the edges being averaged. We show that this construction restores conditional independence of the centered summands, enabling the use of classical probabilistic tools for inference. Under piecewise Lipschitz graphon assumptions and logarithmic degree growth, we derive variance-adaptive concentration inequalities based on Bousquet's inequality and establish Berry-Esseen bounds with explicit rates for the normalized estimation error. These results yield both finite-sample and asymptotic confidence intervals for individual edge probabilities. The same leave-one-out structure also supports an honest cross-validation scheme for tuning parameter selection, for which we prove an oracle inequality. The proposed estimator retains the optimal row-wise mean-squared error rates of classical neighborhood smoothing while providing valid entrywise uncertainty quantification.