We study low-rank estimation of an unknown sparse graphon from sampled network data under operator-norm loss, motivated by targeted interventions in graphon games. Starting from the observed adjacency matrix, we construct low-rank surrogates by singular value thresholding and, for smooth graphons, by block averaging followed by thresholding. We obtain non-asymptotic bounds on both the operator-norm error and the rank of the resulting estimator for stochastic block model, Hölder, and analytic graphons, and we complement these results with minimax lower bounds showing that the rates are essentially sharp for these classes. Our analysis highlights that low rank is valuable here primarily for computation: while it does not improve the minimax operator-norm rate, it yields operator-norm accurate surrogates with substantially smaller rank. We then apply these estimators to linear-quadratic graphon games and derive non-asymptotic stability bounds showing that the welfare loss incurred by using an estimated graphon is controlled by the operator-norm perturbation. This yields near-optimal guarantees for targeted interventions computed from the estimated graphon, together with substantial computational savings. For zero baseline heterogeneity and under a spectral-gap condition, we also establish matching lower bounds for intervention regret. Numerical experiments illustrate the trade-off between statistical accuracy, retained rank, and runtime.

翻译：我们研究了基于算子范数损失从采样网络数据中估计未知稀疏图函数的低秩方法，其动机源于图博弈中的定向干预。从观测到的邻接矩阵出发，我们通过奇异值阈值化构造低秩替代，对于光滑图函数，则采用块平均后阈值化的方法。针对随机块模型、赫尔德连续图函数和解析图函数，我们获得了估计量的算子范数误差和秩的非渐近界，并辅以极小极大下界以表明这些类别的速率本质上是尖锐的。我们的分析强调，低秩在此主要对计算有价值：尽管它不改善极小极大算子范数速率，但能以显著更小的秩产生算子范数精确的替代。随后，我们将这些估计量应用于线性二次型图博弈，并推导非渐近稳定性界，表明使用估计图函数导致的福利损失由算子范数扰动控制。这为基于估计图函数计算的定向干预提供了近乎最优的保证，同时带来显著的计算节省。在零基线异质性和谱间隙条件下，我们还为干预遗憾建立了匹配的下界。数值实验展示了统计精度、保留秩与运行时间之间的权衡。