标题： 关于最小范数插值器的鲁棒性 显示英文标题

标题： On the robustness of minimum-norm interpolators

摘要： 本文开发了一种一般理论，用于在存在加性且可能对抗性误差的线性模型中的最小范数插值估计量。 特别是，对误差没有任何条件限制。 给出了预测误差的定量界，将其与协变量的Rademacher复杂度、误差的最小范数插值器的范数以及真实参数周围次微分的形状相关联。 一般理论通过几个例子进行说明：具有最小$\ell_1$-范数或组Lasso惩罚插值的稀疏线性模型，具有核范数最小化的低秩迹回归模型，以及线性模型中的最小欧几里得范数插值。 在稀疏性或低秩诱导范数的情况下，只要过参数化至少比样本数量大一个对数因子，最小范数插值将产生与平均噪声水平同阶的预测误差。 显示结果接近最优性的下界补充了分析。 显示英文摘要

摘要： This article develops a general theory for minimum-norm interpolated estimators in linear models in the presence of additive, potentially adversarial, errors. In particular, no conditions on the errors are imposed. A quantitative bound for the prediction error is given, relating it to the Rademacher complexity of the covariates, the norm of the minimum norm interpolator of the errors and the shape of the subdifferential around the true parameter. The general theory is illustrated with several examples: the sparse linear model with minimum $\ell_1$-norm or group Lasso penalty interpolation, the low rank trace regression model with nuclear norm minimization, and minimum Euclidean norm interpolation in the linear model. In case of sparsity or low-rank inducing norms, minimum norm interpolation yields a prediction error of the order of the average noise level, provided that the overparameterization is at least a logarithmic factor larger than the number of samples. Lower bounds that show near optimality of the results complement the analysis.