标题： 最小范数插值器和正则化经验风险最小化器的鲁棒性 显示英文标题

标题： On the robustness of minimum norm interpolators and regularized empirical risk minimizers

摘要： 本文提出了一个一般理论，用于线性模型中最小范数插值估计量和正则化经验风险最小化器（RERM），在存在加性且可能对抗性误差的情况下。 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大小联系起来。 该一般理论针对高斯特征和几种范数进行了说明：$\ell_1$，$\ell_2$，组套索和核范数。 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，并且在RERM的情况下正则化参数足够小，最小范数插值器和RERM将产生平均噪声水平量级的预测误差。 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 interpolating estimators and regularized empirical risk minimizers (RERM) 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 size of the subdifferential around the true parameter. The general theory is illustrated for Gaussian features and several norms: The $\ell_1$, $\ell_2$, group Lasso and nuclear norms. In case of sparsity or low-rank inducing norms, minimum norm interpolators and RERM yield 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 and that, in case of RERM, the regularization parameter is small enough. Lower bounds that show near optimality of the results complement the analysis.