We study the learning properties of nonparametric ridge-less least squares. In particular, we consider the common case of estimators defined by scale dependent kernels, and focus on the role of the scale. These estimators interpolate the data and the scale can be shown to control their stability through the condition number. Our analysis shows that are different regimes depending on the interplay between the sample size, its dimensions, and the smoothness of the problem. Indeed, when the sample size is less than exponential in the data dimension, then the scale can be chosen so that the learning error decreases. As the sample size becomes larger, the overall error stop decreasing but interestingly the scale can be chosen in such a way that the variance due to noise remains bounded. Our analysis combines, probabilistic results with a number of analytic techniques from interpolation theory.

翻译：我们研究的是非参数脊脊最少的平方的学习特性。 特别是, 我们考虑由大小依附内核定义的测算员的常见情况, 并关注比例的作用。 这些测算员对数据和比例进行相互推导, 可以通过条件编号来控制其稳定性。 我们的分析显示, 不同的制度取决于样本大小、 其尺寸和问题的平滑性之间的相互作用。 事实上, 当样本大小在数据维度中小于指数时, 比例可以选择, 从而降低学习错误。 随着样本大小的增大, 总体误差会停止减少, 但有趣的是, 比例可以选择, 从而让噪音造成的差异被捆绑起来。 我们的分析将概率结果与来自内推理论的一些分析技术结合起来。