Consider the problem of learning a large number of response functions simultaneously based on the same input variables. The training data consist of a single independent random sample of the input variables drawn from a common distribution together with the associated responses. The input variables are mapped into a high-dimensional linear space, called the feature space, and the response functions are modelled as linear functionals of the mapped features, with coefficients calibrated via ordinary least squares. We provide convergence guarantees on the worst-case excess prediction risk by controlling the convergence rate of the excess risk uniformly in the response function. The dimension of the feature map is allowed to tend to infinity with the sample size. The collection of response functions, although potentially infinite, is supposed to have a finite Vapnik-Chervonenkis dimension. The bound derived can be applied when building multiple surrogate models in a reasonable computing time.
翻译:培训数据包括从共同分布中抽取的单个输入变量的随机样本,这些输入变量被映射成一个高维线性空间,称为特征空间,反应功能以绘图特征的线性功能为模型,通过普通最小方位校准系数。我们通过统一控制响应功能中超重风险的趋同率,为最坏的超重预测风险提供了趋同保证。功能图的尺寸允许与样本大小不尽相同。收集反应功能虽然可能无限,但被认为具有有限的Vapnik-Chervonenkis尺寸。在合理计算时间内建立多个替代模型时,可适用所得出的约束值。