Since statistical guarantees for neural networks are usually restricted to global optima of intricate objective functions, it is not clear whether these theories really explain the performances of actual outputs of neural-network pipelines. The goal of this paper is, therefore, to bring statistical theory closer to practice. We develop statistical guarantees for simple neural networks that coincide up to logarithmic factors with the global optima but apply to stationary points and the points nearby. These results support the common notion that neural networks do not necessarily need to be optimized globally from a mathematical perspective. More generally, despite being limited to simple neural networks for now, our theories make a step forward in describing the practical properties of neural networks in mathematical terms.

翻译：由于神经网络的统计保障通常局限于复杂客观功能的全球选择,因此不清楚这些理论是否真正解释了神经网络管道实际产出的绩效。因此,本文件的目标是使统计理论更接近实践。我们为简单的神经网络制定了统计保障,这些网络与全球选择的对数因素吻合,但适用于固定点和附近点。这些结果支持了一个共同概念,即神经网络不一定需要从数学角度优化全球。 更一般地说,尽管目前仅限于简单的神经网络,但我们的理论在用数学术语描述神经网络的实际性质方面迈出了一步。