In the past decade the mathematical theory of machine learning has lagged far behind the triumphs of deep neural networks on practical challenges. However, the gap between theory and practice is gradually starting to close. In this paper I will attempt to assemble some pieces of the remarkable and still incomplete mathematical mosaic emerging from the efforts to understand the foundations of deep learning. The two key themes will be interpolation and its sibling over-parametrization. Interpolation corresponds to fitting data, even noisy data, exactly. Over-parametrization enables interpolation and provides flexibility to select a suitable interpolating model.As we will see, just as a physical prism separates colours mixed within a ray of light, the figurative prism of interpolation helps to disentangle generalization and optimization properties within the complex picture of modern machine learning. This article is written in the belief and hope that clearer understanding of these issues will bring us a step closer towards a general theory of deep learning and machine learning.

中文翻译：

---

无所畏惧地适应：通过插值棱镜进行深度学习的非凡数学现象

在过去的十年中，机器学习的数学理论远远落后于深度神经网络在实际挑战中取得的胜利。然而，理论与实践之间的差距正在逐渐缩小。在本文中，我将尝试收集一些在理解深度学习基础的努力中出现的非凡且仍然不完整的数学拼图。两个关键主题将是插值及其兄弟过度参数化。插值恰好对应于拟合数据，甚至是噪声数据。过度参数化支持插值并提供选择合适插值模型的灵活性。正如我们将看到的，就像物理棱镜分离光线中混合的颜色一样，插值的比喻性棱镜有助于在现代机器学习的复杂画面中解开泛化和优化属性。写这篇文章的信念是，希望对这些问题的更清晰的理解能让我们更接近深度学习和机器学习的一般理论。