The recent advances in machine learning in various fields of applications can be largely attributed to the rise of deep learning (DL) methods and architectures. Despite being a key technology behind autonomous cars, image processing, speech recognition, etc., a notorious problem remains the lack of theoretical understanding of DL and related interpretability and (adversarial) robustness issues. Understanding the specifics of DL, as compared to, say, other forms of nonlinear regression methods or statistical learning, is interesting from a mathematical perspective, but at the same time it is of crucial importance in practice: treating neural networks as mere black boxes might be sufficient in certain cases, but many applications require waterproof performance guarantees and a deeper understanding of what could go wrong and why it could go wrong. It is probably fair to say that, despite being mathematically well founded as a method to approximate complicated functions, DL is mostly still more like modern alchemy that is firmly in the hands of engineers and computer scientists. Nevertheless, it is evident that certain specifics of DL that could explain its success in applications demands systematic mathematical approaches. In this work, we review robustness issues of DL and particularly bridge concerns and attempts from approximation theory to statistical learning theory. Further, we review Bayesian Deep Learning as a means for uncertainty quantification and rigorous explainability.

翻译：机器学习在各应用领域的近期进展主要归功于深度学习方法与架构的兴起。尽管深度学习是自动驾驶、图像处理、语音识别等关键技术的基石，但其理论基础薄弱以及相关的可解释性与（对抗）鲁棒性问题仍是众所周知的难题。从数学角度理解深度学习相较于其他非线性回归方法或统计学习的特性颇具趣味，同时这一理解在实践中至关重要：将神经网络视为纯粹的黑箱在某些情况下或许足够，但许多应用要求无懈可击的性能保障，并需深入洞察可能出错的原因及机制。公允而言，尽管深度学习作为逼近复杂函数的数学方法已有坚实基础，但它在很大程度上仍像是掌握在工程师和计算机科学家手中的现代炼金术。然而，显然深度学习在应用中取得成功的某些特定机制亟需系统性数学方法。本研究综述了深度学习的鲁棒性问题，尤其连接了从逼近理论到统计学习理论的相关关切与尝试。此外，我们评述了贝叶斯深度学习作为不确定性量化与严谨可解释性的一种手段。