This paper develops fundamental limits of deep neural network learning by characterizing what is possible if no constraints are imposed on the learning algorithm and on the amount of training data. Concretely, we consider Kolmogorov-optimal approximation through deep neural networks with the guiding theme being a relation between the complexity of the function (class) to be approximated and the complexity of the approximating network in terms of connectivity and memory requirements for storing the network topology and the associated quantized weights. The theory we develop establishes that deep networks are Kolmogorov-optimal approximants for markedly different function classes, such as unit balls in Besov spaces and modulation spaces. In addition, deep networks provide exponential approximation accuracy-i.e., the approximation error decays exponentially in the number of nonzero weights in the network-of the multiplication operation, polynomials, sinusoidal functions, and certain smooth functions. Moreover, this holds true even for one-dimensional oscillatory textures and the Weierstrass function-a fractal function, neither of which has previously known methods achieving exponential approximation accuracy. We also show that in the approximation of sufficiently smooth functions finite-width deep networks require strictly smaller connectivity than finite-depth wide networks.

中文翻译：

---

深度神经网络逼近理论


本文通过描述在不限制学习算法和训练数据量的情况下可能发生的情况，提出了深度神经网络学习的基本限制。具体来说，我们考虑通过深度神经网络进行柯尔莫哥洛夫最优逼近，其指导主题是要逼近的函数（类）的复杂性与逼近网络的复杂性之间的关系（在连接性和存储网络拓扑的内存要求方面）以及相关的量化权重。我们开发的理论表明，深度网络是针对明显不同的函数类别（例如贝索夫空间和调制空间中的单位球）的柯尔莫哥洛夫最优近似。此外，深度网络提供乘法运算、多项式、正弦函数和某些平滑函数的指数逼近精度，即逼近误差随网络中非零权重的数量呈指数衰减。此外，即使对于一维振荡纹理和 Weierstrass 函数（分形函数）也是如此，这两种函数以前都没有实现指数逼近精度的已知方法。我们还表明，在足够平滑函数的近似中，有限宽度深度网络需要比有限深度宽网络严格更小的连接性。