Deep Neural Network Approximation Theory
Deep Neural Network Approximation Theory
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DOI：
10.1109/tit.2021.3062161
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发表时间：
2021-05-01
影响因子：
2.5
通讯作者：
Boelcskei, Helmut
中科院分区：
文献类型：
--
作者：
Elbrachter, Dennis;Perekrestenko, Dmytro;Boelcskei, Helmut
本文通过描述如果没有对学习算法和训练数据量施加约束，可能发生的事情来开发深度神经网络学习的基本限制。具体地说，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，即存储网络拓扑和相关量化权重的连接性和内存要求。我们开发的理论建立了深度网络是明显不同的函数类的Kolmogorov最优逼近，例如Besov空间和调制空间中的单位球。此外，深度网络提供指数近似精度-即，近似误差在乘法运算、多项式、正弦函数和某些平滑函数的网络中的非零权重的数量上呈指数衰减。此外，这也适用于一维的振荡纹理和维尔斯特拉斯函数的分形函数，这两个以前已知的方法实现指数近似精度。我们还表明，在足够光滑的函数的近似，有限宽度的深度网络需要严格小于有限深度的宽网络的连接。
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.