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.

本文通过深度神经网络的 Kolmogorov 最优化来发展其基本极限，并阐述了深度网络对于不同函数类的 Kolmogorov 最优逼近性，其提供了指数级的逼近精度，并且在逼近足够光滑的函数时，相较于有限宽深网络，有限宽深层网络需要更小的连通性。