Deep Neural Network Approximation Theory

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...

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Veröffentlicht in:IEEE transactions on information theory Jg. 67; H. 5; S. 2581 - 2623
Hauptverfasser: Elbrachter, Dennis, Perekrestenko, Dmytro, Grohs, Philipp, Bolcskei, Helmut
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York IEEE 01.05.2021
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Schlagworte:
Algorithms
Approximation
Approximation error
Artificial neural networks
Complexity
Complexity theory
Dictionaries
Function approximation
Function space
Functions (mathematics)
Machine learning
Mathematical analysis
metric entropy
Multiplication
Network topologies
Neural networks
nonlinear approximation
Polynomials
rate-distortion theory
Training data
Two dimensional displays
ISSN:0018-9448, 1557-9654
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Zusammenfassung: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.
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ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2021.3062161