VC dimension of neural networks
@inproceedings{Sontag1998VCDO, title={VC dimension of neural networks}, author={Eduardo Sontag}, year={1998} }
This chapter presents a brief introduction to Vapnik-Chervonenkis (VC) dimension, a quantity which characterizes the difficulty of distribution-independent learning. The chapter establishes various elementary results, and discusses how to estimate the VC dimension in several examples of interest in neural network theory.
139 Citations
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References
SHOWING 1-10 OF 58 REFERENCES
Neural Networks with Quadratic VC Dimension
- Computer ScienceJ. Comput. Syst. Sci.
- 1997
It is shown that neural networks which use continuous activation functions have VC dimension at least as large as the square of the number of weightsw, which settles a long-standing open question.
Neural Networks with Quadratic VCDimension 1
- Computer Science
- 1995
This paper shows that neural networks which use continuous activation functions have VC dimension at least as large as the square of the number of weights w. This result settles a long-standing open…
Perspectives of Current Research about the Complexity of Learning on Neural Nets
- Computer Science, Biology
- 1994
This chapter discusses within the framework of computational learning theory the current state of knowledge and some open problems in three areas of research about learning on feedforward neural…
Polynomial bounds for VC dimension of sigmoidal neural networks
- Computer ScienceSTOC '95
- 1995
A new method is introduced for proving explicit upper bounds on the VC Dimension of general functional basis networks, and theVC Dimension of analog neural networks with the sigmoid activation function o(y) = 1/1 + e-y to be bounded by a quadratic polynomial in the number of programmable parameters.
Sample complexity for learning recurrent perceptron mappings
- Computer ScienceIEEE Trans. Inf. Theory
- 1996
This paper provides tight bounds on sample complexity associated to the fitting of recurrent perceptron classifiers to experimental data.
A Theory of Learning and Generalization: With Applications to Neural Networks and Control Systems
- Computer Science
- 1997
This paper presents Vapnik-Chervonenkis and Pollard (Pseudo-) Dimensions, a model of learning based on uniform Convergence of Empirical Means, and applications to Neural Networks and Control Systems, and some Open Problems.
Polynomial Bounds for VC Dimension of Sigmoidal and General Pfaffian Neural Networks
- Computer Science, MathematicsJ. Comput. Syst. Sci.
- 1997
We introduce a new method for proving explicit upper bounds on the VC dimension of general functional basis networks and prove as an application, for the first time, that the VC dimension of analog…
Feedforward Nets for Interpolation and Classification
- Computer ScienceJ. Comput. Syst. Sci.
- 1992
Vapnik-Chervonenkis Dimension of Recurrent Neural Networks
- Computer ScienceDiscret. Appl. Math.
- 1998
Recurrent Neural Networks: Some Systems-Theoretic Aspects
- Computer Science
- 1997
This paper provides an exposition of some recent research regarding system-theoretic aspects of continuous-time recurrent (dynamic) neural networks with sigmoidal activation functions. The class of…