Generalization bound of globally optimal non-convex neural network
training: Transportation map estimation by infinite dimensional Langevin
dynamics
- URL: http://arxiv.org/abs/2007.05824v2
- Date: Mon, 26 Oct 2020 18:10:22 GMT
- Title: Generalization bound of globally optimal non-convex neural network
training: Transportation map estimation by infinite dimensional Langevin
dynamics
- Authors: Taiji Suzuki
- Abstract summary: We introduce a new theoretical framework to analyze deep learning optimization with connection to its generalization error.
Existing frameworks such as mean field theory and neural tangent kernel theory for neural network optimization analysis typically require taking limit of infinite width of the network to show its global convergence.
- Score: 50.83356836818667
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: We introduce a new theoretical framework to analyze deep learning
optimization with connection to its generalization error. Existing frameworks
such as mean field theory and neural tangent kernel theory for neural network
optimization analysis typically require taking limit of infinite width of the
network to show its global convergence. This potentially makes it difficult to
directly deal with finite width network; especially in the neural tangent
kernel regime, we cannot reveal favorable properties of neural networks beyond
kernel methods. To realize more natural analysis, we consider a completely
different approach in which we formulate the parameter training as a
transportation map estimation and show its global convergence via the theory of
the infinite dimensional Langevin dynamics. This enables us to analyze narrow
and wide networks in a unifying manner. Moreover, we give generalization gap
and excess risk bounds for the solution obtained by the dynamics. The excess
risk bound achieves the so-called fast learning rate. In particular, we show an
exponential convergence for a classification problem and a minimax optimal rate
for a regression problem.
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