Infinite-Width Bounded-Norm Networks: A View from Function Space

Nathan Srebro
Computer Science, Toyota Technological Institute

There has been much research in the past four decades on understanding
what functions can be captured, or approximated, by multi-layer neural
networks, with the focus being on how well a function can be
approximated as function of the number of units in the network. But
more recently, we have come to understand that in modern deep learning
the magnitude of the weights, rather then the number of units, play a
more important role in complexity control, and that the models we are
learning are perhaps better thought of as having an unbounded number
of units but bounded overall norm. We make the first steps toward
understanding what kind of functions can be captured by such bounded
norm infinite width networks, and what type of complexity control in
function space does bounding the norm of the weights induce, by
providing a detailed study for one dimensional functions with a
surprisingly simple and satisfying answer.

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