NIPS 96 Workshop on Error Surfaces

###
Towards More Convex Error Surfaces

Talk by
Grace Wahba
at the NIPS Error Surfaces Workshop, December 7, 1996, at Snowmass CO.

Department of Statistics,
University of Wisconsin, Madison.

### Abstract:

In this talk I argued that if the error surface that
you are trying to minimize, as a function of many
variables, is sufficiently nasty, as it frequently
is when fitting a sigmoidal basis function feedforward
neural net, then you should
think about reformulating the optimization problem to
be solved. In keeping with the informality of a workshop
a number of ideas, not all combinations of which have
been tested, were thrown out.
It was argued that sigmoidal basis functions
should be parametrized by (not too many) unknown
scale factors, by a unit vector $\gamma$ and by a distance
$b$ along the unit vector. $b$ tells you how far along you
go along the unit vector to get to the half-power point
of the sigmoidal function, and a scale factor, say
$\alpha$
tell you the slope of the sigmoidal function at the
half power point. I proposed (for fixed scale factors)
the idea of generating a large number of unit vectors and
possibly half-power points via a random number generator,
and using them to generate a large
library of basis functions; then using either a fast rank-one
update to iteratively select one new basis function
at a time to minimize the penalized likelihood function,
possibly with an iterative update of the smoothing parameter(s)
via GCV, perhaps preceeded by a screening of the library
via support vector methods. In general least squares fits
of sigmoidal basis functions are mildly insensitive
to scale factors, it is probably appropriate to allow
only a small discrete number of scale factors, equally
spaced on a log scale, for searching. I also gave a talk in
the NIPS Model Complexity Workshop on December 6, 1996
which provides some related
details and discusses other basis functions and
various penalty functionals. Further information
and references are available via my
nips.96 workshop talks home page. The overhead slides for
the Model Complexity Workshop are
here.

*Key words:*
error surfaces, sigmoidal basis functions,
penalized likelihood, iterative update,
support vector,
generalized cross validation.

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