From: sef@sef-pmax.slisp.cs.cmu.edu
Subject: Re: Help, statistics and learning algorithms
Organization: School of Computer Science, Carnegie Mellon
Date: Wed, 27 Jan 1993 14:01:46 GMT


    From: tstanley@lamar.ColoState.EDU (Thomas R Stanley)

    My question is, what are the statistical equivalents of the
    supervised learning methods commonly used in constructing neural nets? 
    More precisely, are there statistical equivalents to the following (see,
    I did check the FAQ first :) learning methods:
    
    ...
    
    I found one reference that said BP was equivalent to least squares
    fitting.  Is this true of the rest of the methods?  If these methods are
    really just variations on classical statistical procedures, then (and here
    is where I show my ignorance) what do I have to gain by using neural nets? 
    Why should I expect these procedures to perform better than a parametric
    method (let's assume for the sake of argument the assumptions of the
    method (e.g. normality) are met) where there exists maximum likelihood
    estimators (MLE's) for the parameters (i.e. weights at the processing
    element or node)?  MLE's guarantee unbiasedness and efficiency (minimum
    variance), you can't get better than that can you?

Very brief answer: If there were exact statistical equivalents to all these
methods, then we wouldn't be wasting our time making up new neural net
algorithms and architectures.  Well, some of us might, as more
biologically plausible or more easily parallelizable versions of existing
algortihms, but the field would be much less exciting than it is now, since
many of us think we're developing algorithms with new powers.

The real difference is that these neural net models are able to work with a
different (richer, generally more powerful) set of architectures than the
statisticians have been able to use in the past.  We are trying to find an
optimal set of parameters within some model, but the model may involve
cascaded nonlinearities (as in backprop or cascor) or recurrent information
flow and time-dependencies (as in Boltzmann, recurrent cascor,
Williams-Zipser, etc.).  Furthermore, the number, type, and topology of the
basis functions and the number of parameters in the model might themselves
be variable and subject to learning, as in cascor.  (A few statistical
methods, such as MARS, also explore the space of model sizes dynamically.)

Neural nets, then, allow you to fit these more complex models to the data.
Usually the tuning of the parameters is by some sort of procedure that
computes or approximates some gradient of an error measure for each weight,
dE/dw and tries to reduce it iteratively -- it's rare in these kinds of
models to be able to solve for the optimal parmaeter set more directly.
the iterative reduction might be simple gradient descent, or it might be
some accelerated method like conjugate gradient or a pseudo-Newton method.
The error functions might be minimization of sum-squared difference at the
output, or something more complex.

So the parameter estimation methods used in neural nets will look familiar
and perhaps rather primitive to a staticstician.  It is the richness of the
available set of models that gives us some new power.  Whether this
richness buys you anyhting depends on the problem.  If a plane drawn
through a multi-D parameter space is what you need, traditional statistical
methods win.  You can often get some guarantee of optimality, but that's
optimality within the model framework your statistical method handles, and
with the number of parameters you have chosen.  In a number of more complex
situations, neural nets have been shown to give superior generalization.
If you want to think of this as just a new, nonlinear branch of statistics,
feel free.

-- Scott

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