[Martin Taylor 960308 10:40]
Bill Powers (960307.2030 MST)
Martin Taylor 960707 18:20 --
There seems to be something missing from your diagrams of the "wasp-
wasted MLP" -- the system of computations and associated perceptual and
output systems that operates on these nodes to organize them. One
obvious answer to the implied question is "Evolution did it." But is
that a satisfactory answer?
I didn't think it necessary to say more than that it is a standard MLP,
because the algorithms can be looked up in any of many books. There's
really no issue here, and the wasp-waisted MLP itself has been used as
a practical tool to find effective coding schemes for complex inputs
such as speech. And, as I said, there's no need to restrict the net to
being a multilayer perceptron.
I used the MLP as it is the simplest configuration, and is a straight through
feed-forward system. If one wanted to make such a system for a real problem,
one would take advantage of all the different kinds of configurations that
have been found to be advantageous for different kinds of problems. To do
so might make for a more effective device, but it wouldn't make a hair
of difference to the argument.
As to the final model, we seem to have a direct output connection
between the highest level of the learned hierarchy (the X system) and
the highest level of what seems to be proposed as an inherently
hierarchical environment (the Y system).
I haven't shown any hierarchy, at least not in the HPCT sense. There's no
mapping between corresponding levels of the two halves of the pyramid. In
the wasp-waisted MLPs that learn codings, the two halves usually are probably
constructed by the designers to have the same numbers of nodes in levels
equally far from the environment, but that's a convenience, not a requirement.
Rather than a hierarchy, we have a demonstrable (and in a practical sense,
demonstrably useful) way that efficient codings can be learned, codings
that map structured input patterns into a smaller number of variables, with
minimal loss of data, as evidenced by the accuracy of reconstruction of
the original data. That's the original wasp-waisted MLP, doing both jobs.
We then take away from the MLP the job of doing the reconstruction, and
say that whatever it produces at its wasp waist will affect the environment
in some manner it doesn't know (originally). What we ask the remaining
pyramidal MLP to do is to learn how to encode the effects it has on the
environment. This is obviously possible in principle, because ALL the
effect it has on the environment is due to the low-dimensional variations
of signals at the wasp waist, and can thus be recoded back into that same
dimensionality. It is of no relevance whether the Y structure is hierarchic,
analytic, symbolic, or whatever. The Y structure only serves as a possibly
complicated mapping from the low to the high-dimensional space. Lippman's
theorem about the number of hidden levels provides assurance that no matter
how complex the Y mapping, the X structure can learn it if it has enough
nodes and enough levels (where "enough" levels is about 3).
Continuing with the same paragraph of your comment...
This is one possible answer to
the question of why a hierarchy is learned: because the environment is
hierarchically organized. However, this answer raises the question of
how the highest level of the learned hierarchy can communicate with, or
have effects on, the highest level of the environmental hierarchy,
without going through the lowest level of interaction, at the bases of
the pyramids.
When one draws in ASCII, one has certain limitations, both of resolution
and of time. What you call the "top" should be conceived as the output
interface to the environment--the musculature. As I mentioned in the text,
that interface could be anywhere. As drawn the top XXXX should be conceived
as the musculature, but the interface could be anywhere from there to the
middle of the Y structure. What I would have liked to have drawn was
a kind of spoked wheel, with wide X's above wide Y's at 9 o'clock, middle
width X's at 12 o'clock, narrow X's above narrow Y's at 3 o'clock, and
mid-width Y's at 6 o'clock. Then one could slip a radial line in anywhere
between 3 o'clock and, say, 7 o'clock, to represent the effector interface
with the world. The XXXX part refers only to that part of the structure
that adapts (will be "reorganizes" when we develop the HPCT structure).
There is another kind of problem in the middle of your development:
[Picture omitted]
This, apparently, deals implicitly with the possibility of disturbances
that independently affect the input row of Xs. However, if the goal of
this system is to create output X's that exactly match the input X's
(with the sign inverted for negative feedback), then the loop gain is at
most only -1. As a result only half of the effect of any given
disturbance can be counteracted.
True, but there is no reason at this point for the outputs at the musculature
not to be amplified. See below. Remember that the original wasp-waist was
trained in an open-loop configuration, and that the training results will
be a little different if it is done in a closed-loop configuration.
The output cannot maintain the input in a
given state against disturbances unless the output _differs_ from the
state it has when there are no disturbances.
Correct.
The requirement that the
output is to match the input (with either sign, it doesn't matter) is
inconsistent with the requirement that the output _change_ so as to
oppose disturbances.
Correct also. But notice your earlier comment about the loop gain being -1.
If the system is learning _while connected as a loop_ it will inevitably
learn a structure that embodies the required high negative gain, and
it will "discover" (or at least an external analyst will) that it cannot
counter any disturbance completely. However, when it is learning in closed
loop configuration, the learning criterion _cannot_ be that the Y outputs
equal the X inputs, because that equality is set by the direct connection
between them. The criterion _can_ be that the Y outputs are the negative
of the X inputs, because the fact that they are connected to be identical
means that they must be zero.
Remember that this device "knows" what the Y values are, and how they
compare to what they "should" be. And that's not realistic when we are
dealing with the learning problem confronted by a real-life control system.
The learning criterion has to be different from the criterion that builds
the structure of the wasp-waist MLP. That's irrelevant to the argument so
far, but it won't prove irrelevant when we try to develop the HPCT structure
out of it--the criterion then has to relate to reference signals, which are
so far conspicuous by their absence from the structure.
So I am calling your attention to what seem to be two problems with your
proposal: the direct connection between highest levels of two
hierarchies, and the conflict between the "self-teaching" requirement
for matching output to inputs and the requirement (for control) that
output deviate from a match to inputs in order to oppose disturbances.
Any comments?
I hope the above comments are satisfactory. They sum to an apology for not
being more precise and detailed in my original posting. And here I add a
reminder that the connection with HPCT has yet to be made. The structure
as posted is not a full control system, in that its reference values are
only implicit and cannot be changed, and that the learning algorithm that
is suitable for an open-loop MLP is not appropriate when the structure is
seen as a real-life closed loop system with feedback through the environment.
The point of the posting was not to produce an optimum control structure.
It was intended to bring out the fact that the _same_ structure
looked at in different ways can bring reasonable people to talk about
control systems that use model-based input, model-based output, neural
net learning, or (as yet undemonstrated because it is our usual starting
point) scalar-valued hierarchic control. I wanted to try to do something
to quiet what I think has been a sometimes unprofitable conflict between
positions that are really compatible. Just because a control system is
a hierarchy of scalar control units does not mean that it isn't model-based,
nor that it is not a learning neural network on either the perceptual
or the output side. All three can be views on the same elephant.
But, just as a snake is not always an elephant's trunk, nor a tree an
elephant's leg, not all model-based control is a view on a hierarchy.
One has to look further, to determine whether one has a snake of a piece
of an elephant.
Martin