Robustness and commitment

CSGnet Archive CSG2000
1

[From Rick Marken (2000.01.03.1100)]

Here's a status report on my tests to determine whether
increasing the number of control systems at a particular
level in the hierarchy increases the "robustness" of that
set of systems in the sense that it increases the ability of
that set of systems to continue controlling when a lower level
reference signal is fixed ("commited") to a particular value.

The tests were carried out with a version of the spreadsheet
hierarchy model that made it possible to have from 1 to 10
level 2 control systems controlling independent aspects of
the perceptions controlled by a set of from 1 to 10 level 1
control systems. The number of level 1 systems always equalled
the number of level 2 systems. I also made it possible for
the tester to fix any one of the level 1 reference signals to
a constant without typing over an existing value in a cell.

So I have a two level hierarchical control model and I can
let the level 1 references be a function of the outputs of
the level 2 systems (as is normally the case) or I can "commit"
any particular level 2 reference signal to a constant value.

As in the original spreadsheet model, each level 1 perceptual
variable is proportional to a single environmental variable and
each level 2 perceptual signal is a linear function of all the
level 1 perceptions. So if there are 10 level 1 perceptions,
then all level 2 perceptions are a (different) linear combination
of _all_ 10 level 1 perceptions; if there are 2 level 1 perceptions,
the all level 2 perceptions are a (different) linear combination
of the 2 level 1 perceptions.

The spreadsheet was designed to test the hypothesis that
increasing the number of systems at a particular level of
a hierarchy of control systems (level 2 in this case)
increases the ability of all systems at that level to control
in the face of a fixed (commited) lower level reference. The
spreadsheet measures ability to control as the average error
at level 2 of the control hierarchy (which is equivalent to
the amount of error per control system at level 2).

I started my tests with all level 2 references set to 0.
When none of the level 1 references are fixed, the average
error at level 2 (error/control system) is .001, regardless
of the number of level 2 (and, correspondingly, level 1)
control systems in the hierarchy.

The big test was to fix one level 1 reference and see what
happens to averge level 2 error as the number of level 2
(and, correspondingly, level 1) systems is increased. tThe
prediction was that as the number of level 2 systems increased,
the average error/level 2 system would decrease.

The results were not quite so straightforward. It's certainly
true that if there is one level 2 (and level 1) control system
involved, then when you fix the level 1 reference (I always used
1.0 for the fixed reference value) the level 2 control system is
no longer able to control (the average error is maximum; 1.0).

When there is more than one level 2 (and level 1) control system
involved, the average error/control system is _always_ less than
what you get with one level 2 (and level 1) control system. But
there is not always a linear decrease in error/control system
with an increasing number of control systems. In fact, in
some cases (such as going from 2 to 3 control systems) there
was a substantial _increase_ in error/control system with an
_increase_ in the number of control systems.

So my current conclusion is that a level in a hierarchy of
control with more than one control system is more robust than
a level with only one control system. But once you have more
than one control system in some level of the hierarchy, the
robustness of that level depends stongly on the perceptions
(functions of the lower level perceptions) that are being
controlled by all the systems at that level.

The general conclusion is that the robustness of a level
of a hierarchy of control systems (in terms of its ability
to deal with arbitrarily "commited" lower level references)
does _not_ increase with an increasing number of control
systems at that level of the hierarchy once there is more
than one control system at that level.

This means that you can't assume that forcing a child to
commit to some goal (assuming that you _could_ force a child
to do this) will not create much of a control problem for the
child simply because the child is likely to be controlling
_many_ perceptions. at the level above that of the commited
reference.

Best

Rick

···

---
Richard S. Marken Phone or Fax: 310 474-0313
Life Learning Associates e-mail: rmarken@earthlink.net
http://home.earthlink.net/~rmarken/

2

[Martin Taylor 20000103 14:17]

[From Rick Marken (2000.01.03.1100)]

Here's a status report on my tests to determine whether
increasing the number of control systems at a particular
level in the hierarchy increases the "robustness" of that
set of systems in the sense that it increases the ability of
that set of systems to continue controlling when a lower level
reference signal is fixed ("commited") to a particular value.

The tests were carried out with a version of the spreadsheet
hierarchy model that made it possible to have from 1 to 10
level 2 control systems controlling independent aspects of
the perceptions controlled by a set of from 1 to 10 level 1
control systems.

...

As in the original spreadsheet model, each level 1 perceptual
variable is proportional to a single environmental variable and
each level 2 perceptual signal is a linear function of all the
level 1 perceptions.

So this is a pure analogue control hierarchy?

So if there are 10 level 1 perceptions,
then all level 2 perceptions are a (different) linear combination
of _all_ 10 level 1 perceptions; if there are 2 level 1 perceptions,
the all level 2 perceptions are a (different) linear combination
of the 2 level 1 perceptions.

The spreadsheet was designed to test the hypothesis that
increasing the number of systems at a particular level of
a hierarchy of control systems (level 2 in this case)
increases the ability of all systems at that level to control
in the face of a fixed (commited) lower level reference.

I don't remember anyone proposing that hypothesis for an analogue control
hierarchy, or even for a hybrid hierarchy in which the top level has
analogue perceptions.

The only prediction I can remember for an analogue hierarchy was that
if the number of free-to-vary lower systems was smaller than the number
of higher ones, control would fail, and depending on the weighting and
on the exact Excel calculation scheme, the failure might be distributed
across the upper units or might be concentrated in one or two of them.

For a hybrid hierarchy, the prediction was that the higher the number
of units at each level, the higher the proportion of sets of upper-level
reference values that would be achievable when one lower-level reference
was fixed. I cannot remember anyone making a prediction about the average
value of the error with an unachievable set of reference values.

If I understand your message correctly, you didn't test either of the
predictions. But maybe I misread your message.

The
spreadsheet measures ability to control as the average error
at level 2 of the control hierarchy (which is equivalent to
the amount of error per control system at level 2).
...The big test was to fix one level 1 reference and see what
happens to averge level 2 error as the number of level 2
(and, correspondingly, level 1) systems is increased. tThe
prediction was that as the number of level 2 systems increased,
the average error/level 2 system would decrease.

How did you arrive at that prediction? You didn't publish it beforehand,
at last not that I remember seeing.

The results were not quite so straightforward. It's certainly
true that if there is one level 2 (and level 1) control system
involved, then when you fix the level 1 reference (I always used
1.0 for the fixed reference value) the level 2 control system is
no longer able to control (the average error is maximum; 1.0).

That seems to contradict your earlier statement that the level 2
perceptions were linear functions of the level 1 perceptions, and
suggests that maybe your level 2 control units were hybrid units.
Which were they, analogue or hybrid (logical perception derived
from analogue inputs)?

It matters for your general conclusion: "that the robustness of a
level of a hierarchy of control systems (in terms of its ability
to deal with arbitrarily "commited" lower level references)
does _not_ increase with an increasing number of control
systems at that level of the hierarchy once there is more
than one control system at that level."

If you want to test actual predictions, and the spreadsheet actually
has a hybrid top level, you should report how many of the 2^N possible
sets of reference values are achievable as a function of N, when one
lower-level system has a fixed reference value. Or if the spreadsheet
is purely analogue, you should report how the error is distributed among
the upper level units, which indicates whether _any_ of the upper level
units come close to actually controlling, and if any do, how many.

Could you send me a copy of this new spreadsheet. please?

Martin

3

[From Rick Marken (2000.01.03.2050)]

Martin Taylor (2000.01.03 14:17)

So this is a pure analogue control hierarchy?

Yes.

Me:

The spreadsheet was designed to test the hypothesis that
increasing the number of systems at a particular level of
a hierarchy of control systems (level 2 in this case)
increases the ability of all systems at that level to control
in the face of a fixed (commited) lower level reference.

Martin:

I don't remember anyone proposing that hypothesis for an
analogue control hierarchy, or even for a hybrid hierarchy
in which the top level has analogue perceptions.

I was testing a conjecture that was (prehaps incorrectly)
attributed to you by Bill Powers (991228.0726 MDT):

Martin conjectured that the more systems that are involved at
a given level, the more robust the hierarchy will be against
loss of a single intermediate-level control system. I think
your test vindicates his claim.

I thought that that was an interesting conjecture. It seemed
intuitively plausible. So I designed the new spreadsheet so
that I could add systems one at a time and (if the conjecture
was true) watch a smooth decline in the effect of a fixed
lower level (level 1) reference on higher level (level 2)
control (measured as average error/level 2 control system).
What I found is that this "robustness" conjecture is
apparently false: that is, it is apparently not true that
the more systems involved at a given level, the more robust
the hierarchy will be against loss of a single intermediate-
level control system.

I see that you [Martin Taylor (20000103 18:52)] have found
basically the same thing with the "logical" perceptions
(level 3) in the original spreadsheet since you report
getting 4/64 level 3 control failures with six systems and
1 level 2 reference fixed, an average of 3/32 level 3 control
failures with five systems and 1 level 2 reference fixed and
1/16 level 3 control failures with four systems and 1 level 2
reference fixed. So increasing the number of control systems
at a given level (level 3 in this case) does not increase
robustness against loss of a single intermediate-level (level
2 in this case) control system. Increasing the number of level
3 systems from 4 to 6 does not reduce control failures; control
failures are about 1/16 or 6.5% when 1 level 2 reference signal
is fixed, whether there are 4,5 or 6 systems at level 3.

Best

Rick

···

---

Richard S. Marken Phone or Fax: 310 474-0313
Life Learning Associates e-mail: rmarken@earthlink.net
http://home.earthlink.net/~rmarken/

4

[Martin Taylor 20000104 10:01]

[From Rick Marken (2000.01.03.2050)]

I was testing a conjecture that was (prehaps incorrectly)
attributed to you by Bill Powers (991228.0726 MDT):

> Martin conjectured that the more systems that are involved at
> a given level, the more robust the hierarchy will be against
> loss of a single intermediate-level control system. I think
> your test vindicates his claim.

I thought that that was an interesting conjecture.

Yes, but it was about hybrid/logical control levels.

I see that you [Martin Taylor (20000103 18:52)] have found
basically the same thing with the "logical" perceptions
(level 3) in the original spreadsheet

Yes, that's right--more or less. It's not true in the "original
spreadsheet" but it seems to be true in the "symmetric" version I
sent back to you under the title "spreadsheet 4".

In your test of the analogue levels, when you fix one lower reference
level, can any of the higher levels control? In other words, is the error
in the higher levels distributed around all the units, or is it confined
to a subset of conflicting units? In the hybrid level (level 3 in the
spreadsheet) the error always seems to be confined to a group of 4
conflicting units, while the others have no problem controlling.

You report that the level of error is non-monotonic with the number of
units per level. What I found is that the number of configurations of
unachievable reference levels is quite erratic when the perceptions are
not orthogonal. When they are not orthogonal, the number of errors
depends on exactly which intermediate references are fixed (as was
the case with your original spreadsheet). On balance, however, it
seems true that about 1/16 of the configurations are unachievable
whether there are 4, 5, or 6 units per level.

If that is generalizable, and its a big "if", then 93% of the time,
"committing" one reference level will have no effect on control, and
on the other occasions it will cause a major screwup.

One special characteristic of the "symmetric" spreadsheet that may limit
the generalizability of the result is that all the weights connecting
level 2 with level 3 are equal except for sign. This means that when,
say, unit 31 output is trying to reduce the reference at level 24 while
unit 35 output is trying to increase it, nothing happens. If the weights
were unequal, one or other would have the stronger influence. In an
analogue hierarchy this would not result in good control, but in a
hybrid level, the stronger would win if there was even the tiniest
imbalance. Whether this would affect the number of achievable sets of
reference values, I don't know. But if the results are to be generalized
to living systems in which perfect balances never happen, that kind of
test should be done. Stay tuned over the next few days. I may do it.

Martin

5

[From Rick Marken (2000.01.04.1840)]

Martin Taylor (20000104 10:01) --

In your test of the analogue levels, when you fix one lower
reference level, can any of the higher levels control?

If you mean by control "get the perceptual signal to match
the reference signal exactly", then the answer is: sometimes,
depending on the level 2 perceptions controlled. But usually
locking up one level 1 reference input (which is involved in
the control of all level 2 perceptions) reduces control (average
deviation of perception from reference) for all level 2 systems.

On balance, however, it seems true that about 1/16 of the
configurations are unachievable whether there are 4, 5, or 6
units per level.

This particular result (1/16 unachievable) may be dependent
on the particular set of independent perceptions controlled.
I found that the effect of locking up a particular reference
depended on _which_ particular orthogonal set of linear
combinations of lower level perceptions was being controlled.
With some combinations there was less of an effect of the lock
up than there was for other combinations.

If that is generalizable, and its a big "if", then 93% of
the time, "committing" one reference level will have no
effect on control, and on the other occasions it will cause
a major screwup.

If this were true, it would only be true for "binary" perceptions
(perceptions that can be in only two states eg., -1 and 1, as in
the present case). It's not necessarily true that a large
proportion of the control hierarchy controls such variables. So
locking up one reference may have little or no no effect on
control in 93% of the cases where a binary variable is involved.
But 93% of the cases involving binary perceptions may represent
only a small fraction of the controlling done by the entire
hierarchy of control systems.

I think the really interesting discovery so far is that increasing
the number of control systems at any level in a control hierarchy
(beyond one) does _not_ increase the robustness of the systems at
that level of the hierarchy. Whether there are 10 or 10,000
systems at a particular level of the hierarchy controlling
orthogonal perceptions simultaneously, the effect of removing
one functional lower level reference (by fixing or "commiting" it
to a particular value) will be the same; some loss of control by
that level of control systems.

Best

Rick

···

---
Richard S. Marken Phone or Fax: 310 474-0313
Life Learning Associates e-mail: rmarken@earthlink.net
http://home.earthlink.net/~rmarken/

6

[Martin Taylor 20000105 0:42]

[From Rick Marken (2000.01.04.1840)]

> If that is generalizable, and its a big "if", then 93% of
> the time, "committing" one reference level will have no
> effect on control, and on the other occasions it will cause
> a major screwup.

If this were true, it would only be true for "binary" perceptions
(perceptions that can be in only two states eg., -1 and 1, as in
the present case). It's not necessarily true that a large
proportion of the control hierarchy controls such variables.

If Bill P is right, it applies to the category level and above, and to
nothing lower.

I think the really interesting discovery so far is that increasing
the number of control systems at any level in a control hierarchy
(beyond one) does _not_ increase the robustness of the systems at
that level of the hierarchy.

Yes, that's interesting, and it may be a reason why we seem to be built
in a rather modular fashion, both muscularly and in the brain. What we
have been simulating is a set of systems in which all control units at
each level influence the behaviour of all units at the neighbouring
levels. Even under those conditions, in my simulations of the logical
control level, when there is a control problem, it involves exactly
four of the units, no matter whether the level has 4, 5, or 6 units
in all. I'd like to be able to try and see if the same holds true for
wider hierarchies, but even if I were to program it, I think it would
take too long to run on my machine. The time goes up drastically with
the width of the hierarchy. (The number of links to be computed on
each iteration goes up with the square of the width, and the number of
reference patterns goes up exponentially with the width, so an 8-wide
hierarchy should take 1.8 times as long per iteration, and it has 4 times
as many reference patterns to test. So, if a pass with 4 errors takes
nearly an hour on a 6-wide hierarchy, it should take something like 6
hours on an 8-wide hierarchy, even if the number of errors stayed at 4,
ratehr than going up to 16, as we now hypothesise.)

Meanwhile, I am running tests on the 6x6x6 spreadsheet, with the weights
unequal between the various level 3 units and the level 2 units. My
earlier results had all the weights at +1 or -1. Now the + weights are
mostly between 0.6 and 1.4, and similarly for the - weights. The results,
so far, seem to be the same -- 4 failures out of 64 patterns when one
(any one) level 2 reference is fixed at +5, on all the tests I've run
so far.

Whether there are 10 or 10,000
systems at a particular level of the hierarchy controlling
orthogonal perceptions simultaneously, the effect of removing
one functional lower level reference (by fixing or "commiting" it
to a particular value) will be the same; some loss of control by
that level of control systems.

This is only true if the number of available degrees of freedom is
marginal to begin with. If there were 6 higher levels and 7 lower ones,
all interconnected, fixing one reference at the lower level would not
affect the ability of any higher level unit to control. If there were
6 upper and 20 lower, you could commit 14 of the lower-level references
without affecting the ability of the upper ones to control. The problems
arise when the number of uncommitted lower-level reference values less
than the number of higher level perceptions to be controlled.

What you don't know in real life is whether there are already too few
lower-level references, just enough, or plenty to spare.

Martin