**From:** Johnicholas Hines (*johnicholas.hines@gmail.com*)

**Date:** Sun Feb 15 2009 - 15:34:00 MST

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On Tue, Feb 10, 2009 at 4:31 AM, Stuart Armstrong

<dragondreaming@googlemail.com> wrote:

*>> My limited real-world experience with learning machines is that they tend to
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*>> experience a "loss of steam". I'm not sure how best to model that
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*>> pattern using differential equations.
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*>
*

*> Rather easily - take the differential equations of a body falling
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*> through the atmosphere, and take "speed" as "intelligence" (ie there
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*> is a friction factor growing with your level of intelligence).
*

Stuart Armstrong's model, as I understand it:

There is one endogenous variable "intelligence" (i), analogous to speed.

There are two parameters:

First, an "acceleration coefficient" (a), analogous to the force of gravity.

Second, a "friction coefficient" (f), analogous to the friction from the air.

Assuming that we're using a quadratic model of air friction, the

friction force is proportional (f) to the square of the speed, or

intelligence.

The single equation is that the time derivative of intelligence is

proportional to the difference between acceleration and the friction

force. (D[i] = a - f * i^2)

So "terminal intelligence" would be when the acceleration exactly

balances the friction force (a = f * i ^2 or i = sqrt(a/f)).

I'm not sure how to relate the acceleration and friction parameters to

the "phase space" argument about the probability of a random change

being beneficial.

I also created a new model. This one replicates the "power law of

practice", which is observed in human learning.

There is one endogenous variable, the total number of "discoveries" (d).

Each discovery is more difficult than the next - this models the low

hanging fruit phenomenon. The amount of time to make the next

discovery is proportional to the total number of discoveries. The

constant of proportionality is the first parameter (a).

Each discovery increases the capability (c) by a multiplicative

factor. The factor is the second parameter (b).

The time derivative of the discoveries is inversely proportional to

the number of discoveries (D[d]= 1/(a*d)). The capability at each time

is exponential in the number of discoveries (c = b^d).

Combining these, you see the number of discoveries rising over time in

a logarithmic curve. The capability changes over time in a power law

curve, just like the power law of practice.

Johnicholas

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