--- On Wed, 12/3/08, Ben Goertzel <[EMAIL PROTECTED]> wrote: > Well, LTP is definitely real ... and I'm quite sure the scheme you > describe is *not* how learning works in the brain ;-) ,,, > but I'm equally sure that the full story has not yet been > uncovered...
I have attached a program that illustrates how memory can be stored in neurons rather than synapses. In a feed-forward configuration, each neuron is randomly connected to others further back, rather than fully connected. The network is sparse, with around n^(1/2) connections each among n neurons so that there is usually a path between each input and output going through at most one or two intermediate neurons. It also differs from a normal network in that all of the output weights of each neuron are constrained to have the same value. In other words, the activation level x[i] of the i'th neuron is given by x[i] = w[i]/(1+exp(-SUM_j x[j])) where j ranges over the input neurons for x[i]. Note that there is only one weight w[i] per neuron, rather than one per synapse. For the output neurons, w[i] = 1. For all others, the network is trained by adjusting the weights to reduce the RMS output error. This can be done in many ways. For example, you could simulate reinforcement learning by making random changes to w[i] and keeping changes that are followed by a reward computed by comparing desired and actual outputs. I used a more efficient training method, although I kept equivalence to biologically plausible models in mind. I used the following method: select one neuron x[i] at random and calculate the network outputs for w[i], w[i]+d, and w[i]-d for some large delta d like 0.5. Calculate the sum of the squares of the errors (actual - desired)^2 over all the values in the domain of the objective function (the desired behavior). If w[i]+d or w[i]-d gives a smaller total error than w[i], then make that the new weight. Otherwise replace d with d/2 and try again. When d is sufficiently small (like 0.02), then take the best weight and stop. Repeat until the errors are small enough. Neuron centered memory has the same information theoretical constraints as normal Hebbian learning. If your objective function has n bits of complexity, you need at least n neurons (rather than n synapses), because each parameter stores about 1 bit. Also, because each training session communicates about 1 bit, you need at least n sessions, actually O(n log n) to remove the last bit error assuming exponential convergence. In my program, I demonstrate training a neural network to learn a 3 by 3 bit multiplier. In general, a function with NX inputs and NY outputs has NY * 2^NX bits of complexity. For the 3x3 multiplier, NX = NY = 6 = 384 bits. In practice, you need 1.5 to 2 times as much. I used 640 neurons including 6 input and 6 outputs, and 36 random connections per neuron. Training should therefore require 640 * log(640) ~ 6000 sessions, although in practice about 15,000 were needed. The total number of operations is 15000 * 64 * 640 * 36 = 2.2 x 10^10, which took about 20 minutes on my PC. I estimate training a 4x4 multiplier would take about 40 times as long. I do not recommend using neuron-centered networks for solving problems that could be solved using Hebbian learning. This approach is slower by a factor of O(n^(1/2)), in this case 36, not to mention the difficulty of simulating sparse networks on vector processors. The purpose of this program is to show the plausibility of neuron-centered memory in the human brain. The brain would not be affected by the speed penalty because synapse operations are parallel. Furthermore, the model explains most of the discrepancy between Landauer's estimate of 10^9 bits of long term memory and 10^15 synapses. There are 10^11 neurons, a much closer number. Also, this model does not preclude Hebbian learning. Both could occur simultaneously. After all, learning is really a simple idea. The brain is adaptive. You just fiddle with knobs until you get the desired result. You don't have to understand what the knobs do. I believe you could achieve learning in sparse networks by fiddling with just about any neuron-wide parameters. -- Matt Mahoney, [EMAIL PROTECTED] ------------------------------------------- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244&id_secret=120640061-aded06 Powered by Listbox: http://www.listbox.com
nn.cpp
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