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[NN] Added formulas to ease understanding

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Signed-off-by: Jim Martens <github@2martens.de>
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Jim Martens
2018-05-24 11:04:09 +02:00
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45
neural-networks/seminarpaper.tex
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@ -370,23 +370,34 @@ the respective synapse.
\label{tab:mrs-synapse}
\end{table}
Modulated random search means essentially random weight changes. Each synapse
Modulated random search means essentially random weight changes. Each synapse \(i\)
has some parameters that are used (see table \ref{tab:mrs-synapse}). The weight
change probability is the product of the intrinsic weight change probability
and the concentration of the neuromodulator the synapse is sensitive to
at its location. Additionally the maximum neuromodulator sensitivity is
the ceiling for the second part of that product. This means there is a maximum
weight change probability for each synapse. Should a weight change occur a new
weight is chosen randomly from the range of values described by the minimum and
maximum weight of the synapse.
change probability \(p_i^w\) at time \(t\) is the product of the intrinsic weight
change probability \(W_i\) and the concentration of the neuromodulator the synapse
is sensitive to \(c(t, x_i, y_i)\) at its location \((x_i, y_i)\). Additionally
the maximum neuromodulator sensitivity \(M_i\) is the ceiling for the second part
of that product. This means there is a maximum weight change probability for each
synapse. Weight changes can happen at any time step. Therefore the intrinsic weight
change probability has to be very small. Should a weight change occur a new weight
\(w_i\) is chosen randomly from the interval \([W_i^{min}, W_i^{max}]\).
\[
p_i^w = min(M_i, c(t, x_i, y_i)) \cdot W_i,\; 0 < W_i \lll 1
\]
Moreover a synapse can disable or enable itself. The actual disable/enable
probability is the product of the intrinsic value saved as parameter and
the neuromodulator concentration. The concentration is again ceiled by the
maximum sensitivity limit given as parameter. This means there is a maximum
disable/enable probability as well. A disabled synapse is treated as having
weight 0 but the actual value is stored so that it can be restored when the
synapse is enabled again.
probability \(p_i^d\) is the product of the intrinsic value \(D_i\) saved as
parameter and the neuromodulator concentration \(c(t, x_i, y_i)\). The concentration
is again ceiled by the maximum sensitivity limit \(M_i\) given as parameter.
This means there is a maximum disable/enable probability as well. The intrinsic
enable/disable probability must be smaller than the intrinsic weight change probability.
A disabled synapse is treated as having weight 0 but the actual value is stored
so that it can be restored when the synapse is enabled again.
\[
p_i^d = min(M_i, c(t, x_i, y_i) \cdot D_i,\; 0 \leq D_i < W_i
\]
Given a so called neural network structure or substrate this makes it easier
to find different network topologies (structure and weights combined).
@ -401,7 +412,11 @@ but rather the difference to be added to the current weight is sampled from a
normal distribution with a mean of zero and \(\sigma^2\)-variance. The sampled
value could be infinitely large and hence the new weight outside of the given
bounds for it. Therefore the value is sampled until the sum of the
current weight and the sampled value are within the range.
current weight and the sampled value are within the interval \([W_i^{min}, W_i^{max}]\).
\[
w_i (t + 1) = w_i (t) + \Delta w_i \;\text{where}\; \Delta w_i \sim \mathcal{N}(0, \sigma^2)
\]
Toutunji and Pasemann implemented a mechanism for disabling synapses
in the modulated gaussian walk as well but did not make use of it later and