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[NN] Added equation references

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Signed-off-by: Jim Martens <github@2martens.de>
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Jim Martens
2018-06-20 12:25:29 +02:00
parent a51681103a
commit c98f0e1bbf
1 changed files with 23 additions and 23 deletions
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46
neural-networks/seminarpaper.tex
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@ -383,33 +383,33 @@ change probability \(p_i^w\) at time \(t\) is the product of the intrinsic weigh
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 \([W_i^{min}, W_i^{max}]\).
of that product \eqref{eq:weightchangeprob}. 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
\([W_i^{min}, W_i^{max}]\).
The weight change probability \(p_i^w\) tells the network when to learn and leaves
room for variation as it is a probability and not a binary learn/do not learn
situation. Within this example this probability is the so called second environmental
feedback loop.
\[
\begin{equation}\label{eq:weightchangeprob}
p_i^w = min(M_i, c(t, x_i, y_i)) \cdot W_i,\; 0 < W_i \lll 1
\]
\end{equation}
Moreover a synapse can disable or enable itself. The actual disable/enable
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.
parameter and the neuromodulator concentration \(c(t, x_i, y_i)\) \eqref{eq:enableprob}.
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.
\[
\begin{equation}\label{eq:enableprob}
p_i^d = min(M_i, c(t, x_i, y_i) \cdot D_i,\; 0 \leq D_i < W_i
\]
\end{equation}
Given a so called neural network structure or substrate this makes it easier
to find different network topologies (structure and weights combined).
@ -421,14 +421,14 @@ The modulated gaussian walk is introduced by Toutounji and Pasemann. The key dif
start with the parameters. There is no maximum sensitivity for the neuromodulator
concentration. When a weight change occurs the new weight is not chosen randomly
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
normal distribution with a mean of zero and \(\sigma^2\)-variance \eqref{eq:gausswalk}.
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 interval \([W_i^{min}, W_i^{max}]\).
\[
\begin{equation}\label{eq:gausswalk}
w_i (t + 1) = w_i (t) + \Delta w_i \;\text{where}\; \Delta w_i \sim \mathcal{N}(0, \sigma^2)
\]
\end{equation}
Toutounji and Pasemann implemented a mechanism for disabling synapses
in the modulated gaussian walk as well but did not make use of it later and
@ -463,11 +463,11 @@ it when to learn.
How does the actual learning happen? The weight change between two neurons
is dependent on the activation of both neurons, the learning rate and the concentration
of neuromodulators. In short Hebbian learning is employed.
of neuromodulators \eqref{eq:hebbian}. In short Hebbian learning is employed.
\[
\begin{equation}\label{eq:hebbian}
\Delta w_{ij} = \eta \cdot m_i \cdot a_i \cdot a_j
\]
\end{equation}
This explanation should suffice for the general understanding of their method.
The neurons within the vicinity of these sources only update their weights