2martens/uni
1
0
Fork 0
mirror of https://github.com/2martens/uni.git synced 2026-05-06 19:36:26 +02:00
Code Releases Activity

[NN] Added equation references

Browse Source 
Signed-off-by: Jim Martens <github@2martens.de>
This commit is contained in:
Jim Martens
2018-06-20 12:25:29 +02:00
parent a51681103a
commit c98f0e1bbf
1 changed files with 23 additions and 23 deletions
Download Patch File Download Diff File Expand all files Collapse all files

46
neural-networks/seminarpaper.tex
View File

@ -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 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 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 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 of that product \eqref{eq:weightchangeprob}. This means there is a maximum weight
synapse. Weight changes can happen at any time step. Therefore the intrinsic weight change probability for each synapse. Weight changes can happen at any time step.
change probability has to be very small. Should a weight change occur a new weight Therefore the intrinsic weight change probability has to be very small. Should a
\(w_i\) is chosen randomly from the interval \([W_i^{min}, W_i^{max}]\). 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 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 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 situation. Within this example this probability is the so called second environmental
feedback loop. feedback loop.
\begin{equation}\label{eq:weightchangeprob}
\[
p_i^w = min(M_i, c(t, x_i, y_i)) \cdot W_i,\; 0 < W_i \lll 1 p_i^w = min(M_i, c(t, x_i, y_i)) \cdot W_i,\; 0 < W_i \lll 1
\] \end{equation}
Moreover a synapse can disable or enable itself. The actual disable/enable 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 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 parameter and the neuromodulator concentration \(c(t, x_i, y_i)\) \eqref{eq:enableprob}.
is again ceiled by the maximum sensitivity limit \(M_i\) given as parameter. The concentration is again ceiled by the maximum sensitivity limit \(M_i\) given
This means there is a maximum disable/enable probability as well. The intrinsic as parameter. This means there is a maximum disable/enable probability as well.
enable/disable probability must be smaller than the intrinsic weight change probability. The intrinsic enable/disable probability must be smaller than the intrinsic weight
A disabled synapse is treated as having weight 0 but the actual value is stored change probability. A disabled synapse is treated as having weight 0 but the actual
so that it can be restored when the synapse is enabled again. value is stored so that it can be restored when the synapse is enabled again.
\[ \begin{equation}\label{eq:enableprob}
p_i^d = min(M_i, c(t, x_i, y_i) \cdot D_i,\; 0 \leq D_i < W_i p_i^d = min(M_i, c(t, x_i, y_i) \cdot D_i,\; 0 \leq D_i < W_i
\] \end{equation}
Given a so called neural network structure or substrate this makes it easier Given a so called neural network structure or substrate this makes it easier
to find different network topologies (structure and weights combined). 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 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 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 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 normal distribution with a mean of zero and \(\sigma^2\)-variance \eqref{eq:gausswalk}.
value could be infinitely large and hence the new weight outside of the given The sampled value could be infinitely large and hence the new weight outside of
bounds for it. Therefore the value is sampled until the sum of the 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 \([W_i^{min}, W_i^{max}]\). current weight and the sampled value are within the interval \([W_i^{min}, W_i^{max}]\).
\[ \begin{equation}\label{eq:gausswalk}
w_i (t + 1) = w_i (t) + \Delta w_i \;\text{where}\; \Delta w_i \sim \mathcal{N}(0, \sigma^2) w_i (t + 1) = w_i (t) + \Delta w_i \;\text{where}\; \Delta w_i \sim \mathcal{N}(0, \sigma^2)
\] \end{equation}
Toutounji and Pasemann implemented a mechanism for disabling synapses 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 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 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 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.
\[ \begin{equation}\label{eq:hebbian}
\Delta w_{ij} = \eta \cdot m_i \cdot a_i \cdot a_j \Delta w_{ij} = \eta \cdot m_i \cdot a_i \cdot a_j
\] \end{equation}
This explanation should suffice for the general understanding of their method. This explanation should suffice for the general understanding of their method.
The neurons within the vicinity of these sources only update their weights The neurons within the vicinity of these sources only update their weights