andy
236a5eac06
Affected files: .obsidian/app.json .obsidian/appearance.json .obsidian/workspace.json Money/Markets/Commodity.md STEM/AI/Neural Networks/CNN/CNN.md STEM/AI/Neural Networks/CNN/GAN/CycleGAN.md STEM/AI/Neural Networks/CNN/GAN/DC-GAN.md STEM/AI/Neural Networks/CNN/GAN/cGAN.md STEM/AI/Neural Networks/CV/Data Manipulations.md STEM/AI/Neural Networks/MLP/Back-Propagation.md STEM/CS/Code Types.md STEM/CS/Compilers.md STEM/Quantum/Confinement.md
3.2 KiB
3.2 KiB
Error signal graph
- Error Signal
e_j(n)=d_j(n)-y_j(n)
- Net Internal Sum
v_j(n)=\sum_{i=0}^mw_{ji}(n)y_i(n)
- Output
y_j(n)=\varphi_j(v_j(n))
- Instantaneous Sum of Squared Errors
\mathfrak{E}(n)=\frac 1 2 \sum_{j\in C}e_j^2(n)C= o/p layer nodes
- Average Squared Error
\mathfrak E_{av}=\frac 1 N\sum_{n=1}^N\mathfrak E (n)
\frac{\partial\mathfrak E(n)}{\partial w_{ji}(n)}=
\frac{\partial\mathfrak E(n)}{\partial e_j(n)}
\frac{\partial e_j(n)}{\partial y_j(n)}
\frac{\partial y_j(n)}{\partial v_j(n)}
\frac{\partial v_j(n)}{\partial w_{ji}(n)}
From 4
\frac{\partial\mathfrak E(n)}{\partial e_j(n)}=
e_j(n)$$
#### From 1
$$\frac{\partial e_j(n)}{\partial y_j(n)}=-1$$
#### From 3 (note prime)
$$\frac{\partial y_j(n)}{\partial v_j(n)}=
\varphi_j'(v_j(n))$$
#### From 2
$$\frac{\partial v_j(n)}{\partial w_{ji}(n)}=
y_i(n)$$
## Composite
$$\frac{\partial\mathfrak E(n)}{\partial w_{ji}(n)}=
-e_j(n)\cdot
\varphi_j'(v_j(n))\cdot
y_i(n)
\Delta w_{ji}(n)=
-\eta\frac{\partial\mathfrak E(n)}{\partial w_{ji}(n)}$$
$$\Delta w_{ji}(n)=
\eta\delta_j(n)y_i(n)$$
## Gradients
#### Output Local
$$\delta_j(n)=-\frac{\partial\mathfrak E (n)}{\partial v_j(n)}$$
$$=-
\frac{\partial\mathfrak E(n)}{\partial e_j(n)}
\frac{\partial e_j(n)}{\partial y_j(n)}
\frac{\partial y_j(n)}{\partial v_j(n)}$$
$$=
e_j(n)\cdot
\varphi_j'(v_j(n))
Hidden Local
\delta_j(n)=-
\frac{\partial\mathfrak E (n)}{\partial y_j(n)}
\frac{\partial y_j(n)}{\partial v_j(n)}$$
$$=-
\frac{\partial\mathfrak E (n)}{\partial y_j(n)}
\cdot
\varphi_j'(v_j(n))$$
$$\delta_j(n)=
\varphi_j'(v_j(n))
\cdot
\sum_k \delta_k(n)\cdot w_{kj}(n)$$
## Weight Correction
$$\text{weight correction = learning rate $\cdot$ local gradient $\cdot$ input signal of neuron $j$}$$
$$\Delta w_{ji}(n)=\eta\cdot\delta_j(n)\cdot y_i(n)$$
- Looking for partial derivative of error with respect to each weight
- 4 partial derivatives
1. Sum of squared errors WRT error in one output node
2. Error WRT output $y$
3. Output $y$ WRT Pre-activation function sum
4. Pre-activation function sum WRT weight
- Other [weights](../Weight%20Init.md) constant, goes to zero
- Leaves just $y_i$
- Collect 3 boxed terms as delta $j$
- Local gradient
- Weight correction can be too slow raw
- Gets stuck
- Add momentum
- Nodes further back
- More complicated
- Sum of later local gradients multiplied by backward weight (orange)
- Multiplied by differential of activation function at node
## Global Minimum
- Much more complex error surface than least-means-squared
- No guarantees of convergence
- Non-linear optimisation
- Momentum
- $+\alpha\Delta w_{ji}(n-1), 0\leq|\alpha|<1$
- Proportional to the change in weights last iteration
- Can shoot past local minima if descending quickly
$w^+_5=w_5-\eta\cdot\frac{\partial E_{total}}{\partial w_5}$
