112 lines
3.2 KiB
Markdown
112 lines
3.2 KiB
Markdown
Error signal graph
|
|
|
|
|
|
|
|
1. Error Signal
|
|
- $e_j(n)=d_j(n)-y_j(n)$
|
|
2. Net Internal Sum
|
|
- $v_j(n)=\sum_{i=0}^mw_{ji}(n)y_i(n)$
|
|
3. Output
|
|
- $y_j(n)=\varphi_j(v_j(n))$
|
|
4. Instantaneous Sum of Squared Errors
|
|
- $\mathfrak{E}(n)=\frac 1 2 \sum_{j\in C}e_j^2(n)$
|
|
- $C$ = o/p layer nodes
|
|
5. 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
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
$w^+_5=w_5-\eta\cdot\frac{\partial E_{total}}{\partial w_5}$
|
|
|
|
 |