Andy Pack
c3ebefdd64
Affected files: .obsidian/workspace.json Food/Plans & Recipes.canvas Health/ADHD.md Health/Alexithymia.md Health/Anxiety.md Health/BWS.md Health/Worry Tree.md History/Britain.md Media/Listen.md Media/Watch.md Money/Assets/Asset.md Money/Assets/CFD.md Money/Assets/Derivative.md Money/Assets/Financial Instruments.md Money/Assets/Options.md Money/Assets/Security.md Money/Assets/Stocks.md Money/Econ.md Money/Equity.md Money/Inflation.md Money/Markets/Commodity.md Money/Markets/Markets.md Money/Markets/Types.md Money/Prices.md Money/To Learn.md Politcs/British/Prime Ministers.md Politcs/British/Tory.md Politcs/Neoliberalism/Thatcher.md Politcs/Now.md Projects/Dev Options.md STEM/AI/Neural Networks/Activation Functions.md STEM/AI/Neural Networks/MLP/Back-Propagation.md STEM/AI/Neural Networks/SLP/Least Mean Square.md STEM/AI/Pattern Matching/Markov/Markov.md STEM/Maths/Algebra.md STEM/Maths/Tensor.md STEM/Maths/Vector Operators.md Want/Clothes.md Want/House.md Want/Want.md Work/Board.canvas
117 lines
3.3 KiB
Markdown
117 lines
3.3 KiB
Markdown
---
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tags:
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- ai
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- maths
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---
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Error signal graph
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1. Error Signal
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- $e_j(n)=d_j(n)-y_j(n)$
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2. Net Internal Sum
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- $v_j(n)=\sum_{i=0}^mw_{ji}(n)y_i(n)$
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3. Output
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- $y_j(n)=\varphi_j(v_j(n))$
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4. Instantaneous Sum of Squared Errors
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- $\mathfrak{E}(n)=\frac 1 2 \sum_{j\in C}e_j^2(n)$
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- $C$ = o/p layer nodes
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5. Average Squared Error
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- $\mathfrak E_{av}=\frac 1 N\sum_{n=1}^N\mathfrak E (n)$
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$$\frac{\partial\mathfrak E(n)}{\partial w_{ji}(n)}=
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\frac{\partial\mathfrak E(n)}{\partial e_j(n)}
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\frac{\partial e_j(n)}{\partial y_j(n)}
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\frac{\partial y_j(n)}{\partial v_j(n)}
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\frac{\partial v_j(n)}{\partial w_{ji}(n)}
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$$
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#### From 4
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$$\frac{\partial\mathfrak E(n)}{\partial e_j(n)}=e_j(n)$$
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#### From 1
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$$\frac{\partial e_j(n)}{\partial y_j(n)}=-1$$
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#### From 3 (note prime)
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$$\frac{\partial y_j(n)}{\partial v_j(n)}=\varphi_j'(v_j(n))$$
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#### From 2
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$$\frac{\partial v_j(n)}{\partial w_{ji}(n)}=y_i(n)$$
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## Composite
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$$\frac{\partial\mathfrak E(n)}{\partial w_{ji}(n)}=
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-e_j(n)\cdot
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\varphi_j'(v_j(n))\cdot
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y_i(n)
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$$
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$$\Delta w_{ji}(n)=-\eta\frac{\partial\mathfrak E(n)}{\partial w_{ji}(n)}$$
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$$\Delta w_{ji}(n)=\eta\delta_j(n)y_i(n)$$
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## Gradients
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#### Output Local
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$$\delta_j(n)=-\frac{\partial\mathfrak E (n)}{\partial v_j(n)}$$
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$$=-
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\frac{\partial\mathfrak E(n)}{\partial e_j(n)}
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\frac{\partial e_j(n)}{\partial y_j(n)}
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\frac{\partial y_j(n)}{\partial v_j(n)}$$
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$$=
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e_j(n)\cdot
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\varphi_j'(v_j(n))
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$$
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#### Hidden Local
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$$\delta_j(n)=-
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\frac{\partial\mathfrak E (n)}{\partial y_j(n)}
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\frac{\partial y_j(n)}{\partial v_j(n)}$$
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$$=-
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\frac{\partial\mathfrak E (n)}{\partial y_j(n)}
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\cdot
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\varphi_j'(v_j(n))$$
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$$\delta_j(n)=
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\varphi_j'(v_j(n))
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\cdot
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\sum_k \delta_k(n)\cdot w_{kj}(n)$$
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## Weight Correction
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$$\text{weight correction = learning rate $\cdot$ local gradient $\cdot$ input signal of neuron $j$}$$
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$$\Delta w_{ji}(n)=\eta\cdot\delta_j(n)\cdot y_i(n)$$
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- Looking for partial derivative of error with respect to each weight
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- 4 partial derivatives
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1. Sum of squared errors WRT error in one output node
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2. Error WRT output $y$
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3. Output $y$ WRT Pre-activation function sum
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4. Pre-activation function sum WRT weight
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- Other [weights](../Weight%20Init.md) constant, goes to zero
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- Leaves just $y_i$
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- Collect 3 boxed terms as delta $j$
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- Local gradient
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- Weight correction can be too slow raw
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- Gets stuck
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- Add momentum
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- Nodes further back
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- More complicated
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- Sum of later local gradients multiplied by backward weight (orange)
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- Multiplied by differential of activation function at node
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## Global Minimum
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- Much more complex error surface than least-means-squared
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- No guarantees of convergence
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- Non-linear optimisation
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- Momentum
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- $+\alpha\Delta w_{ji}(n-1), 0\leq|\alpha|<1$
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- Proportional to the change in weights last iteration
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- Can shoot past local minima if descending quickly
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$w^+_5=w_5-\eta\cdot\frac{\partial E_{total}}{\partial w_5}$
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