81 lines
2.8 KiB
Markdown
81 lines
2.8 KiB
Markdown
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- To handle overlapping classes
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- Linearity condition remains
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- Linear boundary
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- No hard limiter
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- Linear neuron
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- Cost function changed to error, $J$
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- Half doesn’t matter for error
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- Disappears when differentiating
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$$\mathfrak{E}(w)=\frac{1}{2}e^2(n)$$
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- Cost' w.r.t to weights
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$$\frac{\partial\mathfrak{E}(w)}{\partial w}=e(n)\frac{\partial e(n)}{\partial w}$$
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- Calculate error, define delta
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$$e(n)=d(n)-x^T(n)\cdot w(n)$$
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$$\frac{\partial e(n)}{\partial w(n)}=-x(n)$$
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$$\frac{\partial \mathfrak{E}(w)}{\partial w(n)}=-x(n)\cdot e(n)$$
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- Gradient vector
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- $g=\nabla\mathfrak{E}(w)$
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- Estimate via:
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$$\hat{g}(n)=-x(n)\cdot e(n)$$
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$$\hat{w}(n+1)=\hat{w}(n)+\eta \cdot x(n) \cdot e(n)$$
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- Above is a feedback loop around weight vector, $\hat{w}$
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- Behaves like low-pass filter
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- Pass low frequency components of error signal
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- Average time constant of filtering action inversely proportional to learning-rate
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- Small value progresses algorithm slowly
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- Remembers more
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- Inverse of learning rate is measure of memory of LMS algorithm
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- $\hat{w}$ because it's an estimate of the weight vector that would result from steepest descent
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- Steepest descent follows well-defined trajectory through weight space for a given learning rate
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- LMS traces random trajectory
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- Stochastic gradient algorithm
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- Requires no knowledge of environmental statistics
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## Analysis
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- Convergence behaviour dependent on statistics of input vector and learning rate
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- Another way is that for a given dataset, the learning rate is critical
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- Convergence of the mean
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- $E[\hat{w}(n)]\rightarrow w_0 \text{ as } n\rightarrow \infty$
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- Converges to Wiener solution
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- Not helpful
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- Convergence in the mean square
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- $E[e^2(n)]\rightarrow \text{constant, as }n\rightarrow\infty$
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- Convergence in the mean square implies convergence in the mean
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- Not necessarily converse
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## Advantages
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- Simple
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- Model independent
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- Robust
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- Optimal in accordance with $H^\infty$, minimax criterion
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- _If you do not know what you are up against, plan for the worst and optimise_
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- ___Was___ considered an instantaneous approximation of gradient-descent
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## Disadvantages
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- Slow rate of convergence
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- Sensitivity to variation in eigenstructure of input
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- Typically requires iterations of 10 x dimensionality of the input space
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- Worse with high-d input spaces
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![[slp-mse.png]]
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- Use steepest descent
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- Partial derivatives
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![[slp-steepest-descent.png]]
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- Can be solved by matrix inversion
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- Stochastic
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- Random progress
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- Will overall improve
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![[lms-algorithm.png]]
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$$\hat{w}(n+1)=\hat{w}(n)+\eta\cdot x(n)\cdot[d(n)-x^T(n)\cdot\hat w(n)]$$
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$$=[I-\eta\cdot x(n)x^T(n)]\cdot\hat{w}(n)+\eta\cdot x(n)\cdot d(n)$$
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Where
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$$\hat w(n)=z^{-1}[\hat w(n+1)]$$
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## Independence Theory
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![[slp-lms-independence.png]]
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![[sl-lms-summary.png]]
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