stem/AI/Neural Networks/SLP/Least Mean Square.md

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