42 lines
2.0 KiB
Markdown
42 lines
2.0 KiB
Markdown
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Error-Correcting Perceptron Learning
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- Uses a McCulloch-Pitt neuron
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- One with a hard limiter
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- Unity increment
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- Learning rate of 1
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If the $n$-th member of the training set, $x(n)$, is correctly classified by the weight vector $w(n)$ computed at the $n$-th iteration of the algorithm, no correction is made to the weight vector of the perceptron in accordance with the rule:
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$$w(n + 1) = w(n) \text{ if $w^Tx(n) > 0$ and $x(n)$ belongs to class $\mathfrak{c}_1$}$$
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$$w(n + 1) = w(n) \text{ if $w^Tx(n) \leq 0$ and $x(n)$ belongs to class $\mathfrak{c}_2$}$$
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Otherwise, the weight vector of the perceptron is updated in accordance with the rule
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$$w(n + 1) = w(n) - \eta(n)x(n) \text{ if } w^Tx(n) > 0 \text{ and } x(n) \text{ belongs to class }\mathfrak{c}_2$$
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$$w(n + 1) = w(n) + \eta(n)x(n) \text{ if } w^Tx(n) \leq 0 \text{ and } x(n) \text{ belongs to class }\mathfrak{c}_1$$
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1. _Initialisation_. Set $w(0)=0$. perform the following computations for
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time step $n = 1, 2,...$
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2. _Activation_. At time step $n$, activate the perceptron by applying continuous-valued input vector $x(n)$ and desired response $d(n)$.
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3. _Computation of Actual Response_. Compute the actual response of the perceptron:
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$$y(n) = sgn[w^T(n)x(n)]$$
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where $sgn(\cdot)$ is the signum function.
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4. _Adaptation of Weight Vector_. Update the weight vector of the perceptron:
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$$w(n+1)=w(n)+\eta[d(n)-y(n)]x(n)$$ where
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$$
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d(n) = \begin{cases}
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+1 &\text{if $x(n)$ belongs to class $\mathfrak{c_1}$}\\
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-1 &\text{if $x(n)$ belongs to class $\mathfrak{c_2}$}
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\end{cases}
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$$
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5. _Continuation_. Increment time step $n$ by one and go back to step 2.
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- Guarantees convergence provided
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- Patterns are linearly separable
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- Non-overlapping classes
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- Linear separation boundary
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- Learning rate not too high
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- Two conflicting requirements
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1. Averaging of past inputs to provide stable weight estimates
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- Small eta
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2. Fast adaptation with respect to real changes in the underlying distribution of process responsible for $x$
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- Large eta
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![[slp-separable.png]]
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