The question asks for the correct weight update rule in the perceptron algorithm using a gradient descent procedure. Let's analyze the choices:

Perceptron Algorithm Weight Update Rule

The perceptron algorithm updates the weights based on the misclassified points. The weight update rule in perceptron learning is generally defined as:

$\mathbf{w} \leftarrow \mathbf{w} + \eta \cdot y_i \cdot \mathbf{x}_i$

where:

• $\mathbf{w}$ is the weight vector.
• $\eta$ is the learning rate.
• $y_i$ is the true label of the data point (either +1 or -1).
• $\mathbf{x}_i$ is the input feature vector of the misclassified point.

The update is only applied for misclassified points, so the summation will be over all misclassified points.

Analyzing the Options:

• Option (a): $a(k+1) = a(k) - \eta \sum (\text{for all } y \text{ misclassified}) y$
  This option subtracts the summation of misclassified terms, which does not follow the correct update rule since we should add these terms.

• Option (b): $a(k+1) = a(k) + \eta \sum (\text{for all } y \text{ misclassified}) y$
  This option adds the summation of the misclassified terms with the correct learning rate $\eta$. This is consistent with the perceptron weight update rule.

• Option (c): $a(k+1) = a(k) - \eta \sum (\text{for all } y \text{ classified}) y$
  This option considers correctly classified points instead of misclassified ones, which is not correct.

• Option (d): None of these.
  This would be correct if none of the other options matched the correct rule.

Conclusion:

The correct answer is (b):
$a(k+1) = a(k) + \eta \sum (\text{for all } y \text{ misclassified}) y$

This follows the correct perceptron weight update rule by applying gradient descent for all misclassified points.

Would you like more details or have any further questions?

Related Questions:

1. What is the perceptron learning algorithm?
2. How does the learning rate ($\eta$) affect the convergence of the perceptron algorithm?
3. What is the role of misclassified points in gradient descent?
4. Can the perceptron algorithm work for non-linearly separable data?
5. What modifications can be made to the perceptron algorithm to handle non-linearly separable data?

Tip: Always remember that the perceptron learning rule only updates weights for misclassified data points.