Question

Suppose we modify the Perceptron algorithm as follows: In the update step, instead of performing wᵗ⁺¹ =wᵗ +yᵢxᵢ , whenever we make a mistake, we instead perform wᵗ⁺¹=wᵗ +myᵢ xᵢ , for some η>0;η is sometimes referred to as the learning rate or the step size. Show that this modified Perceptron will perform the same number of iterations as the original Perceptron we studied in class, and that it will converge to a vector that points in the same direction as the output of the vanilla Perceptron. Hint: What can you say about the relationship between the signs of ⟨w,x⟩ and ⟨ηw,x⟩ ?

Answer 1

The modified Perceptron algorithm's convergence in terms of iterations and the direction of the resulting weight vector is unaffected by the introduction of a positive learning rate, since scalar multiplication by a positive value does not change the direction of a vector, only its magnitude.

Step-by-step explanation:

The question at hand involves a modification to the Perceptron algorithm, which is a foundational concept in machine learning, particularly within the study of artificial neural networks. In the scenario given, an update step involves multiplying the error term by a constant, η (eta), which represents the learning rate.

Considering the vector nature of the weights and input, updating the weights with eta times the error (wᵗʹ⁺¹ = wᵗ + ηyᵢ xᵢ) will not change the direction in which the weights are adjusted, only the magnitude of the adjustment. Therefore, irrespective of the values of η, as long as it is positive, it will not affect the convergence of the algorithm in terms of the number of iterations required.

The relationship between ⟨w, x⟩ and ⟨ηw, x⟩ in terms of signs is crucial here; they are the same because scalar multiplication does not affect the sign of the inner product when the scalar is positive. Since the update to the weight vector is merely scaled, and the direction of update remains the same, the modified Perceptron will converge to a vector that points in the same direction as the original Perceptron algorithm would produce, if given the same data and conditions.