Intuition for perceptron weight update rule

Question

I am having trouble understanding the weight update rule for perceptrons:

w(t + 1) = w(t) + y(t)x(t).

Assume we have a linearly separable data set.

• w is a set of weights [w0, w1, w2, ...] where w0 is a bias.
• x is a set of input parameters [x0, x1, x2, ...] where x0 is fixed at 1 to accommodate the bias.

At iteration t, where t = 0, 1, 2, ...,

• w(t) is the set of weights at iteration t.
• x(t) is a misclassified training example.
• y(t) is the target output of x(t) (either -1 or 1).

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Why does this update rule move the boundary in the right direction?

Solution 1

The perceptron's output is the hard limit of the dot product between the instance and the weight. Let's see how this changes after the update. Since

w(t + 1) = w(t) + y(t)x(t),

then

x(t) ⋅ w(t + 1) = x(t) ⋅ w(t) + x(t) ⋅ (y(t) x(t)) = x(t) ⋅ w(t) + y(t) [x(t) ⋅ x(t))].

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Note that:

• By the algorithm's specification, the update is only applied if x(t) was misclassified.
• By the definition of the dot product, x(t) ⋅ x(t) ≥ 0.

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How does this move the boundary relative to x(t)?

• If x(t) was correctly classified, then the algorithm does not apply the update rule, so nothing changes.
• If x(t) was incorrectly classified as negative, then y(t) = 1. It follows that the new dot product increased by x(t) ⋅ x(t) (which is positive). The boundary moved in the right direction as far as x(t) is concerned, therefore.
• Conversely, if x(t) was incorrectly classified as positive, then y(t) = -1. It follows that the new dot product decreased by x(t) ⋅ x(t) (which is positive). The boundary moved in the right direction as far as x(t) is concerned, therefore.

Solution 2

A better derivation of the perceptron update rule is documented here and here. The derivation is using gradient descent.

• Basic premise of gradient descent algorithm is find the error of classification and make your parameters so that error is minimized.

PS: I was trying very hard to get the intuition on why would someone multiply x and y to derive the update for w. Because w is the slope for a single dimension (y = wx+c) and slope w = (y/x) and not y * x.

Comments

[joshreesjones]

Can you expand a little on the last sentence? What happens when x(t) is classified and when x(t) is misclassified?