问题
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).
Why does this update rule move the boundary in the right direction?
回答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))].
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.
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.
回答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.
来源：https://stackoverflow.com/questions/34477827/intuition-for-perceptron-weight-update-rule