Learning to answer Yes or No

What hypothesis set can we use?

question.png

A Simple Hypothesis Set: the “Perceptron”

  • Perceptron in $\mathbb{R}^2$

perceptron

  • features x: points on the plane (or points in $\mathbb{R}^d$ )
  • labels y: (+1), × (-1)
  • hypothesis h: lines (or hyperplanes in $\mathbb{R}^d$ ),positive on one side of a line, negative on the other side
  • different line classifies simples differently
perceptrons <=> linear (binary) classifiers

Select g from $\mathscr{H}$

  • want: $g \approx f$ (hard when f unknown)
  • almost necessary: $g \approx f$ on D, ideally $g(x_n) = f(x_n) = y_n$
  • difficult: H is of infinite size
  • idea: start from some $g_0$, and correct its mistakes on $D$

Perceptron Learning Algorithm

Perceptron Learning Algorithm
PLA01.png
PLA02.png

  • if PLA halts (i.e. no more mistakes),
    (necessary condition) $D$ allows some w to make no mistake
  • call such $D$ linear separable
  • as long as linear separable and correct by mistake
    • inner product of $w_f$ and $w_t$ grows fast; length of wt grows slowly
    • PLA ‘lines’ are more and more aligned with $w_f \rightarrow halts$

Line with Noise Tolerance

$D$ is not linear separable?

not linear separable

Pocket Algorithm

Find the best weights in pocket until enough iterations

pocket algorithm

Question

Since we do not know whether D is linear separable in advance, we may decide to just go with pocket instead of PLA. If D is actually linear separable, what’s the difference between the two?
1 pocket on D is slower than PLA
2 pocket on D is faster than PLA
3 pocket on D returns a better g in approximating f than PLA
4 pocket on D returns a worse g in approximating f than PLA

answer: Because pocket need to check whether $w_{t+1}$ is better than $\hat{w}$ in each iteration, it is slower than PLA. On linear separable D, $w_{POCKET}$ is the same as $w_{PLA}$, both making no mistakes.

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