Perceptron

Let \((x_1, y_1), ..., (x_n, y_n) \in R^d \times \{ ±1 \}\) be labeled training data. The data is said to be linearly separable if there exists a hyperplane that correctly classifies all the examples :

\[\forall t \in n, y_t \langle w^*, x_t \rangle > 0\]

In general, finding \(w^*\) is impossible, but we search for some \(\hat{w}\) that separates the 2 classes. The objective function to optimize is :

\[f(w) = \sum_t 1(y_t \langle w^*, x_t \rangle ≥ 0)\]

This is called a batch objective since it relies on a cumulative fit to data. By our assumption : \(f(w^*) = 0\). There are however many solution hyperplanes if we consider scaling of \(w^*\). To solve this, we fix \(w^*\) to be the smallest-norm vector that guarantees :

\[\forall t \in n, y_t \langle w^*, x_t \rangle ≥ 1\]

How can we solve this? Using the Perceptron recursive update. It has been shown that the perceptron converges to a solution in a finite number of steps.

The algorithm of the Perceptron is the following :

image

Conclusion : That’s it ! I hope this introduction to Online Learning was clear. Don’t hesitate to drop a comment if you have any question.


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