Mistake bound The perceptron algorithm satis es many nice properties. If our input points are \genuinely" linearly separable, it must not matter, for example, what convention we adopt to de ne signpq, or if we interchange the labels of the points and the points. Perceptron是针对线性可分数据的一种分类器，它属于Online Learning的算法。 我在之前的一篇博文中提到了Online Learning模型的Mistake Bound衡量标准。 现在我们就来分析一下Perceptron的Mistake Bound是多少。 The bound holds for any sequence of instance-label pairs, and compares the number of mistakes made by the Perceptron with the cumulative hinge-loss of any fixed hypothesis g ∈ HK, even one defined with prior knowledge of the sequence. A relative mistake bound can be proven for the Perceptron algorithm. one, we obtain a nice guarantee of generalization. rounds. The new al-gorithm performs a Perceptron-style update whenever the margin of an example is smaller than a predefined value. Perceptron的Mistake Bound. with the Perceptron algorithm is 0( kN) mistakes, which comes from the classical Perceptron Convergence Theorem [ 41. good generalization error! Practical use of the Perceptron algorithm 1.Using the Perceptron algorithm with a finite dataset In section 3.2, the authors derive a mistake bound for Perceptron, this time assuming that the dataset is inseparable. For a positive example, the Perceptron update will increase the score assigned to the same input Similar reasoning for negative examples 17 Mistake on positive: 3)*!←3 ... •Variants of Perceptron •Perceptron Mistake Bound 31. • It’s an upper bound on the number of mistakes made by an . Abstract. The bound is after all cast in terms of the number of updates based on mistakes. In section 3.1, the authors introduce a mistake bound for Perceptron, assuming that the dataset is linearly separable. An angular margin of means that a point x imust be rotated about the origin by an angle at least 2arccos() to change its label. As a byproduct we obtain a new mistake bound for the Perceptron algorithm in the inseparable case. We derive worst case mista ke bounds for our algorithm. We also show that the Perceptron algorithm in its basic form can make 2k( N - k + 1) + 1 mistakes, so the bound is essentially tight. online algorithm. •Often these parameters are called weights. on an . Perceptron Mistake Bound Theorem: For any sequence of training examples =( 1, 1,…,( , ) with =max , if there exists a weight vector with =1 and ⋅ ≥ for all 1≤≤, then the Perceptron makes at most 2 2 errors. Maximum margin classifier? What Good is a Mistake Bound? (Upper bound on #mistakes[Perceptron].) of examples • Online algorithms with small mistake bounds can be used to develop classifiers with . Theorem 1. no i.i.d. The mistake bound for the perceptron algorithm is 1= 2 where is the angular margin with which hyperplane w:xseparates the points x i. We have so far used a simple on-line algorithm, the perceptron algorithm, to estimate a arbitrary sequence . Here we’ll prove a simple one, called a mistake bound: if there exists an optimal parameter vector w that can classify all of our examples correctly, then the perceptron algorithm will make at most a small number of mistakes before dis-covering an optimal parameter vector. = min i2[m] jx i:wj (1) 1.1 Perceptron algorithm 1.Initialize w 1 = 0. Perceptron Perceptron is an algorithm for binary classification that uses a linear prediction function: f(x) = 1, wTx+ b ≥ 0-1, wTx+ b < 0 By convention, the slope parameters are denoted w (instead of m as we used last time). assumption and not loading all the data at once! 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