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Presentation transcript:

Lecture 07: Hard-margin SVM CS480/680: Intro to ML Lecture 07: Hard-margin SVM 11/01/2019 Yao-Liang Yu

Outline Maximum Margin Lagrangian Dual Alternative View 11/01/2019 Yao-Liang Yu

Perceptron revisited Two classes: y = 1 or y = -1 Assuming linear separable exist w and b such that for all i, yi(wTxi + b) > 0 Find any such w and b feasibility problem 11/01/2019 Yao-Liang Yu

Margin Take any linear separating hyperplane H for all i, yi(wTxi + b) > 0 Move H until it touches some positive point, H1 increase b say Move H until it touches some negative point, H-1 decrease b say margin = dist(H1,H) ∧ dist(H-1,H) 11/01/2019 Yao-Liang Yu

Put into formula H : wTx + b = 0 H1 : wTx + b = t H-1 : wTx + b = -s What is the distance between H1 and H ? equality is attained 11/01/2019 Yao-Liang Yu

Put into formula H : wTx + b = 0 H1 : wTx + b = t H-1 : wTx + b = -t What is the distance between H1 and H ? 11/01/2019 Yao-Liang Yu

Put into formula H : wTx + b = 0 H1 : wTx + b = 1 H-1 : wTx + b = -1 What is the distance between H1 and H ? 11/01/2019 Yao-Liang Yu

Maximum Margin Important facts. For any f, For positive f, For s. monotone g, 11/01/2019 Yao-Liang Yu

Hard-margin Support Vector Machines Linear programming Quadratic programming margin 11/01/2019 Yao-Liang Yu

Support Vectors Those touch the parallel hyperplanes H1 and H-1 Usually only a handful Entirely determine the hyperplanes! 11/01/2019 Yao-Liang Yu

Existence and uniqueness Always exists a minimizer w and b (if linearly separable) The minimizer w is unique (strong convexity of ) The minimizer b is also unique (why?) 11/01/2019 Yao-Liang Yu

Outline Maximum Margin Lagrangian Dual Alternative View 11/01/2019 Yao-Liang Yu

Lagrangian Primal Lagrangian Lagrangian multiplier [dual variable] [primal variable] 11/01/2019 Yao-Liang Yu

Joseph-Louis Lagrange (1736-1813) 11/01/2019 Yao-Liang Yu

Optimization detour Fermat’s Theorem. Necessarily (Fréchet) Derivative at x. Example. f(x) = xTAx + xTb + c Df(x) = (A+AT)x + b 11/01/2019 Yao-Liang Yu

Just in case 11/01/2019 Yao-Liang Yu

Deriving the dual 11/01/2019 Yao-Liang Yu

The dual problem Dual Rn Only need dot product in the dual ! 11/01/2019 Yao-Liang Yu

Support Vectors 11/01/2019 Yao-Liang Yu

Outline Maximum Margin Dual Alternative View 11/01/2019 Yao-Liang Yu

An dual view 11/01/2019 Yao-Liang Yu

Convex sets and Convex hull Convex set. A point set is convex if the line segment [x,y] connecting any two points x and y in C lies entirely in C. Convex hull. Smallest convex set containing C. 11/01/2019 Yao-Liang Yu

Regularization vs. Constraint Computationally appealing Always true Mild conditions Theoretically appealing can use equality if from top to bottom 11/01/2019 Yao-Liang Yu

From regularization to constraint 11/01/2019 Yao-Liang Yu

Homogeneity 11/01/2019 Yao-Liang Yu

Split 11/01/2019 Yao-Liang Yu

NOW this 11/01/2019 Yao-Liang Yu

Questions? 11/01/2019 Yao-Liang Yu