CSC 411: Lecture 05: Nearest Neighbors
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1 CSC 411: Lecture 05: Nearest Neighbors Raquel Urtasun & Rich Zemel University of Toronto Sep 28, 2015 Urtasun & Zemel (UofT) CSC 411: 05-Nearest Neighbors Sep 28, / 13
2 Today Non-parametric models distance non-linear decision boundaries Urtasun & Zemel (UofT) CSC 411: 05-Nearest Neighbors Sep 28, / 13
3 Classification: Oranges and Lemons Urtasun & Zemel (UofT) CSC 411: 05-Nearest Neighbors Sep 28, / 13
4 Classification: Oranges and Lemons Can$construct$simple$ linear$decision$ boundary:$$$$ $$$y$=$sign(w 0 $+$w 1 x 1 $$$$$ $$$$$$$$+$w 2 x 2 )$ Urtasun & Zemel (UofT) CSC 411: 05-Nearest Neighbors Sep 28, / 13
5 What is the meaning of linear classification Classification is intrinsically non-linear It puts non-identical things in the same class, so a difference in the input vector sometimes causes zero change in the answer Linear classification means that the part that adapts is linear (just like linear regression) z(x) = w T x + w 0 with adaptive w, w 0 The adaptive part is follow by a non-linearity to make the decision y(x) = f (z(x)) What f have we seen so far in class? Urtasun & Zemel (UofT) CSC 411: 05-Nearest Neighbors Sep 28, / 13
6 Classification as Induction Urtasun & Zemel (UofT) CSC 411: 05-Nearest Neighbors Sep 28, / 13
7 Instance-based Learning Alternative to parametric model is non-parametric Simple methods for approximating discrete-valued or real-valued target functions (classification or regression problems) Learning amounts to simply storing training data Test instances classified using similar training instances Embodies often sensible underlying assumptions: Output varies smoothly with input Data occupies sub-space of high-dimensional input space Urtasun & Zemel (UofT) CSC 411: 05-Nearest Neighbors Sep 28, / 13
8 Nearest Neighbors Assume training examples correspond to points in d-dimensional Euclidean space Target function value for new query estimated from known value of nearest training example(s) Distance typically defined to be Euclidean: x (a) x (b) 2 = d (x (a) j j=1 x (b) j ) 2 Algorithm 1. find example (x, t ) closest to the test instance x (q) 2. output y (q) = t Note: we don t need to compute the square root. Why? Urtasun & Zemel (UofT) CSC 411: 05-Nearest Neighbors Sep 28, / 13
9 Nearest Neighbors Decision Boundaries Nearest neighbor algorithm does not explicitly compute decision boundaries, but these can be inferred Decision boundaries: Voronoi diagram visualization show how input space divided into classes each line segment is equidistant between two points of opposite classes Urtasun & Zemel (UofT) CSC 411: 05-Nearest Neighbors Sep 28, / 13
10 k Nearest Neighbors Nearest neighbors sensitive to mis-labeled data ( class noise ) smooth by having k nearest neighbors vote Algorithm: 1. find k examples {x (i), t (i) } closest to the test instance x 2. classification output is majority class y = arg max t (z) k δ(t (z), t (r) ) r=1 Urtasun & Zemel (UofT) CSC 411: 05-Nearest Neighbors Sep 28, / 13
11 k Nearest Neighbors: Issues & Remedies Some attributes have larger ranges, so are treated as more important normalize scale Irrelevant, correlated attributes add noise to distance measure eliminate some attributes or vary and possibly adapt weight of attributes Non-metric attributes (symbols) Hamming distance Brute-force approach: calculate Euclidean distance to test point from each stored point, keep closest: O(dn 2 ). We need to reduce computational burden: 1. Use subset of dimensions 2. Use subset of examples Remove examples that lie within Voronoi region Form efficient search tree (kd-tree), use Hashing (LSH), etc Urtasun & Zemel (UofT) CSC 411: 05-Nearest Neighbors Sep 28, / 13
12 Decision Boundary K-NN Urtasun & Zemel (UofT) CSC 411: 05-Nearest Neighbors Sep 28, / 13
13 K-NN Summary Single parameter (k) how do we set it? Naturally forms complex decision boundaries; adapts to data density Problems: Sensitive to class noise. Sensitive to dimensional scales. Distances are less meaningful in high dimensions Scales with number of examples Inductive Bias: What kind of decision boundaries do we expect to find? Urtasun & Zemel (UofT) CSC 411: 05-Nearest Neighbors Sep 28, / 13
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