CS489/698 Lecture 2: January 8 th, 2018
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1 CS489/698 Lecture 2: January 8 th, 2018 Nearest Neighbour [RN] Sec , [HTF] Sec , [D] Chapt. 3, [B] Sec , [M] Sec CS489/698 (c) 2018 P. Poupart 1
2 Inductive Learning (recap) Induction Given a training set of examples of the form (x, f x ) x is the input, f(x) is the output Return a function h that approximates f h is called the hypothesis CS489/698 (c) 2018 P. Poupart 2
3 Supervised Learning Two types of problems 1. Classification: 2. Regression NB: The nature (categorical or continuous) of the domain (input space) of f does not matter CS489/698 (c) 2018 P. Poupart 3
4 Classification Example Problem: Will you enjoy an outdoor sport based on the weather? Training set: Sky Humidity Wind Water Forecast EnjoySport Sunny Normal Strong Warm Same yes Sunny High Strong Warm Same yes Sunny High Strong Warm Change no Sunny High Strong Cool Change yes Possible Hypotheses: h ' : S = sunny enjoysport = yes h 6 : Wa = cool or F = same enjoysport = yes x f(x) CS489/698 (c) 2018 P. Poupart 4
5 Regression Example Find function h that fits f at instances x CS489/698 (c) 2018 P. Poupart 5
6 More Examples Problem Domain Range Classification / Regression Spam Detection Stock price prediction Speech recognition Digit recognition Housing valuation Weather prediction CS489/698 (c) 2018 P. Poupart 6
7 Hypothesis Space Hypothesis space H Set of all hypotheses h that the learner may consider Learning is a search through hypothesis space Objective: find h that minimizes Misclassification Or more generally some error function with respect to the training examples But what about unseen examples? CS489/698 (c) 2018 P. Poupart 7
8 Generalization A good hypothesis will generalize well i.e., predict unseen examples correctly Usually Any hypothesis h found to approximate the target function f well over a sufficiently large set of training examples will also approximate the target function well over any unobserved examples CS489/698 (c) 2018 P. Poupart 8
9 Inductive Learning Goal: find an h that agrees with f on training set h is consistent if it agrees with f on all examples Finding a consistent hypothesis is not always possible Insufficient hypothesis space: E.g., it is not possible to learn exactly f x = ax + b + xsin(x) when H = space of polynomials of finite degree Noisy data E.g., in weather prediction, identical conditions may lead to rainy and sunny days CS489/698 (c) 2018 P. Poupart 9
10 Inductive Learning A learning problem is realizable if the hypothesis space contains the true function otherwise it is unrealizable. Difficult to determine whether a learning problem is realizable since the true function is not known It is possible to use a very large hypothesis space For example: H = class of all Turing machines But there is a tradeoff between expressiveness of a hypothesis class and the complexity of finding a good hypothesis CS489/698 (c) 2018 P. Poupart 10
11 Nearest Neighbour Classification Classification function h x = y A where y A is the label associated with the nearest neighbour x = argmin A D d(x, x G ) Distance measures: d x, x G L ' : d x, x G L G = x J x J L 6 : d x, x G = L G x J x 6 J J L N : d x, x G L G = x J x N J J J '/6 '/N Weighted dimensions: d x, x G L G = c J x J x N J J '/N CS489/698 (c) 2018 P. Poupart 11
12 Voronoi Diagram Partition implied by nearest neighbor fn h Assuming Euclidean distance CS489/698 (c) 2018 P. Poupart 12
13 K-Nearest Neighbour Nearest neighbour often instable (noise) Idea: assign most frequent label among k- nearest neighbours Let knn x be the k-nearest neighbours of x according to distance d Label: y A mode( y A D x G knn(x) ) CS489/698 (c) 2018 P. Poupart 13
14 Effect of K K controls the degree of smoothing. Which partition do you prefer? Why? CS489/698 (c) 2018 P. Poupart 14
15 Performance of a learning algorithm A learning algorithm is good if it produces a hypothesis that does a good job of predicting classifications of unseen examples Verify performance with a test set 1. Collect a large set of examples 2. Divide into 2 disjoint sets: training set and test set 3. Learn hypothesis h with training set 4. Measure percentage of correctly classified examples by h in the test set CS489/698 (c) 2018 P. Poupart 15
16 The effect of K Best K depends on Problem Amount of training data % correct K CS489/698 (c) 2018 P. Poupart 16
17 Underfitting Definition: underfitting occurs when an algorithm finds a hypothesis h with training accuracy that is lower than the future accuracy of some other hypothesis h Amount of underfitting of h: max {0, max futureaccuracy h trainaccuracy h } YG max {0, max YG Common cause: Classifier is not expressive enough testaccuracy h trainaccuracy h } CS489/698 (c) 2018 P. Poupart 17
18 Overfitting Definition: overfitting occurs when an algorithm finds a hypothesis h with higher training accuracy than its future accuracy. Amount of overfitting of h: max {0, trainaccuracy h futureaccuracy h } max {0, trainaccuracy h testaccuracy h } Common causes: Classifier is too expressive Noisy data Lack of data CS489/698 (c) 2018 P. Poupart 18
19 Choosing K How should we choose K? Ideally: select K with highest future accuracy In practice: select K with highest test accuracy Need to ensure that test accuracy is close to future accuracy Test accuracy becomes more reliable as we increase the size of the test set However, this reduces the amount of data left for training Popular solution: cross-validation CS489/698 (c) 2018 P. Poupart 19
20 Cross-Validation Repeatedly split data in two parts, one for training, one for testing the accuracy of a hypothesis and report the average accuracy. k-fold cross validation: split data in k equal size subsets. Run k experiments, each time testing on one subset the hypothesis trained on the remaining subsets. Compute the average accuracy of the k experiments. Picture: CS489/698 (c) 2018 P. Poupart 20
21 Selecting the Number of Neighbours by Cross-Validation Let k be the number of neighbours Let k be the number of data splits For k = 1 to max # of neighbours For i = 1 to k do h _` train(k, data '..`b',`c'.._ D) acc _` test(h _`, data`) acc _ average( acc _` `) k argmax _ acc _ h _ train(k, data '.._ D) Return k, h _, acc _ CS489/698 (c) 2018 P. Poupart 21
22 Choosing a Hypothesis After determining the optimal number of neighbours k by cross validation, what hypothesis should we return? We could return any of the h _` hypotheses, but we can do better. Instead return a new hypothesis h _ trained with all the data The future accuracy of h _ is likely to be better than the future accuracy of each h _ ` since h _ is trained with all the data acc _ (average test accuracy of each h _ `) is a good (conservative) estimate of the future accuracy of h _ CS489/698 (c) 2018 P. Poupart 22
23 Weighted K-Nearest Neighbour We can often improve K-nearest neighbours by weighting each neighbour based on some distance measure w x, x G 1 distance x, x G Label y A argmax i j w(x, x G ) {A D A D _mm A ioi p D} CS489/698 (c) 2018 P. Poupart 23
24 K-Nearest Neighbour Regression We can also use knn for regression Let y A be a real value instead of a categorical label K-nearest neighbour regression: y A average( y A D x G knn x ) Weighted K-nearest neighbour regression: y A A D _mm A w x, x G y A D w(x, x G ) A D _mm(a) CS489/698 (c) 2018 P. Poupart 24
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