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41 Pages

### cs101.2-08-active-learning

Course: CS 101, Fall 2009
School: Caltech
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Word Count: 1507

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Learning Active and Optimized Information Gathering Lecture 8 Active Learning CS 101.2 Andreas Krause Announcements Homework 1: Due today Office hours Come to office hours before your presentation! Andreas: Monday 3pm-4:30pm, 260 Jorgensen Ryan: Wednesday 4:00-6:00pm, 109 Moore 2 Outline Background in learning theory Sample complexity Key challenges Heuristics for active learning Principled algorithms for...

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Learning Active and Optimized Information Gathering Lecture 8 Active Learning CS 101.2 Andreas Krause Announcements Homework 1: Due today Office hours Come to office hours before your presentation! Andreas: Monday 3pm-4:30pm, 260 Jorgensen Ryan: Wednesday 4:00-6:00pm, 109 Moore 2 Outline Background in learning theory Sample complexity Key challenges Heuristics for active learning Principled algorithms for active learning 3 Spam or Ham? Spam x2 + + o o o o o o + Ham o o o o o o o o x1 label = sign(w0 + w1 x1 + w2 x2) (linear separator) Labels are expensive (need to ask expert) Which labels should we obtain to maximize classification accuracy? 4 Recap: Concept learning Set X of instances, with distribution PX True concept c: X {0,1} Data set D = {(x1,y1),,(xn,yn)}, xi PX, yi = c(xi) Hypothesis h: X {0,1} from H = {h1, , hn, } Assume c H (c also called target hypothesis) errortrue(h) = EX |c(x)-h(x)| errortrain(h) = (1/n) i |c(xi)-h(xi)| If n large enough, errortrue(h) errortrain(h) for all h 5 Recap: PAC Bounds How many samples n to we need to get error with probability 1- ? No noise: n 1/ ( log |H| + log 1/ ) Noise: n 1/2 ( log |H| + log 1/ ) Requires that data is i.i.d.! Today: Mainly no-noise case (more next week) 6 Statistical passive/active learning protocol Data source PX (produces inputs xi) Active learner assembles data set Dn = {(x1,y1),,(xn,yn)} by selectively obtaining labels Learner outputs hypothesis h errortrue(h) = Ex~P[h(x) c(x)] Data set NOT sampled i.i.d.!! 7 Example: Uncertainty sampling Budget of m labels Draw n unlabeled examples Repeat until weve picked m labels Assign each unlabeled data an uncertainty score Greedily pick the most uncertain example One of the most commonly used class of heuristics! 8 Uncertainty sampling for linear separators 9 Active learning bias 10 Active learning bias If we can pick at most m = n/2 labels, with overwhelmingly high probability, US pick points such that there remains a hypothesis with error > .1!!! With standard passive learning, error 0 as n 11 Wish list for active learning Minimum requirement Consistency: Generalization error should go to 0 asymptotically Wed like more than that: Fallback guarantee: Convergence rate of error of active learning at least as good as passive learning What were really after Rate improvement: Error of active learning decreases much faster than for passive learning 12 From passive to active Passive PAC learning 1. 2. 3. Collect data set D of n 1/ ( log |H| + log 1/ ) data points and their labels i.i.d. from PX Output consistent hypothesis h With probability at least 1-, errortrue(h) Key idea Sample n unlabeled data points DX={x1,,xn} i.i.d. Actively query labels until all hypotheses consistent with these labels agree on the labels of all unlabeled data 13 Why might this work? 14 Formalization: Relevant hypothesis Data set D = {(x1,y1),,(xn,yn)}, Hypothesis space H Input data: DX = {x1,,xn} Relevant hypothesis H(DX) = H = Restriction of H on DX Formally: H = {h: DX {0,1} h H s.t. x DX: h(x)=h(x)} 15 Example: Threshold functions 16 Version space Input data DX = {x1,,xn} Partially labeled: Have L = {(xi ,yi ),,(xi ,yi )} 1 1 m m The (relevant) version space is the set of all relevant hypotheses consistent with the labels L Formally: Why useful? Partial labels L imply all remaining labels for DX |V|=1 17 Version space Input data DX = {x1,,xn} Partially labeled: Have L = {(xi ,yi ),,(xi ,yi )} 1 1 m m The (relevant) version space is the set of all relevant hypotheses consistent with the labels L Formally: V(DX,L) = V = {h H(DX): h(xi )=yi for 1 j m} j j Why useful? Partial labels L imply all remaining labels for DX |V|=1 18 Example: Binary thresholds 19 Pool-based active learning with fallback 1. 2. 3. Collect n 1/ ( log |H| + log 1/ ) unlabeled data points DX from PX Actively request labels L until there remains a single hypothesis h H thats consistent with these labels (i.e., |V(H,L)| = 1) Output any hypothesis hH consistent with the obtained labels. With probability 1- errortrue(h) Get PAC guarantees for active learning Bounds on #labels for fixed error carry over from passive to active Fallback guarantee 20 Wish list for active learning Minimum requirement Consistency: Generalization error should go to 0 asymptotically Wed like more than that: Fallback guarantee: Convergence rate of error of active learning at least as good as passive learning What were really after Rate improvement: of Error active learning decreases much faster than for passive learning 21 Pool-based active learning with fallback 1. 2. 3. Collect n 1/ ( log |H| + log 1/ ) unlabeled data points DX from PX Actively request labels L until there remains a single hypothesis h H thats consistent with these labels (i.e., |V(H,L)| = 1) Output any hypothesis hH consistent with the obtained labels. With probability 1- errortrue(h) 22 Example: Threshold functions 23 Generalizing binary search [Dasgupta 04] Want to shrink the version space (number of consistent hypotheses) as quickly as possible. General (greedy) approach: For each unlabeled instance xi compute vi,1 = vi,0 = vi = min {vi,1, vi,0 } Obtain label yi for xi where i = argmaxj {vj} 24 Ideal case 25 Is it always possible to half the version space? 26 Typical case much more benign 27 Query trees A query tree is a rooted, labeled tree on the relevant hypothesis H Each node is labeled with an input x DX Each edge is labeled with {0,1} Each path from root to hypothesis h H is a labeling L such that V(DX,L) = {h} Want query trees of minimum height 28 Example: Threshold functions 29 Example: linear separators (2D) 30 Number of labels needed to identify hypothesis Depends on target hypothesis! Binary thresholds (on n inputs D_X) Optimal query tree needs O(log n) labels! For linear separators in 2D (on n inputs D_X) For some hypotheses, even optimal tree needs n labels On average, optimal query tree needs O(log n) labels! Average-case analysis of active learning 31 Average case query tree learning Query tree T Cost(T) = 1/|H| h` H depth(h,T) Want T* = argminT Cost(T) Superexponential number of query trees Finding the optimal one is hard 32 Greedy construction of query trees [Dasgupta 04] Algorithm GreedyTree(DX, L) V = H(DX) If V={h} return Leaf(h) Else For each unlabeled instance xi compute vi,1 = |V(H,L {(xi,1)}| and vi,0 = |V(H,L {(xi,0)}| vi = min {vi,1, vi,0} Let i = argmaxj {vj} LeftSubTree = GreedyTree(DX, L {(xi,1)}) RightSubTree = GreedyTree(DX, L {(xi,0)}) return Node xi with children LeftSubTree (1) and RightSubTree(0) 33 Near-optimality of greedy tree [Dasgupta 04] Theorem: Let T* = argminT Cost(T) Then GreedyTree constructs a query tree T such that Cost(T) = O(log |H|) Cost(T*) 34 Limitations of this algorithm Often computationally intractable Finding most-disagreeing hypothesis is difficult No-noise assumption Will see how we can relax these assumptions in the talks next week. 35 Bayesian or not Bayesian? Greedy querying needs at most O(log |H|) queries more than optimal query tree on average Assumes prior distribution (uniform) on hypotheses If our assumption is wrong, generalizat...

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