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Machine Learning & Data Mining CS/CNS/EE 155 Lecture 8: Structural SVMs Part 2 & General Structured Prediction 1

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Announcements Homework 2 due next Tuesday – 2pm via Moodle Homework 3 out next week – Due 2 weeks later – (Easier than HW2) Kaggle Mini-Project out next week – Due ~3 weeks later 2

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Kaggle Mini-Project Training set of ~5K labeled data points – ~50 features Test set of unlabeled data points – Submit predictions on test set You choose the methods, loss functions, feature manipulations, etc. – Expected to do cross validation & model selection – Written report Clearly & concisely documenting your process Template will be provided – Groups of up to 3 Due after ~3 weeks 3

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Today Structural SVMs – Recap of Previous Lecture – Training General Structured Prediction – Brief Overview 4

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Input: x = (x 1,…,x M ) Predict: y = (y 1,…,y M ) – Each y i one of L labels. Linear Model w.r.t. pairwise features φ j (a,b|x): Prediction via maximizing F: Recap: 1 st Order Sequential Model 5 POS Tags: Det, Noun, Verb, Adj, Adv, Prep L=6

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Recap: Simple Example “Unary Features” “Pairwise Transition Features” 6

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7 yF(y,x) (N,N) = 2 (N,V) = 4 (V,N) = 3 (V,V) = -2 x = “Fish Sleep” y = (N,V) Prediction:

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Structured Prediction Complex output spaces – All possible Part-of-Speech sequences Naïve prediction is often exponential time: Evaluation is also multivariate: 8 E.g., Hamming Loss

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Examples of Complex Output Spaces Natural Language Parsing – Given a sequence of words x, predict the parse tree y. – Dependencies from structural constraints, since y has to be a tree. The dog chased the cat x S VPNP DetNV NP DetN y 9

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Examples of Complex Output Spaces Information Retrieval – Given a query x, predict a ranking y. – Dependencies between results (e.g. avoid redundant hits) – Loss function over rankings (e.g. Average Precision) SVM x 1.Kernel-Machines 2.SVM-Light 3.Learning with Kernels 4.SV Meppen Fan Club 5.Service Master & Co. 6.School of Volunteer Management 7.SV Mattersburg Online … y 10

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Examples of Complex Output Spaces X Y XY X Y Co-reference Resolution Protein Folding Stereo (binocular) Depth Detection 11 Will see examples later in lecture.

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Training Structured Prediction Models General Form: Or equivalently: Require continuous surrogate of evaluation measure. 12

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Structural SVM 13 Sometimes normalize by M Prediction of Learned Model Consider: Slack is continuous upper bound on Hamming Loss! “Slack”

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14 x i = “Fish Sleep” y i = (N,V) y’F(y’,x i )F(y i,x i ) – F(y’,x i )Loss (N,N)221 (N,V)400 (V,N)132 (V,V)131 Example 1

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15 y’F(y’,x i )F(y i,x i ) – F(y’,x i )Loss (N,N)41 (N,V)300 (V,N)032 (V,V)121 Example 2 x i = “Fish Sleep” y i = (N,V)

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16 y’F(y’,x i )F(y i,x i ) – F(y’,x i )Loss (N,N)221 (N,V)400 (V,N)312 (V,V)131 Example 3 x i = “Fish Sleep” y i = (N,V)

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When is Slack Positive? Whenever margin not big enough! 17 Verify that above definition ≥0 Hamming Hinge Loss

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18 yiyi High Dimensional Point w Structural SVM Geometric Interpretation y’ ≤0 Size of Margin vs Size of Margin Violations (C controls trade-off) (Margin scaled by Hamming Loss)

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Structural SVM Training Strictly convex optimization problem – Same form as standard SVM optimization – Easy right? Intractable # of constraints! 19 Often Exponentially Many!

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Structural SVM Training The trick is to not enumerate all constraints. Only solve the SVM objective over a small subset of constraints (working set). – Efficient! But some constraints might be violated. 20

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y’F(y’,x i )F(y i,x i ) – F(y’,x i )Loss (N,N)221 (N,V) y’F(y’,x i )F(y i,x i ) – F(y’,x i )Loss (N,N)221 (N,V)400 (V,N)312 (V,V)131 Example x i = “Fish Sleep” y i = (N,V)

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Approximate Hinge Loss Choose tolerate ε>0: 22 Prediction of Learned Model Consider: Slack is continuous upper bound on Hamming Loss - ε!

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y’F(y’,x i )F(y i,x i ) – F(y’,x i )Loss (N,N)221 (N,V) y’F(y’,x i )F(y i,x i ) – F(y’,x i )Loss (N,N)221 (N,V)400 (V,N)312 (V,V)131 Example x i = “Fish Sleep” y i = (N,V)

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Structural SVM Training STEP 0: Specify tolerance ε STEP 1: Solve SVM objective function using only working set of constraints W (initially empty). The trained model is w. STEP 2: Using w, find the y’ whose constraint is most violated. STEP 3: If constraint is violated by more than ε, add it to W. Repeat STEP 1-3 until no additional constraints are added. Return most recent model w trained in STEP 1. *This is known as a “cutting plane” method. 24 Constraint Violation Formula:

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25 y’F(y’,x i )F(y i,x i ) – F(y’,x i )LossViol. (N,N)0011 (N,V)0000 (V,N)0022 (V,V)0011 Example x i = “Fish Sleep” y i = (N,V) Init: Solve: Choose ε=0.1 Loss – Slack – ( F(y,x)-F(y’,x) ) = Viol Constraint Violation:

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26 Example x i = “Fish Sleep” y i = (N,V) Update: Solve: y’F(y’,x i )F(y i,x i ) – F(y’,x i )LossViol. (N,N)0011 (N,V)0000 (V,N)0022 (V,V)0011 Choose ε=0.1 Loss – Slack – ( F(y,x)-F(y’,x) ) = Viol Constraint Violation:

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27 Example x i = “Fish Sleep” y i = (N,V) Update: Solve: y’F(y’,x i )F(y i,x i ) – F(y’,x i )LossViol. (N,N) (N,V) (V,N) (V,V) Choose ε=0.1 Loss – Slack – ( F(y,x)-F(y’,x) ) = Viol Constraint Violation:

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28 Example x i = “Fish Sleep” y i = (N,V) Update: Solve: y’F(y’,x i )F(y i,x i ) – F(y’,x i )LossViol. (N,N) (N,V) (V,N) (V,V) Choose ε=0.1 Loss – Slack – ( F(y,x)-F(y’,x) ) = Viol Constraint Violation:

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29 Example x i = “Fish Sleep” y i = (N,V) Update: Solve: y’F(y’,x i )F(y i,x i ) – F(y’,x i )LossViol. (N,N) (N,V)1000 (V,N) (V,V) Choose ε=0.1 Loss – Slack – ( F(y,x)-F(y’,x) ) = Viol Constraint Violation: No constraint is violated by more than ε No constraint is violated by more than ε

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Geometric Interpretation 30 Scoring y corresponds to dot product of high dimensional point. High Dimensional Point Quad. Optimization Problem w/ Linear Constraints!

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Geometric Example Naïve SVM Problem Exponential constraints Most are dominated by a small set of “important” constraints Structural SVM Approach Repeatedly finds the next most violated constraint… …until set of constraints is a good approximation. *This is known as a “cutting plane” method. 31

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Geometric Example Naïve SVM Problem Exponential constraints Most are dominated by a small set of “important” constraints Structural SVM Approach Repeatedly finds the next most violated constraint… …until set of constraints is a good approximation. *This is known as a “cutting plane” method. 32

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Geometric Example Naïve SVM Problem Exponential constraints Most are dominated by a small set of “important” constraints Structural SVM Approach Repeatedly finds the next most violated constraint… …until set of constraints is a good approximation. *This is known as a “cutting plane” method. 33

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Geometric Example Naïve SVM Problem Exponential constraints Most are dominated by a small set of “important” constraints Structural SVM Approach Repeatedly finds the next most violated constraint… …until set of constraints is a good approximation. *This is known as a “cutting plane” method. 34

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Guarantee for any ε>0: Terminates after #iterations: Linear Convergence Rate 35 Proof found in:

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Finding Most Violated Constraint A constraint is violated when: Finding most violated constraint reduces to Highly related to prediction: “Loss augmented inference” 36

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“Augmented” Scoring Function 37 Goal: Additional Unary Feature! Solve Using Viterbi!

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Recap: Structural SVM Define structured scoring function: – E.g., 1 st order sequential model – Efficient prediction algorithm Define error function: – E.g., Hamming Loss: Train by iteratively finding most violated constraint: – Requires efficient algorithm (often same as prediction) 38

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Structural SVMs vs CRFs SVM Objective: CRF Objective: 39 SVM only cares about y’ that violates margin the most! Scales margin by loss of y’ CRF cares about all y’ so that: Incorrect P(y’|x) is minimized Correct P(y|x) is maximized

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General Structured Prediction 40

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More Elaborate Scoring Functions Structure encoded by linear scoring function: 2 nd Order Sequential Model: Classification Model: Efficient Prediction: 41 Remainder of Lecture: Tour of Structured Prediction Models Some Might be Interesting to You… Remainder of Lecture: Tour of Structured Prediction Models Some Might be Interesting to You…

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42 https://www.coursera.org/course/pgm https://piazza.com/cornell/fall2013/btry6790cs6782/resources x1x1 x1x1 x2x2 x2x2 y1y1 y1y1 y2y2 y2y2 yMyM yMyM … x3x3 x3x3 y0y0 y0y0 y3y3 y3y3 x3x3 x3x3 Graph structure encodes structural dependencies between y j ! Graphical Models

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43 https://www.coursera.org/course/pgm https://piazza.com/cornell/fall2013/btry6790cs6782/resources … y0y0 y0y0 Graph structure encodes structural dependencies between y j ! Graphical Models … … x1x1 x1x1 x2x2 x2x2 y1y1 y1y1 y2y2 y2y2 yMyM yMyM x3x3 x3x3 y3y3 y3y3 x3x3 x3x3

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44 x1x1 x1x1 x2x2 x2x2 y1y1 y1y1 y2y2 y2y2 yMyM yMyM … x3x3 x3x3 y0y0 y0y0 Graph structure encodes structural dependencies between y j ! y3y3 y3y3 x3x3 x3x3 Features depend on cliques in graphical model representation. https://www.coursera.org/course/pgm https://piazza.com/cornell/fall2013/btry6790cs6782/resources

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Tree Structured Models 45 x1x1 x1x1 y1y1 y1y1 y2y2 y2y2 y3y3 y3y3 y4y4 y4y4 y5y5 y5y5 y7y7 y7y7 y6y6 y6y6 x3x3 x3x3 x2x2 x2x2 x4x4 x4x4 x5x5 x5x5 x6x6 x6x6 x7x7 x7x7 Child nodes of y j

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Prediction via Dynamic Programming 46 1.Solve partial solutions of Leaves 2.Solve partial sol. of next level up 3.Repeat Step 2 until Root 4.Pick best partial solution of Root *Max-Product Algorithm for Tree Graphical Models *Viterbi = Max-Product for Linear Chain Graphical Models

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Loopy Graphical Models 47 X Y Stereo (binocular) Depth Detection Each y ij is depth of pixel Neighbor pixels are similar Features over pairs of pixels “Loopy” Graphical Model Prediction is NP-Hard! x suppressed for brevity

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String Alignment 48 See Also: x = pair of strings (one from D) y = alignment Predict Folding Structure & Function of Protein Database D of Known Proteins (very well studied) Larger Database G of Homologies (proteins w/ known similarities to D) Train on G: learn how to align any amino acid seq to proteins in D encodes score of different types of substitutions, insertions & deletions

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Ranking Find w that predicts best ranking of search results. Every relevant result should be above every non-relevant result. x = query & set of results y = ranking

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Summary: Structured Prediction Very general setting – Applicable to prediction made jointly over multiple y’s – Prediction in Graphical Models Many learning algorithms for structured prediction – CRFs, SSVMs, Structured Perceptron, Learning Reductions Topic for Entire Class! 50 https://www.sites.google.com/site/spflodd/

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Next Week Decision Trees Bagging Random Forests Boosting Ensemble Selection Often the most accurate methods in practice. – (Hint: try them for the Kaggle mini-project) 51

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