Structured Prediction: A Large Margin Approach Ben Taskar University of Pennsylvania Joint work with: V. Chatalbashev, M. Collins, C. Guestrin, M. Jordan, D. Klein, D. Koller, S. Lacoste-Julien, C. Manning

“Don’t worry, Howard. The big questions are multiple choice.”

Handwriting Recognition brace Sequential structure xy

Object Segmentation Spatial structure xy

Natural Language Parsing The screen was a sea of red Recursive structure xy

Bilingual Word Alignment What is the anticipated cost of collecting fees under the new proposal? En vertu des nouvelles propositions, quel est le coût prévu de perception des droits? xy What is the anticipated cost of collecting fees under the new proposal ? En vertu de les nouvelles propositions, quel est le coût prévu de perception de les droits ? Combinatorial structure

Protein Structure and Disulfide Bridges Protein: 1IMT AVITGACERDLQCG KGTCCAVSLWIKSV RVCTPVGTSGEDCH PASHKIPFSGQRMH HTCPCAPNLACVQT SPKKFKCLSK

Local Prediction Classify using local information  Ignores correlations & constraints! breac

Local Prediction building tree shrub ground

Structured Prediction Use local information Exploit correlations breac

Structured Prediction building tree shrub ground

Outline Structured prediction models Sequences (CRFs) Trees (CFGs) Associative Markov networks (Special MRFs) Matchings Structured large margin estimation Margins and structure Min-max formulation Linear programming inference Certificate formulation

Structured Models Mild assumption: linear combination space of feasible outputs scoring function

Chain Markov Net (aka CRF*) a-z y x *Lafferty et al. 01

Chain Markov Net (aka CRF*) a-z y x *Lafferty et al. 01

Associative Markov Nets Point features spin-images, point height Edge features length of edge, edge orientation yjyj ykyk  jk jj “associative” restriction

CFG Parsing #(NP  DT NN) … #(PP  IN NP) … #(NN  ‘sea’)

Bilingual Word Alignment position orthography association What is the anticipated cost of collecting fees under the new proposal ? En vertu de les nouvelles propositions, quel est le coût prévu de perception de le droits ? j k

Disulfide Bonds: Non-bipartite Matching RSCCPCYWGGCPWGQNCYPEGCSGPKV Fariselli & Casadio `01, Baldi et al. ‘04

Scoring Function RSCCPCYWGGCPWGQNCYPEGCSGPKV RSCCPCYWGGCPWGQNCYPEGCSGPKV amino acid identities phys/chem properties

Structured Models Mild assumptions: linear combination sum of part scores space of feasible outputs scoring function

Supervised Structured Prediction Learning Prediction Estimate w Example: Weighted matching Generally: Combinatorial optimization Data Model: Likelihood (intractable) MarginLocal (ignores structure)

Outline Structured prediction models Sequences (CRFs) Trees (CFGs) Associative Markov networks (Special MRFs) Matchings Structured large margin estimation Margins and structure Min-max formulation Linear programming inference Certificate formulation

We want: Equivalently: OCR Example a lot! … “brace” “aaaaa” “brace”“aaaab” “brace”“zzzzz”

We want: Equivalently: ‘It was red’ Parsing Example a lot! S AB CD S AB DF S AB CD S EF GH S AB CD S AB CD S AB CD … ‘It was red’

We want: Equivalently: ‘What is the’ ‘Quel est le’ Alignment Example a lot! … ‘What is the’ ‘Quel est le’ ‘What is the’ ‘Quel est le’ ‘What is the’ ‘Quel est le’ ‘What is the’ ‘Quel est le’ ‘What is the’ ‘Quel est le’ ‘What is the’ ‘Quel est le’

Structured Loss b c a r e b r o r e b r o c e b r a c e ‘What is the’ ‘Quel est le’ S AE CD S BE AC S BD AC S AB CD ‘It was red’

Large margin estimation Given training examples, we want: Maximize margin Mistake weighted margin: # of mistakes in y *Collins 02, Altun et al 03, Taskar 03

Large margin estimation Eliminate Add slacks for inseparable case

Large margin estimation Brute force enumeration Min-max formulation ‘Plug-in’ linear program for inference

Min-max formulation LP Inference Structured loss (Hamming): Inference discrete optim. Key step: continuous optim.

Outline Structured prediction models Sequences (CRFs) Trees (CFGs) Associative Markov networks (Special MRFs) Matchings Structured large margin estimation Margins and structure Min-max formulation Linear programming inference Certificate formulation

y  z Map for Markov Nets : : : : : 0 a b : z a b : z ab.zab.zab.zab.z

Markov Net Inference LP Has integral solutions z for chains, trees Can be fractional for untriangulated networks normalization agreement

Associative MN Inference LP For K=2, solutions are always integral (optimal) For K>2, within factor of 2 of optimal “associative” restriction

CFG Chart CNF tree = set of two types of parts: Constituents (A, s, e) CF-rules (A  B C, s, m, e)

CFG Inference LP inside outside Has integral solutions z root

Matching Inference LP Has integral solutions z degree What is the anticipated cost of collecting fees under the new proposal ? En vertu de les nouvelles propositions, quel est le coût prévu de perception de le droits ? j k

LP Duality Linear programming duality Variables  constraints Constraints  variables Optimal values are the same When both feasible regions are bounded

Min-max Formulation LP duality

Min-max formulation summary Formulation produces concise QP for Low-treewidth Markov networks Associative MNs (K=2) Context free grammars Bipartite matchings Approximate for untriangulated MNs, AMNs with K>2 *Taskar et al 04

Unfactored Primal/Dual QP duality Exponentially many constraints/variables

Factored Primal/Dual By QP duality Dual inherits structure from problem-specific inference LP Variables  correspond to a decomposition of  variables of the flat case

The Connection b c a r e b r o r e b r o c e b r a c e r c a o c r b 1 e 

Duals and Kernels Kernel trick works: Factored dual Local functions (log-potentials) can use kernels

Simple iterative method Unstable for structured output: fewer instances, big updates May not converge if non-separable Noisy Voted / averaged perceptron [Freund & Schapire 99, Collins 02] Regularize / reduce variance by aggregating over iterations Alternatives: Perceptron

Add most violated constraint Handles more general loss functions Only polynomial # of constraints needed Need to re-solve QP many times Worst case # of constraints larger than factored Alternatives: Constraint Generation [Collins 02; Altun et al, 03; Tsochantaridis et al, 04]

Handwriting Recognition Length: ~8 chars Letter: 16x8 pixels 10-fold Train/Test 5000/50000 letters 600/6000 words Models: Multiclass-SVMs* CRFs M 3 nets *Crammer & Singer CRFs MC–SVMsM^3 nets Test error (average per-character) raw pixels quadratic kernel cubic kernel 45% error reduction over linear CRFs 33% error reduction over multiclass SVMs better

Hypertext Classification WebKB dataset Four CS department websites: 1300 pages/3500 links Classify each page: faculty, course, student, project, other Train on three universities/test on fourth 53% error reduction over SVMs 38% error reduction over RMNs relaxed dual *Taskar et al 02 better loopy belief propagation

3D Mapping Laser Range Finder GPS IMU Data provided by: Michael Montemerlo & Sebastian Thrun Label: ground, building, tree, shrub Training: 30 thousand points Testing: 3 million points

Segmentation results Hand labeled 180K test points ModelAccuracy SVM68% V-SVM73% M3NM3N93%

Fly-through

Word Alignment Results Model*Error Local learning+matching10.0 Our approach8.5 Data: [Hansards – Canadian Parliament] Features induced on  1 mil unsupervised sentences Trained on 100 sentences (10,000 edges) Tested on 350 sentences (35,000 edges) [Taskar+al 05] *Error: weighted combination of precision/recall [Lacoste-Julien+Taskar+al 06] GIZA/IBM4 [Och & Ney 03]6.5 +Our approach+QAP4.5 +Local learning+matching5.4 +Our approach4.9

Outline Structured prediction models Sequences (CRFs) Trees (CFGs) Associative Markov networks (Special MRFs) Matchings Structured large margin estimation Margins and structure Min-max formulation Linear programming inference Certificate formulation

Non-bipartite matchings: O(n 3 ) combinatorial algorithm No polynomial-size LP known Spanning trees No polynomial-size LP known Simple certificate of optimality Intuition: Verifying optimality easier than optimizing Compact optimality condition of wrt ij kl

Certificate for non-bipartite matching Alternating cycle: Every other edge is in matching Augmenting alternating cycle: Score of edges not in matching greater than edges in matching Negate score of edges not in matching Augmenting alternating cycle = negative length alternating cycle Matching is optimal  no negative alternating cycles Edmonds ‘65

Certificate for non-bipartite matching Pick any node r as root = length of shortest alternating path from r to j Triangle inequality: Theorem: No negative length cycle  distance function d exists Can be expressed as linear constraints: O(n) distance variables, O(n 2 ) constraints

Certificate formulation Formulation produces compact QP for Spanning trees Non-bipartite matchings Any problem with compact optimality condition *Taskar et al. ‘05

Disulfide Bonding Prediction Data [Swiss Prot 39] 450 sequences (4-10 cysteines) Features: windows around C-C pair physical/chemical properties [Taskar+al 05] Model*Acc Local learning+matching41% Recursive Neural Net [Baldi+al’04] 52% Our approach (certificate)55% *Accuracy: % proteins with all correct bonds

Formulation summary Brute force enumeration Min-max formulation ‘Plug-in’ convex program for inference Certificate formulation Directly guarantee optimality of

Omissions Kernels Non-parametric models Structured generalization bounds Bounds on hamming loss Scalable algorithms (no QP solver needed) Structured SMO (works for chains, trees) [Taskar 04] Structured ExpGrad (works for chains, trees) [Bartlett+al 04] Structured ExtraGrad (works for matchings, AMNs) [Taskar+al 06]

Open questions Statistical consistency Hinge loss not consistent for non-binary output [See Tewari & Bartlett 05, McAllester 07] Learning with approximate inference Does constant factor approximate inference guarantee anything about learning? No [See Kulesza & Pereira 07] Perhaps other assumptions needed Discriminative structure learning Using sparsifying priors

Conclusion Two general techniques for structured large-margin estimation Exact, compact, convex formulations Allow efficient use of kernels Tractable when other estimation methods are not Efficient learning algorithms Empirical success on many domains

References Y. Altun, I. Tsochantaridis, and T. Hofmann. Hidden Markov support vector machines. ICML03. M. Collins. Discriminative training methods for hidden Markov models: Theory and experiments with perceptron algorithms. EMNLP02 K. Crammer and Y. Singer. On the algorithmic implementation of multiclass kernel-based vector machines. JMLR01 J. Lafferty, A. McCallum, and F. Pereira. Conditional random fields: Probabilistic models for segmenting and labeling sequence data. ICML04 More papers at

Modeling First Order Effects MonotonicityLocal inversionLocal fertility QAP NP-complete Sentences (  30 words,  1k vars)  few seconds (Mosek) Learning: use LP relaxation Testing: using LP, 83.5% sentences, 99.85% edges integral

Segmentation Model  Min-Cut 0 1 Local evidence Spatial smoothness Computing is hard in general, but if edge potentials attractive  min-cut algorithm Multiway-cut for multiclass case  use LP relaxation [Greig+al 89, Boykov+al 99, Kolmogorov & Zabih 02, Taskar+al 04]

Scalable Algorithms Batch and online Linear in the size of the data Iterate until convergence For each example in the training sample Run inference using current parameters (varies by method) Online: Update parameters using computed example values Batch: Update parameters using computed sample values Structured SMO (Taskar et al, 03; Taskar 04) Structured Exponentiated Gradient (Bartlett et al, 04) Structured Extragradient (Taskar et al, 05)

Experimental Setup Standard Penn treebank split (2-21/22/23) Generative baselines Klein & Manning 03 and Collins 99 Discriminative Basic = max-margin version of K&M 03 Lexical & Lexical + Aux Lexical features (on constituent parts only) t s-1 [t s … t e ] t e+1  predicted tags x s-1 [x s … x e ] x e+1 Auxillary features Flat classifier using same features Prediction of K&M 03 on each span

Results for sentences ≤40 words ModelLPLRF1F1 Generative Lexical+Aux* Collins 99* *Trained only on sentences ≤20 words *Taskar et al 04

Example The Egyptian president said he would visit Libya today to resume the talks. Generative model: Libya today is base NP Lexical model: today is a one word constituent