Structured Prediction
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1 Structured Prediction
2 Classification Algorithms Classify objects x X into labels y Y First there was binary: Y = {0, 1} Then multiclass: Y = {1,...,6} The next generation: Structured Labels
3 Structured Prediction Simultaneously predict multiple variables Image segmentation Syntactic parsing John hit the ball
4 Structured Prediction Multi-label document classification Input x Sports Politics Finance x Output y Large state space!
5 Structured Prediction Dependency parsing:, creating another potential agency can raise capital obstacle to Influential -R-/ROOT of members House the and Ways Committee Means legislation introduced would that how restrict new the bailout savings-and-loan government the sal 's amod nsubj prep det nn pobj cc nn conj ROOT dobj nsubj aux rcmod advmod det amod nn nn nsubj aux ccomp dobj p xcomp det amod dobj prep det poss ps pobj p
6 Entity Linking Caroline was the last, best hope for the family, which has had members in the Senate since Jack was elected in Following him: Robert and Ted joined the Senate, both of them with presidential dreams that didn't materialize (because of Bobby's assassination and Teddy's being not as beloved as Bobby and Jack).
7 Why structured? Independent prediction Tree or Grass? Overall labelling should make sense!
8 Why structured? Independent prediction x : John caught the early train y : N V D Adv Adj NV But Adverbs rarely follow Determiners So this is a bad prediction to make. Add constraint that it is invalid. Other constraints: few verbs per sentence, valid combinations of three tags etc.
9 Structured Prediction Prediction task: mapping from input x to output y The label is multivariate: y =[y 1,...,y n ] Overall number of possible labels can be huge Standard multiclass methods cannot be applied The early works: structured perceptron (Collins, 02), conditional random fields (Lafferty, McCallum, Pereira, 02), max margin Markov networks (Taskar, Guestrin, Koller 03)
10 Structured Prediction Key idea: Construct a score function S(x, y). S(x, y) = 1000 Caroline was the last, best hope for the family, which has had members in the Senate since Jack was elected in Following him: Robert and Ted joined the Senate, both of them with presidential dreams that didn't materialize (because of Bobby's assassination and Teddy's being not as beloved as Bobby and Jack. S(x, y) =7 Caroline was the last, best hope for the family, which has had members in the Senate since Jack was elected in Following him: Robert and Ted joined the Senate, both of them with presidential dreams that didn't materialize (because of Bobby's assassination and Teddy's being not as beloved as Bobby and Jack. Predict using: h(x) = arg max y S(x, y)
11 Score Function S(x, y measures how well y fits x. 1,...,y n ) How do we construct it? Need some simplifying assumptions. y 1 y 2 y 3 S(x, y 1,y 2,y 3 )=S 1,2 (x, y 1,y 2 )+S 2,3 (x, y 2,y 3 ) S(x, y) = X c S c (x, y c ) decomposition into parts S(x, y) =S 1,2,3 (y 1,y 2,y 3 )+S 2,3,4 (y 2,y 3,y 4 )
12 Structures for Images For machine vision, typically want to encode constraint that neighboring pixels are similar S(y 1,y 2 )+S(y 1,y 3 )+... y 3 y 1 y 2
13 Two Key Tasks Inference: mapping from input to output h(x) = arg max y Tractable in some cases S(x, y) = arg max y X c S c (x, y c ) NP-hard in the general, interesting, cases. Learning: given a training set {(x m,y m )} M m=1 learn a good score function.
14 Why is it Interesting? Very useful in practice. Almost all of state of the art NLP and machine vision use it. Challenging algorithms and theory: Approximate inference (combinatorial optimization, convex optimization). Learning parameters from data, when inference is hard (learning theory)
15 Inference Want to solve: h(x) = arg max y S(x, y) = arg max y X c S c (x, y c ) Essentially MAP in graphical models. We know a lot about this (BP, LBP, Mean Field, Sampling etc.) Hard in the general case
16 Learning Training sample D: {x m,y m } M m=1 Learn good score function Makes sense to parameterize the score and learn the parameters. S(x, y, w) = X S c (x, y c ) = X c w c c(x, y c ) = w (x, y) Features c(x, y c ) are given. Goal. Find w that results in good classification.
17 Conditional Random Fields (CRF) We want: S(x m,y m ) >S(x m,y), 8y 6= y m Equivalently: w (x m,y m ) > w (x m, ȳ), 8ȳ 6= y m The CRF [LMP 01] approach. Define a distribution: Maximize: X m p(y x, w) = log p(y m x m ; w) 1 ew Z(x; w) (x,y) If you can find w where y m is maximal, you can drive the objective to zero, by scaling w.
18 CRF Loss CRF uses the log-loss, which is also popular in standard multi-class. Here s another interpretation. 1 p(y x, w) = ew (x,y) Z(x; w) Constructs a distribution: Define a distribution: q(y x m )= 1 y = y m 0 y 6= y m log p(y m x m ; w) = X y p m (y x m ) log p(y x m ; w) = Also called crossentropy loss. = D KL [q(y x m ) p(y x m ; w)] + C Will be zero if p is peaked at ym.
19 CRF Limitations The distribution: p(y x, w) = 1 ew Z(x; w) (x,y) Maximizing requires calculating Z and its gradients. i.e., summing over y Easier to maximize over y (and this is what prediction requires). What if we have some loss function on label space. e.g., if you miss one label out of 100, it s not as bad as missing all 100.
20 Max Margin Markov Networks Taskar, Guestrin and Koller Good classification means: S(x m,y m ) >S(x m,y), 8y 6= y m Suppose we have a label loss: (y, y 0 ) Makes sense to require: S(x m,y m ) S(x m,y)+ (y m,y) m Might be impossible to satisfy. Use slack variable m, which we ll want to minimize
21 Max Margin Markov Networks Want to minimize X m m Subject to: m S(x m,y) S(x m,y m )+ (y m,y) Equivalent to minimizing: X m max y This is the structured hinge loss. [S(x m,y) S(x m,y m )+ (y m,y)] Convex if S is linear in the parameters. `h(x m,y m, w)
22 Max Margin Markov Networks Here is an alternative derivation showing that the structured hinge loss is an upper bound on the loss. Denote predicted label: y max (x) = arg max y S(x, y) The true loss is: (y, y max (x)) But non-convex and with little gradient info Easy to show that: (y, y (x)) apple `h(x, y) max
23 Max Margin Markov Networks Goal is to minimize: X m,y max y [S(x m,y; w) S(x m,y m ; w)+ (y m,y)] + kwk 2 2 Regularization added to improve optimization speed and generalization.
24 M3N - Optimization Minimize: X m,y max y [S(x m,y; w) S(x m,y m ; w)+ (y m,y)] + kwk 2 2 Dual algorithms (like SMO) studied extensively, but require variable per data point, and only work when S is linear in w. Primal Stochastic sub gradient descent: sub gradient wrt w requires the argmax of the above.
25 Implementing SGD X m,y max y [S(x m,y; w) S(x m,y m ; w)+ (y m,y)] + kwk 2 2 When the argmax is easy to calculate, SGD is simple. Otherwise, makes sense to replace max by your favorite approximation (also in inference), and take (sub) gradient of that. LP relaxation [FJ 08, MSJG 10] Belief propagation [Schwing, Urtasun 15]
26 Deep Structured Prediction How do we integrate this with deep learning? First stab: replace local scores with deep nets. Linear: S(x, y, w) = X S c (x, y c ) = X c w c c(x, y c ) Deep: S(x, y, w) = X c f(x, y c ; w) But keep inference as before. Works well for image segmentation and parsing Non convex of course
27 Deep Structured Prediction Second approach: for approximate inference iterative algorithms, implement the iterations of the algorithm via neural net. Train end to end. See e.g., for image segmentation. x Model local scores This is a non-convex, but it may work ok in practice. Inference Alg. Beliefs bi(yi) Loss. e.g.: log b i (y i )
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