Probabilistic Graphical Models

Transcription

1 Probabilistic Graphical Models David Sontag New York University Lecture 12, April 23, 2013 David Sontag NYU) Graphical Models Lecture 12, April 23, / 24

2 What notion of best should learning be optimizing? This depends on what we want to do 1 Density estimation: we are interested in the full distribution so later we can compute whatever conditional probabilities we want) 2 Specific prediction tasks: we are using the distribution to make a prediction 3 Structure or knowledge discovery: we are interested in the model itself David Sontag NYU) Graphical Models Lecture 12, April 23, / 24

3 Density estimation for conditional models Suppose we want to predict a set of variables Y given some others X, e.g., for segmentation or stereo vision input: two images! output: disparity! We concentrate on predicting py X), and use a conditional loss function lossx, y, ˆM) = log ˆpy x). Since the loss function only depends on ˆpy x), suffices to estimate the conditional distribution, not the joint David Sontag NYU) Graphical Models Lecture 12, April 23, / 24

4 Density estimation for conditional models CRF: py x) = 1 φ c x, y c ), Zx) = Zx) c C ŷ φ c x, ŷ c ) c C Parameterization as log-linear model: Weights w R d. Feature vectors f c x, y c ) R d. φ c x, y c ; w) = expw f c x, y c )) [ ] Empirical risk minimization with CRFs, i.e. min ˆM E D lossx, y, ˆM) : w ML = arg min w = arg max w = arg max w 1 D x,y) D x,y) D c w x,y) D log py x; w) ) log φ c x, y c ; w) log Zx; w) c ) f c x, y c ) x,y) D log Zx; w) David Sontag NYU) Graphical Models Lecture 12, April 23, / 24

5 Application: Part-of-speech tagging United flies some large jet N V D A N United 1 flies 2 some 3 large 4 jet 5 David Sontag NYU) Graphical Models Lecture 12, April 23, / 24

6 Graphical model formulation of POS tagging Graphical model formulation given: a sentence of length n and a tag set T one variable for each word, takes values in T edge potentials θi 1, i, t, t) foralli n, t, t T example: United 1 flies 2 some 3 large 4 jet 5 T = {A, D, N, V } note: for probabilistic HMM θi 1, i, t, t) = logpw i t)pt t )) David Sontag NYU) Graphical Models Lecture 12, April 23, / 24

7 Features for POS tagging Edge potentials: Fully parameterize T T features and weights), i.e. θ i 1,i t, t) = w T t,t where the superscript T denotes that these are the weights for the transitions Node potentials: Introduce features for the presence or absence of certain attributes of each word e.g., initial letter capitalized, suffix is ing ), for each possible tag T #attributes features and weights) This part is conditional on the input sentence! Edge potential same for all edges. Same for node potentials. David Sontag NYU) Graphical Models Lecture 12, April 23, / 24

8 Structured prediction Often we learn a model for the purpose of structured prediction, in which given x we predict y by finding the MAP assignment: argmax ˆpy x) y Rather than learn using log-loss density estimation), we use a loss function better suited to the specific task One reasonable choice would be the classification error: E x,y) p [1I{ y y s.t. ˆpy x) ˆpy x) }] which is the probability over all x, y) pairs sampled from p that our classifier selects the right labels If p is in the model family, training with log-loss density estimation) and classification error would perform similarly given sufficient data) Otherwise, better to directly go for what we care about classification error) David Sontag NYU) Graphical Models Lecture 12, April 23, / 24

9 Structured prediction Consider the empirical risk for 0-1 loss classification error): 1 1I{ y y s.t. ˆpy x) ˆpy x) } D x,y) D Each constraint ˆpy x) ˆpy x) is equivalent to w f c x, y c) log Zx; w) w f c x, y c ) log Zx; w) c The log-partition function cancels out on both sides. Re-arranging, we have: w f c x, y c) ) f c x, y c ) 0 c c Said differently, the empirical risk is zero when x, y) D and y y, w f c x, y c ) ) f c x, y c) > 0. c c David Sontag NYU) Graphical Models Lecture 12, April 23, / 24 c

10 Structured prediction Empirical risk is zero when x, y) D and y y, w f c x, y c ) ) f c x, y c) > 0. c c In the simplest setting, learning corresponds to finding a weight vector w that satisfies all of these constraints when possible) This is a linear program LP)! How many constraints does it have? D Y exponentially many! Thus, we must avoid explicitly representing this LP This lecture is about algorithms for solving this LP or some variant) in a tractable manner David Sontag NYU) Graphical Models Lecture 12, April 23, / 24

11 Structured perceptron algorithm Input: Training examples D = {x m, y m )} Let fx, y) = c f cx, y c ). Then, the constraints that we want to satisfy are ) w fx m, y m ) fx m, y) > 0, y y m The perceptron algorithm uses MAP inference in its inner loop: MAPx m ; w) = arg max y Y w fxm, y) The maximization can often be performed efficiently by using the structure! The perceptron algorithm is then: 1 Start with w = 0 2 While the weight vector is still changing: 3 For m = 1,..., D 4 y MAPx m ; w) 5 w w + fx m, y m ) fx m, y) David Sontag NYU) Graphical Models Lecture 12, April 23, / 24

12 Structured perceptron algorithm If the training data is separable, the perceptron algorithm is guaranteed to find a weight vector which perfectly classifies all of the data When separable with margin γ, number of iterations is at most where R = max m,y fx m, y) 2 ) 2 2R, γ In practice, one stops after a certain number of outer iterations called epochs), and uses the average of all weights The averaging can be understood as a type of regularization to prevent overfitting David Sontag NYU) Graphical Models Lecture 12, April 23, / 24

13 Allowing slack We can equivalently write the constraints as ) w fx m, y m ) fx m, y) 1, y y m Suppose there do not exist weights w that satisfy all constraints Introduce slack variables ξ m 0, one per data point, to allow for constraint violations: ) w fx m, y m ) fx m, y) 1 ξ m, y y m Then, minimize the sum of the slack variables, min ξ 0 m ξ m, subject to the above constraints David Sontag NYU) Graphical Models Lecture 12, April 23, / 24

14 Structural SVM support vector machine) subject to: min w,ξ ξ m + C w 2 m ) w fx m, y m ) fx m, y) 1 ξ m, m, y y m ξ m 0, m This is a quadratic program QP). Solving for the slack variables in closed form, we obtain ) ) ξm = max 0, max 1 w fx m, y m ) fx m, y) y Y Thus, we can re-write the whole optimization problem as ) ) min max 0, max 1 w fx m, y m ) fx m, y) + C w 2 w y Y m David Sontag NYU) Graphical Models Lecture 12, April 23, / 24

15 Hinge loss )) We can view max 0, max y Y 1 w fx m, y m ) fx m, y) as a loss function, called hinge loss When w fx m, y m ) w fx m, y) for all y i.e., correct prediction), this takes a value between 0 and 1 When y such that w fx m, y) w fx m, y m ) i.e., incorrect prediction), this takes a value 1 Thus, this always upper bounds the 0-1 loss! Minimizing hinge loss is good because it minimizes an upper bound on the 0-1 loss prediction error) David Sontag NYU) Graphical Models Lecture 12, April 23, / 24

16 Better Metrics It doesn t always make sense to penalize all incorrect predictions equally! We can change the constraints to ) w fx m, y m ) fx m, y) y, y m ) ξ m, where y, y m ) 0 is a measure of how far the assignment y is from the true assignment y m This is called margin scaling as opposed to slack scaling) We assume that y, y) = 0, which allows us to say that the constraint holds for all y, rather than just y y m A frequently used metric for MRFs is Hamming distance, where y, y m ) = i V 1I[y i y m i ] y, David Sontag NYU) Graphical Models Lecture 12, April 23, / 24

17 Structural SVM with margin scaling min w m max y Y )) y, y m ) w fx m, y m ) fx m, y) + C w 2 How to solve this? Many methods! 1 Cutting-plane algorithm Tsochantaridis et al., 2005) 2 Stochastic subgradient method Ratliff et al., 2007) 3 Dual Loss Primal Weights algorithm Meshi et al., 2010) 4 Frank-Wolfe algorithm Lacoste-Julien et al., 2013) David Sontag NYU) Graphical Models Lecture 12, April 23, / 24

18 Stochastic subgradient method min w m max y Y )) y, y m ) w fx m, y m ) fx m, y) + C w 2 Although this objective is convex, it is not differentiable everywhere We can use a subgradient method to minimize instead of gradient descent) ) The subgradient of max y Y y, y m ) w fx m, y m ) fx m, y) at w t) is fx m, ŷ) fx m, y m ), where ŷ is one of the maximizers with respect to w t), i.e. ŷ = arg max y Y y, ym ) + w t) fx m, y) This maximization is called loss-augmented MAP inference David Sontag NYU) Graphical Models Lecture 12, April 23, / 24

19 Loss-augmented inference ŷ = arg max y Y y, ym ) + w t) fx m, y) When y, y m ) = i V 1I[y i yi m ], this corresponds to adding additional single-node potentials θ i y i ) = 1 if y i y m, and 0 otherwise If MAP inference was previously exactly solvable by a combinatorial algorithm, loss-augmented MAP inference typically is too The Hamming distance pushes the MAP solution away from the true assignment y m David Sontag NYU) Graphical Models Lecture 12, April 23, / 24

20 Cutting-plane algorithm subject to: w min w,ξ ) fx m, y m ) fx m, y) ξ m + C w 2 m y, y m ) ξ m, m, y Y m ξ m 0, m Start with Y m = {y m }. Solve for the optimal w, ξ Then, look to see if any of the unused constraints that are violated To find a violated constraint for data point m, simply solve the loss-augmented inference problem: ŷ = arg max y Y y, ym ) + w fx m, y) If ŷ Y m, do nothing. Otherwise, let Y m = Y m {ŷ} Repeat until no new constraints are added. Then we are optimal! David Sontag NYU) Graphical Models Lecture 12, April 23, / 24

21 Cutting-plane algorithm Can prove that, in order to solve the structural SVM up to ɛ additive) accuracy, takes a polynomial number of iterations In practice, terminates very quickly David Sontag NYU) Graphical Models Lecture 12, April 23, / 24

22 Block-Coordinate Frank-Wolfe Optimization for Structural SVMs Summary of convergence rates Table 1. Convergence rates given in the number of calls to the oracles for di erent optimization algorithms for the str tural SVM objective 1) in the case of a Markov random field structure, to reach a specific accuracy " measured for di er types of gaps, in term of the number of training examples n, regularization parameter, sizeofthelabelspace Y, m imum feature norm R := max i,y k iy)k 2 some minor terms were ignored for succinctness). Table inspired from Zha et al., 2011). Notice that only stochastic subgradient and our proposed algorithm have rates independent of n. Optimization algorithm Online Primal/Dual Type of guarantee Oracle type # Oracle calls dual extragradient Taskar no primal- dual saddle point gap Bregman projection O nr log Y et al., 2006) " online exponentiated gradient yes dual expected dual error expectation O n+log Y )R 2 Collins et al., 2008) " q excessive gap reduction log Y no primal-dual duality gap expectation O nr Zhang et al., 2011) " nr BMRM Teo et al., 2010) no primal primal error maximization O 2 " 1-slack SVM-Struct Joachims nr no primal-dual duality gap maximization O 2 et al., 2009) " stochastic subgradient R yes primal primal error w.h.p. maximization Õ 2 Shalev-Shwartz et al., 2010a) " this paper: block-coordinate R yes primal-dual expected duality gap maximization O 2 Frank-Wolfe " Thm. 3 lary 3 gives a similar convergence rate as our TheoremR 3. rate in the duality gap as the full Frank-Wolfe meth same Balamurugan as before. et al. 2011) n=number propose toofapprox- imately solve a quadratic problem on each example leads to a simple online algorithm that yields a so trainingfor examples. the dual structural λ is the SVMregularization optimization problem using constant SMO, but correpsonding they do not provide toany 2C/n) rate guarantees. The online-eg method implements a variant for stochastic algorithms: no step-size sequence ne tion to an issue that is notoriously di cult to addr of dual coordinate descent, but it requires an expectation oracle and Collins et al. 2008) estimate its primal ciently computed in closed-form. Further, at the c to be tuned since the optimal step-size can be e convergence Davidat Sontag only ONYU) 1/" 2. Graphical Models of an additional pass Lecture through 12, April the 23, 2013 data which 22 / 24co

23 Application to segmentation & support inference ECCV-12 submission ID Input RGB Surface Normals Aligned Normals Segmentation Input Depth Inpainted Depth 3D Planes Support Relations Image Depth Map 1. Major Surfaces 2. Surface Normals 3. Align Point Cloud Point Cloud RGB Image Regions Features Segmenta Feature Extrac {X i,n i } {R j } {F j } Joint with Nathan Silberman & Rob Fergus) Support Classi ca Structure Labels Support Rela onships Fig. 1. Overview of algorithm. Our algorithm flows from left to right. Given an input image with raw and inpainted depth maps, we compute surface normals and align them to the room by finding three dominant orthogonal directions. We then David Sontag NYU) Graphical Models Lecture 12, April 23, / 24

24 b obj Application to machine translation ly b) Dice and Distance Word alignment between languages: consul r b le un de les grands objectifs de les consultations est de faire en sorte que la relance profite é g a l e m e n t à tous. le un de les grands objectifs de les consultations est de faire en sorte que la relance profite é g a l e m e n t à tous. benefits all. one of the major objectives of these consultations is to make sure that the recovery benefits all. phic, Taskar, and BothShort Lacoste-Julien, Klein d) All 05) features David Sontag NYU) Graphical Models Lecture 12, April 23, / 24