Part 4: Conditional Random Fields

Transcription

1 Part 4: Conditional Random Fields Sebastian Nowozin and Christoph H. Lampert Colorado Springs, 25th June / 39

2 Problem (Probabilistic Learning) Let d(y x) be the (unknown) true conditional distribution. Let D = {(x 1, y 1 ),..., (x N, y N )} be i.i.d. samples from d(x, y). or Find a distribution p(y x) that we can use as a proxy for d(y x). Given a parametrized family of distributions, p(y x, w), find the parameter w making p(y x, w) closest to d(y x). Open questions: What do we mean by closest? What s a good candidate for p(y x, w)? How to actually find w? conceptually, and numerically 2 / 39

3 Principle of Parsimony (Parsimoney, aka Occam s razor) Pluralitas non est ponenda sine neccesitate. William of Ockham We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances. Isaac Newton Make everything as simple as possible, but not simpler. Albert Einstein Use the simplest explanation that covers all the facts. what we ll use 3 / 39

4 1) Define what aspects we consider relevant facts about the data. 2) Pick the simplest distribution reflecting that. Definition (Simplicity Entropy) The simplicity of a distribution p is given by its entropy: H(p) = z Z p(z) log p(z) Definition (Relevant Facts Feature Functions) By φ i : Z R for i = 1,..., D we denote a set of feature functions that express everything we want to be able to model about our data. For example: the grayvalue of a pixel, a bag-of-words histogram of an image, the time of day an image was taken, a flag if a pixel is darker than half of its neighbors. 4 / 39

5 Principle (Maximum Entropy Principle) Let z 1,..., z N be samples from a distribution d(z). Let φ 1,..., φ D be feature functions, and denote by µ i := 1 N n φ i(z n ) their means over the sample set. The maximum entropy distribution, p, is the solution to max H(p) p is a prob.distr. }{{} be as simple as possible Theorem (Exponential Family Distribution) subject to E z p(z) {φ i (z)} = µ i. }{{} be faithful to what we know Under some very reasonable conditions, the maximum entropy distribution has the form p(z) = 1 Z exp ( i w iφ i (z) ) for some parameter vector w = (w 1,..., w D ) and constant Z. 5 / 39

6 Example: Let Z = R, φ 1 (z) = z, φ 2 (z) = z 2. The exponential family distribution is p(z) = 1 Z(w) exp( w 1z + w 2 z 2 ) = b2 a Z(a, b) exp( a( z b ) 2 ) for a = w2, b = w 1 w 2. It s a Gaussian! Given examples z 1,..., z N, we can compute a and b, and derive w. 6 / 39

7 Example: Let Z = {1,..., K}, φ k (z) = z = k, for k = 1,..., K. The exponential family distribution is p(z) = 1 Z(w) exp( w k φ k (z) ) k exp(w 1 )/Z for z = 1, exp(w 2 )/Z for z = 2, =... exp(w K )/Z for z = K. with Z = exp(w 1 ) + + exp(w K ). It s a Multinomial! 7 / 39

8 Example: Let Z = {0, 1} N M image grid, φ i (y) := y i for each pixel i, φ NM (y) = i j y i y j (summing over all 4-neighbor pairs) The exponential family distribution is p(z) = 1 Z(w) exp( w, φ(y) + w y i y j ) i,j It s a (binary) Markov Random Field! 8 / 39

9 Conditional Random Field Learning Assume: a set of i.i.d. samples D = {(x n, y n )},...,N, (x n, y n ) d(x, y) feature functions ( φ 1 (x, y),..., φ D (x, y) ) : φ(x, y) parametrized family p(y x, w) = 1 Z(x,w) exp( w, φ(x, y) ) Task: adjust w of p(y x, w) based on D. Many possible technique to do so: Expectation Matching Maximum Likelihood Best Approximation MAP estimation of w Punchline: they all turn out to be (almost) the same! 9 / 39

10 Maximum Likelihood Parameter Estimation Idea: maximize conditional likelihood of observing outputs y 1,..., y N for inputs x 1,..., x N w = argmax w R D p(y 1,..., y N x 1,..., x N, w) i.i.d. = argmax w R D log( ) = argmin w R D N p(y n x n, w) N log p(y n x n, w) }{{} negative conditional log-likelihood (of D) 10 / 39

11 Best Approximation Idea: find p(y x, w) that is closest to d(y x) Definition (Similarity between conditional distributions) For fixed x X : KL-divergence measure similarity KL cond (p d)(x) := y Y d(y x) log d(y x) p(y x, w) For x d(x), compute expectation: } KL tot (p d) : = E x d(x) {KL cond (p d)(x) = x X d(x, y) log d(y x) p(y x, w) y Y 11 / 39

12 Best Approximation Idea: find p(y x, w) of minimal KL tot -distance to d(y x) w = argmin w R D d(x, y) log x X y Y d(y x) p(y x, w) w R D (x,y) X Y drop const. = argmin d(x, y) log p(y x, w) N (x n,y n ) d(x,y) argmin w R D log p(y n x n, w) }{{} negative conditional log-likelihood (of D) 12 / 39

13 MAP Estimation of w Idea: Treat w as random variable; maximize posterior probability p(w D) p(w D) Bayes = p(x1, y 1,..., x n, y n w)p(w) i.i.d. = p(w) p(d) N p(w): prior belief on w (cannot be estimated from data). p(y n x n, w) p(y n x n ) w = argmax w R D = argmin w R D [ = argmin w R D [ ] p(w D) = argmin log p(w D) w R D [ N log p(w) log p(w) N ] log p(y n x n, w) + log p(y n x n ) }{{} indep. of w ] log p(y n x n, w) 13 / 39

14 Choices for p(w): p(w) : const. w [ = argmin log p(w) w R D w [ = argmin w R D N log p(y n x n, w) ] (uniform; in R D not really a distribution) N log p(y n x n, w) + const. ] }{{} negative conditional log-likelihood p(w) := const. e 1 2σ 2 w 2 w = argmin w R D [ 1 (Gaussian) N 2σ 2 w 2 + log p(y n x n, w) }{{} regularized negative conditional log-likelihood + const. ] 14 / 39

15 Probabilistic Models for Structured Prediction - Summary Negative (Regularized) Conditional Log-Likelihood (of D) L(w) = 1 2σ 2 w 2 N [ w, φ(x n, y n ) log e w,φ(xn,y) ] (σ 2 makes it unregularized) y Y Probabilistic parameter estimation or training means solving w = argmin L(w). w R D Same optimization problem as for multi-class logistic regression. 15 / 39

16 Negative Conditional Log-Likelihood (Toy Example) negative log likelihood σ 2 =0.01 negative log likelihood σ 2 = negative log likelihood σ 2 = negative log likelihood σ / 39

17 Steepest Descent Minimization minimize L(w) input tolerance ɛ > 0 1: w cur 0 2: repeat 3: v w L(w cur ) 4: η argmin η R L(w cur ηv) 5: w cur w cur ηv 6: until v < ɛ output w cur Alternatives: L-BFGS (second-order descent without explicit Hessian) Conjugate Gradient We always need (at least) the gradient of L. 17 / 39

18 L(w) = 1 2σ 2 w 2 N [ w, φ(x n, y n ) + log e w,φ(xn,y) ] y Y w L(w) = 1 N σ 2 w [ φ(x n, y n y Y ) e w,φ(xn,y) φ(x n, y) ] ȳ Y e w,φ(xn,ȳ) = 1 N σ 2 w [ φ(x n, y n ) p(y x n, w)φ(x n, y) ] y Y = 1 N σ 2 w [ φ(x n, y n ) E y p(y x n,w)φ(x n, y) ] L(w) = 1 N σ 2 Id D D + [E y p(y x n,w)φ(x n, y)][e y p(y x n,w)φ(x n, y)] 18 / 39

19 L(w) = 1 2σ 2 w 2 N [ w, φ(x n, y n ) + log e w,φ(xn,y) ] y Y C -differentiable on all R D. 19 / 39

20 w L(w) = 1 N σ 2 w [ φ(x n, y n ) E y p(y x n,w)φ(x n, y) ] For σ : E y p(y x n,w)φ(x n, y) = φ(x n, y n ) w L(w) = 0, criticial point of L (local minimum/maximum/saddle point). Interpretation: We aim for expectation matching: E y p φ(x, y) = φ(x, y obs ) but discriminatively: only for x {x 1,..., x n }. 20 / 39

21 L(w) = 1 N σ 2 Id D D + [E y p(y x n,w)φ(x n, y)][e y p(y x n,w)φ(x n, y)] positive definite Hessian matrix L(w) is convex w L(w) = 0 implies global minimum. 21 / 39

22 Milestone I: Probabilistic Training (Conditional Random Fields) p(y x, w) log-linear in w R D. Training: many probabilistic derivations lead to same optimization problem minimize negative conditional log-likelihood, L(w) L(w) is differentiable and convex, gradient descent will find global optimum with w L(w) = 0 Same structure as multi-class logistic regression. For logistic regression: this is where the textbook ends. we re done. For conditional random fields: we re not in safe waters, yet! 22 / 39

23 Task: Compute v = w L(w cur ), evaluate L(w cur + ηv): L(w) = 1 2σ 2 w 2 N [ w, φ(x n, y n ) + log e w,φ(xn,y) ] y Y w L(w) = 1 N σ 2 w [ φ(x n, y n ) p(y x n, w)φ(x n, y) ] y Y Problem: Y typically is very (exponentially) large: binary image segmentation: Y = ranking N images: Y = N!, e.g. N = 1000: Y We must use the structure in Y, or we re lost. 23 / 39

24 Solving the Training Optimization Problem Numerically w L(w) = 1 N σ 2 w [ φ(x n, y n ) E y p(y x n,w)φ(x n, y) ] Computing the Gradient (naive): O(K M ND) L(w) = 1 2σ 2 w 2 N [ w, φ(x n, y n ) + log Z(x n, w) ] Line Search (naive): O(K M ND) per evaluation of L N: number of samples D: dimension of feature space M: number of output nodes 100s to 1,000,000s K: number of possible labels of each output nodes 2 to 100s 24 / 39

25 Probabilistic Inference to the Rescue Remember: in a graphical model with factors F, the features decompose: φ(x, y) = ( ) φ F (x, y F ) F F ( ) E y p(y x,w) φ(x, y) = E y p(y x,w) φ F (x, y F ) F F = ( E yf p(y F x,w)φ F (x, y F ) ) F F E yf p(y F x,w)φ F (x, y F ) = Factor marginals µ F = p(y F x, w) p(y F x, w) φ }{{} F (x, y F ) y F Y F }{{} factor marginals K F terms are much smaller than complete joint distribution p(y x, w), can be computed/approximated, e.g., with (loopy) belief propagation. 25 / 39

26 Solving the Training Optimization Problem Numerically w L(w) = 1 N σ 2 w [ φ(x n, y n ) E y p(y x n,w)φ(x n, y) ] Computing the Gradient: O(K M nd), O(MK Fmax ND): L(w) = 1 2σ 2 w 2 N [ w, φ(x n, y n ) + log e w,φ(xn,y) ] y Y Line Search: O(K M nd), O(MK Fmax ND) per evaluation of L N: number of samples 10s to 1,000,000s D: dimension of feature space M: number of output nodes K: number of possible labels of each output nodes 26 / 39

27 What, if the training set D is too large (e.g. millions of examples)? Simplify Model Train simpler model (smaller factors) Less expressive power results might get worse rather than better Subsampling Create random subset D D. Train model using D Ignores all information in D \ D Parallelize Train multiple models in parallel. Merge the models. Follows the multi-core/gpu trend How to actually merge? (or?) Doesn t really save computation. 27 / 39

28 What, if the training set D is too large (e.g. millions of examples)? Stochastic Gradient Descent (SGD) Keep maximizing p(w D). In each gradient descent step: Create random subset D D, often just 1 3 elements! Follow approximate gradient w L(w) = 1 σ 2 w (x n,y n ) D [ φ(x n, y n ) E y p(y x n,w)φ(x n, y) ] Line search no longer possible. Extra parameter: stepsize η SGD converges to argmin w L(w)! (if η chosen right) SGD needs more iterations, but each one is much faster more: see L. Bottou, O. Bousquet: The Tradeoffs of Large Scale Learning, NIPS also: 28 / 39

29 Solving the Training Optimization Problem Numerically w L(w) = 1 N σ 2 w [ φ(x n, y n ) E y p(y x n,w)φ(x n, y) ] Computing the Gradient: O(K M nd), O(MK 2 ND) (if BP is possible): L(w) = 1 2σ 2 w 2 N [ w, φ(x n, y n ) + log e w,φ(xn,y) ] y Y Line Search: O(K M nd), O(MK 2 ND) per evaluation of L N: number of samples D: dimension of feature space: φ i,j 1 10s, φ i : 100s to 10000s M: number of output nodes K: number of possible labels of each output nodes 29 / 39

30 Typical feature functions in image segmentation: φ i (y i, x) R 1000 : local image features, e.g. bag-of-words w i, φ i (y i, x) : local classifier (like logistic-regression) φ i,j (y i, y j ) = y i = y j R 1 : test for same label w ij, φ ij (y i, y j ) : penalizer for label changes (if w ij > 0) combined: argmax y p(y x) is smoothed version of local cues original local classification local + smoothness 30 / 39

31 Typical feature functions in handwriting recognition: φ i (y i, x) R 1000 : image representation (pixels, gradients) w i, φ i (y i, x) : local classifier if x i is letter y i φ i,j (y i, y j ) = e yi e yj R : letter/letter indicator w ij, φ ij (y i, y j ) : encourage/suppress letter combinations combined: argmax y p(y x) is corrected version of local cues local classification local + correction 31 / 39

32 Typical feature functions in pose estimation: φ i (y i, x) R 1000 : local image representation, e.g. HoG w i, φ i (y i, x) : local confidence map φ i,j (y i, y j ) = good fit(y i, y j ) R 1 : test for geometric fit w ij, φ ij (y i, y j ) : penalizer for unrealistic poses together: argmax y p(y x) is sanitized version of local cues original local classification local + geometry [V. Ferrari, M. Marin-Jimenez, A. Zisserman: Progressive Search Space Reduction for Human Pose Estimation, CVPR 2008.] 32 / 39

33 Typical feature functions for CRFs in Computer Vision: φ i (y i, x): local representation, high-dimensional w i, φ i (y i, x) : local classifier φ i,j (y i, y j ): prior knowledge, low-dimensional w ij, φ ij (y i, y j ) : penalize outliers learning adjusts parameters: unary wi : learn local classifiers and their importance binary w ij : learn importance of smoothing/penalization argmax y p(y x) is cleaned up version of local prediction 33 / 39

34 Solving the Training Optimization Problem Numerically Idea: split learning of unary potentials into two parts: local classifiers, their importance. Two-Stage Training pre-train f y i (x) ˆ= log p(y i x) use φ i (y i, x) := f y i (x) RK (low-dimensional) keep φ ij (y i, y j ) are before perform CRF learning with φ i and φ ij Advantage: lower dimensional feature space during inference faster f y i (x) can be stronger classifiers, e.g. non-linear SVMs Disadvantage: if local classifiers are bad, CRF training cannot fix that. 34 / 39

35 Solving the Training Optimization Problem Numerically CRF training methods is based on gradient-descent optimization. The faster we can do it, the better (more realistic) models we can use: w L(w) = w σ 2 N [ φ(x n, y n ) y Y p(y x n, w) φ(x n, y) ] R D A lot of research on accelerating CRF training: problem solution method(s) Y too large exploit structure (loopy) belief propagation smart sampling contrastive divergence use approximate L e.g. pseudo-likelihood N too large mini-batches stochastic gradient descent D too large trained φ unary two-stage training 35 / 39

36 Summary CRF Learning Given: training set {(x 1, y 1 ),..., (x n, y n )} X Y, feature function φ : X R D. (x n, y n ) i.i.d. d(x, y) 1 Task: find parameter vector w such that Z exp( w, φ(x, y) ) d(y x). CRF solution derived by minimizing negative conditional log-likelihood: w = argmin w 1 2σ 2 w 2 N [ w, φ(x n, y n ) log e w,φ(xn,y) ] y Y convex optimization problem gradient descent works training needs repeated runs of probabilistic inference 36 / 39

37 Sebastian Nowozin and Christoph Lampert Structured Models in Computer Vision Part 4. Conditional Random Fields Extra I: Beyond Fully Supervised Learning So far, training was fully supervised, all variables were observed. In real life, some variables can be unobserved even during training. missing labels in training data latent variables, e.g. part location latent variables, e.g. part occlusion latent variables, e.g. viewpoint 37 / 39

38 Graphical model: three types of variables x X always observed, y Y observed only in training, z Z never observed (latent). Marginalization over Latent Variables Construct conditional likelihood as usual: p(y, z x, w) = Derive p(y x, w) by marginalizing over z: p(y x, w) = 1 exp( w, φ(x, y, z) ) Z(x, w) 1 Z(x, w) p(y, z x, w) z Z 38 / 39

39 Negative regularized conditional log-likelihood: L(w) = λ w 2 = λ w 2 = λ w 2 + N log p(y n x n, w) N log p(y n, z x n, w) z Z N log exp( w, φ(x n, y n, z) ) z Z N log z Z y Y exp( w, φ(x n, y, z) ) L is not convex in w can have local minima no agreed on best way for treating latent variables 39 / 39