Transfer Learning Algorithms for Image Classification

Transcription

1 Transfer Learning Algorithms for Image Classification Ariadna Quattoni MIT, CSAIL Advisors: Michael Collins Trevor Darrell 1

2 Motivation Goal: We want to be able to build classifiers for thousands of visual categories. We want to exploit rich and complex feature representations. Problem: We might only have a few labeled samples per category. Word: team Word: actress Word: president 2

3 Thesis Contributions We study efficient transfer algorithms for image classification which can exploit supervised training data from a set of related tass. Learn an image representation using supervised data from auxiliary tass automatically derived from unlabeled images + meta-data. A feature sharing transfer algorithm based on joint regularization. An efficient algorithm for training jointly sparse classifiers in high dimensional feature spaces. 3

4 Outline A joint sparse approximation model for transfer learning. Asymmetric transfer experiments. An efficient training algorithm. Symmetric transfer image annotation experiments. 4

5 Transfer Learning: A brief overview The goal of transfer learning is to use labeled data from related tass to mae learning easier. Two settings: Asymmetric transfer: Resource: Large amounts of supervised data for a set of related tass. Goal: Improve performance on a target tas for which training data is scarce. Symmetric transfer: Resource: Small amount of training data for a large number of related tass. Goal: Improve average performance over all classifiers. 5

6 Transfer Learning: A brief overview Three main approaches: Learning intermediate latent representations: [Thrun 1996, Baxter 1997, Caruana 1997, Argyriou 2006, Amit 2007] Learning priors over parameters: [Raina 2006, Lawrence et al ] Learning relevant shared features via joint sparse regularization: [Torralba 2004, Obozinsy 2006] 6

7 Feature Sharing Framewor: Wor with a rich representation: Complex features, high dimensional space Some of them will be very discriminative (hopefully) Most will be irrelevant Related problems may share relevant features. If we new the relevant features we could: Learn from fewer examples Build more efficient classifiers We can train classifiers from related problems together using a regularization penalty designed to promote joint sparsity. 7

8 Church Airport Grocery Store Flower-Shop 8

9 Church Airport Grocery Store Flower-Shop 9

10 Related Formulations of Joint Sparse Approximation Torralba et al. [2004] developed a joint boosting algorithm based on the idea of learning additive models for each class that share wea learners. Obozinsi et al. [2006] proposed L 1-2 joint penalty and developed a blocwise boosting scheme based on Boosted-Lasso.. 10

11 Our Contribution A new model and optimization algorithm for training jointly sparse classifiers in high dimensional feature spaces. Previous approaches to joint sparse approximation (Torralba et al., 2004, Obozinsi et al.,2006;) have relied on greedy coordinate descent methods. We propose a simple an efficient global optimization algorithm with guaranteed convergence rates 1 O 2 ε Our algorithm can scale to large problems involving hundreds of problems and thousands of examples and features. We test our model on real image classification tass where we observe improvements in both asymmetric and symmetric transfer settings. We show that our algorithm can successfully recover jointly sparse solutions. 11

12 Notation D 1 D = D 2 D, D,..., D { 1 2 m } Dm Collection of Tass D = { D, D2,..., D 1 m } D = {( x1, y1 ),...,( x n, y n )} Joint Sparse Approximation d x R y { + 1, 1} W w w 1,1 2,1 M w w 1, 2 2, 2 O L L O w w 1, m 2, m M w d,1 w d, 2 L w d, m 12

13 Single Tas Sparse Approximation Consider learning a single sparse linear classifier of the form: f ( x) = w x We want a few features with non-zero coefficients Recent wor suggests to use L 1 regularization: min arg w l( f ( x), y) + Q ( x, y) D j= 1 d w j Classification error L 1 penalizes non-sparse solutions Donoho [2004] proved (in a regression setting) that the solution with smallest L 1 norm is also the sparsest solution. 13

14 Joint Sparse Approximation Setting : Joint Sparse Approximation f ( x) = w x arg min 1 m w,,..., l ( f ( x), y) + Q R( w1, w 2,..., w m) 1 w 2 wm = 1 D ( x, y) D Average Loss on training set penalizes solutions that utilize too many features 14

15 Joint Regularization Penalty How do we penalize solutions that use too many features? W W W 1,1 2,1 M d,1 R( W ) W W W 1,2 2,2 O d,2 L L O L =# non W W W zero 1, m 2, m M d, m rows Would lead to a hard combinatorial problem. Coefficients for for feature 2 Coefficients for classifier 2 15

16 Joint Regularization Penalty We will use a L 1- norm [Tropp 2006] R( W ) = d i= 1 max( W i ) This norm combines: The L norm on each row promotes non-sparsity on the rows. Share features An L 1 norm on the maximum absolute values of the coefficients across tass promotes sparsity. Use few features The combination of the two norms results in a solution where only a few features are used but the features used will contribute in solving many classification problems. 16

17 Joint Sparse Approximation Using the L 1- norm we can rewrite our objective function as: min W m 1 l ( x), y) + Q = 1 D ( x, y) D i= 1 ( f d max( W i ) For any convex loss this is a convex objective. For the hinge loss: l( f ( x), y) = max(0,1 yf ( x)) the optimization problem can be expressed as a linear program. 17

18 Joint Sparse Approximation Linear program formulation (hinge loss): Objective: min Max value constraints: for : =1: m and for : i =1: [ W, ε, t] d m 1 D D ε + j Q = 1 j= 1 i= 1 t i w i t i d t i Slac variables constraints: for : =1: m and for : j = 1: D y j f ( x j ε j ) 1 ε 0 j 18

19 Outline A joint sparse approximation model for transfer learning. Asymmetric transfer experiments. An efficient training algorithm. Symmetric transfer image annotation experiments. 19

20 Setting: Asymmetric Transfer SuperBowl Sharon Danish Cartoons Academy Awards Australian Open Trapped Miners Golden globes Grammys Figure Sating Iraq Train a classifier for the 10 th held out topic using the relevant features R only. Learn a representation using labeled data from 9 topics. Learn the matrix W using our transfer algorithm. Define the set of relevant features to be: R = r { r : max ( w ) > 0} 20

21 Results Baseline Representation Transfered Representation Asymmetric Transfer Average AUC # training samples 21

22 An efficient training algorithm The LP formulation can be optimized using standard LP solvers. The LP formulation is feasible for small problems but becomes intractable for larger data-sets with thousands of examples and dimensions. We might want a more general optimization algorithm that can handle arbitrary convex losses. 22

23 Outline A joint sparse approximation model for transfer learning. Asymmetric transfer experiments. An efficient training algorithm. Symmetric transfer image annotation experiments. 23

24 L 1- Regularization: Constrained Convex Optimization Formulation arg min W m 1 D l = 1 ( x, y) D ( f ( x), y) A convex function s. t. d i= 1 max( W i ) C Convex constraints We will use a Projected SubGradient method. Main advantages: simple, scalable, guaranteed convergence rates. Projected SubGradient methods have been recently proposed: L 2 regularization, i.e. SVM [Shalev-Shwartz et al. 2007] L 1 regularization [Duchi et al. 2008] 24

25 Euclidean Projection into the L 1- ball Snapshot of the idea: We map the projection to a simpler problem which involves finding new maximums for each feature across tass and using them to truncate the original matrix. The total mass removed from a feature across tass should be the same for all features whose coefficients don t become zero. 25

26 Euclidean Projection into the L 1- ball 26

27 Characterization of the solution 1 27

28 Characterization of the solution 2 Feature I Feature II Feature III Feature VI μ 1 θ = A 2, j A 2, j > μ 2 μ 2 μ 4 μ 2 μ 3 28

29 Mapping to a simpler problem We can map the projection problem to the following problem which finds the optimal maximums μ: 29

30 Efficient Algorithm for:, in pictures 4 Features, 6 problems, C=14 max( d i= 1 A i ) = 29 μ 1 μ 4 μ 2 μ 3 30

31 Complexity The total cost of the algorithm is dominated by a sort of the entries of A The total cost is in the order of: O( dmlog( dm)) Notice that we only need to consider non-zero entries of A, so the computational cost is dominated by the number of nonzero. 31

32 Outline A joint sparse approximation model for transfer learning. Asymmetric transfer experiments. An efficient training algorithm. Symmetric transfer image annotation experiments. 32

33 Synthetic Experiments Generate a jointly sparse parameter matrix W: For every tas we generate pairs: ( x i, y ) where y = sign( w i t x i ) i We compared three different types of regularization (i.e. projections): L 1 projection L2 projection L1 projection 33

34 Synthetic Experiments Test Error Performance on predicting relevant features 50 Synthetic Experiments Results: 60 problems 200 features 10% relevant 100 Feature Selection Performance Error L2 L1 L1-LINF # training examples per tas Precision L1-INF Recall L1 Precision L1 Recall L1-INF # training examples per tas 34

35 Dataset: Image Annotation Word: president Word: actress Word: team 40 top content words Raw image representation: Vocabulary Tree (Grauman and Darrell 2005, Nister and Stewenius 2006) dimensions 35

36 Experiments: Vocabulary Tree representation Find patches Map each patch to a feature vector. Perform hierarchical -means To compute a representation for an image: Find patches. Map each patch to its closest cluster in each level l l x = [# c, # c2,..., c,,...# c1, # c2,...,# c 1 p1 l p l ] 36

37 Results 37

38 Results

39 Results: 39

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41 Summary of Thesis Contributions We presented a method that learns efficient image representations using unlabeled images + meta-data. We developed a feature sharing transfer based on performing a joint loss minimization over the training sets of related tass with a shared regularization. Previous approaches to joint sparse approximation have relied on greedy coordinate descent methods. We propose a simple an efficient global optimization algorithm for training joint models with L 1 constraints. We provide a tool that maes implementing a joint sparsity regularization penalty as easy and almost as efficient as implementing the standard L1 and L2 penalties. We show the performance of our transfer algorithm on real image classification tass for both an asymmetric and symmetric transfer setting. 41

42 Future Wor Online Optimization. Tas Clustering. Combining feature representations. Generalization properties of L 1 regularized models. 42

43 Thans! 43