Applied Statistics and Machine Learning

Transcription

1 Applied Statistics and Machine Learning MDL, e-l2boosting, Sparse Graphical Model Estimation, and Ridge Estimation Bin Yu, IMA, June 24, /24/13

2 Minimum Description Length Principle MDL is a modern articulation of Occam s Razor through Shannon s Information Theory. Jorma Rissanen (1978): One should choose the model that gives the shortest description of data. Conceptually, the MDL framework removes the need for the postulation of a true model. Rissanen, J. (1978). "Modeling by shortest data description". Automatica 14 (5): doi: / (78) Hansen, M. and Yu. B (2001). Model selection and the principle of Minimum Description Length, JASA, vol. 96, pp More at 2 6/24/13

3 Information Theoretic Foundation of MDL Example of a prefix code on the alphabet of {a, b, c}: a maps into 0, b maps into 10 and c into /24/13

4 Information Theoretic Foundation of MDL 4 6/24/13

5 Any distribution can be viewed as a codelength For any probability distribution on Q on the alphabet, - log Q(a) could be viewed a code length. Q does not have to be the data generating probability distribution. When it is, the code is most efficient in the Shannon sense. 5 6/24/13

6 A valid code to be used in MDL Rissenan (1978) generalized Shannon s source coding theorem to a universal source coding theorem. That is, he showed that if a data generating distribution is known to belong to a nice parametric family of dim k, then the redundancy of any universal code is lower bounded by k/2 log n for n data points, for all members of the parametric family except for a zero measure set. This means that one has to pay a price of k/2 log n for not knowing the particular member of a parametric family. If a code achieves this lower bound, we call it valid to be used in MDL to compare model classes. 6 6/24/13

7 Different valid forms of code lengths to use with MDL in parametric models (0) two-stage code (0) 7 6/24/13

8 Two-stage code length } BIC balances fitting to data with model complexity and so does AIC. The penalty in AIC is motivated from the complexity penalty on prediction for a model while that in BIC is code length of the model both go up with the complexity of the model. } Using two stage coding MDL with sparse regression model will lead to something quite close to LS with a log form of penalty (Mairal s lecture) 8 6/24/13

9 Predictive MDL Two other forms of valid codes (or best to the first order after entropy term): 9 Predictive MDL is closely related to on-line learning. 6/24/13

10 Predictive MDL: example Example: predictive MDL for regression model selection with known noise variance: 10 6/24/13

11 Predictive MDL for combining predictors in audio compression Question: how to compress, in a perceptually lossless manner, mixed audio signals (music and speech, songs) with low delay? Application: internet radio, musicians playing together longdistance 11 6/24/13

12 Predictive MDL for combining predictors in audio compression Key steps in Schuller et al (2002): 1. Perceptual pre-filter banks (related to wavelets) 2. Quantization on the filtered signals 3. Lossless predictive coding for quantized signals 12 6/24/13

13 Predictive MDL for combining predictors in audio compression WCLMS predictor (weighted cascading LMS) in Step3 : Basic predictor P1: normalized LMS (Least Mean Square minimization with one data point at a time: a version of stochastic gradient) (LMS is an old algorithm from control theory) with lag L1. Fitting another N-LMS (with lag L2) to the residual time series to arrive at a new predictor P2; Repeating this with lag L3 gives P3. This cascading is really Tukey s twicing or L2boosting in some sense. For each of the 3 predictors, we get a weight at time i (c=2, mu=0.9) to linearly combine the three predictors: 13 6/24/13

14 Compression results (bit rates) of LMS and WCLMS with different lags 14 6/24/13

15 Comments on compression results Single LMS does not work well compared to WCLMS. For WCLMS, it is better to have long lags (200, 80, 40) to remove more harmonic signals first. 15 6/24/13

16 Mixture form of MDL 16 6/24/13

17 Boosting: computation and estimation in one Boosting was originated from Valiant s Probability Approximate Correct (PAC) learning (Schapire, 1991), but made practical by Freund and Schapire (1996) (AdaBoost). It has enjoyed impressive empirical successes. Boosting idea: choose a base procedure and apply it sequentially to modified data and then linearly combine the resulted estimators. Cross-validation or a test set are commonly used to stop the iterations early. June 24, 2013 page 17

18 Boosting: a gradient view An important step forward for understanding boosting was made via the gradient view of Boosting by Breiman, Mason et al, Friedman et al, Friedman (01): L2 boosting or boosting with squared loss in the regression case: refitting of residuals (Twicing, Tukey, 1972). Number of iteration steps is the regularization parameter. Minimax optimality of L2 boosting in 1-dim case for Sobolev classes (Buhlmann & Yu, 03) (see Yu s last lecture) June 24, 2013 page 18

19 Stepwise regression: selected refs Efroymson, MA (1960) "Multiple regression analysis." In Ralston, A. and Wilf, HS, editors, Mathematical Methods for Digital Computers. Wiley. Hocking, R. R. (1976) "The Analysis and Selection of Variables in Linear Regression," Biometrics, 32. Roughly, forward selection methods use tests (e.g. F-test) to find the most significant predictor to add and then the new model is refitted via OLS. Backward-selection or backward elimination is done similarly. June 24, 2013 page 19

20 Epsilon-L2Boosting (with fixed small steps) Regression set-up: standardized data X i : a p-dimensional predictor Y i : response variable We consider loss function and let j index the jth predictor for j=1,, p. } L 2 Boosting with a fixed step size (Forward Stagewise Fitting, Efron et al, 2004): June 24, 2013 page 20

21 News media text data June 24, 2013 page 21

22 Comparative summarization (CCS) of text June 24, 2013 page 22

23 The need for automatic text summarization June 24, 2013 page 23

24 Concise and Comparative Summarization (CCS) on a given topic (Miratrix et al, 2012) } Make the summarization problem predictive: automatic labeling of documents into positive and negative by thresholding the count of relevant words on the topic. (e.g. if the topic is china, count words china, chinese, ) } Predictors are counts of word phrases (uni-grams, bi-grams, tri-grams ) what are very high dimension (1 million) and very sparse. } Epsilon-L2 Boosting for picking a fixed number of word phrases Our algorithm is a modification of BBR (Genkins et al, 2007) from L1penalized logistic regression to Lasso for sparse, short and fat matrix text data. June 24, 2013 page 24

25 Human comparison experiment Human experiment findings: Lasso with L2 column rescaling works better overall for different document sizes (articles vs paragraphs) than tf-idf normalization and stop-word removal and co-occurrence. tf-idf works well for articles. June 24, 2013 page 25

26 Some CCS results from NYT international section in 2009 June 24, 2013 page 26

27 How does e-l2boosting compare with Lasso? } Lasso (Least Absolute Sum of Squares Operator): where is a regularization parameter. e-l2boosting also gives sparse models. Are they the same? June 24, 2013 page 27

28 Striking Similarity (Efron et al, 2004) Lasso e-l2 Boosting (with small steps) The paths are not always the same. (These plots are reverse of Mairal s path plots that start with lambda=0. June 24, 2013 page 28

29 e-l2boosting could be very different from Lasso L 2 Boosting Lasso June 24, 2013 page 29

30 Lasso trade-off between empirical loss and penalty If we run coordinate gradient descent with a fixed step size on the Lasso loss, at each iteration t: Trade-off June 24, 2013 page 30

31 Forward and Backward Two ways one can reduce the Lasso loss: } Forward Step -- Reduce Empirical Loss (L2 boosting) } Backward Step Reduce Penalty Let s start with a forward step to get, and use an initial June 24, 2013 page 31

32 Blasso or Stagewise Lasso (Zhao &Yu, JMLR, 2004) } Find the backward direction that leads to the minimal empirical loss } Take the direction if it leads to a decrease in the Lasso loss } Otherwise force a forward step } Relax λ whenever necessary T. Zhang s work on backward-forward algorithms: see Mairal s lecture. June 24, 2013 page 32

33 BLasso Converges to Lasso Path as ε goes to zero (Zhao and Y, 2004) BLasso Lasso June 24, pa

34 Gaussian graphical model (cf. Rocha et al, 08, ) 34 6/24/13

35 Gaussian graphical model 35 6/24/13

36 Gaussian graphical model 36 6/24/13

37 Relating regression coefficients with inverse covariance matrix It follows that the precision matrix has the same off-diagonal sparsity patterns as B. 37 6/24/13

38 Gaussian Graphical Model Gaussian inverse covariance characterizes conditional independencies (and hence graphical model) Inverse " Covariance Graph June 24, 2013 page 38

39 L1 penalized -log Gaussian Likelihood Given n iid observations of X with Yuan and Lin (06): Banerjee et al (08), Friedman et al (07): fast algorithms based on block descent. An earlier work was Meinshause and Buhlmann (2006) who estimated the sparsity pattern of the precision matrix via Lasso by regressing one variable at a time against the rest. Rocha et al (2008) integrated the many Lasso problems into one via pseudo-likelihood (SPLICE). June 24, 2013 page 39

40 Example: voting clusters via sparse graphical model estimation Banerjee, El Ghaoui, d Aspremont (08) June 24, 2013 page 40

41 Nuclear norm penalty to induce low-rank in matrix estimation The nuclear norm of a matrix is the sum of its singular values and it is the same as trace norm for symmetric matrices. It has been used as a surrogate for rank in the rank minimization problem to complete a matrix (e.g. Netflix) with missing entries and noise. See lectures by Smola and Mairal and Yu s V4 modeling lecture where the nuclear norm is used with data. 41 6/24/13

42 AdaBoost L2Boosting is an extension of AdaBoost for binary classification to regression with the L2 loss function. AdaBoost was invented by Freund, Yoav; Schapire, Robert E. (1996). A Decision-Theoretic Generalization of on-line Learning and an Application to Boosting. It often improved the classification accuracy of a simple classifier such as stumps by 30% or more via repeatedly application of the same classifier to modified data and then a combination the resulting classifiers. The iteration is stopped early by CV or test data. It was observed that for many data sets even after the training error hit zero, the generalization error kept improving. Is AdaBoost resistant to overfitting? 42 6/24/13

43 AdaBoost Recall that L2Boosting is an extension of AdaBoost for binary classification to regression with the L2 loss function. AdaBoost was invented by Freund, Yoav; Schapire, Robert E. (1995). A Decision-Theoretic Generalization of on-line Learning and an Application to Boosting. It often improved the classification accuracy of a simple classifier such as stumps by 30% or more via repeatedly application of the same classifier to modified data and then a combination the resulting classifiers. The iteration is stopped early by CV or test data. It was observed that for many data sets even after the training error hit zero, the generalization error kept improving. Is AdaBoost resistant to overfitting? 43 6/24/13

44 AdaBoost for a given base (weak) learner with early stopping via test or CV For data (x i,y i ) n, to obtain a classification function F i=1 given a base learner that is better than random guessing, update as follows: w 0 (i) =1/n F 0 (x) =0 f 0 (x) is the classifier learned from original data using the base learner. For m=0, 1,, let Sample data with weights: 44 err m be the error rate of F m+1 (x) =F m (x)+ 1 2 log 1 err m err m f m (x) f m (x) w m+1 (i) w m (i) exp(log 1 err m err m )I [yi =f m (x i )]) Apply the base learner to the sampled data to get 6/24/13 f m+1 (x)

45 AdaBoost: a gradient descent view AdaBoost was much better understood statistically after the gradient descent point of view of it was developed by Breiman, Manson et al and Friedman et al all in It could be viewed as taking gradient descent on an exponential loss function of the margin yf(x): exp (-yf(x)), which is an upper bound on the 0-1 loss function of classification. The log likelihood function of the logistic model can also been seen as a different upper bound. 45 6/24/13

46 Different upper bounds of the 0-1 loss function (Friedman et al, 2000, ADDITIVE LOGISTIC REGRESSION: A STATISTICAL VIEW OF BOOSTING, Annals of Stats, p ) June 24, 2013 page 46

47 AdaBoost is an ensemble method Other ensemble methods include bagging, random forests (RF), model averaging, Such methods average different models/classifiers to come up with the final model/classifier. See Hansen s lecture on CART (decision trees) and RF June 24, 2013 page 47

48 Ridge regression method: L2 penalized LS Ridge regression is an old method, but still very useful. Tikhonov (1943) used it for the first time and this method is also called Tikhonov regularization. June 24, 2013 page 48

49 Ridge regression method: L2 penalized LS June 24, 2013 page 49

50 Ridge regression method: L2 penalized LS which can always be made true if we choose appropriately alpha in (0,1). June 24, 2013 page 50

51 Ridge covariance estimation X X + λi can also be viewed as a regularized covariance estimation. This is the method that we used to reconstruct the movies based on encoding models and fmri signals. We also tried sparse graphical method for precision matrix estimation and the ridge method worked better. June 24, 2013 page 51

52 Back to reconstructing movies Using e-l2boosting or Lasso, for each voxel, we have a good linear prediction rule for fmri responses corresponding to any movie clis. What else do we have? A large movie clip database from movie trailers, YouTube, etc. An external memory to approximately replace our internal memory? 52 6/24/13

53 Given an observed fmri vector Y over selected informative voxels clip 1 Lasso-model > predicted fmri vector Y1 clip > predicted fmri vector Y2 clip > predicted fmri vector Y3 53 : : Rank clip 1, clip 2, clip 3, depending on how close Y1,, Y3 are to Y in terms of a weighted LS metric. 6/24/13

54 Decoding details } Task A: Identify the stimulus the subject has seen } Main strategy: 1. Fit encoding models 2. Choose r=2700 best voxels (based on CV) 3. For each candidate stimulus: 1. Generate prediction of brain activity 2. Give score by how well it fits observed activity Notation Im,Im i,im k represent images. Im i,i 1...n t training images, Im k,k =1...N candidates. Y i R r f(im i ) R p (averaged) observed responses to Im i is the vector of features, β, ( ˆβ) R p r (fitted) linear coefficients. 54 6/24/13

55 Simplest solution square distance } In other words, for a fixed response, we want to find an input that gives the prediction through the encoding model that is closest to the response. } It is natural to use square distance d 2 (Ŷ,Y)=Ŷ Y 2 Problem: Best voxels tend to estimate the same thing. The same signal (part of the image) dominates the reconstruction. Need to down-weigh voxel-groups 55 6/24/13

56 Gaussian model (second order) Given an observed fmri response vector, we postulate a regression model with its linear response function estimated on training data and e-l2boosting or Lasso: Y = f(im)β + w, w N(0, Σ w ) Then the likelihood function for the kth image in the validation set, given the observed fmri vector Y, is exp[ (Y f(im k )β) Σ 1 w (Y f(im k )β)] Wed like to pick the image k in validation set that is closest to the observed fmri signal in a metric defined by the inverse covariance (precision or concentration) matrix. ˆk = arg min k N (Y f(im k )β) Σ 1 w (Y f(im k )β) 56 6/24/13

57 Estimating } } Σ 1 ω Take β = ˆβ, soω represents observed residuals estimated from residuals (of data set aside) ˆΣ ω = Cov(Y i f(im i ) ˆβ) } Since } r n we need some regularization to invert } (Banerjee et al 2009) } (Ledoit and Wolf 2003) 57 6/24/13

58 Results on a simpler image identification data set from the Gallant Lab (2 Repeats) (13 Repeats) 58 6/24/13

59 Reconstruction of movies Nishimoto, Vu, Naselaris, Benjamini, Yu, and Gallant (2011) 1. Download N videos/images from the internet to construct an empirical prior (e.g. movie trailers, youtube videos, ) 2. Using residuals from fitted encoding models, we take into account correlation between voxels through a ridge regularized covariance estimate ˆΣ ω 3. Given an fmri vector Y on voxels, for some image feature g(im), we can calculate its posterior mean: g(im k )exp[ (Y f(im k )β) ˆΣ 1 w (Y f(im k )β))] k N g s represent red, green, blue values at each pixel, frame. Recover color based on the prior, not part of the model. 59 6/24/13

60 Some issues with reconstruction } Weighing images by prior does not work well: the posterior weights are very similar, or not differentiating, possibly because in 26K-dim space, everything is quite far part. } Instead, we used equal weights for top 100 images -- the ordering of posterior weights is informative. 60 6/24/13

61 Reconstruction algorithm on one slide A. Record data B. Fit encoding models (p=26220, n=7200; 20-35K voxels) B. Sample prior and rank fit based on posterior (using chosen voxels) B. Pick top 100 ranks B. Average in color space Averaging in color-scale is quite rough 61 6/24/13

62 Analyzing static image data: comparing Lasso and ridge, nonlinear sparse model and power transform for prediction Episode 1: linear } Fitting details: pre-selection by correlation localization Regularization parameter selected by 5-fold cross validation } e-l2 boosting applied to all 10,000+ features -- L2 boosting is the method of choice in Gallant Lab } Other methods applied to 500 features pre-selected by correlation } E- June 24, 2013 page 62

63 Other methods 1 β = argmin b n Y Xb α Σk/2 b 2 β =(ˆΣ + ασ k ) 1 X T Y/n } } } k = 1: semi OLS (theoretically better than OLS) k = 0: ridge k = -1: semi OLS (inverse) k =1,-1: semi-supervised because of the use of population cov., which is estimated from the whole image data base. June 24, 2013 page 63

64 Validation correlation voxel OLS L2 boost Semi OLS Ridge Semi OLS inverse (k=-1) June 24, 2013 page 64

65 Comparison of the feature locations Semi methods e-l2boost June 24, 2013 page 65

66 Our Story (cont) Episode 2: nonlinear via iv-spam (machine learning stage) } Additive Models (Hastie and Tibshirani, 1990): } Sparse: f p Y i = f j ( X ij ) + " i, i = 1,, j= 1 j 0 for most j n } High dimensional: p >>> n SpAM (Sparse Additve Models) By Ravikumar, Lafferty, Liu, Wasserman (2007) Related work: COSSO, Lin and Zhang (2006) June 24, 2013 page 66 page 66

67 SpAM V1 encoding model (1331 voxels from V1) (Ravikumar, Vu, Gallant Lab, BY, NIPS08, AoAS11) For each voxel, Start with complex cells (features) Pre-selection of 500 complex cells via correlation Fitting of SpAM to 500 complex cells with AIC stopping Pooling of SpAM fitted complex cells according to location and frequency to form pooled complex cells Fitting SpAM to 500 complex cells and pooled complex cells with AIC stopping June 24, 2013 page 67 page 67

68 page 68 June 24, 2013 page 68 Prediction performance (R 2 ) sparse linear pooling model SpAM V1 Model (mean = 14, SD = 17, median = 12, IQR = 17) relative improvement (%) Frequency median improvement 12%. inset region

69 Nonlinearities Compressive effect (finite energy supply) Common nonlinearity for each voxel? June 24, 2013 page 69 page 69

70 Power transformation of features Episode 3: nonlinearity via power transformations (classical stage) Plot a predictor vs. the response: The feature is skewed The response is Gaussian. Response Feature After transformation (4 th root): Response Get better correlation 24 June 2013 June 24, 2013 Feature page 70 page 70

71 Make it Gaussian } Histograms of Gabor wavelets, by spatial frequency. Transformations Frequency None Square root 4 th root Log June 24, 2013 page 71 page 71

72 Gaussianization of features improves prediction } Gaussianized features used to train (using L2 epsilon-boosting) F T = F 1/2 if F in levels 2,3 (4 x 4 and 8 x 8) F T = F 1/4 if F in levels 4,5 (16 x 16 and 32 x 32) June 24, 2013 page 72 page 72