Large-Margin Thresholded Ensembles for Ordinal Regression

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1 Large-Margin Thresholded Ensembles for Ordinal Regression Hsuan-Tien Lin and Ling Li Learning Systems Group, California Institute of Technology, U.S.A. Conf. on Algorithmic Learning Theory, October 9, 2006 H.-T. Lin and L. Li (Learning Systems Group) Large-Margin Thresholded Ensembles 2006/0/09 / 8

2 Ordinal Regression Problem Ordinal Regression what is the age-group of the person in the picture? rank: a finite ordered set of labels Y = {, 2,, K } ordinal regression: given training set {(x n, y n )} N n=, find a decision function g that predicts the ranks of unseen examples well e.g. ranking movies, ranking by document relevance, etc. matching human preferences: applications in social science and info. retrieval H.-T. Lin and L. Li (Learning Systems Group) Large-Margin Thresholded Ensembles 2006/0/09 2 / 8

3 Ordinal Regression Problem Properties of Ordinal Regression regression without metric: possibly metric underlying (age), but not encoded in {, 2, 3, 4} monotonic invariance relabel by {2, 3, 5, 7} should not change results general regression deteriorates without metric classification with ordered categories: small mistake classify a teenager as a child; big mistake classify an infant as an adult no shuffle invariance relabel by {3,, 2, 4} lose information general classification cannot use ordering information ordinal regression resides uniquely between classification and regression H.-T. Lin and L. Li (Learning Systems Group) Large-Margin Thresholded Ensembles 2006/0/09 3 / 8

4 Ordinal Regression Problem Error Functions for Ordinal Regression two aspects of ordinal regression: determine the category discrete nature or at least have a close prediction ordering preference categorical prediction: classification error L C (g, x, y) = [ g(x) y ] close prediction: absolute error L A (g, x, y) = g(x) y neither perfect; both common H.-T. Lin and L. Li (Learning Systems Group) Large-Margin Thresholded Ensembles 2006/0/09 4 / 8

5 Ordinal Regression Problem Our Contributions new model for ordinal regression: thresholded ensemble model combines thresholding and ensemble learning new generalization bounds for thresholded ensembles theoretical guarantee of performance new algorithms for constructing thresholded ensembles simple and efficient Figure: target; traditional regression; our ordinal regression promising experimental results H.-T. Lin and L. Li (Learning Systems Group) Large-Margin Thresholded Ensembles 2006/0/09 5 / 8

6 Thresholded Ensemble Model Thresholded Model commonly used in previous work: thresholded perceptrons (PRank, Crammer and Singer, 2005) thresholded SVMs (SVOR, Chu and Keerthi, 2005) prediction procedure: compute a potential function H(x) (e.g. raw perceptron output) 2 quantize H(x) by some ordered θ to get g(x) g(x) x x H(x) θ θ2 θ3 thresholded model: g(x) g H,θ (x) = min {k : H(x) < θ k } H.-T. Lin and L. Li (Learning Systems Group) Large-Margin Thresholded Ensembles 2006/0/09 6 / 8

7 Thresholded Ensemble Model Thresholded Ensemble Model g(x) x x H(x) θ θ2 θ3 the potential function H(x) is a weighted ensemble H(x) H T (x) = T t= w th t (x) intuition: combine preferences to estimate the overall confidence e.g. if many people, h t, say a movie x is good, the confidence of the movie H(x) should be high good theoretical and algorithmic properties inherited from ensemble learning for classification H.-T. Lin and L. Li (Learning Systems Group) Large-Margin Thresholded Ensembles 2006/0/09 7 / 8

8 New Large-Margin Bounds of Thresholded Ensembles Margins of Thresholded Ensembles g(x) x x H(x) θ θ 2 θ 3 ρ ρ2 ρ 3 margin: safe from the boundary normalized margin for thresholded ensemble { } /( ) HT (x) θ ρ(x, y, k) = k, if y > k T K w θ k H T (x), if y k t + θ k t= k= negative margin wrong prediction K ] [ ρ(x, y, k) 0 g(x) y = L A (g, x, y) k= H.-T. Lin and L. Li (Learning Systems Group) Large-Margin Thresholded Ensembles 2006/0/09 8 / 8

9 New Large-Margin Bounds of Thresholded Ensembles New Large-Margin Bounds for the Model core results: if (x n, y n ) i.i.d. from D, with prob. > δ, > 0, N K E (x,y) D L A (g, x, y) N [ ρ(xn, y n, k) ] ( ) ) + O K N + log δ E (x,y) D L C (g, x, y) 2 N n= k= N y n n= k=y n [ ρ(xn, y n, k) ] + O sketch of the proof (to be illustrated with L A ): ( N ( log 2 N 2 ( log 2 N 2 + log δ reduce ordinal regression examples to dependent binary examples 2 extract i.i.d. binary examples; apply existing classification bounds 3 bound the deviation caused by the i.i.d. extraction large-margin thresholded ensembles could generalize ) ) H.-T. Lin and L. Li (Learning Systems Group) Large-Margin Thresholded Ensembles 2006/0/09 9 / 8

10 New Large-Margin Bounds of Thresholded Ensembles Reduction to Binary Classification g(x) x x H(x) θ K binary classification problems w.r.t. each θ k encode (x, y, k) as ( (X) k, (Y ) k ) = ( (x, k ), sign(y k 0.5) ) : ρ(x, y, k) (Y ) k ( HT (x) θ, k ) = bin. classifier margin ρ C ( (X)k, (Y ) k ) key observation: K ] E (x,y) D L A (g, x, y) = E (x,y) D [ ρ(x, y, k) 0 k= ] = (K )E (x,y) D,k K [ ρ(x, y, k) 0 [ ( ) ] = (K )E ((X)k,(Y ) k ) ˆD ρc (X)k, (Y ) k 0 ordinal regression problem = one big joint binary classification problem H.-T. Lin and L. Li (Learning Systems Group) Large-Margin Thresholded Ensembles 2006/0/09 0 / 8

11 New Large-Margin Bounds of Thresholded Ensembles Extraction of Independent Examples E (x,y) D L A (g, x, y) = (K )E ((X)k,(Y ) k ) ˆD testing distribution ˆD of ( (X) k, (Y ) k ) : derived from (x, y, k) D K [ ρc ( (X)k, (Y ) k ) 0 ] extended training examples Ŝ = {( (X n ) k, (Y n ) k )} : not i.i.d. from ˆD ; cannot be directly used in existing bounds i.i.d. subset of Ŝ: randomly choose k n for each n apply ensemble learning bound (Schapire et al., 998): if (x n, y n, k n ) i.i.d. from D K, with prob. > δ, > 0, E (x,y) D L A (g, x, y) K N N [ ρ(xn, y n, k n ) ] ( + O K n= can we obtain a deterministic RHS? N ( log 2 N 2 + log δ H.-T. Lin and L. Li (Learning Systems Group) Large-Margin Thresholded Ensembles 2006/0/09 / 8 ) )

12 New Large-Margin Bounds of Thresholded Ensembles Deviation from the Extraction E (x,y) D L A (g, x, y) K N N [ ρ(xn, y n, k n ) ] ( + O K n= N ( log 2 N 2 + log δ let b n = [ ρ(x n, y n, k n ) ] : binary independent r.v. with mean µ n = K K [ ρ(xn, y n, k) ] k= extended Chernoff bound: with prob. > δ, K N N b n N n= N K [ ρ(xn, y n, k) ] + O n= k= connection between bound and algorithm design? boosting ( ) N log δ H.-T. Lin and L. Li (Learning Systems Group) Large-Margin Thresholded Ensembles 2006/0/09 2 / 8 ) )

13 New Large-Margin Algorithms for Thresholded Ensembles Boosting for Large-Margin Thresholded Ensembles existing algorithm (RankBoost, Freund et al., 2003): construct H T iteratively with some margin concepts, but no θ our work: RankBoost-AE: extended RankBoost for ordinal regression obtain θ by minimizing training L A using dynamic programming ORBoost: new boosting formulation for ordinal regression ORBoost: simpler and faster than existing approaches; connects well to large-margin bounds H.-T. Lin and L. Li (Learning Systems Group) Large-Margin Thresholded Ensembles 2006/0/09 3 / 8

14 New Large-Margin Algorithms for Thresholded Ensembles ORBoost: Ordinal Regression Boosting inspired from AdaBoost: operationally N min exp ( ρ(x n, y n ) ) max softmin n ρ(x n, y n ) n= ORBoost: N min exp ( ρ(x n, y n, k) ) N [ const. ρ(xn, y n, k) ] n= k n= k ORBoost-LR k {y n, y n } connects to bound on L C ORBoost-All k {, 2,, K } connects to bound on L A algorithmic derivation based on theoretical bounds H.-T. Lin and L. Li (Learning Systems Group) Large-Margin Thresholded Ensembles 2006/0/09 4 / 8

15 New Large-Margin Algorithms for Thresholded Ensembles Advantages of ORBoost ensemble learning: combine simple preferences to approximate complex targets thresholding: adaptively estimating scales to perform ordinal regression benefits inherited from AdaBoost simple implementation if h t good enough: guarantee on rapidly minimizing [ ρ(xn, y n, k) ] n,k decision function g improves with T ORBoost not very vulnerable to overfitting in practice H.-T. Lin and L. Li (Learning Systems Group) Large-Margin Thresholded Ensembles 2006/0/09 5 / 8

16 Experimental Results ORBoost v.s. RankBoost test absolute error RankBoost ORBoost Results (ORBoost-All) significantly better than RankBoost (best existing boosting approach) simpler to implement and less vulnerable to overfitting 0 py ma bo ab ba co ca ce dataset ORBoost: promising boosting approach for ordinal regression H.-T. Lin and L. Li (Learning Systems Group) Large-Margin Thresholded Ensembles 2006/0/09 6 / 8

17 Experimental Results ORBoost v.s. SVOR test absolute error SVOR ORBoost Results (ORBoost-All) comparable to SVOR (state-of-the-art algorithm) much faster in training ( hour v.s. 2 days on 6000 examples) 0 py ma bo ab ba co ca ce dataset ORBoost: could be especially useful for large-scale tasks H.-T. Lin and L. Li (Learning Systems Group) Large-Margin Thresholded Ensembles 2006/0/09 7 / 8

18 Conclusion Conclusion thresholded ensemble model: useful for ordinal regression theoretical reduction: new large-margin bounds algorithmic reduction: new learning algorithms ORBoost: simplicity and better performance over existing boosting algorithm comparable performance to state-of-the-art algorithms fast training and not very vulnerable to overfitting broader reduction view: many more bounds/algorithms and more general error functions (Li and Lin, NIPS 2006) Thank you. Questions? H.-T. Lin and L. Li (Learning Systems Group) Large-Margin Thresholded Ensembles 2006/0/09 8 / 8

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