1 / 21

Regression trees and regression graphs: Efficient estimators for Generalized Additive Models

Regression trees and regression graphs: Efficient estimators for Generalized Additive Models . Adam Tauman Kalai TTI-Chicago. Outline. Generalized Additive Models (GAM) Computationally efficient regression Model Thm: Regression graph algorithm efficiently learns GAMs

Sophia
Télécharger la présentation

Regression trees and regression graphs: Efficient estimators for Generalized Additive Models

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Regression trees and regression graphs:Efficient estimators for Generalized Additive Models Adam Tauman Kalai TTI-Chicago

  2. Outline • Generalized Additive Models (GAM) • Computationally efficient regression • Model Thm: Regression graph algorithm efficiently learns GAMs • Regression tree algorithm • Regression graph algorithm Correlation boosting • [Valiant] [Kearns&Schapire] New [Mansour&McAllester] New

  3. Generalized Additive Models[Hastie & Tibshirani] • e.g., Generalized linear models • u( w¢x ), monotonic u • linear/logistic models • e.g., f(x) = e–||x||2 = e–x(1)2–x(2)2…–x(d)2 Dist.  over X£Y = Rd£R f(x) = E[y|x] = u(f1(x(1))+f2(x(2))+…+fd(x(d))) monotonic u: R!R, arbitrary fi: R!R

  4. Non-Hodgkin’s Lymphoma International Prognostics Index [NEJM ‘93] Risk Factors age>60, # sites>1, perf. status>1, LDH>normal, stage>2

  5. Setup  X £Y .1 1 1 1 0 .4 .4 0 1 0 0 .3 0 0 1 1 1 0 0 1 .1 1 1 1 0 0 .2 1 1 0 1 regression algorithm 0 1 1 0 .3 1 1 1 1 0 0 1 .8 0 1 .3 0 1 1 1 .4 1 .4 0 1 1 1 0 .5 0 1 1 .7 1 1 0 0 0 0 1 0 0 0 1 0 1 1 1 0 1 0 0 .3 1 1 1 1 1 1 0 1 1 “training error” (h,train) = i(h(xi)-y)2 0 0 0 0 0 .2 0 0 1 0 0 0 .02 1 0 0 1 1 .5 0 0 0 0 1 1 .4 0 0 0 0 1 .2 0 0 0 1 0 1 0 0 1 1 .2 0 1 0 1 .3 1 n 0 0 h: X! [0,1] “true error” (h) = E[(h(x)-y)2] X = RdY = [0,1] training sample: (x1,y1),…,(xn,yn)

  6. X £ [0,1] h: X! [0,1] Computationally-efficient regression [Kearns&Schapire] Family of target functions Definition: A efficiently learns F: f(x) = E[y|x] 2F, 8 with probability 1-, >0 E[(h(x)-y)2] · E[(f(x)-y)2]+(term)/nc n examples true error (h) poly(|f|,1/) Learning Algorithm A A’s runtime must be poly(n,|f|)

  7. Outline • Generalized Additive Models (GAM) • Computationally efficient regression • Model Thm: Regression graph algorithm efficiently learns GAMs • Regression tree algorithm • Regression graph algorithm Correlation boosting • [Valiant] [Kearns&Schapire] New [Mansour&McAllester] New

  8. New Results for GAM’s 1 .1 1 0 0 .6 0 0 0 .7 0 1 1 Regression Graph Learner 0 0 .8 1 .4 0 1 .2 1 1 1 1 0 1 1 0 1 h: Rd ![0,1] n samples 2 X £ [0,1] X µRd Thm:reg. graph learner efficiently learns GAMs • 8dist  over X£Y with E[y|x] = f(x) 2 GAM • E[(h(x)-y)2] · E[(f(x)-y)2] + O(LV log(dn/)) • runtime = poly(n,d) 8  with probability 1-, n1/7

  9. New Results for GAM’s • f(x) = u(i fi(x(i))) • u: R!R, monotonic, L-Lipschitz (L=max |u’(z)|) • fi: R!R, bounded total variationV = i s |fi’(z)|dz Thm:reg. graph learner efficiently learns GAMs • 8dist  over X£Y with E[y|x]=f(x) 2 GAM • E[(h(x)-y)2] · E[(f(x)-y)2] + O(LV log(dn/)) • runtime = poly(n,d) n1/7

  10. New Results for GAM’s 1 .1 0 0 .6 0 0 1 0 .7 0 1 1 Regression Tree Learner 0 0 .8 1 .4 0 1 .2 1 1 1 1 0 1 1 0 1 h: Rd ![0,1] n samples 2 X £ [0,1] X µRd Thm:reg. tree learner inefficiently learns GAMs • 8dist  over X£Y with E[y|x]=f(x) 2 GAM • E[(h(x)-y)2] · E[(f(x)-y)2] + O(LV) • runtime = poly(n,d) ( ) 1/4 log(d) log(n)

  11. Regression Tree Algorithm • Regression tree RT: Rd! [0,1] • Training sample (x1,y1),(x2,y2),…,(xn,yn) 2Rd£ [0,1] (x1,y1), (x2,y2), … avg(y1,y2,…,yn)

  12. Regression Tree Algorithm • Regression tree RT: Rd! [0,1] • Training sample (x1,y1),(x2,y2),…,(xn,yn) 2Rd£ [0,1] x(j) ¸ ? (xi,yi): x(j) <  (xi,yi): x(j) ¸ avg(yi: xi(j)<) avg(yi: xi(j)¸)

  13. Regression Tree Algorithm • Regression tree RT: Rd! [0,1] • Training sample (x1,y1),(x2,y2),…,(xn,yn) 2Rd£ [0,1] x(j) ¸ ? (xi,yi): x(j) <  x(j’) ¸’ ? avg(yi: xi(j)<) (xi,yi): x(j) ¸  andx(j’)< ’ (xi,yi): x(j) ¸  andx(j’) ¸’ avg(yi: x(j)¸Æx(j’)¸’) avg(yi: x(j)¸Æx(j’)<’)

  14. Regression Tree Algorithm • n = amount of training data • Put all data into one leaf • Repeat until size(RT)=n/log2(n): • Greedily choose leaf and split x(j) · to minimize (RT,train) =  (RT(xi)-yi)2/n • Divide data in split node into two new leaves Equivalent to “Gini”

  15. Regression Graph Algorithm [Mansour&McAllester] • Regression graph RG: Rd! [0,1] • Training sample (x1,y1),(x2,y2),…,(xn,yn) 2Rd£ [0,1] x(j) ¸ ? x(j’’) ¸’’ ? x(j’) ¸’ ? (xi,yi): x(j) < andx(j’’)< ’’ (xi,yi): x(j) <  andx(j’’) ¸’’ (xi,yi): x(j) ¸  andx(j’)< ’ (xi,yi): x(j) ¸  andx(j’) ¸’ avg(yi: x(j)¸Æx(j’)¸’) avg(yi: x(j)<Æx(j’’)<’’) avg(yi: x(j)¸Æx(j’)<’) avg(yi: x(j)<Æx(j’’)¸’’)

  16. Regression Graph Algorithm [Mansour&McAllester] • Regression graph RG: Rd! [0,1] • Training sample (x1,y1),(x2,y2),…,(xn,yn) 2Rd£ [0,1] x(j) ¸ ? x(j’’) ¸’’ ? x(j’) ¸’ ? (xi,yi): x(j) < andx(j’’)< ’’ (xi,yi): x(j) < andx(j’’) ¸’’ or x(j) ¸ and x(j’) < ’ (xi,yi): x(j) ¸  andx(j’) ¸’ avg(yi: x(j)¸Æx(j’)¸’) avg(yi: x(j)<Æx(j’)<’) avg(yi: (x(j)<Æx(j’’)¸’’)Ç(x(j)¸Æx(j’)<’))

  17. Regression Graph Algorithm [Mansour&McAllester] • Put all n training data into one leaf • Repeat until size(RG)=n3/7: • Split: greedily choose leaf and split x(j) · to minimize (RG,train) =  (RG(xi)-yi)2/n • Divide data in split node into two new leaves • Let  be the decrease in (RG,train) from this split • Merge(s): • Greedily choose two leaves whose merger increases (RG,train) as little as possible • Repeat merging while total increase in (RG,train) from merges is ·/2

  18. Two useful lemmas • Uniform generalization bound for any n: • Existence of a correlated split:There always exists a split I(x(i) ·) s.t., regression graph R probability over training sets (x1,y1),…,(xn,yn)

  19. Motivating natural example • X = {0,1}d, f(x) = (x(1)+x(2)+…+x(d))/d, uniform  • Size(RT) ¼ exp(Size(RG)c), e.g. d=4: x(1)>½ x(1)>½ x(2)>½ x(2)>½ x(2)>½ x(2)>½ x(3)>½ x(3)>½ x(3)>½ x(4)>½ x(4)>½ x(4)>½ x(4)>½ x(3)>½ x(3)>½ x(3)>½ x(3)>½ 0 .25 .5 .75 1 x(4)>½ x(4)>½ x(4)>½ x(4)>½ x(4)>½ x(4)>½ x(4)>½ x(4)>½ .25 .5 .5 .75 .5 .75 .75 1 .25 .5 .5 .75 0 .25 .25 .5

  20. Regression boosting • Incremental learning • Suppose you find something of positive correlation with y, then reg. graphs make progress • “Weak regression” implies strong regression, i.e. small correlations can efficiently be combined to get correlation near 1 (error near 0) • Generalizes binary classification boosting[Kearns&Valiant, Schapire, Mansour&McAllester,…]

  21. Conclusions • Generalized additive models are very general • Regression graphs, i.e., regression trees with merging, provably estimate GAMs using polynomial data and runtime • Regression boosting generalizes binary classification boosting • Future work • Improve algorithm/analysis • Room for interesting work in statistics Å computational learning theory

More Related