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AB.MHSVM
balance6.0±4.010.0±2.0
blood22.0±4.021.0± 5.0
wdbc3.0± 2.02.0±3.0
breast34.0 ±13.037.0±8.0
ecoli15.0±6.016.0±6.0
iris7.0±6.05.0±6.0
pima24.0±5.024.0±4.0
sonar13.0±10.014.0±8.0
wine2.0±3.03.0±4.0
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MethodisoletletteroptdigitspendigitsUSPS
ADABoosT.MH w Hamming trees3.5±0.52.1±0.22.0±0.32.1±0.34.5±0.5
ADABoosT.MH w Hamming prod. (Kégl & Busa-Fekete, 2009)4.2±0.52.5±0.22.1 ± 0.42.1±0.24.4±0.5
AOSOLOGITBoOST J= 20,v = 0.1 (Sun et al.,2012)3.5 ± 0.52.3±0.22.1 ± 0.32.4± 0.34.9±0.5
ABCLOGITB0OSTJ= 20,v= 0.1 (Li,2009b)4.2±0.52.2±0.23.1± 0.42.9±0.34.9±0.5
ABCMARTJ= 20,v=0.1(Li,2009a)5.0±0.62.5±0.22.6±0.43.0± 0.35.2±0.5
LOGITB0OST J= 20,v = 0.1 (Li,2009b)4.7± 0.52.8±0.33.6±0.43.1± 0.35.8±0.5
SAMME w single-label trees (Zhu etal., 2009)2.3±0.22.5 ± 0.3
ADABoosT.MH w single-label trees (Zhu et al., 2009)2.6± 0.32.8±0.3
ADABoosT.MM (Mukherjee & Schapire,2013)2.5± 0.22.7 ± 0.3
ADABoosT.MH w single-label trees (Mukherjee & Schapire,2013)9.0±0.57.0± 0.4
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1 (a,v,φ(·)) ←BASE(X,Y,W)
2S ← PRIORITYQUEUE→ O(log N) insertion and extraction of maximum key
3INSERT(S,(v,φ(.),X,Y,NULL,O),γ(ν,𝜑,X,Y,W))> key = edge γ
4$←(）Dinitialize classifier list
5for j←1 to N
6ζ ←τj ←NULLinitialize child indices
7(Vj,j(:),Xj,Yj,·,jp) ← ExTRACTMAX(S)>best node in the priority queue
8if·=- then lj, ← j else if·= + then rj, ← j>child index of parent
9$ ←APPEND(S,Vjj(·))> adding hj(·) =Vjφj(-) to S
10(X_,Y_,W_,X+,Y+,W+) ← CUTDATASET(Xj,Yj,W,φj())
11
12for· ∈ {-,+}→insert children into priority queue
13(a.,v.,φ.(·)) ←BASE(X.,Y.,W.) INSERT(S,(v.,φ.(-),X.,Y.,.,j),γ(v.,γ.,X.,Y.,W.) - γ(vj,φj,X.,Y.W.))
14D key= edge improvement over parent edge 1+γ(h,W) α D standard coefficient of the full tree classifier h1 (14)
15log 1-γ(61,W) return (a,5,[,t)
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STUMPBASE(X, Y,W)
1 forl←1 to K>for all classes n > 2 20↑ Wi,lyi,l> classwise edges (1O) of constant classifier h(O)(x) = 1
3 for j←1 to di=1 D all (numerical) features
4s ← SORT(𝑥x(),..,x)） sort the jth column of X
5(vj,bj,γj) ← BEsTSTUMP(s,Y,W,γ(0))D best stump per feature
6 αj←1+Yj log base coefficient (12) 2 1-Yj
7 j* ← arg min Z(αjVjj,b,W)> best stump across features
J return (αj*,Vj*,j*,b*(·))
BESTSTUMP(s,Y,W, γ(0))
1γ*← γ(0）best edge vector
2γ ← γ(o）initial edge vector
3for i← 1to n-1 >for all points in order s1 ≤...≤ sn-1
4for l ← 1 to K>for all classes
5
Ye ← Ye - 2wi,lyi,l > update classwise edges of stump with v = 1
6if si Si+1 then> no threshold if identical coordinates Si = Si+1
7if∑=1 lel >∑ε-1l|thenfound beter stump
8γ* ←γupdate best edge vector
96*←Si+si+1 update best threshold 2
10forl ←1 to Kfor all classes
11U ← sign(γe) D set vote vector according to (13)
12ifγ*=γ(0） did not beat the constant classifier
13return (v*,-O,llγ*lli）D constant classifier with optimal votes
14else
15
return (v*,b*,llγ*ll1）)>best stump
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