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LTR4L のブースティング
アルゴリズム
8/25/2018
カムエラ ラウ (Kamuela Lau)
1
自己紹介
• ロンウイットに2017年12月入社
• LTR4L において LTR のアルゴリズムを実装
• ブログ
• https://jp.kamulau.com 

(English: en.kamulau.com)
• ツイッター: @kamu_lau
2
強い学習機
ブースティングとは
• アンサンブル学習の一種
• 複数の弱い(低性能の)学習機を組み合わせ、
強い(高性能の)学習機を生成する
弱い学習機 1 弱い学習機 2 弱い学習機 n+ + +. . .
3
弱い学習機 (weak learner)
身長
閾値 :165cm
女性 男性
身長 165身長 < 165
4
弱い学習機 その2
スカート・ドレス過去5年間
の履く回数
閾値 : 3
男性 女性
回数 3回数 < 3
5
ブースティングアルゴリズムの異な
る点
• 弱い学習機の構造
• 弱い学習機の求め方
• 弱い学習機のスコア、重み計算
6
RankBoost
• AdaBoost に基づいたアルゴ
リズム
• 学習時がペアワイズ

(判別時はポイントワイズ)
• 弱い学習機のイメージは右
と同様
身長
閾値 :165cm
女性 男性
身長 165身長 < 165
7
RankBoost 初期化
ペアワイズの分布を用意
D (x0, x1) = c · max {0, (x0, x1)}<latexit sha1_base64="AXzZC+0rGDBZaxOl/KNGL6p4J1o=">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</latexit><latexit sha1_base64="AXzZC+0rGDBZaxOl/KNGL6p4J1o=">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</latexit><latexit sha1_base64="AXzZC+0rGDBZaxOl/KNGL6p4J1o=">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</latexit><latexit sha1_base64="AXzZC+0rGDBZaxOl/KNGL6p4J1o=">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</latexit>
X
x0,x1
D (x0, x1) = 1
<latexit sha1_base64="DCkf8z3npseqsEpsHKXEitYVvhE=">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</latexit><latexit sha1_base64="DCkf8z3npseqsEpsHKXEitYVvhE=">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</latexit><latexit sha1_base64="DCkf8z3npseqsEpsHKXEitYVvhE=">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</latexit><latexit sha1_base64="DCkf8z3npseqsEpsHKXEitYVvhE=">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</latexit>
例えば以下の3個の文書があ
るとしたら、、、(y, x)
(0, xl)
(1, x2)
(2, x3)
分布は以下のようになる

D[x1, x2] = 1/3
D[x1, x3] = 1/3
D[x2, x1] = 0
D[x2, x3] = 1/3
D[x3, x1] = 0
D[x3, x2] = 0
(x0, x1) =
⇢
+1 if y0 < y1
1 if y0 > y1<latexit sha1_base64="Jut8A4aDeyglofK511/WjuThqIQ=">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</latexit><latexit sha1_base64="Jut8A4aDeyglofK511/WjuThqIQ=">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</latexit><latexit sha1_base64="Jut8A4aDeyglofK511/WjuThqIQ=">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</latexit><latexit sha1_base64="Jut8A4aDeyglofK511/WjuThqIQ=">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</latexit>
8
になるように c を選ぶ
RankBoost のイタレーション
• D を用いて r が最大化されるように弱い学習機
(hi(x))を求める
• 以下のルールで分布を変更(Z は正規化因子)

h(x) =
8
<
:
1 if fi(x) > ✓
0 if fi(x)  ✓
q der if fi(x) =?<latexit sha1_base64="tjdsi1n4HgZTKz8/OAc9fVzdhNI=">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</latexit><latexit sha1_base64="tjdsi1n4HgZTKz8/OAc9fVzdhNI=">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</latexit><latexit sha1_base64="tjdsi1n4HgZTKz8/OAc9fVzdhNI=">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</latexit><latexit sha1_base64="tjdsi1n4HgZTKz8/OAc9fVzdhNI=">AAADUnicjVHNahRBEK7Z8SdZo1njRfDSuCTEy1ITBEVQgl48Jlk3CWTC0jNbs9tkdmbS07tkM+xFb76AB08RRMTH8OID6CGPIOIpgiAerJmdICao6aa7vv6qvuqqbi8JVWoQD62Kfe78hYtT09VLM5evzNauzq2n8UD71PLjMNabnkwpVBG1jDIhbSaaZN8LacPbeZT7N4akUxVHT8wooe2+7EYqUL40TLVrX3tiUeyJW+K+cEMKjJsJ16OuijKptRyNRSZCnrl1eF9g6xraM2xVwEQg2jlkNEnzgN09MpIJ12UPnkXEN++e0O0WIceqDml2jM+Si/tISCd5Hoo6x124WnV7ptGu1bGBxRCngVOCOpRjJa69ARc6EIMPA+gDQQSGcQgSUp5b4ABCwtw2ZMxpRqrwE4yhytoBRxFHSGZ3eO/yaatkIz7nOdNC7fMtIS/NSgHz+Anf4hF+wHf4GX/+NVdW5MhrGbH1JlpK2rPPrze//1fVZ2ug91v1z5oNBHC3qFVx7UnB5F34E/1w/8VR897afLaAr/AL13+Ah/ieO4iG3/zXq7T2Eqr8Ac7J5z4N1pcaDjac1dv15YflV0zBDbgJi/zed2AZHsMKtMC3mtbIemo9q3ys/LAt256EVqxScw3+GPbML4Nn07s=</latexit>
r =
X
x0,x1
D (x0, x1) (h (x1) h (x0))
<latexit sha1_base64="9BXPpa1NNo+LNDH6sbaBnBn/P1s=">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</latexit><latexit sha1_base64="9BXPpa1NNo+LNDH6sbaBnBn/P1s=">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</latexit><latexit sha1_base64="9BXPpa1NNo+LNDH6sbaBnBn/P1s=">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</latexit><latexit sha1_base64="9BXPpa1NNo+LNDH6sbaBnBn/P1s=">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</latexit>
Dt+1 (x0, x1) =
Dt (x0, x1) exp (↵t (ht (x0) ht (x1)))
Zt<latexit sha1_base64="/MxwSm99FLx4baui938rfrkKgOw=">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</latexit><latexit sha1_base64="/MxwSm99FLx4baui938rfrkKgOw=">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</latexit><latexit sha1_base64="/MxwSm99FLx4baui938rfrkKgOw=">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</latexit><latexit sha1_base64="/MxwSm99FLx4baui938rfrkKgOw=">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</latexit>
数式は Yoav Freund, et, al. の「An Efficient Boosting Algorithm for Combining Preferences」より抜粋。
9
fi(x) = x の i特徴量
θ = 閾値
LTR4L における RankBoost
"features" : [ 22, 38, 22, 38, 23, …],
"thresholds" : [0.6, 0.6, 0.5, 0.5, 0.6, … ],
"weights" : [ 1.9261765640212982,
1.7517158411371152, 1.1027753871492012,
1.9348789644695028, 1.2068653815682175, … ]
fid: 22
thresh: 0.6
weight: 1.9
0 1
fid: 38
thresh: 0.6
weight: 1.75
0 1
fid: 22
thresh: 0.5
weight: 1.1
0 1
…
10
LambdaMART
スカート・ドレス過去5年間
の履く回数
閾値 : 3
男性 女性
回数 3回数 < 3
身長
閾値 :165cm
女性
身長 165身長 < 165
MART と LambdaRankを
組み合わせたアルゴリズム
11
LambdaMART 決定木の求め方
• 以下を最小化するために特
徴量と閾値を選び決定木を
作成(MART)
Sj ⌘
X
i2L
(yi µL)
2
+
X
i2R
(yi µR)
2
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feature:j
threshold:t
(y1, x1)
(y2, x2)

.

.

.

(ym, xm)
(y1, x1)
(y2, x2)

.

.

.

(yn, xn)
L R
µL =
1
m
X
i2L
yi
<latexit sha1_base64="ngI3VYY/ZYHT6cNuhrnTFHyylto=">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</latexit><latexit sha1_base64="ngI3VYY/ZYHT6cNuhrnTFHyylto=">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</latexit><latexit sha1_base64="ngI3VYY/ZYHT6cNuhrnTFHyylto=">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</latexit><latexit sha1_base64="ngI3VYY/ZYHT6cNuhrnTFHyylto=">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</latexit>
µR =
1
n
X
i2R
yi
<latexit sha1_base64="VllFTpaHYkWm2mLa13EuJlnaG/k=">AAACi3ichVFNSxtBGH7cWk3TatL2UvCyNCg9hXe1UAkVQkXoUWMTBTcsu+tEB/eL/QjEJX+g9O7Bk0IpxVuv7a0X/4AHf4J4jNBLD32zWSg1WN9hZp555n3eeWbGChwZxUSXE8qDyYdT04VHxcdPZmZL5afPWpGfhLZo2r7jh9uWGQlHeqIZy9gR20EoTNdyxJZ1sDrc3+qKMJK+9yHuBaLtmnue7EjbjJkyygu6mxgNdUXVO6Fpp1o/9fqqHiWukUpVl57a6Ks9QxrlClUpC3UcaDmoII91v/wFOnbhw0YCFwIeYsYOTETcdqCBEDDXRspcyEhm+wJ9FFmbcJbgDJPZAx73eLWTsx6vhzWjTG3zKQ73kJUq5umCvtKAzumMruj3nbXSrMbQS49na6QVgVH6+GLz170ql+cY+39V//Uco4PlzKtk70HGDG9hj/Tdw6PBZq0xny7QKV2z/xO6pJ98A697Y3/eEI1jFPkDtNvPPQ5ai1WNqtrG60r9Xf4VBczhJV7xe79BHe+xjiaf+wnf8B0/lBllSakpb0epykSueY5/Qln7AyX4l84=</latexit><latexit sha1_base64="VllFTpaHYkWm2mLa13EuJlnaG/k=">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</latexit><latexit sha1_base64="VllFTpaHYkWm2mLa13EuJlnaG/k=">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</latexit><latexit sha1_base64="VllFTpaHYkWm2mLa13EuJlnaG/k=">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</latexit>
数式は Christopher J.C. Burges の「From RankNet to LambdaRank to LambdaMART: An Overview」より抜粋。
12
xj
t<latexit sha1_base64="MlyQyAc8NXpIQBG8stpjYGZCQOs=">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</latexit><latexit sha1_base64="MlyQyAc8NXpIQBG8stpjYGZCQOs=">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</latexit><latexit sha1_base64="MlyQyAc8NXpIQBG8stpjYGZCQOs=">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</latexit><latexit sha1_base64="MlyQyAc8NXpIQBG8stpjYGZCQOs=">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</latexit>
xj
< t<latexit sha1_base64="z/YaJOieDIezYfKEkReP68q3aiA=">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</latexit><latexit 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sha1_base64="bs8QlLljyYde7WbNFuyzJJm+ZMk=">AAACVXichVG7SgNBFD1ZX3F9JMFGsAmGiFW4a6NYCTaWeZgHxBB2N6Mu2ewuu5tADP5AWgsLKwUR8TNs/AGL9DZiGcHGwrubgKiod5iZM2fm3DlzR3NMw/OJBhFpYnJqeiY6K8/NywuLsfh8ybPbri6Kum3abkVTPWEalij6hm+KiuMKtaWZoqw1d4P9cke4nmFb+37XEbWWemQZh4au+kxl6/EUZSiM5E+gjEEK47DjNzhAAzZ0tNGCgAWfsQkVHrcqFBAc5mroMecyMsJ9gVPIrG3zKcEnVGabPB7xqjpmLV4HOb1QrfMtJneXlUmk6ZFuaUgPdEfP9P5rrl6YI/DS5VkbaYVTj/WXC2//qlo8+zj+VP3p2cchtkKvBnt3QiZ4hT7Sd07Oh4XtfLq3Rlf0wv4vaUD3/AKr86pf50T+AjLXX/le7Z+gtJFRKKPkCFGsYBXrXOZN7GAPWRT5ugb6OIs8SbKUGP2TFBl/WAJfQlr6AAYJh7c=</latexit><latexit 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sha1_base64="OP5UZCZNADBb/YRzEHUPgZQj0Uc=">AAACX3ichVG7SgNBFD1Z3/GRmEpJExTFKty1UcRCsLE0ahIhPthdx7hmX+xOgjH4A3ZWgqkURMTPsPEHLPwE0S6CjYU3m4BoUO8wM2fOzLlzZq7uWWYgiZ4iSld3T29f/0B0cGh4JBYfHcoFbtk3RNZwLdff1LVAWKYjstKUltj0fKHZuiXyemm5uZ+vCD8wXWdDVj2xbWtFx9w3DU0ylT/aOUwtpuRufJLSFEaqE6htMIl2rLrxG2xhDy4MlGFDwIFkbEFDwK0AFQSPuW3UmPMZmeG+wAmirC3zKcEnNGZLPBZ5VWizDq+bOYNQbfAtFneflSlM0SPdUoMe6I6e6ePXXLUwR9NLlWe9pRXebux0bP39X5XNs8TBl+pPzxL7mA+9muzdC5nmK4yWvnJ83lhfWJuqTdMVvbD/S3qie36BU3kzrjNirY4oF0D9+d2dIDebVimtZgj9SGICM/zNc1jCClaRDU2d4QL1yKuSUMZbpVIi7Zol8C2U5CdEN4sW</latexit><latexit 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sha1_base64="z/YaJOieDIezYfKEkReP68q3aiA=">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</latexit><latexit sha1_base64="z/YaJOieDIezYfKEkReP68q3aiA=">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</latexit><latexit sha1_base64="z/YaJOieDIezYfKEkReP68q3aiA=">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</latexit><latexit sha1_base64="z/YaJOieDIezYfKEkReP68q3aiA=">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</latexit><latexit sha1_base64="z/YaJOieDIezYfKEkReP68q3aiA=">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</latexit>
LambdaMART のスコア計算
• LambdaRank の λ でスコア
を計算する
ij =
| Zij|
1 + e (si sj )
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i =
X
j:{i,j}2I
ij
X
j:{j,i}2I
ij
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@2C
@s2
i<latexit sha1_base64="adKZCbBiL2USW2qNDA2sL2jNLg4=">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</latexit><latexit sha1_base64="adKZCbBiL2USW2qNDA2sL2jNLg4=">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</latexit><latexit sha1_base64="adKZCbBiL2USW2qNDA2sL2jNLg4=">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</latexit><latexit sha1_base64="adKZCbBiL2USW2qNDA2sL2jNLg4=">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</latexit>
数式は Christopher J.C. Burges の「From RankNet to LambdaRank to LambdaMART: An Overview」より抜粋。
13
LTR4L における LambdaMART
14
LTR4L における LambdaMART
{
"config" : null,
"leafIds" : [ 0, 1, 3, 7, 8, 4, 2 ],
"featureIds" : [ 38, 38, 39, -1, -1, -1, -1 ],
"thresh" : [ 0.748092, 0.523628, 0.4, "-Infinity",
"-Infinity", "-Infinity", "-Infinity" ],
"scores" : [ 0.0, 0.0, 0.0, 0.04757469250124157,
0.044148895268034685, -0.026720292627365028,
0.045137089557086 ]
}
15
まとめ
• ブースティングでは、どの特徴量が損失に影響
を与えているかモデルでわかる
• 特徴量選択
• ニューラルネットワークとの組み合わせ
• 特徴量を増やすとどうなるか調査しやすい
16
今後の開発
• SVM
• Click Model
17

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LearningToRank meetup-vol-2-pt2

  • 2. 自己紹介 • ロンウイットに2017年12月入社 • LTR4L において LTR のアルゴリズムを実装 • ブログ • https://jp.kamulau.com 
 (English: en.kamulau.com) • ツイッター: @kamu_lau 2
  • 4. 弱い学習機 (weak learner) 身長 閾値 :165cm 女性 男性 身長 165身長 < 165 4
  • 7. RankBoost • AdaBoost に基づいたアルゴ リズム • 学習時がペアワイズ
 (判別時はポイントワイズ) • 弱い学習機のイメージは右 と同様 身長 閾値 :165cm 女性 男性 身長 165身長 < 165 7
  • 8. RankBoost 初期化 ペアワイズの分布を用意 D (x0, x1) = c · max {0, (x0, x1)}<latexit sha1_base64="AXzZC+0rGDBZaxOl/KNGL6p4J1o=">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</latexit><latexit sha1_base64="AXzZC+0rGDBZaxOl/KNGL6p4J1o=">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</latexit><latexit sha1_base64="AXzZC+0rGDBZaxOl/KNGL6p4J1o=">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</latexit><latexit sha1_base64="AXzZC+0rGDBZaxOl/KNGL6p4J1o=">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</latexit> X x0,x1 D (x0, x1) = 1 <latexit sha1_base64="DCkf8z3npseqsEpsHKXEitYVvhE=">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</latexit><latexit sha1_base64="DCkf8z3npseqsEpsHKXEitYVvhE=">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</latexit><latexit sha1_base64="DCkf8z3npseqsEpsHKXEitYVvhE=">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</latexit><latexit sha1_base64="DCkf8z3npseqsEpsHKXEitYVvhE=">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</latexit> 例えば以下の3個の文書があ るとしたら、、、(y, x) (0, xl) (1, x2) (2, x3) 分布は以下のようになる
 D[x1, x2] = 1/3 D[x1, x3] = 1/3 D[x2, x1] = 0 D[x2, x3] = 1/3 D[x3, x1] = 0 D[x3, x2] = 0 (x0, x1) = ⇢ +1 if y0 < y1 1 if y0 > y1<latexit sha1_base64="Jut8A4aDeyglofK511/WjuThqIQ=">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</latexit><latexit sha1_base64="Jut8A4aDeyglofK511/WjuThqIQ=">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</latexit><latexit sha1_base64="Jut8A4aDeyglofK511/WjuThqIQ=">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</latexit><latexit sha1_base64="Jut8A4aDeyglofK511/WjuThqIQ=">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</latexit> 8 になるように c を選ぶ
  • 9. RankBoost のイタレーション • D を用いて r が最大化されるように弱い学習機 (hi(x))を求める • 以下のルールで分布を変更(Z は正規化因子)
 h(x) = 8 < : 1 if fi(x) > ✓ 0 if fi(x)  ✓ q der if fi(x) =?<latexit sha1_base64="tjdsi1n4HgZTKz8/OAc9fVzdhNI=">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</latexit><latexit sha1_base64="tjdsi1n4HgZTKz8/OAc9fVzdhNI=">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</latexit><latexit sha1_base64="tjdsi1n4HgZTKz8/OAc9fVzdhNI=">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</latexit><latexit sha1_base64="tjdsi1n4HgZTKz8/OAc9fVzdhNI=">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</latexit> r = X x0,x1 D (x0, x1) (h (x1) h (x0)) <latexit 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sha1_base64="/MxwSm99FLx4baui938rfrkKgOw=">AAADWnicjZHPTxNBFMffdv0BRaX8OJB4mdBgMELz1pBgTEgIevDIDwvELmlmh2k7Ybq72Z02QNMDV/gDOHjCxBjjn+HFf8ADif+A4YjGiwdft1sNJVVns/PefN/7vHkz44VaxQbx3MrYN27euj00nB25c/feaG5sfDMOGpGQRRHoINr2eCy18mXRKKPldhhJXve03PL2nnXiW00ZxSrwX5qDUO7UedVXFSW4Iamc+/6clVmLGfaIOazNXC0rZpbtJyKSMJf6STBS1Zp5yJaYW4m4ILUH/yfoyv2wl+lyHdZ4P18bXLBXZH5Q0pWd+myb4q9+Y+1yLo8FTAa77jipk4d0rAa5d+DCLgQgoAF1kOCDIV8Dh5i+EjiAEJK2Ay3SIvJUEpfQhiyxDcqSlMFJ3aO5SqtSqvq07tSME1rQLpr+iEgGM/gZ3+MlfsIP+BV/DqzVSmp0ejkg63VZGZZHj6c2fvyTqpM1UPtD/bVnAxV4kvSqqPcwUTqnEF2+eXh6ufF0fab1AN/gBfV/huf4kU7gN7+Jt2ty/TVk6QGc/uu+7mw+LjhYcNYW8ssr6VMMwX2Yhlm670VYhhewCkUQVsk6so6tk8wXO2MP2yPd1IyVMhNwZdiTvwBtGNMQ</latexit> 数式は Yoav Freund, et, al. の「An Efficient Boosting Algorithm for Combining Preferences」より抜粋。 9 fi(x) = x の i特徴量 θ = 閾値
  • 10. LTR4L における RankBoost "features" : [ 22, 38, 22, 38, 23, …], "thresholds" : [0.6, 0.6, 0.5, 0.5, 0.6, … ], "weights" : [ 1.9261765640212982, 1.7517158411371152, 1.1027753871492012, 1.9348789644695028, 1.2068653815682175, … ] fid: 22 thresh: 0.6 weight: 1.9 0 1 fid: 38 thresh: 0.6 weight: 1.75 0 1 fid: 22 thresh: 0.5 weight: 1.1 0 1 … 10
  • 11. LambdaMART スカート・ドレス過去5年間 の履く回数 閾値 : 3 男性 女性 回数 3回数 < 3 身長 閾値 :165cm 女性 身長 165身長 < 165 MART と LambdaRankを 組み合わせたアルゴリズム 11
  • 12. LambdaMART 決定木の求め方 • 以下を最小化するために特 徴量と閾値を選び決定木を 作成(MART) Sj ⌘ X i2L (yi µL) 2 + X i2R (yi µR) 2 <latexit sha1_base64="0QYNlnFUG/uU45TVrjhlMDqiP0k=">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</latexit><latexit sha1_base64="0QYNlnFUG/uU45TVrjhlMDqiP0k=">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</latexit><latexit sha1_base64="0QYNlnFUG/uU45TVrjhlMDqiP0k=">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</latexit><latexit sha1_base64="0QYNlnFUG/uU45TVrjhlMDqiP0k=">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</latexit> feature:j threshold:t (y1, x1) (y2, x2)
 .
 .
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 (ym, xm) (y1, x1) (y2, x2)
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 .
 .
 (yn, xn) L R µL = 1 m X i2L yi <latexit sha1_base64="ngI3VYY/ZYHT6cNuhrnTFHyylto=">AAACi3ichVHLShxBFD22z0zUGZNNwE3joLgabhtBEQOSEHDhwteoYA9Nd6dmUtgv+jEwNvMDIfssskpAQnCXbdxlkx9wMZ8Qshwhmyxyu6dBdEhyi6o6deqeW6eqrMCRUUzUG1FGx8YnJqcelB5Oz8yWK3OPjiI/CW1Rt33HD08sMxKO9EQ9lrEjToJQmK7liGPr7EW2f9wWYSR97zDuBKLhmi1PNqVtxkwZlSXdTYwd9ZmqN0PTTrVu6nZVPUpcI5WqLj11p6t2DGlUqlSjPNRhoBWgiiJ2/con6HgFHzYSuBDwEDN2YCLidgoNhIC5BlLmQkYy3xfoosTahLMEZ5jMnvHY4tVpwXq8zmpGudrmUxzuIStVLNI1faY+fadL+kG//1orzWtkXjo8WwOtCIzymycHv/6rcnmO8fpW9U/PMZpYz71K9h7kTHYLe6Bvn7/rH2zsL6ZL9JF+sv8P1KNvfAOvfWNf7In99yjxB2j3n3sYHK3UNKppe6vVrefFV0xhHgtY5vdewxa2sYs6n/sWX/AVV8qM8lTZUDYHqcpIoXmMO6G8/AMLAJfB</latexit><latexit 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sha1_base64="VllFTpaHYkWm2mLa13EuJlnaG/k=">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</latexit><latexit 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sha1_base64="VllFTpaHYkWm2mLa13EuJlnaG/k=">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</latexit><latexit sha1_base64="VllFTpaHYkWm2mLa13EuJlnaG/k=">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</latexit> 数式は Christopher J.C. Burges の「From RankNet to LambdaRank to LambdaMART: An Overview」より抜粋。 12 xj t<latexit sha1_base64="MlyQyAc8NXpIQBG8stpjYGZCQOs=">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</latexit><latexit 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  • 13. LambdaMART のスコア計算 • LambdaRank の λ でスコア を計算する ij = | Zij| 1 + e (si sj ) <latexit sha1_base64="2+3acnls2R5eUWjyDzUTNsjYHZs=">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</latexit><latexit 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sha1_base64="2+3acnls2R5eUWjyDzUTNsjYHZs=">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</latexit> i = X j:{i,j}2I ij X j:{j,i}2I ij <latexit 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sha1_base64="adKZCbBiL2USW2qNDA2sL2jNLg4=">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</latexit><latexit sha1_base64="adKZCbBiL2USW2qNDA2sL2jNLg4=">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</latexit> 数式は Christopher J.C. Burges の「From RankNet to LambdaRank to LambdaMART: An Overview」より抜粋。 13
  • 15. LTR4L における LambdaMART { "config" : null, "leafIds" : [ 0, 1, 3, 7, 8, 4, 2 ], "featureIds" : [ 38, 38, 39, -1, -1, -1, -1 ], "thresh" : [ 0.748092, 0.523628, 0.4, "-Infinity", "-Infinity", "-Infinity", "-Infinity" ], "scores" : [ 0.0, 0.0, 0.0, 0.04757469250124157, 0.044148895268034685, -0.026720292627365028, 0.045137089557086 ] } 15
  • 16. まとめ • ブースティングでは、どの特徴量が損失に影響 を与えているかモデルでわかる • 特徴量選択 • ニューラルネットワークとの組み合わせ • 特徴量を増やすとどうなるか調査しやすい 16