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ニューラルネットワーク
のアルゴリズム
4/18/2018
カムエラ ラウ (Kamuela Lau)
1
自己紹介
• ロンウイットに12月に入社
• 米国ハワイ州出身
• Williams College(ウィリアムズ大学)
• 東北復興支援・ボランティア活動
• ツイッター:@kamu_lau
2
ランキング学習
Lucene/Solr/ES
クエリ1
文書a1
文書b1
:
ランキング1
クエリ2
文書a2
文書b2
:
ランキング2
クエリn
文書an
文書bn
:
ランキングn
・・・ ランキング学習
モデル
クエリx
文書ax
文...
損失関数
• 結果を期待(ラベル)と比較
• 二重誤差(Square Error)
• 交差エントロピー誤差
C =
X
i
P0
i log Pi
<latexit sha1_base64="BtY2KgOiGEY3Xe4gHynu3EDJo...
精度評価指標
• 損失ではランク全体の精度が見えな
い
• アルゴリズムの性能を評価するもの
• 例:NDCG、MAP、MRR など
DCG@k =
kX
j=1
G(yj)⌘(j)
G(yj) = 2yj
1
⌘(j) =
1
log(j +...
パーセプトロンとPRank
パーセプトロン(Perceptron)
• パーセプトロンは、生き物の神経メカニズムを単純に
モデルしたもの
• 複数の信号を受け取り、ひとつの信号を出力
出力
入力
PRank
• 入力層のノード数は文書の特徴量
...
PRank
xd
w
Abstract
ument is used to produce the equations used in the Learning-
sentation.
ons
~xd = x1 . . . xn
~! = !1 ...
PRank 重みの更新
b1 b2 b3 b4 b5
w
x
w
x
8
PRank NDCG
MQ2007 Fold1
9
PRank の利点と欠点
• 単純パーセプトロンはシン
プルで学習ができる
• 不安定な評価と損失
• 線形非分離な問題を解けな
い
モデル
10
OAP-BPM
• 複数のPRank
• bernoulli number という確率で
文書の特徴を各PRankに入力
• PRank に文書の特徴を入力し
た場合、OAP を以下の式で重
みを更新
PRank1
PRank2
PRank3
...
OAP-BPM NDCG
MQ2007 Fold1
12
ニューラルネットワークと
多層パーセプトロン
• 複数のノードの層で構造
• 各ノードは相前後する層の全てのノードにリ
ンク
• リンクは重み付き
• 隠れ層と出力層の入力信号はwx+b
• 一つか複数の入力
• 一つの出力
• 活性化関数
•...
多層パーセプトロンの
アルゴリズム
共通点
• フォーワードプロパゲーション
• 最急降下法・確率的勾配降下法とバックプロパゲーションで損失を
最小化
異なる点
• ネットワークの構造
• 損失関数
• 重みの更新手法・頻度
14
最急降下法と
確率的勾配降下法
• 最急降下法(Gradient Descent; Batch Gradient Descent)
• 関数の勾配ベクトルと逆の傾きへ向かうと、極小値に近づく
• エポック(全データ)ごとに重みを更新→ 時間がか...
バックプロパゲーション
隠れ層 2
出力層
(oi = A(Ii))<latexit sha1_base64="e/bqynqxifSovS9ZB3kQYM9PG/Q=">AAACcXichVHLSsNAFD2N7/qqulHchBalIpQb...
NNRank
• ポイントワイズ
• 出力層のノード数はクラス(分
類)の数
• 出力活性化関数:Sigmoid
• ネットワークは直接文書を分類
• しきい値が初めて出力を超える
ノードの番号が予測の星の数
入力層
隠れ層 1
出力層
しきい...
NNRank
• 損失関数:二重誤差
• 文書ごとに重みを更新:
• 特徴を入力、
forwardProp()
• ネットワークの出力により
分類
• ラベルと違えば、ラベルを
出力層に入力、backProp()
• 重みを更新
入力層
隠れ層...
NNRank NDCG
MQ2008 Fold1
19
RankNet
• Microsoft が開発
• Bing で使われている
• 様々なアルゴリズムのベース
• ペアワイズ
• 入力次元:文書1個の特徴量
• 出力ノード1個
• 出力活性化関数:Sigmoid
入力層
隠れ層
出力層
20
RankNet
• 交差エントロピー誤差
• バックプロパゲーションに
は2つの文書が必要
oij ⌘ f(~xi) f( ~xj)<latexit sha1_base64="BHpjhXe1W9x38YpjP/MTY7w+IuQ=">AAAC...
RankNet 重みの更新
1. 文書i で forwardProp
2. 文書j で forwardProp
3. o を計算
4. で backProp
5. 文書i で forwardProp
6. で backProp
7. updat...
RankNet NDCG
MQ2008 Fold1
23
FRankNet (Factored Rank Net)
• ペアワイズ(だが、リスト
の情報も必要)
• ネットワークの構造は
RankNet と同じ
• 重みの更新手法が異なる点
入力層
隠れ層
出力層
24
FRankNet 重みの更新
• クエリ k に対して
1. 全部の文書ペアで
forwardProp、λi を計算
2. 各文書において
forwardProp、backProp
3. 重みを更新*
• λの計算にはリストが必要
• Mini...
FRankNet NDCG
MQ2008 Fold1
26
LambdaRank
• FRankNet のλを改良したア
ルゴリズム
• 勾配ベクトル(λ)を直接定
義
• 損失関数を最大化する x1
x2
@C
@o1
>
@C
@o2<latexit sha1_base64="iHxeis5ptrr...
LambdaRank NDCG
MQ2008 Fold1
28
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Learning-to-Rank for Apache Lucene

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learningtorank-meetup-vol-1-pt2

  1. 1. ニューラルネットワーク のアルゴリズム 4/18/2018 カムエラ ラウ (Kamuela Lau) 1
  2. 2. 自己紹介 • ロンウイットに12月に入社 • 米国ハワイ州出身 • Williams College(ウィリアムズ大学) • 東北復興支援・ボランティア活動 • ツイッター:@kamu_lau 2
  3. 3. ランキング学習 Lucene/Solr/ES クエリ1 文書a1 文書b1 : ランキング1 クエリ2 文書a2 文書b2 : ランキング2 クエリn 文書an 文書bn : ランキングn ・・・ ランキング学習 モデル クエリx 文書ax 文書bx : ランキング? クエリx 文書ax 文書bx : ランキング 推定値 教師データ作成 リランキング特徴抽出 特徴抽出 :LTR4Lが提供 3
  4. 4. 損失関数 • 結果を期待(ラベル)と比較 • 二重誤差(Square Error) • 交差エントロピー誤差 C = X i P0 i log Pi <latexit sha1_base64="BtY2KgOiGEY3Xe4gHynu3EDJobY=">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</latexit><latexit sha1_base64="BtY2KgOiGEY3Xe4gHynu3EDJobY=">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</latexit><latexit sha1_base64="BtY2KgOiGEY3Xe4gHynu3EDJobY=">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</latexit><latexit sha1_base64="BtY2KgOiGEY3Xe4gHynu3EDJobY=">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</latexit> 4 C = X i (yi xi)2 <latexit sha1_base64="6AUnMNHHv6V37hcTV93XTA5a/Wc=">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</latexit><latexit sha1_base64="6AUnMNHHv6V37hcTV93XTA5a/Wc=">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</latexit><latexit sha1_base64="6AUnMNHHv6V37hcTV93XTA5a/Wc=">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</latexit><latexit sha1_base64="6AUnMNHHv6V37hcTV93XTA5a/Wc=">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</latexit>
  5. 5. 精度評価指標 • 損失ではランク全体の精度が見えな い • アルゴリズムの性能を評価するもの • 例:NDCG、MAP、MRR など DCG@k = kX j=1 G(yj)⌘(j) G(yj) = 2yj 1 ⌘(j) = 1 log(j + 1)<latexit sha1_base64="y7maz3GxJVki3qvh90hNgaHRl2s=">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</latexit><latexit 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  6. 6. パーセプトロンとPRank パーセプトロン(Perceptron) • パーセプトロンは、生き物の神経メカニズムを単純に モデルしたもの • 複数の信号を受け取り、ひとつの信号を出力 出力 入力 PRank • 入力層のノード数は文書の特徴量 • 出力ノード1個 • ランキング順位数と同じ数のしきい値を用意 • 出力を初めて超えるしきい値の番号が予測値(クラス) • 符号に基づいて重みを更新 6
  7. 7. PRank xd w Abstract ument is used to produce the equations used in the Learning- sentation. ons ~xd = x1 . . . xn ~! = !1 !2 . . . !n (1) ~! · ~x = 0 B B B B B B @ !1 !2 . . . !n 1 C C C C C C A x1 x2 . . . xn (2) 𝝉 = クラスの符号の数 = i番目のしきい値(符号のみ)bi ~! = ~! + (⌧ ⇥ ~x) bi = bi 1 ~b = b1 . . . bm ~! = !1 . . . !n ~xd = x1 . . . xn ~! = 0 . . . 0 ~b = 0 . . . 1 重みとしきい値の初期値 7
  8. 8. PRank 重みの更新 b1 b2 b3 b4 b5 w x w x 8
  9. 9. PRank NDCG MQ2007 Fold1 9
  10. 10. PRank の利点と欠点 • 単純パーセプトロンはシン プルで学習ができる • 不安定な評価と損失 • 線形非分離な問題を解けな い モデル 10
  11. 11. OAP-BPM • 複数のPRank • bernoulli number という確率で 文書の特徴を各PRankに入力 • PRank に文書の特徴を入力し た場合、OAP を以下の式で重 みを更新 PRank1 PRank2 PRank3 OAP 文書 b b b ~W = ~W + ~wi N ~W<latexit sha1_base64="eCP6JyRHmvy5ZPoGKXu1EmqkSxY=">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</latexit><latexit sha1_base64="eCP6JyRHmvy5ZPoGKXu1EmqkSxY=">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</latexit><latexit sha1_base64="eCP6JyRHmvy5ZPoGKXu1EmqkSxY=">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</latexit><latexit sha1_base64="eCP6JyRHmvy5ZPoGKXu1EmqkSxY=">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</latexit> 11
  12. 12. OAP-BPM NDCG MQ2007 Fold1 12
  13. 13. ニューラルネットワークと 多層パーセプトロン • 複数のノードの層で構造 • 各ノードは相前後する層の全てのノードにリ ンク • リンクは重み付き • 隠れ層と出力層の入力信号はwx+b • 一つか複数の入力 • 一つの出力 • 活性化関数 • g(x) = 出力 入力層 隠れ層 1 隠れ層 2 出力層 13
  14. 14. 多層パーセプトロンの アルゴリズム 共通点 • フォーワードプロパゲーション • 最急降下法・確率的勾配降下法とバックプロパゲーションで損失を 最小化 異なる点 • ネットワークの構造 • 損失関数 • 重みの更新手法・頻度 14
  15. 15. 最急降下法と 確率的勾配降下法 • 最急降下法(Gradient Descent; Batch Gradient Descent) • 関数の勾配ベクトルと逆の傾きへ向かうと、極小値に近づく • エポック(全データ)ごとに重みを更新→ 時間がかかる • 確率的勾配降下法(Stochastic Gradient Descent, SGD) • 勾配ベクトルをデータの一部(mini-batch)で近似 ~! = ~! ⌘~r · C(~!)<latexit sha1_base64="C+ojduwYEID5BhfpgaV7/WhHJBI=">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</latexit><latexit sha1_base64="C+ojduwYEID5BhfpgaV7/WhHJBI=">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</latexit><latexit sha1_base64="C+ojduwYEID5BhfpgaV7/WhHJBI=">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</latexit><latexit sha1_base64="C+ojduwYEID5BhfpgaV7/WhHJBI=">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</latexit> -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 -1 1 2 3 4 15
  16. 16. バックプロパゲーション 隠れ層 2 出力層 (oi = A(Ii))<latexit sha1_base64="e/bqynqxifSovS9ZB3kQYM9PG/Q=">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</latexit><latexit 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  17. 17. NNRank • ポイントワイズ • 出力層のノード数はクラス(分 類)の数 • 出力活性化関数:Sigmoid • ネットワークは直接文書を分類 • しきい値が初めて出力を超える ノードの番号が予測の星の数 入力層 隠れ層 1 出力層 しきい値: 0.5 0.9 0.7 0.4 2 0.9 0.4 0.7 1 0.4 0.6 0.7 0星の数 17
  18. 18. 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sha1_base64="ei2Fw21EH4eJMeAhnr9TLdkGfNo=">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</latexit> ターゲット 18
  19. 19. NNRank NDCG MQ2008 Fold1 19
  20. 20. RankNet • Microsoft が開発 • Bing で使われている • 様々なアルゴリズムのベース • ペアワイズ • 入力次元:文書1個の特徴量 • 出力ノード1個 • 出力活性化関数:Sigmoid 入力層 隠れ層 出力層 20
  21. 21. RankNet • 交差エントロピー誤差 • バックプロパゲーションに は2つの文書が必要 oij ⌘ f(~xi) f( ~xj)<latexit sha1_base64="BHpjhXe1W9x38YpjP/MTY7w+IuQ=">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</latexit><latexit 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  22. 22. RankNet 重みの更新 1. 文書i で forwardProp 2. 文書j で forwardProp 3. o を計算 4. で backProp 5. 文書i で forwardProp 6. で backProp 7. updateWeights** (sj)<latexit sha1_base64="maLPbcvr7hn91Ka5EbwWb6KaxwU=">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</latexit><latexit 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sha1_base64="4beQqzidsGlNBOPduniRdvAWFYs=">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</latexit><latexit 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sha1_base64="4beQqzidsGlNBOPduniRdvAWFYs=">AAACb3ichVG7SgNBFD1ZXzE+ErVQECQYFC0Md0VQxEKwsfQVFYyE3XWMo/tidxPQkB/wA7Sw8AEi4mfY+AMWfoJYiYKNhTebBVFR7zAzZ87cc+fMjO6a0g+IHmJKQ2NTc0u8NdHW3tGZTHV1r/pOyTNEznBMx1vXNV+Y0ha5QAamWHc9oVm6Kdb0vbna/lpZeL507JVg3xWblla05bY0tICp/Fjel0VLK1TkbrWQylCWwkj/BGoEMohiwUldIY8tODBQggUBGwFjExp8bhtQQXCZ20SFOY+RDPcFqkiwtsRZgjM0Zvd4LPJqI2JtXtdq+qHa4FNM7h4r0xiie7qmF7qjG3qk919rVcIaNS/7POt1rXALycO+5bd/VRbPAXY+VX96DrCNqdCrZO9uyNRuYdT15YPjl+XppaHKMF3QE/s/pwe65RvY5VfjclEsnSDBH6B+f+6fYHU8q1JWXZzIzM5EXxFHPwYxwu89iVnMYwE5PtfFEU5xFntWepUBJV1PVWKRpgdfQhn9AED4jqw=</latexit><latexit sha1_base64="4beQqzidsGlNBOPduniRdvAWFYs=">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</latexit> @C @oi = 1 1 + e oij = @C @oj<latexit 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sha1_base64="O6DpxFueQCYYFkV7ZTa15SFMxM0=">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</latexit><latexit sha1_base64="O6DpxFueQCYYFkV7ZTa15SFMxM0=">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</latexit> 22
  23. 23. RankNet NDCG MQ2008 Fold1 23
  24. 24. FRankNet (Factored Rank Net) • ペアワイズ(だが、リスト の情報も必要) • ネットワークの構造は RankNet と同じ • 重みの更新手法が異なる点 入力層 隠れ層 出力層 24
  25. 25. FRankNet 重みの更新 • クエリ k に対して 1. 全部の文書ペアで forwardProp、λi を計算 2. 各文書において forwardProp、backProp 3. 重みを更新* • λの計算にはリストが必要 • Mini-batch i = X {i,j}2I ij SSX j:{i,j}2I ij <latexit sha1_base64="xl5Y9BC+MC/PWcSG3J3f0anU6ng=">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</latexit><latexit sha1_base64="xl5Y9BC+MC/PWcSG3J3f0anU6ng=">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</latexit><latexit sha1_base64="xl5Y9BC+MC/PWcSG3J3f0anU6ng=">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</latexit><latexit sha1_base64="xl5Y9BC+MC/PWcSG3J3f0anU6ng=">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</latexit> !k = X i i <latexit sha1_base64="qiXFRxSFv//JCRbcA7wSEo0WIAI=">AAAChnichVHLShxBFD22RseJ0YnZCNl0MiiuhtuSoASEgWxc+sioYEtT3VNOiql+0F0zYIasBX/ARVYRREJwaz7AjT+QhZ8gWSq4ySJ3ehpClCS3qKpTp+65darKT7TKDNHVkDU88mh0rDRefjzxZHKq8nR6M4s7aSAbQazjdNsXmdQqkg2jjJbbSSpF6Gu55bff9ve3ujLNVBy9M/uJ3A1FK1J7KhCGKa/ywm1KbYQbh7IlvLa9bLtZJ/SU7Wou0hSe8ipVqlEe9kPgFKCKIlbjyilcNBEjQAchJCIYxhoCGbcdOCAkzO2ix1zKSOX7Eh9RZm2HsyRnCGbbPLZ4tVOwEa/7NbNcHfApmnvKShuz9J2+0A1d0le6pp9/rdXLa/S97PPsD7Qy8aYOZzbu/qsKeTZ4/1v1T88Ge1jKvSr2nuRM/xbBQN/9cHSz8WZ9tjdHx/SD/X+mK7rgG0Td2+BkTa5/Qpk/wLn/3A/B5kLNoZqz9qparxdfUcJzvMQ8v/ci6ljBKhp87gHOcI5vVsmqWa+txUGqNVRonuGPsOq/AD4JliA=</latexit><latexit sha1_base64="qiXFRxSFv//JCRbcA7wSEo0WIAI=">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</latexit><latexit sha1_base64="qiXFRxSFv//JCRbcA7wSEo0WIAI=">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</latexit><latexit sha1_base64="qiXFRxSFv//JCRbcA7wSEo0WIAI=">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</latexit> ij = 1 1 + e oij <latexit sha1_base64="9gvufywzNFEvs4GFHjCeosgKBHU=">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</latexit><latexit sha1_base64="9gvufywzNFEvs4GFHjCeosgKBHU=">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</latexit><latexit sha1_base64="9gvufywzNFEvs4GFHjCeosgKBHU=">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</latexit><latexit sha1_base64="9gvufywzNFEvs4GFHjCeosgKBHU=">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</latexit> 25
  26. 26. FRankNet NDCG MQ2008 Fold1 26
  27. 27. LambdaRank • FRankNet のλを改良したア ルゴリズム • 勾配ベクトル(λ)を直接定 義 • 損失関数を最大化する x1 x2 @C @o1 > @C @o2<latexit sha1_base64="iHxeis5ptrrn4QqMgMO5h4Gy9k4=">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</latexit><latexit sha1_base64="iHxeis5ptrrn4QqMgMO5h4Gy9k4=">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</latexit><latexit sha1_base64="iHxeis5ptrrn4QqMgMO5h4Gy9k4=">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</latexit><latexit sha1_base64="iHxeis5ptrrn4QqMgMO5h4Gy9k4=">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</latexit> ij = | NDCG| 1 + eoij <latexit sha1_base64="Zd0wMYDGOG47oidv3HuX7kPA1Ew=">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</latexit><latexit sha1_base64="Zd0wMYDGOG47oidv3HuX7kPA1Ew=">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</latexit><latexit sha1_base64="Zd0wMYDGOG47oidv3HuX7kPA1Ew=">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</latexit><latexit sha1_base64="Zd0wMYDGOG47oidv3HuX7kPA1Ew=">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</latexit> wk = wk + ⌘ @C @wk<latexit sha1_base64="0/S3gMPqsR3lLta26nYnz1Ojxyo=">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</latexit><latexit sha1_base64="0/S3gMPqsR3lLta26nYnz1Ojxyo=">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</latexit><latexit sha1_base64="0/S3gMPqsR3lLta26nYnz1Ojxyo=">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</latexit><latexit sha1_base64="0/S3gMPqsR3lLta26nYnz1Ojxyo=">AAAClXichVHdShtBGD1ua6up1rS9UOjN0qAIQvhWCkpBCFXEO3+jgpEwu050yGZ32Z2kpCEv4APohVcKItJX6F2p+AK98BGKlwreeOG3mwWxUv2GmTlz5jvfnJmxA1dFmuiiy3jxsvvV657ezJu+/rcD2XfvVyO/Hjqy6PiuH67bIpKu8mRRK+3K9SCUoma7cs2uTsf7aw0ZRsr3VnQzkJs1se2pinKEZqqcpW/lqjllxuOYWZJamKVKKJxWKRChVsI1p9v3mLPa5WyO8pSE+RhYKcghjQU/e4IStuDDQR01SHjQjF0IRNw2YIEQMLeJFnMhI5XsS7SRYW2dsyRnCGarPG7zaiNlPV7HNaNE7fApLveQlSaG6Q+d0hWd0w/6S7f/rdVKasRemjzbHa0MygO7Q8s3z6pqPGvs3Kue9KxRwWTiVbH3IGHiWzgdfeP7/tXyl6Xh1ggd0SX7P6QL+sU38BrXzvGiXDpAhj/A+ve5H4PV8bxFeWvxc67wNf2KHnzEJ4zye0+ggDksoMjn7uEnfuPMGDSmjBljtpNqdKWaD3gQxvwdj0mb5g==</latexit> x1 x2 x1 x2 RankNet LambdaRank )<latexit sha1_base64="rHOo/+l3OdLDYPu2bZIJuPGst98=">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</latexit><latexit sha1_base64="rHOo/+l3OdLDYPu2bZIJuPGst98=">AAACbnichVFdKwRRGH52fK31tSgpiWzkantHilyJG5csi1rSzDh2T2Znppmzq7X5A+7lQhElyc9w4w+48BPkRlFuXHh3dkoI73TmPOc57/Oe57zH9GwZKKKHmNbQ2NTcEm9NtLV3dHYlu3tWA7fkWyJrubbrr5tGIGzpiKySyhbrni+MommLNXN3vra/VhZ+IF1nRVU8sVk08o7ckZahmMptZGS+oAzfd/e2kilKUxjDP4EegRSiWHSTV9jANlxYKKEIAQeKsQ0DAX856CB4zG2iypzPSIb7AgdIsLbEWYIzDGZ3+Z/nVS5iHV7Xagah2uJTbB4+K4cxSvd0TS90Rzf0SO+/1qqGNWpeKjybda3wtroO+5ff/lUVeVYofKr+9Kywg+nQq2TvXsjUbmHV9eX945flmcxodYwu6In9n9MD3fINnPKrdbkkMidI8APo39v9E6xOpHVK60uTqdm56CniGMAIxrnfU5jFAhaRDTt2hFOcxZ61Pm1QG6qnarFI04svoY1/AJxwjmk=</latexit><latexit sha1_base64="rHOo/+l3OdLDYPu2bZIJuPGst98=">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</latexit><latexit sha1_base64="rHOo/+l3OdLDYPu2bZIJuPGst98=">AAACbnichVFdKwRRGH52fK31tSgpiWzkantHilyJG5csi1rSzDh2T2Znppmzq7X5A+7lQhElyc9w4w+48BPkRlFuXHh3dkoI73TmPOc57/Oe57zH9GwZKKKHmNbQ2NTcEm9NtLV3dHYlu3tWA7fkWyJrubbrr5tGIGzpiKySyhbrni+MommLNXN3vra/VhZ+IF1nRVU8sVk08o7ckZahmMptZGS+oAzfd/e2kilKUxjDP4EegRSiWHSTV9jANlxYKKEIAQeKsQ0DAX856CB4zG2iypzPSIb7AgdIsLbEWYIzDGZ3+Z/nVS5iHV7Xagah2uJTbB4+K4cxSvd0TS90Rzf0SO+/1qqGNWpeKjybda3wtroO+5ff/lUVeVYofKr+9Kywg+nQq2TvXsjUbmHV9eX945flmcxodYwu6In9n9MD3fINnPKrdbkkMidI8APo39v9E6xOpHVK60uTqdm56CniGMAIxrnfU5jFAhaRDTt2hFOcxZ61Pm1QG6qnarFI04svoY1/AJxwjmk=</latexit> 損失が下がる 27
  28. 28. LambdaRank NDCG MQ2008 Fold1 28

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