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MMDs 12.3 SVM

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Mining of Massive Datasets
12.3 SVM

Published in: Technology
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MMDs 12.3 SVM

  1. 1. MMDs 12.3
 SVM
  2. 2. Support Vector Machine (SVM) • 境界面と境界面に最も近いデータ点との距離を最大化 するように境界面を決定 
  3. 3. 定式化 (未完) • とりあえず,線形分離可能な場合について • 訓練集合 (yi = 1 or +1)
 が与えられたとき,全てのiについて
 を満たすγのうち,最大のもとを求めたい • の両辺を定数倍するとγはいくらでも大 きくなる
  4. 4. 以下のように境界面をとるようにする
  5. 5. 定式化 • 適当な定数を掛ける事でサポートベクタに対し
 (or -1 ) となるようにできる • 図より,        , より -1=
  6. 6. 定式化 • 結局, γの最大化 → の最小化問題に • 解くべき問題は,
 最小化
 制約条件 (i = 1,2,…,n)

  7. 7. 問題例 制約条件: 最小化: ||w|| = p u2 + v2
  8. 8. もう少し一般の場合 • 先ほどは完全に線形分離できる場合だった • 一般にはノイズ等の影響でだいたい分離できても100% はできないことがある • また,100%分離できたとして,過学習してしまう可能 性がある
  9. 9. 完全に線形分離できない例
  10. 10. 完全に線形分離できない場合 • 分類に失敗したものに対してペナルティを与える • 以下の関数を制約条件のもとで最小化する
 
 
 
 (C: 定数)
 第一項: ||w|| についての関数
 第二項: 損失関数 (ヒンジ関数)
  11. 11.     を最小化する理由 • ||w|| の単調関数ならなんでもいい • ||w|| はルートを含むので計算しにくい • だと2次式で表現され,さらに
 =
  12. 12. ヒンジ関数
  13. 13. SVMの解法 • よく使われるのは,2次計画問題(QP, 目的関数が2次関 数で制約条件が1次関数の最適か問題) として解く方法
 アルゴリズムとしてはSMOなど • 大規模なデータに対しては,最急降下法 (Gradient Descent) が有効 (逐次計算なのでメモリに優しい)
  14. 14. 最急降下法によるSVMの解法 • f の勾配は,
 
 
 • 以下の式で w を更新していく (η: 定数)
  15. 15. 最急降下法によるSVM解法の例
  16. 16. 最急降下法によるSVM解法の例 • C = 0.1, η = 0.2
  17. 17. 確率的最急降下法 (Stochastic Gradient Descent) • 実際に大規模データを扱うときは,全ての訓練集合を 扱うと量が膨大なので,一部を抽出して学習すること がある
 => SGD

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