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2009.10.08

天気予報から機械学習、金融工学まで

もう随分経ちますが,先日CompView秋の学校というのに行き,2泊3日みっちり機会学習を勉強してきました.講師陣は豪華でどの話も面白かったのですが特にElad Hazanによる"Prediction in the dark: the multi-armed bandit problem"が非常に面白かったです.

その話を説明するために,まず簡単ながら驚くべき性能を達成するアルゴリズムを紹介しましょう.

解きたい問題は,毎日,次の日の天気が晴れか雨かを予想する問題です.t日目が晴れの場合 y(t)=1, 雨の場合 y(t)=0と表すことにしましょう.t日目にy(t+1)を予想するわけです.

さて、自分は天気の専門家ではないので,自分で予報せずに,専門家に頼ることにしてみます.M人の天気予報士がいて,それぞれが独自に次の日の天気を予想しています.i人目の天気予報士のt日目の予報をp(i,t)∈{0,1}とします.

これらの天気予報士の予報を組み合わせて,次の日の予報をしてみます.

まず,全予報士に重み(信頼度)を設定します.m(i,t)がi番目の予報士に対するt日目の信頼度とします.初期値は1,つまりm(i,1)=1 for all iです.

次に,予想が外れた予報士の重みは下げることにします.つまりp(i,t+1)≠y(t+1)だった場合,m(i,t+1)=βm(i,t)とします.βは0<β<1で,どれくらい重みを減らすかの係数です.予報が当たった場合は重みはそのままとします.

最後に肝心の自分の予想をどう決めるかですが,これは重み付き多数決で決めます.晴れと予想した予報士の重みの合計と雨と予想した予報士の重みの合計を比較し,大きい方を採用します.

これが全アルゴリズムです.

さて、このアルゴリズムで,N日目までに自分が間違えた回数をwとします.また,ここまで,一番当たっている予報士i'が予想を間違えた回数をw'とおきましょう.このときwとw'がどういう関係になっているかを解析してみます.

まず,t回目の時点での全予報士の重みの合計をφ(t)とおきます.φ(t)=∑_i m(i,t)です.
次に,自分が間違えた時は重みが過半数な予報を採用しているはずなので
φ(t+1)<=(φ(t)/2) + (βφ(t)/2) = ((1+β)/2)φ(t)
が成り立ちます.
自分がw回間違えており,φ(t)=Mであることから,
φ(N)<=M((1+β)/2)^w (1)
が成り立ちます.
(x^yはxのy乗を意味します)
また,一番当たっている予報士の重みはw'回間違っているので,m(i',N)=β^{w'}です。
よって,
φ(N)=∑_i m(i,N)>=m(i',N)=β^{w'} (2)
が成り立ちます.(m(i,t)は全て正より)

さて,(1)と(2)を組み合わせると
β^{w'} <= M((1+β)/2)^w
がでてきます.これをwとw'について解くと
w <= (w' log(1/β) + logM) / (log(2/(1+β))
が求められます.例えばβ→1の時,大体w <= 2w' + logMとなるので,一番当たっている予報士の二倍以内で
予想できることがわかります.例えば,1万人ぐらい予報士がいれば一人ぐらいは殆ど当たっている人がいるでしょう.その場合でも,このアルゴリズムは誰がそれだけ当たるかをあらかじめ知らないながら、その人の二倍以内(logMの項はw,w'が大きくなれば無視できる)で予想できるということを意味しているので驚異的です.

さらに出力や予報に関して一切の仮定やモデルをおいていないことに注意しましょう.例えばこれまでよく当たっている人が次の日に当たるという仮定(いわゆるi.i.d.)は使っていません.悪意がある神様が重みが大きい方とは違う天気の方を出しまくっても上の定理は成り立ちます.
(自分も不思議に思ったが,もしそのようにした場合,全予報士の性能が悪くなっていくので一番当たっている予報士の正解率が悪くなっていくと思えばよい)

---

これが一番単純なアルゴリズムで,もっと一般的な問題に対するアルゴリズムが数多く提案されています.このようなアルゴリズムは基本的には,凸である損失関数と予想領域が凸集合である場合に,入力列を順に予想し損失関数の和を最小化するような問題としてみることができます.このような問題は統計的決定理論やゲーム理論,金融工学,情報理論(データ圧縮),機械学習など多くの分野で独立に提唱されてきたものであり,似た概念やアルゴリズムが提唱されています.それらの話が以下の本でまとまっています.

(一応ざっとは読んでみました.面白いですが,ポップな表紙とは裏腹に,読むには数学の知識がそこそこ必要です.分野毎に分かれているのでどこかの分野を知っていると読みやすいかも)

例えば,機械学習の文脈で言い直すと,決定するのはパラメータ関数(重みベクトルw)であり,凸であるような損失関数が毎回与えられた場合に,全損失関数の和を最小にするようなwをオンライン学習で求めている問題としてみることができます.また,情報理論(データ圧縮)の文脈で言い直すと,損失関数がlog-loss関数で,決定するのは次の文字の予測確率となり,全体の情報量を最小化する問題となります.金融工学では,決定するのはポートフォリオで損失関数は損失額そのものになります

このようなオンライン凸最適化は近年,様々な分野を包含し研究が進んでいます.例えば,パーセプトロン,最急降下法,multiplicative updateなどがRegularized Follow the Leader(RTFL)と呼ばれるアルゴリズムの特殊化としてみることができるようになっています[1].

[1] "The convex optimization approach to regret minimization." , E. Hazan

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Comments

w <= (w' log(1/β) + logM) / (log(2/(1+β))
がβ→1の時に大体
w <= 2w' + logM
とできる理由がわかりません。
もう少し詳しい説明か、ヒントでもいただけないでしょうか。

Posted by: k | 2009.11.30 07:40 PM

kさん

この記述は正確ではありませんでした。
β→1の時、log(1/β)とlog(2/(1+β))はともに
0に近づきますが、f(x)=log(1-x)とでもして、テイラー展開をしてあげるとlog(1/β)/log(2/(1+β))は2に近づくことがわかります
それに対し、logMの係数は1/log(2/1+β))なので非常に大きくなってしまいますす。ですので、ここはO(logM)とでもかいて、非常に大きい試行回数の場合は2w'に比べてO(logM)は無視できる程度だとかくべきでした。

ご指摘ありがとうございます

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If you are going backcountry camping outdoors, you should most likely have a snake mouthful package inside your gear. The very best snake bite packages are the type that utilize suction power. Some kits have scalpels and the flow of blood constrictors inside them. Scalpels can certainly cut the poison in the blood stream speedier, and constrictors may be deadly if not applied correctly.

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Will not ingest your homemade juices too quickly. Because you are skipping the move of chewing the fresh vegetables, the juices requirements time for you to mix along with your saliva to aid in digestive function and also to protect against a lot of glucose from going into your blood stream simultaneously.

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Think about getting a dog in order to relieve stress. Using a canine waiting around for you in the home regarding his tail wagging can be a wonderful improve after having a difficult, demanding work day. Even jut possess a species of fish can actually improve your mood. But don't take on a dog if you're not proceeding in order to take care of it.

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E-mail marketing marches on, as well as the classic newsletter is becoming mainly outdated. As opposed to delivering all around a general circular for your clients, use up the time and effort to customize e-mail to the likes and dislikes of their specific users. There are lots of robust applications and providers which will help you do this immediately.

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Posted by: Floydreoke | 2017.11.15 07:46 PM

Posted by: | 2017.12.24 03:46 PM

Wow, she is one of the most beautiful girls I have seen
http://www.youtube.com/watch?v=xmxdAu6RQtM

Posted by: arthurclin | 2017.12.24 03:47 PM

Posted by: | 2017.12.24 03:47 PM

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