notebooks/BANDIT_General_Eluder_dimension.ipynb

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    "# バンディットとEluder Dimension \n",
    "\n",
    "参考\n",
    "* [Eluder dimension and the sample complexity of optimistic exploration](https://papers.nips.cc/paper_files/paper/2013/hash/41bfd20a38bb1b0bec75acf0845530a7-Abstract.html)\n"
   ]
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    "**表記と設定**\n",
    "\n",
    "* 行動の集合: $\\mathcal{A}$\n",
    "* 実数関数の集合: $\\mathcal{F}=\\{f_\\rho : \\mathcal{A}\\to \\mathbb{R}\\mid \\rho \\in \\Theta\\}$\n",
    "    * $\\rho \\in \\Theta$はパラメータです\n",
    "    * 任意の$f \\in \\mathcal{F}$について$f(a) \\in [0, C]$とします．\n",
    "    * $R_t-f_\\theta(A_t)$は$\\eta$-sub-gaussianとします． \n",
    "    * 確率変数$\\theta$は真の報酬関数$f_\\theta$を表すとします．つまり，$\\mathbb{E}[R_t \\mid H_t, \\theta, A_t]=f_\\theta(A_t)$とします．\n",
    "* 時刻$t$で，報酬$R_t$を観測した後，エージェントはランダムな行動の集合$\\mathcal{A}_t \\subseteq \\mathcal{A}$から行動を一つ選択します．\n",
    "* $H_t$は履歴$(\\mathcal{A}_1, A_1, R_1, \\dots, \\mathcal{A}_{t-1}, A_{t-1}, R_{t-1}, \\mathcal{A}_t)$を表します．\n",
    "* $\\pi=\\{\\pi_t\\mid t\\in \\mathbb{N}\\}$はエージェントの方策を表します．$\\pi_t: \\mathcal{H} \\to \\Delta(\\mathcal{A})$とします．\n",
    "* $\\arg\\max_{a\\in \\mathcal{A}_t} f(a)$は空ではないとします．\n",
    "* リグレット: $\\mathcal{R}(T, \\pi)=\\sum^T_{t=1}[f_\\theta(A^*_t) - f_\\theta(A_t)]$とします．\n",
    "    * ここで，$A^*_t$は$\\arg\\max_{a\\in \\mathcal{A}_t} f_\\theta(a)$としました．\n",
    "\n",
    "**モチベーション**\n",
    "\n",
    "上の問題設定は文脈付きバンディットや線形バンディットを内包します．\n",
    "実際，[線形バンディット](./BANDIT_General_contextual.ipynb)では，$d$次元のベクトル$\\theta$について，\n",
    "報酬関数が$\\mathbb{E}[R_t \\mid H_t, \\theta, A_t]=\\langle \\theta, \\phi_t(A_t)\\rangle$であるような状況を考えました．また，各時刻で行動の集合$\\mathcal{A}_t$が与えられます．\n",
    "\n",
    "線形バンディットでは$\\widetilde{O}(d\\sqrt{T})$が最適リグレットでした．より一般の報酬関数のクラスを考えることはできないでしょうか？\n",
    "例えば，VC次元はどうでしょうか？\n"
   ]
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    "## VC次元（ここの説明かなり分かりづらい．．．TODO: 図など）\n",
    "\n",
    "VC次元は特に二値判別問題のための仮説集合に対して定義される複雑度です．\n",
    "今回は多値判別問題を考えます．つまり，$n$個の行動の中から一つ最適なものを選択する判別問題です．\n",
    "\n",
    "* 二値関数の集合$\\mathcal{F}=\\{f_\\rho: \\mathcal{A}\\to \\{0, 1\\}\\mid \\rho \\in \\{1,\\dots, n\\}\\}$を考えましょう．\n",
    "* 行動の集合は$\\mathcal{A}=\\{1,\\dots, n\\}$とします．\n",
    "* 報酬関数はパラメータ$\\rho$に対して$f_\\rho(a)=\\mathbb{1}(\\rho=a)$を考えます．つまり，真の$a$に対する指示関数です．\n",
    "\n",
    "このとき，バンディットのVC次元を次のように定義しましょう（後で定義するEluder dimensionの説明に即しています）：\n",
    "\n",
    "---\n",
    "\n",
    "**VC独立とVC次元**\n",
    "\n",
    "次を満たすとき， **「与えられた$\\mathcal{F}$に対して行動$a$は$\\widetilde{\\mathcal{A}}\\subseteq \\mathcal{A}$に対してVC-独立である」** とします．\n",
    "* 任意の$f, \\widetilde{f}\\in \\mathcal{F}$について，次を満たす$\\bar{f}\\in \\mathcal{F}$が存在する：\n",
    "    * $\\bar{f}(a)=f(a)$であり，$\\bar{f}(\\widetilde{a})=\\widetilde{f}(\\widetilde{a})$ for all $\\widetilde{a} \\in \\widetilde{A}$\n",
    "\n",
    "例として，$n=3$を考えましょう．\n",
    "* つまり，$\\mathcal{A}=\\{1, 2, 3\\}$，$\\rho \\in \\{1, 2, 3\\}$，$\\mathcal{F}=\\{f_1, f_2, f_3\\}$です．それぞれ以下のとおりです．\n",
    "    * $f_1(1)=1, f_1(2)=0, f_1(3)=0$\n",
    "    * $f_2(1)=0, f_2(2)=1, f_2(3)=0$\n",
    "    * $f_3(1)=0, f_3(2)=0, f_3(3)=1$\n",
    "* $a=1$, $\\widetilde{\\mathcal{A}}=\\{1, 2\\}$とします\n",
    "  * $f=f_1$, $\\widetilde{f}=f_2$のとき，\n",
    "    * $f(1)=1$なので，$a$で$f=\\bar{f}$を満たすのは$\\bar{f}=f_1$だけです．一方，$\\bar{f}(2)=0\\neq \\widetilde{f}(2)=1$なので，\n",
    "「与えられた$\\mathcal{F}$に対して行動$a$は$\\widetilde{\\mathcal{A}}\\subseteq \\mathcal{A}$に対してVC-依存」です．\n",
    "* $a=1$, $\\widetilde{\\mathcal{A}}=\\{2, 3\\}$とすると，\n",
    "  * $f=f_1$, $\\widetilde{f}=f_2$のとき，\n",
    "    * $f(1)=1$なので，$a$で$f=\\bar{f}$を満たすのは$\\bar{f}=f_1$だけです．一方，$\\bar{f}(2)=0\\neq \\widetilde{f}(2)=1$なので，\n",
    "    「与えられた$\\mathcal{F}$に対して行動$a$は$\\widetilde{\\mathcal{A}}\\subseteq \\mathcal{A}$に対してVC-依存」です．\n",
    "\n",
    "上の例では，$\\bar{f}=f_1$を$\\widetilde{\\mathcal{A}}=\\{2, 3\\}$や$\\widetilde{\\mathcal{A}}=\\{1, 2\\}$で調べることで，$a=1$での$f$の値を特定することができます．\n",
    "つまり，$\\widetilde{A}$についてVC-依存な$a$では，$f(\\widetilde{a})$の値を調べることで，$a$で発生する値の集合を制限することができます．\n",
    "これを使って，VC次元は次のように定義されます：\n",
    "\n",
    "**「与えられた$\\mathcal{F}$の$\\mathcal{A}$上でのVC次元は，全ての$a\\in \\widetilde{\\mathcal{A}}$が$\\widetilde{\\mathcal{A}}\\setminus \\{a\\}$についてVC-独立であるような 最大の集合$\\widetilde{A}\\subseteq \\mathcal{A}$の要素数です．**\n",
    "\n",
    "---\n",
    "\n",
    "つまり，VC次元が大きいほど，$a=1$での$f$の値を特定しづらいわけですね．\n",
    "上の例では，VC次元は１になります．\n",
    "しかし，上の例では明らかに$n$が大きくなると問題が難しくなりそうです．つまり，VC次元はバンディット問題の一般的な複雑度を考える上では適していないかもしれません．\n",
    "次に見るEluder Dimensionでは，上の例のEluder Dimensionはちゃんと$n$になります．\n"
   ]
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   "source": [
    "## VC次元\n",
    "\n",
    "VC次元は特に二値判別問題のための仮説集合に対して定義される複雑度です．\n",
    "[MATH_complexity_of_hypothesis.ipynb](MATH_complexity_of_hypothesis.ipynb)を復習しましょう．\n",
    "仮説集合$\\mathcal{F}$と入力空間$\\mathcal{X}$に対して，\n",
    "$$\n",
    "\\Pi_\\mathcal{H}(x_1, \\cdots, x_n)=|\\{(h(x_1), \\cdots, h(x_n))\\in \\mathcal{Y}^n\\mid h \\in \\mathcal{H}\\}|\n",
    "$$\n",
    "は入力の集合に対する出力の集合の要素数とします．\n",
    "すると，VC次元は\n",
    "$$\n",
    "\\operatorname{VC}(\\mathcal{H}) = \\max\\left\\{n \\in \\mathbb{N}\\mid \\max_{x_1, \\cdots, x_n\\in \\mathcal{X}} \\Pi_{\\mathcal{H}}(x_1, \\cdots, x_n) = 2^n\\right\\}\n",
    "$$\n",
    "として定義されました．\n",
    "\n",
    "ここで改めて$n$腕バンディット問題を考えましょう．つまり行動の集合は$\\mathcal{A}=\\{1,\\dots, n\\}$です．\n",
    "\n",
    "そして，\n",
    "$n$個のアームのうち，ひとつだけ１の報酬を取り，それ以外は０であるとします．すなわち，次のような報酬関数の仮説集合を考えれば十分です．\n",
    "\n",
    "* 二値関数の集合$\\mathcal{F}=\\{f_\\rho: \\mathcal{A}\\to \\{0, 1\\}\\mid \\rho \\in \\{1,\\dots, n\\}\\}$\n",
    "として，\n",
    "$f_\\rho(a)=\\mathbb{1}(\\rho=a)$とします．つまり，$f_\\rho$は真の$a$に対する指示関数です．\n",
    "\n",
    "つまり仮説集合は$|\\mathcal{F}|=n$であり，そのVC次元は1です．\n",
    "（**注意** VC次元が$n$になるのは，$\\mathcal{F}$の集合のサイズが$2^n$になる場合です．）\n",
    "しかし，このバンディット問題は明らかに$n$が大きくなると問題が難しくなります．つまり，VC次元はバンディット問題の一般的な複雑度を考える上では適していないかもしれません．\n",
    "次に見るEluder Dimensionでは，上の例のEluder Dimensionはちゃんと$n$になります．\n"
   ]
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   "source": [
    "## Eluder Dimension\n",
    "\n",
    "政治家Aが，報道陣を避けながらその政治家人生を過ごす状況を考えましょう．\n",
    "Aは，なるべく自分の政治的な立ち位置（どの政党に組み入れているか?）を明らかにしないまま過ごしたいと思っています．\n",
    "しかし，自身の政治姿勢について報道の際にはなんらかの新しい情報を提供しなければならないとします．\n",
    "このとき，Aはどれくらい長く立ち位置がバレずに過ごせるでしょうか？\n",
    "\n",
    "これがEluder dimensionの根本的なアイディアです（報道陣から逃げてる政治家をイメージするとわかりやすいですね）．\n",
    "Eluder Dimensionは次で定義されます：\n",
    "\n",
    "---\n",
    "\n",
    "次を満たす時，行動$a$は，集合$\\{a_1, \\dots, a_n\\} \\subseteq \\mathcal{A}$と$\\mathcal{F}$について$\\epsilon$-依存であるといいます．\n",
    "\n",
    "$$\n",
    "\\sqrt{\\sum_{j=1}^{i-1}\\left(f(a_i)-\\widetilde{f}(a_i)\\right)^2} \\leq \\epsilon \n",
    "$$\n",
    "を満たす任意の$f, \\widetilde{f} \\in \\mathcal{F}$が$f(a) - \\widetilde{f}(a) \\leq \\epsilon$を満たす．\n",
    "\n",
    "**直感**\n",
    "\n",
    "行動$a$は，$\\{a_1, \\dots, a_n\\}$では同じような予測をするが，$a$でだけ$\\epsilon$以上に出力を変える関数$f, \\widetilde{f}\\in \\mathcal{F}$が存在するとき，$\\epsilon$-独立になります．\n",
    "例えば下の図では$n=10$について$a=8$で，$f$と$g$はdisagreeします．\n",
    "\n",
    "これを使って，Eluder dimensionは次のように定義されます．\n",
    "\n",
    "**「与えられた$\\mathcal{F}$の$\\mathcal{A}$上での$\\epsilon$に対してのEluder dimensionは，全ての$a\\in \\widetilde{\\mathcal{A}}$が$\\widetilde{\\mathcal{A}}\\setminus \\{a\\}$について$\\epsilon'$-独立であるような$\\epsilon' \\geq \\epsilon$が存在する最大の集合$\\widetilde{A}\\subseteq \\mathcal{A}$の要素数です．**\n",
    "\n",
    "Eluder dimensionは，関数クラス$\\mathcal{F}$の中の関数$f$を他の$g$と区別するために「間違える回数」の最悪ケースを表しています．\n",
    "これはVC次元とよく似ていますね．[仮説集合の複雑度](MATH_statistical_learning_theory.ipynb)の回を思い出しましょう．\n",
    "\n",
    "また，ベクトル空間の次元との対応を考えてもわかりやすいです．\n",
    "ベクトル空間の次元が$d$なのは「ベクトルが独立になるような要素の系列の最大長」でした．\n",
    "例えば$(1, 0, 0)$と$(0, 1, 0)$によって張られる空間の次元は２ですね．Eluder dimensionはこの一般化であり，線形な構造を持つ必要がないため，より一般的な問題を扱うことができます．\n"
   ]
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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "x = np.linspace(0, 1.0, 10)\n",
    "\n",
    "f = x ** 2\n",
    "g = x ** 2 + 0.1\n",
    "\n",
    "g[7] += 3\n",
    "\n",
    "plt.plot(x, f, label=\"f\")\n",
    "plt.plot(x, g, label=\"g\")\n",
    "\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "TODO: アルゴリズムについて"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "language_info": {
   "name": "python"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}