notebooks/RL_UCB_VI_regret_proof.ipynb

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                "# UCB-VIアルゴリズムのリグレットの証明\n",
                "\n",
                "参考\n",
                "* [Reinforcement Learning: Theory and Algorithms](https://rltheorybook.github.io/)\n",
                "* [Covering with $\\varepsilon$-nets](https://erdogdu.github.io/csc2532/lectures/lecture05.pdf)\n",
                "* [Notes on Exploration in Linear MDPs](https://nanjiang.cs.illinois.edu/files/cs542f22/note8.pdf)\n",
                "* [Useful Inequalities Cheat Sheet](https://www.lkozma.net/inequalities_cheat_sheet/ineq.pdf)\n",
                "* [Upper Bound of $\\sum_{k=1}^{n} \\frac{1}{\\sqrt{k}}$](https://math.stackexchange.com/questions/3745324/upper-bound-of-sum-k-1n-frac1-sqrtk)\n",
                "* [Upper Bound for Harmonic Number](https://proofwiki.org/wiki/Upper_Bound_for_Harmonic_Number)\n",
                "* [講義スライド](https://wensun.github.io/CS6789_data_fall_2021/UCBVI_annotated.pdf)\n",
                "* [Minimax Regret Bounds for Reinforcement Learning](https://arxiv.org/abs/1703.05449)\n",
                "* [Near-optimal Regret Bounds for Reinforcement Learning](https://www.jmlr.org/papers/volume11/jaksch10a/jaksch10a.pdf)\n",
                "\n",
                "[前回](BANDIT_UCB_regret_proof.ipynb)の拡張として、RLにおけるUCB-VIアルゴリズムの擬リグレットを簡潔に証明します。\n",
                "今回の導出ではHigh-probability リグレットを考えますが、期待リグレットのは今回の導出について、$\\delta$以下で成立するBadイベントを考えます。\n",
                "面倒なので定数係数は全部$\\square$でまとめます。"
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                "## Tabular MDP\n",
                "\n",
                "次の問題設定を考えます。\n",
                "\n",
                "問題設定\n",
                "\n",
                "* エピソードの数：$K$\n",
                "* $k$回目の方策：$\\pi^k=\\left\\{\\pi_0^k, \\ldots, \\pi_{H-1}^k\\right\\}$\n",
                "* ホライズン：$H$\n",
                "* 軌跡：$\\tau^k=\\left\\{s_h^k, a_h^k\\right\\}_{h=0}^{H-1}$\n",
                "* 行動：$a_h \\sim \\pi_h^k (s_h^k)$\n",
                "* 遷移確率：$s_{h+1}^k \\sim P^\\star_h\\left(\\cdot \\mid s_h^k, a_h^k\\right)$\n",
                "* 次状態遷移の回数：$N_h^k\\left(s, a, s^{\\prime}\\right)=\\sum_{i=0}^{k-1} \\mathbf{1}\\left\\{\\left(s_h^i, a_h^i, s_{h+1}^i\\right)=\\left(s, a, s^{\\prime}\\right)\\right\\}$\n",
                "* 状態行動の訪問回数：$N_h^k(s, a)=\\sum_{i=0}^{k-1} \\mathbf{1}\\left\\{\\left(s_h^i, a_h^i\\right)=(s, a)\\right\\}, \\forall h, s, a$.\n",
                "* 遷移確率の推定：$\\widehat{P}_h^k\\left(s^{\\prime} \\mid s, a\\right)=\\frac{N_h^k\\left(s, a, s^{\\prime}\\right)}{N_h^k(s, a)}, \\forall h, s, a, s^{\\prime}$.\n",
                "* (擬)リグレット：$\\sum_{k=0}^{K-1}\\left(V^{\\star}\\left(s_0\\right)-V^{\\pi^k}\\left(s_0\\right)\\right)$"
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                "### UCB-VIアルゴリズムの証明\n",
                "\n",
                "\\begin{aligned}\n",
                "& \\text { Algorithm } 5 \\text { UCBVI } \\\\\n",
                "& \\text { Input: reward function } r \\text { (assumed to be known), confidence parameters } \\\\\n",
                "& \\text { 1: for } k=0 \\ldots K-1 \\text { do } \\\\\n",
                "& \\text { 2: } \\quad \\text { Compute } \\widehat{P}_h^k \\text { as the empirical estimates, for all } h \\\\\n",
                "& \\text { 3: } \\quad \\text { Compute reward bonus } b_h^k \\text { for all } \\mathrm{h} \\\\\n",
                "& \\text { 4: } \\quad \\text { Run Value-Iteration on }\\left\\{\\widehat{P}_h^k, r+b_h^k\\right\\}_{h=0}^{H-1} \\\\\n",
                "& \\text { 5: } \\quad \\text { Set } \\pi^k \\text { as the returned policy of VI. } \\\\\n",
                "& \\text { 6: end for }\n",
                "\\end{aligned}\n",
                "\n",
                "さらにアルゴリズム特有の記法を導入します。\n",
                "\n",
                "* $L:=\\ln (S A H K / \\delta)$\n",
                "* ボーナス項：$b_h^k(s, a)=2 H \\sqrt{\\frac{L}{N_h^k(s, a)}}$\n",
                "* 価値反復法：\n",
                "\n",
                "\\begin{equation}\n",
                "\\begin{aligned}\n",
                "& \\widehat{V}_H^k(s)=0, \\forall s \\\\\n",
                "& \\widehat{Q}_h^k(s, a)=\\min \\left\\{r_h(s, a)+b_h^k(s, a)+\\widehat{P}_h^k(\\cdot \\mid s, a) \\cdot \\widehat{V}_{h+1}^k, H\\right\\} \\\\\n",
                "& \\widehat{V}_h^k(s)=\\max _a \\widehat{Q}_h^k(s, a), \\pi_h^k(s)=\\operatorname{argmax}_a \\widehat{Q}_h^k(s, a), \\forall h, s, a .\n",
                "\\end{aligned}\n",
                "\\end{equation}\n",
                "\n",
                "これはモデルベースなアルゴリズムですね。\n",
                "このアルゴリズムに対してリグレットの上界を証明していきます。\n",
                "\n",
                "参考\n",
                "* [Reinforcement Learning: Theory and Algorithms](https://rltheorybook.github.io/)\n"
            ]
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                "\n",
                "#### 1. 遷移確率の推定 (弱め)\n",
                "\n",
                "ここの議論は次の資料が参考になります：\n",
                "* [Near-optimal Regret Bounds for Reinforcement Learning](https://www.jmlr.org/papers/volume11/jaksch10a/jaksch10a.pdf)の式（４４）あたり\n",
                "* [Minimax Regret Bounds for Reinforcement Learning](https://arxiv.org/abs/1703.05449)の６ページあたり\n",
                "\n",
                "後で$\\left|\\left(\\widehat{P}_h^k\\left(\\cdot \\mid s_h, a_h\\right)-P^{\\star}\\left(\\cdot \\mid s_h, a_h\\right)\\right) \\cdot \\widehat{V}_{h+1}^{\\pi^k}\\right|$をバウンドする必要があるので、推定した遷移確率の誤差のバウンドを確認しましょう。\n",
                "ここで注意することがあります。\n",
                "* $\\widehat{P}_h^k$と$\\widehat{V}_{h+1}^{\\pi^k}$のどちらも確率変数。\n",
                "    * $\\widehat{V}_{h+1}^{\\pi^k}$の計算に使うサンプルは$\\widehat{P}_{h+1}^k$に依存しています。\n",
                "        * $\\widehat{P}_{h+1}^k$の計算の際に使ったサンプルは$s_{h, k}$に依存しています。\n",
                "    * $\\widehat{P}_{h}^k$の計算の際に使ったサンプルは全ての$(s, a)$について$N_h^k(s, a)$に依存しています。\n",
                "        * $N_h^k(s, a)$は$s_{h, k}$に依存しています。\n",
                "    * つまり、$\\widehat{P}_h^k$と$\\widehat{V}_{h+1}^{\\pi^k}$は独立ではありません。\n",
                "    * よって、独立ではない確率変数の積なので、$\\left(\\widehat{P}_h^k\\left(\\cdot \\mid s_h, a_h\\right)-P^{\\star}\\left(\\cdot \\mid s_h, a_h\\right)\\right) \\cdot \\widehat{V}_{h+1}^{\\pi^k}$の期待値が０かどうかはわかりません。\n",
                "\n",
                "なので普通にAzuma-Hoeffdingを使ってはいけないんですね。\n",
                "その代わり、$\\widehat{P}$に独立な任意の関数$f:\\mathcal{S} \\mapsto [0, H]$について、$\\left|\\left(\\widehat{P}_h^k(\\cdot \\mid s, a)-P_h^{\\star}(\\cdot \\mid s, a)\\right)^{\\top} f\\right|$の上界を導出します。\n",
                "まず定義より、\n",
                "\n",
                "$$\n",
                "\\widehat{P}_h^k(\\cdot \\mid s, a)^{\\top} f - {P}_h^\\star(\\cdot \\mid s, a)^{\\top} f =\n",
                "\\frac{1}{N_h^k(s, a)}\n",
                "\\sum_{i=0}^{k-1} \\left(\\mathbf{1}\\left\\{s_h^i, a_h^i=s, a\\right\\} f\\left(s_{h+1}^i\\right)\n",
                "- \\mathbf{1}\\left\\{s_h^i, a_h^i=s, a\\right\\} \\mathbb{E}_{s^{\\prime} \\sim P_h^{\\star}(s, a)} f\\left(s^{\\prime}\\right)\\right)\n",
                "$$\n",
                "\n",
                "です。\n",
                "\n",
                "\n",
                "ここで、和の中身について、$X_i=\\mathbf{1}\\left\\{s_h^i, a_h^i=s, a\\right\\} f\\left(s_{h+1}^i\\right)-\\mathbf{1}\\left\\{s_h^i, a_h^i=s, a\\right\\} \\mathbb{E}_{s^{\\prime} \\sim P_h^*}(s, a) f\\left(s^{\\prime}\\right)$とします。\n",
                "また、$\\mathcal{H}_{h, <i}$を$h$での$i$を含まない$i$までの履歴とすると、決定的なアルゴリズムについては$\\mathbb{E}\\left[X_i \\mid \\mathcal{H}_{h, <i}\\right]=0$なので、$\\sum_{i=0}^{k-1} X_i$はマルチンゲールです。\n",
                "また、$\\left(s_h^i, a_h^i\\right) \\neq(s, a)$では$\\left|X_i\\right|=0$, $\\left(s_h^i, a_h^i\\right)=(s, a)$では$\\left|X_i\\right| \\leq 2H$であることに気をつければ、\n",
                "次のAzuma-Hoeffdingのバウンドが使えます：\n",
                "\n",
                "---\n",
                "**定理：Azuma-Hoeffdingの不等式**\n",
                "\n",
                "$\\left(\\mathcal{F}_n\\right)_{n=1}^N$に対して可測な実数値確率過程$\\left(X_n\\right)_{n=1}^N$を考える. \n",
                "$X_n \\in\\left[l_n, u_n\\right]$ かつ $\\mathbb{E}_n\\left[X_n\\right]=0$ が任意の$n$でa.s.で成立するとする. \n",
                "このとき、\n",
                "$$\n",
                "P\\left(\\sum_{n=1}^N X_n \\geq \\sqrt{\\sum_{n=1}^N \\frac{\\left(u_n-l_n\\right)^2}{2} \\log \\frac{1}{\\delta}}\\right) \\leq \\delta\n",
                "$$\n",
                "\n",
                "---\n",
                "\n",
                "これを使うと、固定された$a$, $s$, $h$, $k$, そして$f$について、$1-\\delta$以上で次が成立します。\n",
                "\n",
                "$$\n",
                "\\left|\\sum_{i=0}^{k-1} X_i\\right|=\n",
                "\\left|\\sum_{i=0}^{k-1} \\left(\\mathbf{1}\\left\\{s_h^i, a_h^i=s, a\\right\\} f\\left(s_{h+1}^i\\right)\n",
                "- \\mathbf{1}\\left\\{s_h^i, a_h^i=s, a\\right\\} \\mathbb{E}_{s^{\\prime} \\sim P_h^{\\star}(s, a)} f\\left(s^{\\prime}\\right)\\right)\\right|\n",
                "\\leq \\square H \\sqrt{N_h^k(s, a) \\ln (1 / \\delta)}\n",
                "$$\n",
                "\n",
                "そして$s \\in \\mathcal{S}, a \\in \\mathcal{A}, h \\in[0, \\ldots, H-1], k \\in[0, \\ldots, K-1]$にUnion boundを取れば、任意の$s, a, h, k$について\n",
                "$$\n",
                "P\\left(\\left|\\widehat{P}_h^k(\\cdot \\mid s, a)^{\\top} f - {P}_h^\\star(\\cdot \\mid s, a)^{\\top} f\\right|\\geq \\square H \\sqrt{\\frac{\\ln (SAKH / \\delta)}{N_h^k(s, a)}}\\right) \\leq {\\delta}\n",
                "$$\n",
                "です。\n",
                "\n",
                "ここでまでの話では、まだ$f$についてのUnion boundを取っていません。バンディットの場合は$f$が報酬関数で固定されていたのでUnion Boundを取らなくても良かったわけですが、今回は$f$が$f:\\mathcal{S} \\mapsto [0, H]$で動き回ります。\n",
                "一方で、この$f$は全部で$\\infty$個存在するので、単純にUnion Boundを取ることができません。そこで、$\\epsilon$-net法を使います。詳しくは[Covering with $\\varepsilon$-nets](https://erdogdu.github.io/csc2532/lectures/lecture05.pdf)参照。\n",
                "\n",
                "まず任意の$f: \\mathcal{S} \\mapsto[0, H]$について、$\\|f\\|_2 \\leq H \\sqrt{S}$であることに注意しましょう。\n",
                "詳細は[Covering with $\\varepsilon$-nets](https://erdogdu.github.io/csc2532/lectures/lecture05.pdf)に任せますが、このときどんな$f\\in[0, H]^S$に対しても、$\\left\\|f-f^{\\prime}\\right\\|_2 \\leq \\epsilon$を満たすような$f'\\in \\mathcal{N}_\\varepsilon$が存在できる$\\mathcal{N}_\\varepsilon$のサイズは$\\left|\\mathcal{N}_\\epsilon\\right| \\leq(1+2 H \\sqrt{S} / \\epsilon)^S$になります (リンク先のFigure2が分かりやすいです)。この$\\mathcal{N}_\\varepsilon$のサイズを``Covering Number``と呼びます。\n",
                "\n",
                "これを使って$s, a, k, h, f \\in \\mathcal{N}_\\varepsilon$についてUnion Boundを取ると、\n",
                "\n",
                "$$\n",
                "P\\left(\\left|\\widehat{P}_h^k(\\cdot \\mid s, a)^{\\top} f' - {P}_h^\\star(\\cdot \\mid s, a)^{\\top} f'\\right|\\geq \\square H \\sqrt{\\frac{\\ln (SAKH(1+2 H \\sqrt{S} / \\epsilon) / \\delta)}{N_h^k(s, a)}}\\right) \\leq {\\delta}\n",
                "$$\n",
                "\n",
                "最後に任意の$f \\in[0, H]^S$についてのバウンドに変形します。次のバウンドが確率$1-\\delta$で成立します。\n",
                "\n",
                "$$\n",
                "\\begin{aligned}\n",
                "\\left|\\widehat{P}_h^k(\\cdot \\mid s, a)^{\\top} f - {P}_h^\\star(\\cdot \\mid s, a)^{\\top} f\\right|\n",
                "&=\\left|\n",
                "\\widehat{P}_h^k(\\cdot \\mid s, a)^{\\top} f' - {P}_h^\\star(\\cdot \\mid s, a)^{\\top} f'\n",
                "+ \\widehat{P}_h^k(\\cdot \\mid s, a)^{\\top}(f - f')\n",
                "+ \\widehat{P}_h^\\star(\\cdot \\mid s, a)^{\\top}(f' - f)\n",
                "\\right|\\\\\n",
                "&\\leq\\left|\\widehat{P}_h^k(\\cdot \\mid s, a)^{\\top} f' - {P}_h^\\star(\\cdot \\mid s, a)^{\\top} f'\\right|\n",
                "+ \\left|\\widehat{P}_h^k(\\cdot \\mid s, a)^{\\top}(f - f')\\right|\n",
                "+ \\left|\\widehat{P}_h^\\star(\\cdot \\mid s, a)^{\\top}(f' - f)\\right|\\\\\n",
                "&\\leq \\square H \\sqrt{\\frac{\\ln (SAKH(1+2 H \\sqrt{S} / \\varepsilon) / \\delta)}{N_h^k(s, a)}} + 2\\varepsilon\n",
                "\\end{aligned}\n",
                "$$\n",
                "\n",
                "最後の不等式では$\\|f-f^{\\prime} \\|_2 \\leq \\epsilon$ならば$\\left|f(s)-f^{\\prime}(s)\\right| \\leq \\epsilon, \\forall s \\in \\mathcal{S}$であることを使いました。([Useful Inequalities Cheat Sheet](https://www.lkozma.net/inequalities_cheat_sheet/ineq.pdf)のMean参照)\n",
                "\n",
                "あとは$\\varepsilon$を決めるだけです。\n",
                "よく見ると$\\varepsilon$は$\\ln$の中で登場していますね。つまり、我々が対数の中身を気にしない限り、$\\varepsilon$は多項式であれば何でも良いわけですね。\n",
                "また、$N_h^k(s, a) \\leq K$と上のバウンドで$1\\varepsilon$が出てきていることも踏まえると、$\\varepsilon=1/K$として良いでしょう。([Notes on Exploration in Linear MDPs](https://nanjiang.cs.illinois.edu/files/cs542f22/note8.pdf)を参考にしましたが、やや自信ないです。)\n",
                "\n",
                "以上をまとめると、\n",
                "\n",
                "\n",
                "$$\n",
                "\\left|\\widehat{P}_h^k(\\cdot \\mid s, a)^{\\top} f - {P}_h^\\star(\\cdot \\mid s, a)^{\\top} f\\right|\n",
                "\\leq \\square H \\sqrt{\\frac{S \\ln (S A K H(1+2 H K \\sqrt{S}) / \\delta)}{N_h^k(s, a)}}+\\frac{\\square}{K}\n",
                "\\leq \\square H \\sqrt{\\frac{S \\ln (H S K A / \\delta)}{N_h^k(s, a)}},\n",
                "$$\n",
                "\n",
                "になり、これが欲しかったバウンドになります。\n",
                "\n",
                "実はこれはそんなに良いバウンドではありません。が、Covering Numberは大事な概念なので載せておきました。"
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                "#### 2. 遷移確率＋$V^\\star$の推定\n",
                "\n",
                "ここでは$V^\\star$が固定されていることを利用すれば、1.の議論からすぐに次の結果が得られます。\n",
                "任意の$s, a, h, k$について\n",
                "$$\n",
                "P\\left(\\left|\\widehat{P}_h^k(\\cdot \\mid s, a)^{\\top} V_{h+1}^\\star - {P}_h^\\star(\\cdot \\mid s, a)^{\\top} V^\\star_{h+1}\\right|\\geq \\square H \\sqrt{\\frac{\\ln (SAKH / \\delta)}{N_h^k(s, a)}}\\right) \\leq {\\delta}\n",
                "$$\n",
                "が成り立ちます。"
            ]
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            "source": [
                "#### 3. $V^{\\star}\\left(s_0\\right) \\leq \\widehat{V}^{k}\\left(s_0\\right)$の証明\n",
                "\n",
                "* 条件：1.と2.のバウンドが成り立っているとき\n",
                "\n",
                "リグレットをバウンドするために、UCBアルゴリズムと似たようなことをしていきます。\n",
                "\n",
                "リグレットは\n",
                "$\\sum_{k=0}^{K-1}\\left(V^{\\star}\\left(s_0\\right)-V^{\\pi^k}\\left(s_0\\right)\\right)$だったので、\n",
                "$V^{\\star}\\left(s_0\\right)-V^{\\pi^k}\\left(s_0\\right)$がバウンドできればほぼゴールです。\n",
                "\n",
                "これをバウンドするためには、UCBのときと似た感じで\n",
                "$V^{\\star}\\left(s_0\\right)-V^{\\pi^k}\\left(s_0\\right) \\leq \\widehat{V}^{k}\\left(s_0\\right)-V^{\\pi^k}\\left(s_0\\right)$\n",
                "が成り立つことを利用したいわけですね。\n",
                "\n",
                "これは再帰的に確認できます。まず、$h=H$については\n",
                "$\\widehat{V}^k_H(s) = \\widehat{V}^\\star_H(s)=0$なのでOKです。$h+1$のときにOKだとすると、任意の$(s, a)\\in \\mathcal{S}\\times \\mathcal{A}$について、\n",
                "\n",
                "$$\n",
                "\\begin{aligned}\n",
                "\\widehat{Q}_h^k(s, a)-Q_h^{\\star}(s, a) & =b_h^k(s, a)+\\widehat{P}_h^k(\\cdot \\mid s, a) \\cdot \\widehat{V}_{h+1}^k-P_h^{\\star}(\\cdot \\mid s, a) \\cdot V_{h+1}^{\\star} \\\\\n",
                "& \\geq b_h^k(s, a)+\\widehat{P}_h^k(\\cdot \\mid s, a) \\cdot V_{h+1}^{\\star}-P_h^{\\star}(\\cdot \\mid s, a) \\cdot V_{h+1}^{\\star} \\\\\n",
                "& =b_h^k(s, a)+\\left(\\widehat{P}_h^k(\\cdot \\mid s, a)-P_h^{\\star}(\\cdot \\mid s, a)\\right) \\cdot V_{h+1}^{\\star} \\\\\n",
                "& \\geq b_h^k(s, a)-\\square H \\sqrt{\\frac{\\ln (S A H K / \\delta)}{N_h^k(s, a)}} \\geq 0 .\n",
                "\\end{aligned}\n",
                "$$\n",
                "\n",
                "が成立します。"
            ]
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                "#### 4. $\\sum_{k=0}^{K-1} \\sum_{h=0}^{H-1} \\frac{1}{\\sqrt{N_h^k\\left(s_h^k, a_h^k\\right)}}$のバウンド\n",
                "\n",
                "遷移確率のバウンドで出てきた$\\frac{1}{\\sqrt{N_h^k(s, a)}}$は後々バウンドしないといけません。今のうちにバウンドしておきましょう。\n",
                "\n",
                "$$\n",
                "\\begin{aligned}\n",
                "& \\sum_{k=0}^{K-1} \\sum_{h=0}^{H-1} \\frac{1}{\\sqrt{N_h^k\\left(s_h^k, a_h^k\\right)}}=\\sum_{h=0}^{H-1} \\sum_{k=0}^{K-1} \\frac{1}{\\sqrt{N_h^k\\left(s_h^k, a_h^k\\right)}}=\\sum_{h=0}^{H-1} \\sum_{s, a \\in \\mathcal{S} \\times \\mathcal{A}} \\sum_{i=1}^{N_h^K(s, a)} \\frac{1}{\\sqrt{i}} \\\\\n",
                "& \\leq 2 \\sum_{h=0}^{H-1} \\sum_{s, a \\in \\mathcal{S} \\times \\mathcal{A}} \\sqrt{N_h^K(s, a)} \\leq \\sum_{h=0}^{H-1} \\sqrt{S A \\sum_{s, a} N_h^K(s, a)}=H \\sqrt{S A K},\n",
                "\\end{aligned}\n",
                "$$\n",
                "\n",
                "１つ目の不等式は[Upper Bound of $\\sum_{k=1}^{n} \\frac{1}{\\sqrt{k}}$](https://math.stackexchange.com/questions/3745324/upper-bound-of-sum-k-1n-frac1-sqrtk)参考。２つ目の不等式はコーシー・シュワルツか平均の不等式を使います。[Useful Inequalities Cheat Sheet](https://www.lkozma.net/inequalities_cheat_sheet/ineq.pdf)参考。"
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            "metadata": {},
            "source": [
                "#### 5. Simulation Lemmaを使って$\\widehat{V}_0^k\\left(s_0\\right)-V_0^{\\pi^k}\\left(s_0\\right)$をバウンド\n",
                "\n",
                "* 条件：1.と2.のバウンドが成り立っているとき\n",
                "\n",
                "1と2で求めたバウンドが成り立っている事象を考えているので、3の結果を使うと、\n",
                "\n",
                "$$\n",
                "V^{\\star}\\left(s_0\\right)-V^{\\pi^k}\\left(s_0\\right) \\leq \\widehat{V}_0^k\\left(s_0\\right)-V_0^{\\pi^k}\\left(s_0\\right)\n",
                "$$\n",
                "\n",
                "が成り立っています。\n",
                "\n",
                "ここで、Simulation Lemmaってやつでこれを更に深く掘り下げていきます（[講義スライド](https://wensun.github.io/CS6789_data_fall_2021/UCBVI_annotated.pdf)参照）\n",
                "\n",
                "---\n",
                "**補題：Simulation Lemma**\n",
                "$$\n",
                "\\begin{aligned}\n",
                "& \\widehat{V}_0^n\\left(s_0\\right)-V_0^{\\pi^n}\\left(s_0\\right)=\\widehat{Q}_0^n\\left(s_0, \\pi^n\\left(s_0\\right)\\right)-Q_0^{\\pi^n}\\left(s_0, \\pi^n\\left(s_0\\right)\\right) \\\\\n",
                "& \\leq r_0\\left(s_0, \\pi^n\\left(s_0\\right)\\right)+b_h^n\\left(s_0, \\pi^n\\left(s_0\\right)\\right)+\\widehat{P}_0^n\\left(\\cdot \\mid s_0, \\pi^n\\left(s_0\\right)\\right) \\cdot \\widehat{V}_1^n-r_0\\left(s_0, \\pi^n\\left(s_0\\right)\\right)-P_0\\left(\\cdot \\mid s_0, \\pi^n\\left(s_0\\right)\\right) \\cdot V_1^{\\pi^n} \\\\\n",
                "& =b_h^n\\left(s_0, \\pi^n\\left(s_0\\right)\\right)+\\widehat{P}_0^n\\left(\\cdot \\mid s_0, \\pi^n\\left(s_0\\right)\\right) \\cdot \\widehat{V}_1^n-P_0\\left(\\cdot \\mid s_0, \\pi^n\\left(s_0\\right)\\right) \\cdot V_1^{\\pi^n} \\\\\n",
                "& =b_h^n\\left(s_0, \\pi^n\\left(s_0\\right)\\right)+\\left(\\widehat{P}_0^n\\left(\\cdot \\mid s_0, \\pi^n\\left(s_0\\right)\\right)-P_0\\left(\\cdot \\mid s_0, \\pi^n\\left(s_0\\right)\\right)\\right) \\cdot \\widehat{V}_1^n+P_0\\left(\\cdot \\mid s_0, \\pi^n\\left(s_0\\right)\\right) \\cdot\\left(\\widehat{V}_1^n-V_1^{\\pi^n}\\right) \\\\\n",
                "& =\\sum_{h=0}^{H-1} \\mathbb{E}_{s, a \\sim d_h^n}\\left[b_h^n(s, a)+\\left(\\widehat{P}_h^n(\\cdot \\mid s, a)-P_h(\\cdot \\mid s, a)\\right) \\cdot \\widehat{V}_{h+1}^n\\right]\n",
                "\\end{aligned}\n",
                "$$\n",
                "\n",
                "ここで、最初の不等式は$\\widehat{V}$が$H$でclipされているせいです。\n",
                "\n",
                "---\n",
                "\n",
                "これを使って、\n",
                "\n",
                "$$\n",
                "V^{\\star}\\left(s_0\\right)-V^{\\pi^k}\\left(s_0\\right) \\leq \\widehat{V}_0^k\\left(s_0\\right)-V_0^{\\pi^k}\\left(s_0\\right) \\leq \\sum_{h=0}^{H-1} \\mathbb{E}_{s_h, a_h \\sim d_h^{\\pi^k}}\\left[b_h^k\\left(s_h, a_h\\right)+\\left(\\widehat{P}_h^k\\left(\\cdot \\mid s_h, a_h\\right)-P^{\\star}\\left(\\cdot \\mid s_h, a_h\\right)\\right) \\cdot \\widehat{V}_{h+1}^{\\pi^k}\\right]\n",
                "$$\n",
                "\n",
                "を得ます。"
            ]
        },
        {
            "attachments": {},
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "#### 6. リグレットをバウンド\n",
                "\n",
                "ここまで来たらリグレットをバウンドするだけですね。\n",
                "5の結果の和の中身をバウンドしていきます。まず、定義よりボーナス項は$b_h^k(s, a)=2 H \\sqrt{\\frac{L}{N_h^k(s, a)}}$です。\n",
                "\n",
                "一方で、\n",
                "$$\n",
                "\\left|\\left(\\widehat{P}_h^k\\left(\\cdot \\mid s_h, a_h\\right)-P^{\\star}\\left(\\cdot \\mid s_h, a_h\\right)\\right) \\cdot \\widehat{V}_{h+1}^{\\pi^k}\\right| \\leq\\left\\|\\hat{P}_h^k\\left(\\cdot \\mid s_h, a_h\\right)-P^{\\star}\\left(\\cdot \\mid s_h, a_h\\right)\\right\\|_1\\left\\|\\widehat{V}_{h+1}^{\\pi^k}\\right\\|_{\\infty} \\leq \\square H \\sqrt{\\frac{S L}{N_h^k(s, a)}},\n",
                "$$\n",
                "\n",
                "が1の結果より出てきます。ここで、１つ目の不等式はHolderの不等式を使ってます。１で議論したように、直接Azuma-Hoeffdingを使ってはいけません。\n",
                "\n",
                "最後に、\n",
                "\n",
                "$$\n",
                "\\begin{aligned}\n",
                "V^{\\star}\\left(s_0\\right)-V^{\\pi^k}\\left(s_0\\right) & \\leq \\sum_{h=0}^{H-1} \\mathbb{E}_{s_h, a_h \\sim d_h^{\\pi^k}}\\left[b_h^k\\left(s_h, a_h\\right)+\\square H \\sqrt{S L / N_h^k\\left(s_h, a_h\\right)}\\right] \\\\\n",
                "& \\leq \\sum_{h=0}^{H-1} \\mathbb{E}_{s_h, a_h \\sim d_h^{\\pi_h^k}}\\left[\\square H \\sqrt{S L / N_h^k\\left(s_h, a_h\\right)}\\right] \\\\\n",
                "& =\\square H \\sqrt{S L} \\cdot \\mathbb{E}\\left[\\sum_{h=0}^{H-1} \\frac{1}{\\sqrt{N_h^k\\left(s_h^k, a_h^k\\right)}} \\mid \\mathcal{H}_{<k}\\right],\n",
                "\\end{aligned}\n",
                "$$\n",
                "\n",
                "なので、\n",
                "$$\n",
                "\\begin{aligned}\n",
                "\\sum_{k=0}^{K-1} V^{\\star}\\left(s_0\\right)-V^{\\pi^k}\\left(s_0\\right)\n",
                "&\\leq\\square H \\sqrt{S L} \\cdot \\mathbb{E}\\left[\\sum_{k=0}^{K-1}\\sum_{h=0}^{H-1} \\frac{1}{\\sqrt{N_h^k\\left(s_h^k, a_h^k\\right)}}\\right]\\\\\n",
                "&\\leq\\square H^2 S\\sqrt{AK \\ln (S A H K / \\delta)}\n",
                "\\end{aligned}\n",
                "$$\n",
                "が成り立ちます。\n"
            ]
        },
        {
            "attachments": {},
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "\n",
                "#### 7.1. 遷移確率の推定 (強め)\n",
                "\n",
                "実は１で証明した推定誤差のバウンドはそんなに良くありません。\n",
                "１では$\\widehat{P}_h^k(\\cdot | s, a) - P_h^\\star(\\cdot|s, a)$の信頼区間をAzuma-Hoeffdingで導出していました。\n",
                "これは$S$次元の信頼区間を作らないといけないので、代わりに$Q(s, a)$の信頼区間を導出します。こうすれば$1$次元の信頼区間になるためです。\n",
                "\n",
                "この導出にはBernsteinの不等式を使います。さっそくやっていきましょう。\n",
                "\n",
                "---\n",
                "**定理：Azuma-Bernsteinの不等式**\n",
                "\n",
                "$\\left(\\mathcal{F}_n\\right)_{n=1}^N$に対して可測な実数値確率過程$\\left(X_n\\right)_{n=1}^N$を考える. \n",
                "$X_n \\in\\left[l_n, u_n\\right]$ かつ $\\mathbb{E}_n\\left[X_n\\right]=0$ が任意の$n$でa.s.で成立するとする. \n",
                "また，$V^{\\prime}:=\\sum_{n=1}^N \\mathbb{E}_n\\left[X_n^2\\right]$とする。\n",
                "このとき、\n",
                "\n",
                "$$\n",
                "\\mathbb{P}\\left(\\sum_{n=1}^N X_n \\geq \\frac{u_n - l_n}{3} \\log \\frac{1}{\\delta}+\\sqrt{2 V \\log \\frac{1}{\\delta}}\\ \\mathrm{ and } \\ V^{\\prime} \\leq V\\right) \\leq \\delta\n",
                "$$\n",
                "\n",
                "\n",
                "**UCB-VIの証明での直感（多分）**\n",
                "\n",
                "先にAzuma-HoeffdingとAzuma-Bernsteinを使ったときの結果を比較しておきます\n",
                "\n",
                "* Hoeffdingを使ったとき： \n",
                "    * $\\left|\\widehat{P}_h^k(\\cdot \\mid s, a)^{\\top} f - {P}_h^\\star(\\cdot \\mid s, a)^{\\top} f\\right|\\leq \\square \\sqrt{\\frac{H^2SL}{N_h^k(s, a)}}$\n",
                "    * $\\sum_{h=0}^{H-1} \\sum_{k=0}^{K-1} \\frac{1}{\\sqrt{N_h^k\\left(s_h^k, a_h^k\\right)}}=\\sum_{h=0}^{H-1} \\sum_{s, a \\in \\mathcal{S} \\times \\mathcal{A}} \\sum_{i=1}^{N_h^K(s, a)} \\frac{1}{\\sqrt{i}} \\leq H \\sqrt{S A K}$\n",
                "\n",
                "* Bernsteinを使ったとき： \n",
                "    * $\\left|\\widehat{P}_h^k(\\cdot \\mid s, a)^{\\top} f - {P}_h^\\star(\\cdot \\mid s, a)^{\\top} f\\right|\\leq \\mathbb{E}_{s^{\\prime} \\sim P_h^{\\star}(\\cdot \\mid s, a)} \\frac{f\\left(s^{\\prime}\\right)}{H}+\\square\\frac{H^2 S L}{N_h^k(s, a)}$\n",
                "    * $\\sum_{h=0}^{H-1} \\sum_{k=0}^{K-1} \\frac{1}{N_h^k\\left(s_h^k, a_h^k\\right)}=\\sum_{h=0}^{H-1} \\sum_{s, a \\in \\mathcal{S} \\times \\mathcal{A}} \\sum_{i=1}^{N_h^K(s, a)} \\frac{1}{i} \\leq HSA \\ln{(K)}$\n",
                "\n",
                "Bernsteinの方の２個めのバウンドは[Upper Bound for Harmonic Number](https://proofwiki.org/wiki/Upper_Bound_for_Harmonic_Number)を使うと出てきます。\n",
                "\n",
                "Bernsteinの方では第二項目は$\\ln(K)$に比例するため、最終的には第一項目の$\\mathbb{E}_{s^{\\prime} \\sim P_h^{\\star}(\\cdot \\mid s, a)} \\frac{f\\left(s^{\\prime}\\right)}{H}$に関する項だけ気にすれば良くなります。\n",
                "そして第一項目は最終的にボーナス項の和の形になり、$\\sum_{h=0}^{H-1} \\sum_{k=0}^{K-1} \\frac{1}{\\sqrt{N_h^k\\left(s_h^k, a_h^k\\right)}}\\leq H \\sqrt{S A K}$っぽいものが出てきます。\n",
                "つまりHoeffdingと比較すると、定数項を犠牲にして$\\sqrt{S}$の係数を消しているわけですね。\n",
                "\n",
                "---\n",
                "\n",
                "以上を踏まえて、推定した遷移確率の誤差のバウンドを確認しましょう。\n",
                "固定された$(s, a, k, h, s')$について、$\\left|\\widehat{P}_h^k(s' \\mid s, a)-P_h^{\\star}(s' \\mid s, a)\\right|$の上界を導出します(１では$s'$は$f$との内積で消していました)。\n",
                "まず定義より、\n",
                "\n",
                "$$\n",
                "\\begin{aligned}\n",
                "\\widehat{P}_h^k(s'\\mid s, a) - {P}_h^\\star(s' \\mid s, a) &=\n",
                "\\frac{1}{N_h^k(s, a)}\n",
                "\\sum_{i=0}^{k-1} \\left(\\mathbf{1}\\left\\{s_h^i, a_h^i, s_{h+1}^i=s, a, s'\\right\\}\n",
                "- \\mathbf{1}\\left\\{s_h^i, a_h^i=s, a\\right\\} \\mathbb{E}_{s^{\\prime\\prime} \\sim P_h^{\\star}(s, a)} \\mathbf{1}\\left\\{s''=s^{\\prime}\\right\\}\\right)\\\\\n",
                "&=\n",
                "\\frac{1}{N_h^k(s, a)}\n",
                "\\sum_{i=0}^{k-1} \\left(\\mathbf{1}\\left\\{s_h^i, a_h^i, s_{h+1}^i=s, a, s'\\right\\}\n",
                "- \\mathbf{1}\\left\\{s_h^i, a_h^i=s, a\\right\\} P_h^{\\star}(s'|s, a)\\right)\\\\\n",
                "\\end{aligned}\n",
                "$$\n",
                "です。\n",
                "\n",
                "\n",
                "ここで、和の中身について、$X_i=\\mathbf{1}\\left\\{s_h^i, a_h^i, s_{h+1}^i=s, a, s'\\right\\} - \\mathbf{1}\\left\\{s_h^i, a_h^i=s, a\\right\\} P_h^{\\star}(s'|s, a)$とします。\n",
                "また、$\\mathcal{H}_{h, i}$を$h$での$i$を含む$i$までの履歴とすると、決定的なアルゴリズムについては$\\mathbb{E}\\left[X_i \\mid \\mathcal{H}_{h, i}\\right]=0$なので、$\\sum_{i=0}^{k-1} X_i$はマルチンゲールです。\n",
                "また、$\\left(s_h^i, a_h^i\\right) \\neq(s, a)$では$\\left|X_i\\right|=0$, $\\left(s_h^i, a_h^i\\right)=(s, a)$では$\\left|X_i\\right| \\leq 1$であることに気をつけましょう。\n",
                "そして、$X_i$の第二項目は$\\mathcal{H}_{h, i}$が与えられたときは定数なので、$X_i$の分散を\n",
                "\n",
                "$$\n",
                "\\mathbb{E}\\left[X_i^2 \\mid \\mathcal{H}_{h, i}\\right]=\\mathbf{1}\\left\\{s_h^i, a_h^i=s, a\\right\\} P_h^{\\star}\\left(s^{\\prime} \\mid s, a\\right)\\left(1-P_h^{\\star}\\left(s^{\\prime} \\mid s, a\\right)\\right)\n",
                "$$\n",
                "\n",
                "として出すことができます(これはベルヌーイ分布の分散を考えると出ます)。\n",
                "これをBernsteinの不等式に当てはめると、\n",
                "\n",
                "$$\n",
                "\\begin{aligned}\n",
                "\\left|\\widehat{P}_h^k(s'\\mid s, a) - {P}_h^\\star(s' \\mid s, a)\\right| & \\leq \\sqrt{\\frac{2 P_h^{\\star}\\left(s^{\\prime} \\mid s, a\\right)\\left(1-P_h^{\\star}\\left(s^{\\prime} \\mid s, a\\right)\\right) \\ln (1 / \\delta)}{N_h^k(s, a)}}+\\frac{2 \\ln (1 / \\delta)}{N_h^k(s, a)} \\\\\n",
                "& \\leq \\square\\sqrt{\\frac{P_h^{\\star}\\left(s^{\\prime} \\mid s, a\\right) \\ln (1 / \\delta)}{N_h^k(s, a)}}+\\frac{\\square\\ln (1 / \\delta)}{N_h^k(s, a)}\n",
                "\\end{aligned}\n",
                "$$\n",
                "\n",
                "最後に$s, a, h, k, s'$についてUnion Boundを取って、\n",
                "\n",
                "$$\n",
                "\\left|\\widehat{P}_h^k\\left(s^{\\prime} \\mid s, a\\right)-P_h^{\\star}\\left(s^{\\prime} \\mid s, a\\right)\\right| \\leq \\sqrt{\\frac{2 P_h^{\\star}\\left(s^{\\prime} \\mid s, a\\right) L}{N_h^k(s, a)}}+\\frac{2 L}{N_h^k(s, a)},\n",
                "$$\n",
                "\n",
                "を得ます。\n",
                "\n",
                "#### 7.2. 任意の$f$についての遷移確率のバウンド\n",
                "\n",
                "まだ$f$については議論していなかったので、$f$が混ざったバウンドも導出します。\n",
                "任意の$f:S\\mapsto [0, H]$について、\n",
                "\n",
                "$$\n",
                "\\begin{aligned}\n",
                "& \\left|\\left(\\widehat{P}_h^k(\\cdot \\mid s, a)-P_h^{\\star}(\\cdot \\mid s, a)\\right)^{\\top} f\\right| \\\\\n",
                "& \\leq \\sum_{s^{\\prime} \\in \\mathcal{S}}\\left|\\widehat{P}_h^k\\left(s^{\\prime} \\mid s, a\\right)-P_h^{\\star}(\\cdot \\mid s, a)\\right| f\\left(s^{\\prime}\\right) \\leq \\sum_{s^{\\prime} \\in \\mathcal{S}} \\sqrt{\\frac{2 L P_h^{\\star}\\left(s^{\\prime} \\mid s, a\\right) f^2\\left(s^{\\prime}\\right)}{N_h^k(s, a)}}+\\sum_{s^{\\prime} \\in \\mathcal{S}} \\frac{2 L f\\left(s^{\\prime}\\right)}{N_h^k(s, a)} \\\\\n",
                "& \\leq \\sum_{s^{\\prime} \\in \\mathcal{S}} \\sqrt{\\frac{2 L P_h^{\\star}\\left(s^{\\prime} \\mid s, a\\right) f^2\\left(s^{\\prime}\\right)}{N_h^k(s, a)}}+\\frac{2 S H L}{N_h^k(s, a)} \\leq \\sqrt{S} \\sqrt{\\frac{\\sum_{s^{\\prime} \\in \\mathcal{S}} 2 L P_h^{\\star}\\left(s^{\\prime} \\mid s, a\\right) f^2\\left(s^{\\prime}\\right)}{N_h^k(s, a)}}+\\frac{2 S H L}{N_h^k(s, a)} \\\\\n",
                "& =\\sqrt{\\frac{2 H^2 S L}{N_h^k(s, a)} \\cdot \\frac{\\sum_{s^{\\prime} \\in \\mathcal{S}} P_h^{\\star}\\left(s^{\\prime} \\mid s, a\\right) f^2\\left(s^{\\prime}\\right)}{H^2}}+\\frac{2 S H L}{N_h^k(s, a)} \\leq \\frac{H^2 S L}{N_h^k(s, a)}+\\frac{\\mathbb{E}_{s^{\\prime} \\sim P_h^{\\star}(\\cdot \\mid s, a)} f^2\\left(s^{\\prime}\\right)}{H^2}+\\frac{2 S H L}{N_h^k(s, a)} \\\\\n",
                "& \\leq \\frac{\\mathbb{E}_{s^{\\prime} \\sim P_h^{\\star}(\\cdot \\mid s, a)} f\\left(s^{\\prime}\\right)}{H}+\\frac{\\square H^2 S L}{N_h^k(s, a)}\n",
                "= \\frac{\\mathbb{E}_{s^{\\prime} \\sim P_h^{\\star}(\\cdot \\mid s, a)} f\\left(s^{\\prime}\\right)}{H}+\\xi_h^i(s, a), \\\\\n",
                "&\n",
                "\\end{aligned}\n",
                "$$\n",
                "\n",
                "が成り立ちます。\n",
                "これは$\\|f\\|_\\infty \\leq H$やコーシー・シュワルツの不等式、そして$a b \\leq\\left(a^2+b^2\\right) / 2$を使うと出てきます。\n",
                "最後の等式では$\\xi_h^i(s, a) := \\frac{\\square H^2 S L}{N_h^k(s, a)}$を導入しました。\n",
                "\n",
                "\n",
                "#### 7.3. $f:=\\widehat{V}_{h+1}^k-V_{h+1}^{\\star}$のときのバウンド\n",
                "\n",
                "このとき、7.2より\n",
                "\n",
                "$$\n",
                "\\left|\\left(\\widehat{P}_h^k(\\cdot \\mid s, a)-P_h^{\\star}(\\cdot \\mid s, a)\\right)^{\\top} f\\right| \\leq \\frac{\\mathbb{E}_{s^{\\prime} \\sim P_h^{\\star}(\\cdot \\mid s, a)}\\left(\\widehat{V}_{h+1}^k\\left(s^{\\prime}\\right)-V_{h+1}^{\\star}\\left(s^{\\prime}\\right)\\right)}{H}+\\xi_h^k(s, a)\n",
                "$$\n",
                "\n",
                "が成り立ってます。これを再帰的にバウンドしていきます。まず、丁寧に分解すると\n",
                "\n",
                "$$\n",
                "\\begin{aligned}\n",
                "\\widehat{V}_{h+1}^k\\left(s^{\\prime}\\right)-V_{h+1}^{\\star}\\left(s^{\\prime}\\right) \\leq & \\widehat{V}_{h+1}^k\\left(s^{\\prime}\\right)-\\left(r\\left(s^{\\prime}, \\pi^k\\left(s^{\\prime}\\right)\\right)+P_{h+1}^{\\star}\\left(\\cdot \\mid s^{\\prime}, \\pi^k\\left(s^{\\prime}\\right)\\right)^{\\top} V_{h+2}^{\\star}\\right) \\\\\n",
                "\\leq & r\\left(s^{\\prime}, \\pi^k\\left(s^{\\prime}\\right)\\right)+\\widehat{P}_{h+1}^k\\left(\\cdot \\mid s^{\\prime}, \\pi^k\\left(s^{\\prime}\\right)\\right)^{\\top} \\widehat{V}_{h+2}^k-\\left(r\\left(s^{\\prime}, \\pi^k\\left(s^{\\prime}\\right)\\right)+P_{h+1}^{\\star}\\left(\\cdot \\mid s^{\\prime}, \\pi^k\\left(s^{\\prime}\\right)\\right)^{\\top} V_{h+2}^{\\star}\\right) \\\\\n",
                "= & \\widehat{P}_{h+1}^k\\left(\\cdot \\mid s^{\\prime}, \\pi^k\\left(s^{\\prime}\\right)\\right)^{\\top} \\widehat{V}_{h+2}^k-P_{h+1}^{\\star}\\left(\\cdot \\mid s^{\\prime}, \\pi^k\\left(s^{\\prime}\\right)\\right)^{\\top} \\widehat{V}_{h+2}^k+P_{h+1}^{\\star}\\left(\\cdot \\mid s^{\\prime}, \\pi^k\\left(s^{\\prime}\\right)\\right)^{\\top}\\left(\\widehat{V}_{h+2}^k-V_{h+2}^{\\star}\\right) \\\\\n",
                "= & \\left(\\widehat{P}_{h+1}^k\\left(\\cdot \\mid s^{\\prime}, \\pi^k\\left(s^{\\prime}\\right)\\right)-P_{h+1}^{\\star}\\left(\\cdot \\mid s^{\\prime}, \\pi^k\\left(s^{\\prime}\\right)\\right)\\right)^{\\top}\\left(\\widehat{V}_{h+2}^k-V_{h+2}^{\\star}\\right) \\\\\n",
                "& +\\left(\\widehat{P}_{h+1}^k\\left(\\cdot \\mid s^{\\prime}, \\pi^k\\left(s^{\\prime}\\right)\\right)-P_{h+1}^{\\star}\\left(\\cdot \\mid s^{\\prime}, \\pi^k\\left(s^{\\prime}\\right)\\right)\\right)^{\\top} V_{h+2}^{\\star} \\\\\n",
                "& +P_{h+1}^{\\star}\\left(\\cdot \\mid s^{\\prime}, \\pi^k\\left(s^{\\prime}\\right)\\right)^{\\top}\\left(\\widehat{V}_{h+2}^k-V_{h+2}^{\\star}\\right)\n",
                "\\end{aligned}\n",
                "$$\n",
                "\n",
                "が出てきます(これはベルマン方程式などを丁寧にいじれば簡単にわかります)。\n",
                "\n",
                "ここで、\n",
                "\n",
                "1. 同じことをして、$\\left(\\widehat{P}_{h+1}^k\\left(\\cdot \\mid s^{\\prime}, \\pi^k\\left(s^{\\prime}\\right)\\right)-P_{h+1}^{\\star}\\left(\\cdot \\mid s^{\\prime}, \\pi^k\\left(s^{\\prime}\\right)\\right)\\right)^{\\top}\\left(\\widehat{V}_{h+2}^k-V_{h+2}^{\\star}\\right) \\leq \\frac{P_{h+1}^{\\star}\\left(\\cdot \\mid s^{\\prime}, \\pi^k\\left(s^{\\prime}\\right)\\right)^{\\top}\\left(\\widehat{V}_{h+2}^k-V_{h+2}^{\\star}\\right)}{H}+\\xi_{h+1}^k\\left(s^{\\prime}, \\pi^k\\left(s^{\\prime}\\right)\\right)$\n",
                "2. 2. 遷移確率＋$V^\\star$の推定より$\\left(\\widehat{P}_{h+1}^k\\left(\\cdot \\mid s^{\\prime}, \\pi^k\\left(s^{\\prime}\\right)\\right)-P_{h+1}^{\\star}\\left(\\cdot \\mid s^{\\prime}, \\pi^k\\left(s^{\\prime}\\right)\\right)\\right)^{\\top} V_{h+2}^{\\star} \\leq b_{h+1}^k\\left(s^{\\prime}, \\pi^k\\left(s^{\\prime}\\right)\\right)$\n",
                "\n",
                "なので、まとめると、\n",
                "\n",
                "$$\n",
                "\\left(\\widehat{P}_{h+1}^k\\left(\\cdot \\mid s^{\\prime}, \\pi^k\\left(s^{\\prime}\\right)\\right)-P_{h+1}^{\\star}\\left(\\cdot \\mid s^{\\prime}, \\pi^k\\left(s^{\\prime}\\right)\\right)\\right)^{\\top}\\left(\\widehat{V}_{h+2}^k-V_{h+2}^{\\star}\\right) \\leq \\frac{P_{h+1}^{\\star}\\left(\\cdot \\mid s^{\\prime}, \\pi^k\\left(s^{\\prime}\\right)\\right)^{\\top}\\left(\\widehat{V}_{h+2}^k-V_{h+2}^{\\star}\\right)}{H}+\\xi_{h+1}^k\\left(s^{\\prime}, \\pi^k\\left(s^{\\prime}\\right)\\right) .\n",
                "$$\n",
                "\n",
                "が出てきます。\n",
                "\n",
                "これを再帰的に繰り返すと、\n",
                "\n",
                "$$\n",
                "\\begin{aligned}\n",
                "\\widehat{V}_{h+1}^k\\left(s^{\\prime}\\right)-V_{h+1}^{\\star}\\left(s^{\\prime}\\right) & \\leq \\mathbb{E}\\left[\\sum_{\\tau=h+1}^{H-1}\\left(1+\\frac{1}{H}\\right)^{\\tau-h-1}\\left(\\xi_\\tau^k\\left(s_\\tau, a_\\tau\\right)+b_\\tau^k\\left(s_\\tau, a_\\tau\\right)\\right) \\mid s_{h+1}=s^{\\prime}, \\pi^k\\right] \\\\\n",
                "& \\leq e \\mathbb{E}\\left[\\sum_{\\tau=h+1}^{H-1}\\left(\\xi_\\tau^k\\left(s_\\tau, a_\\tau\\right)+b_\\tau^k\\left(s_\\tau, a_\\tau\\right)\\right) \\mid s_{h+1}=s^{\\prime}, \\pi^k\\right],\n",
                "\\end{aligned}\n",
                "$$\n",
                "\n",
                "をえます。ここで、$(1+1 / H)^x \\leq e$を使いました (これは$(1+1 / x)^x$が$x$について単調増加であることと、$e$の定義を考えるとすぐに出てきます)。\n",
                "\n",
                "よって、\n",
                "\n",
                "$$\n",
                "\\left|\\left(\\widehat{P}_h^k(\\cdot \\mid s, a)-P_h^{\\star}(\\cdot \\mid s, a)\\right)^{\\top} \\left(\\widehat{V}_{h+1}^k-V_{h+1}^{\\star}\\right)\\right| \\leq \\xi_h^k(s, a)+\\frac{\\square}{H} \\mathbb{E}\\left[\\sum_{\\tau=h+1}^{H-1}\\left(\\xi_\\tau^k\\left(s_\\tau, a_\\tau\\right)+b_\\tau^k\\left(s_\\tau, a_\\tau\\right)\\right) \\mid s_h=s, a_h=a, \\pi^k\\right]\n",
                "$$\n",
                "\n",
                "#### 7.4 リグレットの導出\n",
                "\n",
                "あとは３で出した$V^{\\star}\\left(s_0\\right)-V^{\\pi^k}\\left(s_0\\right) \\leq \\widehat{V}_0^k\\left(s_0\\right)-V_0^{\\pi^k}\\left(s_0\\right)$を使ってリグレットの形に直すだけです。\n",
                "まずは３と7.3の結果を使って：\n",
                "\n",
                "$$\n",
                "\\begin{aligned}\n",
                "& V^{\\star}\\left(s_0\\right)-V^{\\pi^k}\\left(s_0\\right) \\leq \\widehat{V}_0^k\\left(s_0\\right)-V_0^{\\pi^k}\\left(s_0\\right) \\leq \\sum_{h=0}^{H-1} \\mathbb{E}_{s_h, a_h \\sim d_h^{\\pi k}}\\left[b_h^k\\left(s_h, a_h\\right)+\\left(\\widehat{P}_h^k\\left(\\cdot \\mid s_h, a_h\\right)-P^{\\star}\\left(\\cdot \\mid s_h, a_h\\right)\\right) \\cdot \\widehat{V}_{h+1}^k\\right] \\\\\n",
                "&= \\sum_{h=0}^{H-1} \\mathbb{E}_{s_h, a_h \\sim d_h^{\\pi_h^k}}\\left[b_h^k\\left(s_h, a_h\\right)+\\left(\\widehat{P}_h^k\\left(\\cdot \\mid s_h, a_h\\right)-P^{\\star}\\left(\\cdot \\mid s_h, a_h\\right)\\right)^{\\top}\\left(\\widehat{V}_{h+1}^k-V_{h+1}^{\\star}\\right)+\\left(\\widehat{P}_h^k\\left(\\cdot \\mid s_h, a_h\\right)-P^{\\star}\\left(\\cdot \\mid s_h, a_h\\right)\\right)^{\\top} V_{h+1}^{\\star}\\right] \\\\\n",
                "& \\leq \\sum_{h=0}^{H-1} \\mathbb{E}_{s_h, a_h \\sim d_h^{\\pi^k}}\\left[2 b_h^k\\left(s_h, a_h\\right)+\\left(\\widehat{P}_h^k\\left(\\cdot \\mid s_h, a_h\\right)-P^{\\star}\\left(\\cdot \\mid s_h, a_h\\right)\\right)^{\\top}\\left(\\widehat{V}_{h+1}^k-V_{h+1}^{\\star}\\right)\\right] \\\\\n",
                "& \\leq \\sum_{h=0}^{H-1} \\mathbb{E}_{s_h, a_h \\sim d_h^{\\pi^k}}\\left[2 b_h^k\\left(s_h, a_h\\right)+\\xi_h^k\\left(s_h, a_h\\right)+\\frac{e}{H} \\mathbb{E}\\left[\\sum_{\\tau=h+1}^{H-1}\\left(\\xi_\\tau^k\\left(s_\\tau, a_\\tau\\right)+b_\\tau^k\\left(s_\\tau, a_\\tau\\right)\\right) \\mid \\pi^k\\right]\\right] \\\\\n",
                "& \\leq e \\sum_{h=0}^{H-1} \\mathbb{E}_{s_h, a_h \\sim d_h^\\pi}\\left[2 b_h^k\\left(s_h, a_h\\right)+\\xi_h^k\\left(s_h, a_h\\right)\\right]\n",
                "= e \\mathbb{E}\\left[\\sum_{k=0}^{K-1} \\sum_{h=0}^{H-1}\\left[2 b_h^k\\left(s_h^k, a_h^k\\right)+\\xi_h^k\\left(s_h^k, a_h^k\\right)\\right]\\right]\n",
                "\\end{aligned}\n",
                "$$\n",
                "\n",
                "が成り立ちます。\n",
                "\n",
                "そして、\n",
                "\n",
                "* $\\sum_{k=0}^{K-1} \\sum_{h=0}^{H-1} 2 b_h^k\\left(s_h^k, a_h^k\\right) \\leq 4 H \\sqrt{L} \\sum_k \\sum_h \\frac{1}{\\sqrt{N_h^k\\left(s_h^k, a_h^k\\right)}} \\leq 8 H^2 \\sqrt{S A K L}$.\n",
                "* $\\begin{aligned} \\sum_{k=0}^{K-1} \\sum_{h=0}^{H-1} \\xi_h^k\\left(s_h^k, a_h^k\\right) & =2 H^2 S L \\sum_{k=0}^{K-1} \\sum_{h=0}^{H-1} \\frac{1}{N_h^k\\left(s_h^i, a_h^k\\right)}=2 H^2 S L \\sum_{h=0}^{H-1} \\sum_{s, a} \\sum_{i=1}^{N_h^K(s, a)} \\frac{1}{i} \\\\ & \\leq 2 H^2 S A L H S \\ln (K)=2 H^3 S^2 A L \\ln (K),\\end{aligned}$\n",
                "\n",
                "が簡単に出せるので、最終的なリグレットは $\\tilde{O}\\left(H^2 \\sqrt{S A K}+H^3 S^2 A\\right)$になります。"
            ]
        },
        {
            "attachments": {},
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "完成！実験してみましょう。[RL_reward_free.ipynb](RL_reward_free.ipynb)で使ったDouble Chain MDPを使います。"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 2,
            "metadata": {},
            "outputs": [
                {
                    "name": "stderr",
                    "output_type": "stream",
                    "text": [
                        "No GPU/TPU found, falling back to CPU. (Set TF_CPP_MIN_LOG_LEVEL=0 and rerun for more info.)\n"
                    ]
                }
            ],
            "source": [
                "import numpy as np\n",
                "from functools import partial\n",
                "from typing import Optional, NamedTuple\n",
                "import jax.numpy as jnp\n",
                "\n",
                "def create_rew_P(S: int, A: int):\n",
                "    \"\"\" double_chain MDP (報酬関数, 遷移確率) を作成します\n",
                "    詳しくは６章参照\n",
                "\n",
                "    Args:\n",
                "        S (int): 状態空間のサイズ\n",
                "        A (int): 行動空間のサイズ\n",
                "\n",
                "    Returns:\n",
                "        rew (float): 報酬関数\n",
                "        P (float): 遷移確率\n",
                "    \"\"\"\n",
                "    assert A == 2\n",
                "\n",
                "    # 一番右側の状態で1がもらえます\n",
                "    rew = np.zeros((S, 2))\n",
                "    rew[-1] = 1.0\n",
                "\n",
                "    # 1を選ぶと右に, 0を選ぶと左に動きます。0.1の確率で逆方向に動きます。\n",
                "    P = np.zeros((S, A, S))\n",
                "    for s in range(S):\n",
                "        P[s, 0, max(0, s-1)] = 0.9 \n",
                "        P[s, 0, min(s+1, S-1)] = 0.1 \n",
                "\n",
                "        P[s, 1, max(0, s-1)] = 0.1 \n",
                "        P[s, 1, min(s+1, S-1)] = 0.9 \n",
                "    np.testing.assert_almost_equal(P.sum(axis=-1), 1)\n",
                "    return jnp.array(rew), jnp.array(P)\n",
                "\n",
                "\n",
                "class MDP(NamedTuple):\n",
                "    S_array: np.array  # 状態空間\n",
                "    A_array: np.array  # 行動空間\n",
                "    gamma: float  # 割引率\n",
                "    horizon: int  # ホライゾン\n",
                "    rew: float  # 報酬関数\n",
                "    P: float  # 遷移確率\n",
                "    optimal_Q: Optional[np.ndarray] = None  # 最適Q値\n",
                "    optimal_Q_idx: Optional[np.ndarray] = None  # 後でplot用に使います\n",
                "\n",
                "    @property\n",
                "    def S(self) -> int:  # 状態空間のサイズ\n",
                "        return len(self.S_array)\n",
                "\n",
                "    @property\n",
                "    def A(self) -> int:  # 行動空間のサイズ\n",
                "        return len(self.A_array)\n",
                "\n",
                "\n",
                "S = 30\n",
                "A = 2\n",
                "S_array = np.arange(S)\n",
                "A_array = np.arange(A)\n",
                "gamma = 1\n",
                "horizon = 20\n",
                "\n",
                "# Double chain MDP を作ります\n",
                "rew, P = create_rew_P(S, A)\n",
                "double_chain = MDP(S_array, A_array, gamma, horizon, rew, P)"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "# 実験に便利な関数\n",
                "\n",
                "実験の前に、いくつか便利な関数を導入します。\n",
                "\n",
                "1. ``compute_greedy_policy``: Q関数 ($S \\times A \\to \\mathcal{R}$) の貪欲方策を返します\n",
                "2. ``compute_optimal_Q``: MDPの最適Q関数 $q_* : S \\times A \\to \\mathcal{R}$ を返します。\n",
                "3. ``compute_policy_Q``: 方策 $\\pi$ のQ関数 $q_\\pi : S \\times A \\to \\mathcal{R}$ を返します。\n",
                "4. ``compute_optimality_gap``: 方策 $\\pi$ と最適解との差 $\\|q_* - q_{\\pi}\\|_\\infty$ を返します。この値が小さいほど、$\\pi$ はほぼ全ての状態行動対で最適な方策になっています。\n",
                "5. ``sample_next_state``: $(s, a)$の次の状態をサンプルします。\n",
                "6. ``sample_eps_greedy_act``: 与えられたQ値から、epsilon-greedy方策に従って行動を返します"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 3,
            "metadata": {},
            "outputs": [],
            "source": [
                "import jax\n",
                "from functools import partial\n",
                "from tqdm import tqdm\n",
                "import random\n",
                "\n",
                "\n",
                "# >>>> 1 >>>>\n",
                "@jax.jit\n",
                "def compute_greedy_policy(Q: np.ndarray):\n",
                "    greedy_policy = jnp.zeros_like(Q)\n",
                "    S, A = Q.shape\n",
                "    greedy_policy = greedy_policy.at[jnp.arange(S), Q.argmax(axis=1)].set(1)\n",
                "    assert greedy_policy.shape == (S, A)\n",
                "    return greedy_policy\n",
                "# <<<< 1 <<<<\n",
                "\n",
                "\n",
                "# >>>> 2 >>>>\n",
                "@partial(jax.jit, static_argnames=(\"S\", \"A\"))\n",
                "def _compute_optimal_Q(mdp: MDP, S: int, A: int):\n",
                "\n",
                "    def backup(optimal_Q):\n",
                "        greedy_policy = compute_greedy_policy(optimal_Q)\n",
                "        max_Q = (greedy_policy * optimal_Q).sum(axis=1)\n",
                "        next_v = mdp.P @ max_Q\n",
                "        assert next_v.shape == (S, A)\n",
                "        return mdp.rew + mdp.gamma * next_v\n",
                "    \n",
                "    optimal_Q = jnp.zeros((S, A))\n",
                "    body_fn = lambda i, Q: backup(Q)\n",
                "    return jax.lax.fori_loop(0, mdp.horizon, body_fn, optimal_Q)\n",
                "\n",
                "compute_optimal_Q = lambda mdp: _compute_optimal_Q(mdp, mdp.S, mdp.A)\n",
                "# <<<< 2 <<<<\n",
                "\n",
                "\n",
                "# >>>> 3 >>>>\n",
                "def compute_policy_Q(mdp: MDP, policy: np.ndarray):\n",
                "    S, A = policy.shape\n",
                "\n",
                "    def backup(policy_Q):\n",
                "        max_Q = (policy * policy_Q).sum(axis=1)\n",
                "        next_v = mdp.P @ max_Q\n",
                "        assert next_v.shape == (S, A)\n",
                "        new_Q = mdp.rew + mdp.gamma * next_v\n",
                "        return jnp.minimum(mdp.horizon, new_Q)\n",
                "    \n",
                "    policy_Q = jnp.zeros((S, A))\n",
                "    body_fn = lambda i, Q: backup(Q)\n",
                "    return jax.lax.fori_loop(0, mdp.horizon, body_fn, policy_Q)\n",
                "# <<<< 3 <<<<\n",
                "\n",
                "\n",
                "# >>>> 4 >>>>\n",
                "@jax.jit\n",
                "def compute_optimality_gap(mdp: MDP, policy: np.ndarray):\n",
                "    policy_Q = compute_policy_Q(mdp, policy)\n",
                "    optimality_gap = jnp.abs(mdp.optimal_Q - policy_Q).max()\n",
                "    return optimality_gap\n",
                "# <<<< 4 <<<<\n",
                "\n",
                "\n",
                "# >>>> 5 >>>>\n",
                "def sample_next_state(mdp: MDP, s: int, a: int):\n",
                "    \"\"\" sample next state according to the transition matrix P\n",
                "    Args:\n",
                "        mdp: MDP\n",
                "        s: int\n",
                "        a: int\n",
                "    Returns:\n",
                "        next_s: int\n",
                "    \"\"\"\n",
                "    probs = np.array(mdp.P[s, a])\n",
                "    return np.random.choice(mdp.S_array, p=probs)\n",
                "# <<<< 5 <<<<\n",
                "\n",
                "\n",
                "# >>>> 6 >>>>\n",
                "def sample_eps_greedy_act(mdp: MDP, q_s: np.array, eps: float):\n",
                "    if random.random() < eps:\n",
                "        return random.randint(0, mdp.A-1)\n",
                "    else:\n",
                "        return q_s.argmax()\n",
                "# <<<< 6 <<<<\n",
                "\n",
                "double_chain = double_chain._replace(optimal_Q=compute_optimal_Q(double_chain))"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 4,
            "metadata": {},
            "outputs": [
                {
                    "name": "stderr",
                    "output_type": "stream",
                    "text": [
                        "100%|██████████| 1000/1000 [00:31<00:00, 31.45it/s]\n",
                        "100%|██████████| 1000/1000 [00:38<00:00, 26.16it/s]"
                    ]
                },
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "1.8055249778189661\n"
                    ]
                },
                {
                    "name": "stderr",
                    "output_type": "stream",
                    "text": [
                        "\n"
                    ]
                }
            ],
            "source": [
                "import matplotlib.pyplot as plt\n",
                "\n",
                " \n",
                "def EpsilonGreedyVI(mdp: MDP, K: int):\n",
                "    S, A, H = mdp.S, mdp.A, mdp.horizon\n",
                "\n",
                "    # MDPが定常なのでHは省略します\n",
                "    visit_SA = np.zeros((S, A))\n",
                "    visit_SAS = np.zeros((S, A, S))\n",
                "    P_approx = np.ones((S, A, S)) / S\n",
                "    policy = np.ones((S, A)) / A\n",
                "\n",
                "    init_s = int((mdp.S - 1) / 2)\n",
                "    optimal_V = mdp.optimal_Q[init_s].max()\n",
                "    regrets = []\n",
                "    regret = 0\n",
                "\n",
                "    for k in tqdm(range(K)):\n",
                "        s = init_s\n",
                "\n",
                "        # 探索をします　\n",
                "        for h in range(H):\n",
                "            a = np.random.randint(A)\n",
                "            visit_SA[s, a] += 1\n",
                "            next_s = sample_next_state(mdp, s, a)\n",
                "            visit_SAS[s, a, next_s] += 1\n",
                "            P_approx[s, a] = visit_SAS[s, a] / visit_SA[s, a]\n",
                "            s = next_s\n",
                "\n",
                "        # 方策を更新します\n",
                "        mdp_approx = mdp._replace(P=P_approx, rew=mdp.rew)\n",
                "        mdp_approx = mdp_approx._replace(optimal_Q=compute_optimal_Q(mdp_approx))\n",
                "        policy = compute_greedy_policy(mdp_approx.optimal_Q)\n",
                "\n",
                "        # リグレットを計算します\n",
                "        policy_V = compute_policy_Q(mdp, policy)[init_s].max()\n",
                "        regret += optimal_V - policy_V\n",
                "        regrets.append(regret)\n",
                "\n",
                "    return regrets, visit_SA\n",
                "\n",
                "\n",
                " \n",
                "def UCB_VI(mdp: MDP, delta: float, K: int):\n",
                "    S, A, H = mdp.S, mdp.A, mdp.horizon\n",
                "    L = np.log(S*A*H*K / delta)\n",
                "\n",
                "    # MDPが定常なのでHは省略します\n",
                "    visit_SA = np.zeros((S, A))\n",
                "    visit_SAS = np.zeros((S, A, S))\n",
                "    P_approx = np.ones((S, A, S)) / S\n",
                "    policy = np.ones((S, A)) / A\n",
                "\n",
                "    init_s = int((mdp.S - 1) / 2)\n",
                "    optimal_V = mdp.optimal_Q[init_s].max()\n",
                "    regrets = []\n",
                "    regret = 0\n",
                "\n",
                "    for k in tqdm(range(K)):\n",
                "        s = init_s\n",
                "\n",
                "        # 探索をします　\n",
                "        for h in range(H):\n",
                "            a = sample_eps_greedy_act(mdp, policy[s], 0)\n",
                "            visit_SA[s, a] += 1\n",
                "            next_s = sample_next_state(mdp, s, a)\n",
                "            visit_SAS[s, a, next_s] += 1\n",
                "            P_approx[s, a] = visit_SAS[s, a] / visit_SA[s, a]\n",
                "            s = next_s\n",
                "\n",
                "        # 方策を更新します\n",
                "        # 0.1はbonusが大きくなりすぎるのを防ぐためです。\n",
                "        bonus = 0.1 * H * np.sqrt(L / (visit_SA + 1e-3))  # 1e-3はzero division errorを防ぐためです\n",
                "        mdp_approx = mdp._replace(P=P_approx, rew=mdp.rew + bonus)\n",
                "        mdp_approx = mdp_approx._replace(optimal_Q=compute_optimal_Q(mdp_approx))\n",
                "        policy = compute_greedy_policy(mdp_approx.optimal_Q)\n",
                "\n",
                "        # リグレットを計算します\n",
                "        policy_V = compute_policy_Q(mdp, policy)[init_s].max()\n",
                "        regret += optimal_V - policy_V\n",
                "        regrets.append(regret)\n",
                "\n",
                "    print(bonus.max())\n",
                "    return regrets, visit_SA\n",
                "\n",
                "\n",
                "delta = 0.1\n",
                "K = 1000\n",
                "eps_greedy_regrets, eps_greedy_visit_SA = EpsilonGreedyVI(double_chain, K)\n",
                "ucb_regrets, ucb_visit_SA = UCB_VI(double_chain, delta, K)\n"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 5,
            "metadata": {},
            "outputs": [
                {
                    "data": {
                        "text/plain": [
                            "<matplotlib.legend.Legend at 0x7f317458aa00>"
                        ]
                    },
                    "execution_count": 5,
                    "metadata": {},
                    "output_type": "execute_result"
                },
                {
                    "data": {
                        "image/png": 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",
                        "text/plain": [
                            "<Figure size 1600x400 with 2 Axes>"
                        ]
                    },
                    "metadata": {},
                    "output_type": "display_data"
                }
            ],
            "source": [
                "plt.figure(figsize=(16, 4))\n",
                "plt.subplot(1, 2, 1)\n",
                "plt.plot(eps_greedy_regrets, label=\"Epsilon Greedy\")\n",
                "plt.plot(ucb_regrets, label=\"UCB VI\")\n",
                "plt.ylabel(\"Regret\")\n",
                "plt.xlabel(\"K\")\n",
                "plt.legend()\n",
                "\n",
                "plt.subplot(1, 2, 2)\n",
                "plt.plot(eps_greedy_visit_SA.sum(axis=1), label=\"Epsilon Greedy\")\n",
                "plt.plot(ucb_visit_SA.sum(axis=1), label=\"UCB VI\")\n",
                "plt.ylabel(\"Number of visits\")\n",
                "plt.xlabel(\"S\")\n",
                "plt.legend()"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": null,
            "metadata": {},
            "outputs": [],
            "source": []
        }
    ],
    "metadata": {
        "kernelspec": {
            "display_name": "Python 3.9.12 ('shumi-VTLwuKSy-py3.9')",
            "language": "python",
            "name": "python3"
        },
        "language_info": {
            "codemirror_mode": {
                "name": "ipython",
                "version": 3
            },
            "file_extension": ".py",
            "mimetype": "text/x-python",
            "name": "python",
            "nbconvert_exporter": "python",
            "pygments_lexer": "ipython3",
            "version": "3.9.12"
        },
        "orig_nbformat": 4,
        "vscode": {
            "interpreter": {
                "hash": "d6b7cac5e0d2ff733f340da4d53ae5ecfef7f7ad39623f5982b029a09306b36b"
            }
        }
    },
    "nbformat": 4,
    "nbformat_minor": 2
}