notebooks/BANDIT_UCB_regret_proof.ipynb

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                "# UCB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",
                "* [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",
                "* [Useful Inequalities Cheat Sheet](https://www.lkozma.net/inequalities_cheat_sheet/ineq.pdf)\n",
                "\n",
                "探索が関わる強化学習アルゴリズムの証明はだいぶややこしいです。\n",
                "その準備として、バンディット問題におけるUCB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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                "## K-Armed Bandit\n",
                "\n",
                "次の問題設定を考えます。\n",
                "\n",
                "問題設定\n",
                "\n",
                "* アームの数：$K$\n",
                "* 報酬：アーム$a \\in {1,2,\\ldots K}$について、報酬 $r_a \\in [-1,1] \\sim R(a) \\in \\Delta([-1,1])$。\n",
                "* 平均報酬：$\\mathbb{E}_{r_a \\sim R(a)}\\left[r_a\\right]=\\mu_a$であり、($\\mu_a \\in [-1,1]$)。\n",
                "* (擬)リグレット： アーム$A_t \\in [1,2,\\ldots K]$に対して、\n",
                "$$\n",
                "R_T=T \\cdot \\max _i \\mu_i-\\sum_{t=0}^{T-1} \\mu_{A_t}\n",
                "$$"
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                "### UCBアルゴリズムの証明\n",
                "\n",
                "\\begin{aligned}\n",
                "& \\text { Algorithm } 3 \\text { UCB } \\\\\n",
                "& \\text { 1: Play each arm once and denote received reward as } r_a \\text { for all } a \\in\\{1,2, \\ldots K\\} \\\\\n",
                "& \\text { 2: for } t=1 \\rightarrow T-K \\text { do } \\\\\n",
                "& \\text { 3: } \\quad \\text { Execute arm } A_t=\\arg \\max _{a \\in[K]}\\left(\\hat{\\mu}_a^t+\\sqrt{\\frac{\\log (T K / \\delta)}{N_a^t}}\\right) \\\\\n",
                "& \\text { 4: } \\quad \\text { Observe } r_t:=r_{A_t} \\\\\n",
                "& \\text { 5: end for }\n",
                "\\end{aligned}\n",
                "\n",
                "[BANDIT_basics.ipynb](BANDIT_basics.ipynb)でも簡単な証明は与えましたが、ちゃんと書きます。\n",
                "簡単のために、時刻$t=0$で全てのアームを一気に引くとして、$t>0$ではUCBを走らせるとします（こうしても一般性を失いません）。\n",
                "\n",
                "参考\n",
                "* [Reinforcement Learning: Theory and Algorithms](https://rltheorybook.github.io/)\n",
                "\n",
                "#### 1. アームの報酬の推定\n",
                "\n",
                "K-Armed Banditでは基本的にそれぞれの**アームの報酬の平均値**を推定します。このバウンドを導出しましょう。\n",
                "\n",
                "* 時刻$t$時点のアーム$a$を引いた回数：$N_a^t=1+\\sum_{i=1}^{t-1} \\mathbf{1}\\left\\{A_i=a\\right\\}$\n",
                "* 平均報酬の推定($r_a$は最初の回で引いた報酬。)：$\\widehat{\\mu}_a^t=\\frac{1}{N_a^t}\\left(r_a+\\sum_{i=1}^{t-1} \\mathbf{1}\\left\\{A_i=a\\right\\} r_i\\right)$\n",
                "\n",
                "まず、アーム$a$の推定値の幅を集中不等式で導出します。\n",
                "\n",
                "$$\n",
                "\\begin{aligned}\n",
                "\\widehat{\\mu}_a^t - \\mu_a \n",
                "&= \\frac{1}{N_a^t}\\left(r_a+\\sum_{i=1}^{t-1} \\mathbf{1}\\left\\{A_i=a\\right\\} r_i - N_a^t \\mu_a \\right)\\\\\n",
                "&= \\frac{1}{N_a^t}\\left(r_a - \\mu_a + \\sum_{i=1}^{t-1} \\mathbf{1}\\left\\{A_i=a\\right\\} (r_i - \\mu_a) \\right)\\\\\n",
                "\\end{aligned}\n",
                "$$\n",
                "\n",
                "ここで、和の中身について、$X_0 = r_a - \\mu_a$, $X_1 = \\mathbf{1}\\left\\{A_K=a\\right\\} (r_K - \\mu_a)$, ..., $X_i = \\mathbf{1}\\left\\{A_{i}=a\\right\\} (r_{i} - \\mu_a)$ とします。\n",
                "また、$\\mathcal{H}_{<i}$を$i$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$\\mathbb{E}[X_i|\\mathcal{H}_{<i}]=0$なので、$\\sum_{i=0}^{t-1} X_i$はマルチンゲールです。\n",
                "\n",
                "このようなマルチンゲールの系列に対して、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$a$と$t$について、$1-\\delta$以上で次が成立します。\n",
                "$$\n",
                "\\left|\\widehat{\\mu}_a^t-\\mu_a\\right|=\\frac{1}{N_a^t}\\left|\\sum_{i=0}^{t-1} X_i\\right|\\leq \\square \\sqrt{\\frac{\\ln (1 / \\delta)}{N_a^t}}\n",
                "$$\n",
                "\n",
                "つまり、\n",
                "$$\n",
                "P\\left(\\left|\\widehat{\\mu}_a^t-\\mu_a\\right|\\geq \\square \\sqrt{\\frac{\\ln (TK / \\delta)}{N_a^t}}\\right) \\leq \\frac{\\delta}{TK}\n",
                "$$\n",
                "です (ホントは$\\square$をちゃんと書かないとダメです)。\n",
                "\n",
                "\n",
                "最後にUnion boundを$0\\leq t \\leq T$と$a\\in \\{1, \\dots, K\\}$で取れば、任意の$a$と$t$について\n",
                "$$\n",
                "P\\left(\\left|\\widehat{\\mu}_a^t-\\mu_a\\right|\\geq \\square \\sqrt{\\frac{\\ln (TK / \\delta)}{N_a^t}}\\right) \\leq {\\delta}\n",
                "$$\n",
                "です。\n",
                "\n",
                "#### 2. リグレットのバウンド\n",
                "\n",
                "* 条件：1.のバウンドが成り立っているとき\n",
                "\n",
                "ここまで来たらあとはリグレットをバウンドするだけです。\n",
                "リグレットは$$\\sum_{t=0}^{T-1} \\mu_{a^{\\star}}-\\mu_{A_t}$$だったので、$\\mu_{a^{\\star}}-\\mu_{A_t}$をバウンドすればほとんどゴールです。\n",
                "\n",
                "\n",
                "１で求めたバウンドが成り立っている事象を考えましょう。つまり、任意の$a$と$t$について、\n",
                "$$\n",
                "\\mu_a \\leq \\widehat{\\mu}_a^t + \\square \\sqrt{\\frac{\\ln (TK / \\delta)}{N_a^t}}\n",
                "$$\n",
                "が確率$1-\\delta$で成り立っています。\n",
                "\n",
                "ここで、アルゴリズムに戻ってみましょう。アルゴリズムの３行目を見直すと\n",
                "$$\n",
                "\\text { 3: } \\quad \\text { Execute arm } A_t=\\arg \\max _{a \\in[K]}\\left(\\hat{\\mu}_a^t+\\sqrt{\\frac{\\log (T K / \\delta)}{N_a^t}}\\right)\n",
                "$$\n",
                "より、時刻$t$で引かれるアームはUCBスコア($\\widehat{\\mu}_a^t + \\square \\sqrt{\\frac{\\ln (TK / \\delta)}{N_a^t}}\n",
                "$)が一番大きいものでした。\n",
                "さらに事象$G$が成立してるので、時刻$t$で引かれるアーム$A_t$については\n",
                "\n",
                "$$\n",
                "\\mu_{a^\\star} \\leq \\widehat{\\mu}_{a^\\star}^t + \\square \\sqrt{\\frac{\\ln (TK / \\delta)}{N_{a^\\star}^t}} \\leq \\widehat{\\mu}_{A_t}^t + \\square \\sqrt{\\frac{\\ln (TK / \\delta)}{N_{A_t}^t}}\n",
                "$$\n",
                "\n",
                "が成り立っています。変形して、\n",
                "\n",
                "$$\n",
                "\\mu_{a^\\star} - {\\mu}_{A_t} \n",
                "\\leq \\widehat{\\mu}_{A_t}^t - \\mu_{A_t} + \\square \\sqrt{\\frac{\\ln (TK / \\delta)}{N_{A_t}^t}}\n",
                "\\leq \\square \\sqrt{\\frac{\\ln (TK / \\delta)}{N_{A_t}^t}}\n",
                "$$\n",
                "です。最後の不等式では$2\\square$を$\\square$にまとめました。\n",
                "\n",
                "最後に、\n",
                "$$\n",
                "\\sum_{t=0}^{T-1} \\mu_{a^{\\star}}-\\mu_{A_t} \\leq \\square\\sqrt{\\ln (T K / \\delta)} \\sum_{t=0}^{T-1} \\sqrt{\\frac{1}{N_{A_t}^t}} \n",
                "$$\n",
                "であり、ここで、$$\\sum_{t=0}^{T-1} \\sqrt{\\frac{1}{N_{A_t}^t}}=\\sum_a \\sum_{i=1}^{N_a^T} \\frac{1}{\\sqrt{i}} \\leq 2\\sum_a \\sqrt{{N_a^T}} \\leq 2\\sqrt{K \\sum_a {N_a^T}}$$です (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)。\n",
                "このようなバウンドはPotential lemmaと呼ばれます（[RL_General_Foundation_of_RL.ipynb](RL_General_Foundation_of_RL.ipynb)）．\n",
                "\n",
                "よって、\n",
                "$$\n",
                "\\sum_{t=0}^{T-1} \\mu_{a^{\\star}}-\\mu_{A_t} \\leq \\square \\sqrt{\\ln (T K / \\delta)} \\sqrt{K \\sum_a N_a^T}\n",
                "\\leq \\square \\sqrt{KT \\ln (T K / \\delta)}\n",
                "$$\n",
                "が導かれます。\n",
                "\n",
                "最初に$K$回アームを引くことを仮定したので、結局リグレットバウンドは\n",
                "\n",
                "$$\n",
                "R_T \\leq \\mathcal{O}\\left(\\sqrt{KT \\ln (T K / \\delta)} + K\\right)\n",
                "$$\n",
                "になります。\n",
                "\n",
                "\n",
                "#### 3. Gapに依存したバウンド\n",
                "\n",
                "最適アームと他のアームのGapが大きい場合、バンディット問題がやさしくなるような気がしますね。Gapに依存したバウンドは求まらないでしょうか？\n",
                "まず、Gapを定義します\n",
                "* ギャップ：$a^{\\star}=\\operatorname{argmax}_i \\mu_i$について$\\Delta_a = \\mu_{a^{\\star}} - \\mu_a$\n",
                "\n",
                "このとき、リグレットは\n",
                "\n",
                "$$\n",
                "R_T=T \\cdot \\max _i \\mu_i-\\sum_{t=0}^{T-1} \\mu_{A_t} \\leq \\sum_{a\\neq a^*} N_a^T \\Delta_a\n",
                "$$\n",
                "\n",
                "なので、$N_a^T$のバウンドを出せば良いことがわかります。\n",
                "一方で、２の途中の議論から、任意の$t$で\n",
                "$$\n",
                "\\Delta_{A_t} = \\mu_{a^\\star} - {\\mu}_{A_t} \n",
                "\\leq \\square \\sqrt{\\frac{\\ln (TK / \\delta)}{N_{A_t}^t}}\n",
                "\\Rightarrow \n",
                "N_{a}^T \\leq \\square {\\frac{\\ln (TK / \\delta)}{\\Delta_a^2}}\n",
                "$$\n",
                "なので、\n",
                "$$\n",
                "R_T\\leq \\square \\sum_{a\\neq a^*}{\\frac{\\ln (TK / \\delta)}{\\Delta_a}}\n",
                "$$\n",
                "であり、２と合体させると、\n",
                "\n",
                "$$\n",
                "R_T \\leq \\mathcal{O}\\left(\\min\\left(\\sqrt{KT \\ln (T K / \\delta)}, \\sum_{a\\neq a^*}{\\frac{\\ln (TK / \\delta)}{\\Delta_a}}\\right) + K\\right)\n",
                "$$\n",
                "になります。"
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                "簡単にPythonで確認してみましょう"
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                        "  0%|          | 0/30 [00:00<?, ?it/s]/tmp/ipykernel_79306/3047913655.py:29: RuntimeWarning: divide by zero encountered in divide\n",
                        "  ucb = np.where(num_arm_pulled == 0, np.infty, avg_rewards + np.sqrt(np.log(T*K / delta) / num_arm_pulled))\n",
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                            "Text(0.5, 1.0, 'Regret w.r.t. K')"
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",
                        "text/plain": [
                            "<Figure size 1200x400 with 2 Axes>"
                        ]
                    },
                    "metadata": {},
                    "output_type": "display_data"
                }
            ],
            "source": [
                "import numpy as np\n",
                "import matplotlib.pyplot as plt\n",
                "from tqdm import tqdm\n",
                "Array = np.ndarray\n",
                "\n",
                "\n",
                "class GaussianBandit:\n",
                "    def __init__(self, rew_mean: Array, noise_scale: Array):\n",
                "        self.rew_mean = rew_mean\n",
                "        self.noise_scale = noise_scale\n",
                "        self.optimal_mean = np.max(self.rew_mean)\n",
                "\n",
                "    def pull(self, arm: int):\n",
                "        mean = self.rew_mean[arm]\n",
                "        noise = np.random.randn() * self.noise_scale[arm]\n",
                "        reward = mean + noise\n",
                "        return mean, reward\n",
                "\n",
                "\n",
                "def upper_confidence_bound(bandit, K, T, delta, seeds=10):\n",
                "    avg_rewards = np.zeros(K)\n",
                "    num_arm_pulled = np.zeros(K)\n",
                "\n",
                "    pseudo_regret = 0\n",
                "    regrets = []\n",
                "    for _ in tqdm(range(seeds)):\n",
                "        for i in range(T):\n",
                "            # UCBスコアの計算と行動選択\n",
                "            ucb = np.where(num_arm_pulled == 0, np.infty, avg_rewards + np.sqrt(np.log(T*K / delta) / num_arm_pulled))\n",
                "            act = np.argmax(ucb)\n",
                "            mean, rew = bandit.pull(act)\n",
                "\n",
                "            # 平均値と探索結果の記録\n",
                "            num_pulled = num_arm_pulled[act]\n",
                "            avg_rewards[act] += (rew - avg_rewards[act]) / (num_pulled + 1)\n",
                "            num_arm_pulled[act] += 1\n",
                "\n",
                "            # 結果の記録\n",
                "            pseudo_regret += bandit.optimal_mean - mean\n",
                "        regrets.append(pseudo_regret)\n",
                "    return np.mean(regrets)\n",
                "\n",
                "\n",
                "# Tを変化した比較\n",
                "plt.figure(figsize=(12, 4))\n",
                "plt.subplot(1, 2, 1)\n",
                "K = 10\n",
                "delta = 0.1\n",
                "regrets = []\n",
                "gap = 0.5\n",
                "mean = np.zeros(K)\n",
                "mean[0] = gap\n",
                "noise_scale = np.ones(K)\n",
                "bandit = GaussianBandit(mean, noise_scale)\n",
                "Ts = [10, 30, 50, 100, 300, 500, 1000, 3000, 5000, 10000, 30000, 50000]\n",
                "\n",
                "for T in Ts:\n",
                "    regrets.append(upper_confidence_bound(bandit, K, T, delta, seeds=30))\n",
                "\n",
                "plt.plot(Ts, regrets)\n",
                "plt.ylabel(\"Regret\")\n",
                "plt.xlabel(\"T\")\n",
                "plt.title(\"Regret w.r.t. T\")\n",
                "\n",
                "\n",
                "# Kを変化した比較\n",
                "plt.subplot(1, 2, 2)\n",
                "T = 1000\n",
                "delta = 0.1\n",
                "regrets = []\n",
                "gap = 0.5\n",
                "mean = np.zeros(K)\n",
                "mean[0] = gap\n",
                "noise_scale = np.ones(K)\n",
                "bandit = GaussianBandit(mean, noise_scale)\n",
                "Ks = [2, 5, 7, 10, 20, 30, 50, 70, 100]\n",
                "\n",
                "for K in Ks:\n",
                "    mean = np.zeros(K)\n",
                "    mean[0] = gap\n",
                "    noise_scale = np.ones(K)\n",
                "    bandit = GaussianBandit(mean, noise_scale)\n",
                "    regrets.append(upper_confidence_bound(bandit, K, T, delta, seeds=30))\n",
                "\n",
                "plt.plot(Ks, regrets)\n",
                "plt.ylabel(\"Regret\")\n",
                "plt.xlabel(\"K\")\n",
                "plt.title(\"Regret w.r.t. K\")"
            ]
        },
        {
            "attachments": {},
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "Regretは$T$については$\\sqrt{T}$で、$K$については線形に増えていきます。"
            ]
        }
    ],
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