02.06 Unbiased Constant-Step-Size Trick.ipynb

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   "source": [
    "# Unbiased Constant-Step-Size Trick"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Author:** ZHENG Wenjie\n",
    "\n",
    "**Last Update:** 2021-08-13"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This notebook is related to Exercise 2.7 of the book.\n",
    "\n",
    "In the book, the authors claimed that the **unbiased constant-step size trick** avoids the initial bias. That is, the intial action value estimates do not have a roll in performance (not just *asympototic* performance). The trick consists in starting with a step size of 1 and converging to a constant step size of $\\alpha$.\n",
    "\n",
    "Here, I will show you that this trick *cannot* avoid the initial bias (although the initial bias can still varnish asymptotically).\n",
    "\n",
    "For the plot renderer, I used 'notebook_connected' to reduce the file size.\n",
    "For personal use, replace it with 'notebook'."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from tqdm import trange\n",
    "import plotly.graph_objects as go\n",
    "from plotly.subplots import make_subplots\n",
    "import plotly.io as pio\n",
    "pio.renderers.default = 'notebook_connected' # or 'notebook' for personal use\n",
    "\n",
    "from chapter02 import Bandit, NonStationaryGreedy, play"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## M times rerun on the same bandit"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here, we have a *single* bandit machine, and we run various algorithms on it over and over.\n",
    "Of course, each run has different random numbers. This experiment does not exist in the book."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "100%|██████████| 1000/1000 [00:26<00:00, 37.38it/s]\n"
     ]
    }
   ],
   "source": [
    "M = 1000\n",
    "K = 10\n",
    "T = 1000\n",
    "σ = 1\n",
    "average_reward = np.zeros((2, T))\n",
    "average_hit = np.zeros((2, T))\n",
    "for m in trange(M):\n",
    "    reward = np.zeros((2, T))\n",
    "    hit = np.zeros((2, T))\n",
    "    for i, player in enumerate([NonStationaryGreedy(K, seed=m), NonStationaryGreedy(K, initial=1, seed=m)]):\n",
    "        np.random.seed(0)\n",
    "        bandit = Bandit(np.random.randn(K), σ, seed=m)\n",
    "        reward[i], hit[i] = play(bandit, player, T)\n",
    "    average_reward += reward\n",
    "    average_hit += hit\n",
    "average_reward /= M\n",
    "average_hit /= M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
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   ],
   "source": [
    "fig = make_subplots(rows=2, cols=1, shared_xaxes=True, vertical_spacing=0.05, subplot_titles=['Average Reward', '% Optimal Pull'])\n",
    "fig.update_layout(height=700, width=700, title=f'{M} time(s) rerun on the same bandit')\n",
    "fig.add_trace(go.Scatter(x=np.arange(T), y=average_reward[0, :], legendgroup=0, name='initial=0'), row=1, col=1)\n",
    "fig.add_trace(go.Scatter(x=np.arange(T), y=average_reward[1, :], legendgroup=1, name='initial=1'), row=1, col=1)\n",
    "fig.add_trace(go.Scatter(x=np.arange(T), y=average_hit[0, :], legendgroup=0, showlegend=False, marker_color='#636EFA'), row=2, col=1)\n",
    "fig.add_trace(go.Scatter(x=np.arange(T), y=average_hit[1, :], legendgroup=1, showlegend=False, marker_color='#EF553B'), row=2, col=1)\n",
    "fig.show()"
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Discussion"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can observe that the initial estimates do play an important role in the beginning."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this notebook, we arrive at a different conclusion from the one in the book. Sutton and Barto made a mistake because they falsely believed that these estimates will be immediately overwritten after their respective first pulls. Nevertheless, in fact, they are already in use (for comparision) before they are ever updated."
   ]
  }
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