Lecture 3 - Model-Free Policy Evaluation.ipynb

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   "source": [
    "# Model-Free Policy Evaluation\n",
    "\n",
    "provided by [Stanford CS234](https://www.youtube.com/watch?v=FgzM3zpZ55o)\n",
    "\n",
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-info\">\n",
    "Table of Contents: <br>\n",
    "    \n",
    "<ul>\n",
    "    <li>1. <a href=\"#1.-Introduction\">Introduction</a></li>\n",
    "    <li>2. <a href=\"#2.-Monte-Carlo-(MC)-Policy-Evaluation\">Monte Carlo (MC) Policy Evaluation</a></li>\n",
    "    <li>3. <a href=\"#3.-Temporal-Difference-(TD)-Learning\">Temporal Difference (TD) Learning</a></li>\n",
    "    <li>4. <a href=\"#4.-Comparing-Approaches\">Comparing Approaches</a></li>\n",
    "    <li>5. <a href=\"#5.-Resource\">Resource</a></li>\n",
    "</ul>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 1. Introduction"
   ]
  },
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   "metadata": {},
   "source": [
    "Last time we covered policy evaluation & control with a true model of the world (environment). This lecture will cover policy evaluation (no control!) without known dynamics & reward models. Next time, we will cover control for this case."
   ]
  },
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   "cell_type": "markdown",
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   "source": [
    "Here are our definitions:\n",
    "\n",
    "$$\n",
    "G_{t} = r_{t} + \\gamma r_{t + 1} + ... \\\\\n",
    "V^{\\pi}(s) = \\mathbb{E}_{\\pi}[G_{t}~|~s_{t} = s] = \\mathbb{E}_{\\pi}[r_{t} + \\gamma r_{t + 1} + ...~|~s_{t} = s] \\\\\n",
    "Q^{\\pi}(s, a) = \\mathbb{E}_{\\pi}[G_{t}~|~s_{t} = s, a_{t} = a] = \\mathbb{E}_{\\pi}[r_{t} + \\gamma r_{t + 1} + ...~|~s_{t} = s, a_{t} = a]\n",
    "$$\n",
    "\n",
    "Remember the dynamic programming approach to evaluating a policy for a finite MDP."
   ]
  },
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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![dp_mdp.PNG](attachment:dp_mdp.PNG) <br>\n",
    "_Figure 1. Dynamic Programming for evaluation._"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can think of $V^{\\pi}_{k}(s)$ as an exact value of the k-horizon value of state $s$ under policy $\\pi$. And we can say it's an estimate of the infinite horizon for state $s$ under policy $\\pi$.\n",
    "\n",
    "$$\n",
    "V^{\\pi}(s) = \\mathbb{E}_{\\pi}[G_{t}~|~s_{t} = s] \\approx \\mathbb{E}_{\\pi}[r_{t} + \\gamma V_{k - 1}~|~s_{t} = s]\n",
    "$$\n",
    "\n",
    "This is formalized in the above equation."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<center><img src=\"https://imgur.com/DnDgOZJ.jpg\" alt=\"drawing\"/></center>\n",
    "_Figure 2. DP tree for evaluation._"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can think of the DP approach for evaluation as a tree where a state is followed by an action which can lead to a variable number of other states (which also have their corresponding actions). The point here is that we are __bootstrapping__. This means we are replacing the expected return by its estimate. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 2. Monte Carlo (MC) Policy Evaluation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Okay, notation is about to get a little nuanced. But this isn't the end of the road! Most notations can be omitted. They are there to highlight something. With that, let's take a look at the value function under a certain policy $\\pi$. \n",
    "\n",
    "$$\n",
    "V^{\\pi}(s) = \\mathbb{E}_{T \\sim \\pi}[G_{t}~|~s_{t} = s] \\hspace{1em} (Eq.~1)\\\\\n",
    "$$\n",
    "\n",
    "It is the same as before, but now we specify that we sample a trajectory following policy $\\pi$.\n",
    "\n",
    "> __Monte Carlo Policy Evaluation__ : a model-free (model here means the ground truth dynamics model of the environment) policy evaluation method\n",
    "\n",
    "Requirements for MC Policy Evaluation:\n",
    "\n",
    "* trajectories/episodes need to be finite (need to be episodic)\n",
    "* no bootstrapping (just sampling)\n",
    "* does not assume state is markov\n",
    "\n",
    "Let's take a look at 2 different monte carlo methods for model-free policy evaluation.\n",
    "\n",
    "> __First-Visit MC__ : a version of monte carlo policy evaluation that updates the value function with the first time $t$ that state $s$ is visited in episode $i$\n",
    "\n",
    "> __Every-Visit MC__ : a version of monte carlo policy evaluation that updates the value function with the with every time $t$ that state $s$ is visited in episode $i$\n",
    "\n",
    "Below is the algorithm for First-Visit:\n",
    "\n",
    "Initialize $N(s) = 0, G(s) = 0 ~~ \\forall s \\in S$ <br>\n",
    "Loop <br>\n",
    "$\\quad$ Sample episode $i = s_{i, 1}, a_{i, 1}, r_{i, 1}, s_{i, 2}, a_{i, 2}, r_{i, 2}, ..., s_{i, T_{i}}$\n",
    "<br>\n",
    "$\\quad$ Define $G_{i, t} = r_{i, t} + \\gamma r_{i, t + 1} + \\gamma^{2} r_{i, t + 2} + ... \\gamma^{T_{i} - 1} r_{i, T_{i}}$ for the $ith$ episode at time step $t$ <br>\n",
    "$\\quad$ For each state $s$ visited in episode $i$ <br>\n",
    "$\\quad\\quad$ For __first__ time $t$ that state $s$ is visited in episode $i$ <br>\n",
    "$\\quad\\quad\\quad$ Increment counter of total first visits: $N(s) = N(s) + 1$ <br>\n",
    "$\\quad\\quad\\quad$ Increment total return $G(s) = G(s) + G_{i, t}$ <br>\n",
    "$\\quad\\quad\\quad$ Update estimate $V^{\\pi}(s) = G(s)/N(s)$ <br>\n",
    "<br>\n",
    "_Algorithm 1. First-Visit Monte Carlo_"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<center><img src=\"https://imgur.com/mopfDoZ.jpg\" alt=\"drawing\"/></center>\n",
    "\n",
    "_Figure 1. Bias and variance and MSE._"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Simply put, the bias is the difference between the expected value of the statistic $\\hat{\\theta}$ and the true statistic $\\theta$. The variance is the expected squared difference between the $\\hat{\\theta}$ and the expected $\\hat{\\theta}$. The MSE is simply a combination of these 2.\n",
    "\n",
    "* $V^{\\pi}$ is an _unbiased_ estimator of true $\\mathbb{E}_{\\pi}[G_{t}~|~s_{t}=s]$. <br>\n",
    "* By the law of large numbers, as the count for First-Visit MC approaches infinity, the value estimates would also approach the expected value estimates under that same policy: $N(s) \\rightarrow \\infty, V^{\\pi}(s) \\rightarrow \\mathbb{E}_{\\pi}[G_{t}~|~s_{t} = s]$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Initialize $N(s) = 0, G(s) = 0 ~~ \\forall s \\in S$ <br>\n",
    "Loop <br>\n",
    "$\\quad$ Sample episode $i = s_{i, 1}, a_{i, 1}, r_{i, 1}, s_{i, 2}, a_{i, 2}, r_{i, 2}, ..., s_{i, T_{i}}$\n",
    "<br>\n",
    "$\\quad$ Define $G_{i, t} = r_{i, t} + \\gamma r_{i, t + 1} + \\gamma^{2} r_{i, t + 2} + ... \\gamma^{T_{i} - 1} r_{i, T_{i}}$ for the $ith$ episode at time step $t$ <br>\n",
    "$\\quad$ For each state $s$ visited in episode $i$ <br>\n",
    "$\\quad\\quad$ For __every__ time $t$ that state $s$ is visited in episode $i$ <br>\n",
    "$\\quad\\quad\\quad$ Increment counter of total first visits: $N(s) = N(s) + 1$ <br>\n",
    "$\\quad\\quad\\quad$ Increment total return $G(s) = G(s) + G_{i, t}$ <br>\n",
    "$\\quad\\quad\\quad$ Update estimate $V^{\\pi}(s) = G(s)/N(s)$ <br>\n",
    "<br>\n",
    "_Algorithm 2. Every-Visit Monte Carlo_"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is the exact same as the First Visit MC except it is done for every visit. In this case:\n",
    "\n",
    "* $V^{\\pi}$ for every-visit MC is _biased_ because now a state in one episode that occurs too often will be given a lot more priority. \n",
    "*It often has better MSE than first-visit and is a consistent estimator."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> __Incremental MC__ : an approach that can be layered on top of the previous MC methods to incrementally update the value estimate function\n",
    "\n",
    "After each episode $i$ <br>\n",
    "$\\quad$ Define $G_{i, t}$ <br>\n",
    "$\\quad$ For each state $s$ visited in episode $i$ <br>\n",
    "$\\quad\\quad$ For a time $t$ that state $s$ is visited in episode $i$ <br>\n",
    "$\\quad\\quad\\quad$ Increment counter of total first visits: $N(s) = N(s) + 1$ <br>\n",
    "$\\quad\\quad\\quad$ Update estimate $V^{\\pi}(s) = V^{\\pi}(s) + \\alpha(G_{i, t} - V^{\\pi}(s))$ <br>\n",
    "<br>\n",
    "_Algorithm 3. Incremental Monte Carlo._"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* $\\alpha = \\frac{1}{N(s)}$: every-visit MC\n",
    "* $\\alpha > \\frac{1}{N(s)}$: forget older data, good for non-stationary domains (when the MDP is constantly changing)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The general limitations of MC methods:\n",
    "* they require episodes to terminate at some point\n",
    "* they generally have high variance\n",
    "\n",
    "MC methods:\n",
    "* don't bootstrap and instead, samples\n",
    " * converges to true value under some assumptions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 3. Temporal Difference (TD) Learning"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> __Temporal Difference (TD) Learning__ : combines MC methods (sampling) and dynamic programming methods (bootstrapping)\n",
    "\n",
    "* the TD family of methods are model-free (much like the MC methods)\n",
    "* they bootstrap and sample (so they do what dynamic programming and MC do)\n",
    "* can be used in both episodic and infinite-horizon settings (unlike MC which can be used only in episodic settings)\n",
    "* updates for eery state, action, reward, next_state tuple"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Set $\\alpha$ <br>\n",
    "Initialize $V^{\\pi}(s) = 0, \\forall s \\in S$ <br>\n",
    "Loop <br>\n",
    "$\\quad$ Sample __tuple__ $(s_{t}, a_{t}, r_{t}, s_{t + 1})$ <br>\n",
    "$\\quad$ $V^{\\pi}(s_{t}) = V^{\\pi}(s_{t}) + \\alpha([r_{t} + \\gamma V^{\\pi}(s_{t + 1})] - V^{\\pi}(s_{t}))$ <br>\n",
    "\n",
    "_Algorithm 4. TD Learning TD(0)._"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We call it TD(0) because we take the initial reward (via sampling every-visit if $\\alpha = \\frac{1}{N(s)}$) and we bootstrap. Notice how similar it is to the dynamic programming approach and the MC approach. \n",
    "\n",
    "The __TD error__ is:\n",
    "\n",
    "$$\n",
    "\\delta_{t} = [r_{t} + \\gamma V^{\\pi}(s_{t + 1})] - V^{\\pi}(s_{t})\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 4. Comparing Approaches"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<center><img src=\"https://imgur.com/sgeZy75.jpg\" alt=\"drawing\"/></center>\n",
    "\n",
    "_Figure 2. Comparing different approaches._"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We pick different model-free policy evaluation algorithms based on:\n",
    "\n",
    "* bias/variance trade-offs\n",
    "* data efficiency\n",
    "* computational efficiency\n",
    "\n",
    "MC is:\n",
    "* unbiased\n",
    "* high variance\n",
    "* converges to true even with function approximation\n",
    "\n",
    "TD is:\n",
    "* moderate bias\n",
    "* lower variance\n",
    "* converges if tabular representation"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<center><img src=\"https://imgur.com/MaMUCeC.jpg\" alt=\"drawing\"/></center>\n",
    "\n",
    "_Figure 3. Data & Computational efficiency of model-free policy evaluation algorithms._"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 5. Resource"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If you missed the link right below the title, I'm providing the resource here again along with the course website.\n",
    "\n",
    "- [Stanford CS234](https://www.youtube.com/watch?v=FgzM3zpZ55o)\n",
    "- [Course Website](http://web.stanford.edu/class/cs234/index.html)\n",
    "\n",
    "This is a series of 15 lectures provided by Stanford.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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