1_Introduction.ipynb

{
  "nbformat": 4,
  "nbformat_minor": 0,
  "metadata": {
    "colab": {
      "name": "1_Introduction.ipynb",
      "provenance": [],
      "collapsed_sections": [],
      "include_colab_link": true
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
    },
    "language_info": {
      "name": "python"
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
      },
      "source": [
        "<a href=\"https://colab.research.google.com/github/Tikquuss/GflowNets_Tutorial/blob/main/1_Introduction.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "@author TODO"
      ],
      "metadata": {
        "id": "1v3D9FKodtfJ"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "\n",
        "\n",
        "We start by defining the type of problem we want to solve. Let consider the set $\\mathcal{X} \\subset \\mathbb{N}$, and let define on $\\mathcal{X}$ the reward function $R(x) = \\lambda^{- \\beta \\times min \\{x-n*q_n(x) \\text{, } n*(q_n(x)+1)-x \\}} \\in [0, 1]$ where $q_n(x)$ is the quotient of the Euclidean division of $x$ by $n$ ($n \\in \\mathbb{N}^* $), $\\lambda > 1$ and $\\beta > 0$  (the closer you are to a multiple of $n$, the greater the reward).  \n",
        "\n",
        "The intuition behind this reward function is the following : a number $x$ is always surrounded by two multiples of another number $n$, $n*q_n(x) \\le x < n*(q_n(x)+1)$. Among these two multiples, we choose the closest one to $x$, then we compute the distance $d = min \\{x-n*q_n(x) \\text{, } n*(q_n(x)+1)-x \\}$ between it and this mutiple. To transform the distance into a reward, we just do $\\lambda^{- \\beta \\times d}, \\ \\lambda > 1$ and $\\beta > 0$.\n",
        "\n",
        "Let $n=5$, $\\lambda = 2$ and $\\beta = 1$, then :\n",
        "$$R(x) = \\left\\{\n",
        "    \\begin{array}{ll}\n",
        "        1 & \\mbox{if } x = 5k \\\\\n",
        "        2^{-1} & \\mbox{if } x=5k+1 \\mbox{ or } x=5k+4=5(k+1)-1 \\\\\n",
        "        2^{-2} & \\mbox{if } x=5k+2 \\mbox{ or } x=5k+3=5(k+1)-2\n",
        "    \\end{array} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (k \\in \\mathbb{N})\n",
        "\\right.$$\n",
        "\n",
        "We can see the distribution $R$ in the figure below for $\\mathcal{X} = \\{0, ....,100\\}$ (*we will consider this set for the rest*).\n",
        "\n",
        "\n",
        "\n"
      ],
      "metadata": {
        "id": "DPLWBF2mXQil"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "import matplotlib.pyplot as plt\n",
        "\n",
        "def reward_function(x, a, lambda_ = 2, beta = 1, R_0 = 1e-6):\n",
        "    q = x // a \n",
        "    return lambda_**-(beta*min(x-a*q, a*(q+1)-x)) + R_0\n",
        "\n",
        "X = list(range(100+1))\n",
        "R = [reward_function(x, a=5) for x in X]\n",
        "\n",
        "figsize=(25,5)\n",
        "fig, ax1 = plt.subplots(1, 1, sharex=False, sharey=False, figsize = figsize)\n",
        "\n",
        "ax1.plot(X, R, label=\"Rewards distribution.\")\n",
        "ax1.set(xlabel='states', ylabel='rewards')\n",
        "#ax1.set_title('...')\n",
        "ax1.legend()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 351
        },
        "id": "mr-9tP1SXN8c",
        "outputId": "a9f7383e-0665-4ad0-9c98-56b68277c421"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<matplotlib.legend.Legend at 0x7f9e61b87a90>"
            ]
          },
          "metadata": {},
          "execution_count": 4
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 1800x360 with 1 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Let $Z = \\sum_{x \\in \\mathcal{X}}R(x) = 51$ (the canonical partition function) and let also define on $\\mathcal{X}$, a probability distribution $\\pi$ given by :\n",
        "\n",
        "$$\\pi(x) = \\frac{R(x)}{Z} = \\left\\{\n",
        "    \\begin{array}{ll}\n",
        "        \\frac{1}{51} & \\mbox{if } x = 5k \\\\\n",
        "        \\frac{1}{102} & \\mbox{if } x=5k+1 \\mbox{ or } x=5k+4=5(k+1)-1 \\\\\n",
        "        \\frac{1}{204} & \\mbox{if } x=5k+2 \\mbox{ or } x=5k+3=5(k+1)-2\n",
        "    \\end{array} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (k \\in \\mathbb{N})\n",
        "\\right.$$\n",
        "\n",
        "I call $Z$ the canonical partition function in reference to statistical physics, because our reward can be written $R(x) = e^{-{\\frac{E(x)}{k_B \\ T}}}$, where $E(x)$ is the Hamiltonian/energy of the microstate $x$ (of the physical system), $T$ the temperature and $k_B$ the Boltzmann constant. In this case, $\\pi(x) = \\frac{R(x)}{Z}$ becomes the Boltzmann factor at $x$ (probability for the system to be present in the microstate $x$).\n",
        "\n",
        "We would like to have an algorithm that, just from the vocabulary of $\\mathcal{X}$, ie $\\{0, 1, ..., 9\\}$, generates numbers from $\\mathcal{X}$ such that the probability of an $x \\in \\mathcal{X}$ being generated is equal to $\\pi(x)$. In other words, if the algorithm generates a list of $N$ elements, we would like to have a number $N*\\pi(x)$ of each $x \\in \\mathcal{X}$ in the list (ie a proportion $\\pi(x)$ of each $x$ in the list). *__This is called sampling from a probability distribution__*, and it is different from solving $arg\\_max_{x} R(x)$ directly. The high reward or high probability (and therefore low energy) states will therefore be more generated than low reward (high energy) states."
      ],
      "metadata": {
        "id": "wDxnFug_Khiu"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "\n",
        "\n",
        "---\n",
        "If $\\pi$ is explicitly known (by its mass function or repartition function), we can use methods like Inverse Transform Sampling or Acceptance-Rejection sampling (presented below). But, in many problems, $Z$, the denominator of $\\pi$ is (too) hard to compute. More precisely, we must be able to sample from $\\pi$ just by knowing $R$ (or a function of $R$). This situation arises in\n",
        "several contexts:\n",
        "* In Bayesian inference, we have $\\pi(\\theta | x) = \\frac{\\pi(\\theta)p(x | \\theta)}{p(x)}$ where $\\pi(\\theta)p(x | \\theta)$ is easy to compute but $p(x)= \\int_{\\theta}{} \\pi(\\theta)p(x | \\theta) d\\theta$\n",
        "can be very difficult or impossible to compute. So we need to be able to sample from the posterior $\\pi(\\theta | x)$ just by knowing $\\pi(\\theta)p(x | \\theta)$, ie the prior $\\pi(\\theta)$ and the likelihood $p(x | \\theta)$.\n",
        "* In models from statistical physics (e.g. the Ising model), we only know $R(x) = e^{−\\frac{E(x)}{k_B \\ T}}$, where $E$ is the Hamiltonian, $k$ is the Boltzmann’s constant and $T$ the temperature. $Z \\propto \\int_{\\mathcal{X}} R(x)dx$ is often difficult to compute for a system with many possible microstates (often infinite) or many parameters.\n",
        "\n",
        "There are several methods to solve this problem. Among them we have Markov Chain Monte Carlo (MCMC) methods and GflowNets.\n",
        "\n",
        "MCMC methods try to find a (reversible) Markov chain of transition matrix $P$ such that $\\pi$ is its stationary law, i.e. $\\pi P = \\pi$ (note that this constraint remains valid if we multiply the two sides by a constant, $Z \\pi P=Z \\pi=R$ , and this is where the MCMC methods take their advantage). If such a Markov chain is found, it will be enough to simulate it up to some step $n \\rightarrow \\infty$ (ie simulate a random walk of infinite length on the chain, $\\pi_{n+1}= \\pi_n P $ with $n \\rightarrow \\infty$) and take the last elements obtained  (the first steps in generaly call the **burn-in** phase). Given the difficulty of finding $\\pi$ and/or $P$ in practice, MCMC methods find alternatives to simulate a random walk without explicitly having $\\pi$, but just $R$. In this family of algorithm we have Gibbs sampling, Metropolis-Hasting, Metropolis-adjusted Langevin (MCMC with langevin dynamics)... \n",
        "\n",
        "These methods are generally too expensive and generally do only local exploration (if $\\pi$ has several modes separated by deserts of probabilities, these algorithms can get stuck in the same regions during the random walk). \n",
        "\n",
        "On the other hand, the space $\\mathcal{X}$ on which the reward function is defined is very complex in many problems (and not a simple list of numbers as in the example above), for example the set of molecules, the set of sentences, the set of images ... And sampling requires very complex algorithms or models to build objects to be evaluated. In this kind of case, an algorithm that tries to successively build the objects and test them (like MCMC methods) would be very expensive. GflowNets reduce this cost because it only has to do the *build-test* process during training. Once the training is done, GflowNets can be used to build objects that follow its training reward or distribution, unlike other methods where there is no training : each time we need an object, we have to do a search (usually expensive) until we find the right one (rejection sampling) or (theoretically) up to infinity (MCMC).\n",
        "\n",
        "\n",
        "GFlowNets are generative models that learn a stochastic policy that iteratively constructs the sampled object through a sequence of simpler steps, such that the probability of generating an object is proportional to its reward.\n",
        "\n",
        "GFlowNets are interesting when the problem to solve has these properties: \n",
        "\n",
        "* Objects can be constructed sequentially\n",
        "\n",
        "A molecule for example can be built by combining several atoms, a sentence can be constructed by adding each time a word after the sentence already constructed.\n",
        "\n",
        "An example is shown in the figure below for the construction of a $3 \\times 3$ binary image. We start with an empty image (initial state), then at each step, we have $9-n$ ways to choose the cell to color ($n$ represents the number of cell already colored), and 2 ways (black or white) to choose the color to assign to the chosen cell. The process continues until there are no more empty cells (final state).\n",
        "\n"
      ],
      "metadata": {
        "id": "oOPl98DfRGBF"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "![1.png](data:image/png;base64,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)\n",
        "\n",
        "Figure from [4]"
      ],
      "metadata": {
        "id": "238clRTb9zUS"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "\n",
        "\n",
        "\n",
        "\n",
        "* The same object can be built (and destroyed, as GflowNets learns to build and deconstruct at the same time) in several different ways. \n",
        "\n",
        "In this case, we are dealing with a DAG (Directed Acyclic Graph). For example, on figure above, there are several paths that lead from the initial state to the final state in red.\n",
        "\n",
        "The DAG can in many cases be reduced to a simple tree. For example, considering that the final sentence `[BOS] a b c [EOS]` (`[BOS]` for *Beginning Of Sentence* and `[EOS]` for *End Of Sentence*) has been constructed in an autoregressive way, it is obvious that the sentence that led to this state is `[BOS] a b c`, and the one that led to `[BOS] a b c` is `[BOS] a b`, ..., `[BOS]` (initial state). \n",
        "\n",
        "On the other hand, if the sentences are not constructed in an autoregressive way (for example if at each step one must first choose a position in the sentence before choosing the word to be placed at this position), one has more to do with a tree: `[BOS] a b c [EOS]` can come from `[BOS] a b c` or `[BOS] a b [EOS]` ...\n",
        "\n",
        "* The reward function has several modes (local maximums, even though this term is not very appropriate in our case).\n",
        "\n",
        "As an example, consider that you are looking for all multiples of a number $n$ of length less than $l > 2$. I assume here that the numbers are constructed autoregressively starting from an initial state (like `[BOS]`) and adding each time a digit between 0 and 9 at the end of the already constructed number. The process continues until a special symbol is generated (like `[EOS]`) or until the number is of length $l$. If we have instead a neural network that directly returns a number (like the generator of a GAN), then we cannot speak of a notion of flow. As a reward function, we can decide to give a score of $1$ to the multiple of $n$, and a lower and lower score when the considered number is far from a multiple of $n$. A suitable reward function in this case is the reward function introduced at the beginning of this notebook, $R(x) = \\lambda^ {- \\beta \\times min \\{x-n*q_n(x) \\text{, } n*(q_n(x)+1)-x \\}}$ where $q_n(x)$ is the quotient of the Euclidean division of $x$ by $n$, $\\lambda > 1$ and $\\beta > 0$ : the larger $\\lambda$ or/and $\\beta$ or/and $n$ are, the more difficult the problem becomes because of the probability desert that is created between the modes.\n",
        "\n",
        "* The oracle (the real reward function in other words) is difficult to call. \n",
        "\n",
        "In the case of the multiples of $n$ above, the analytical form of the reward function is known (moreover, with a little mathematics one can find all $x$ such that $R(x) = r, \\ r \\in \\mathbb{R}^+$).\n",
        "\n",
        "But let's imagine the case where we have to find a molecule that kills the given virus, and that has other properties like its time of action (if the molecule kills the virus very slowly a sick person can take the drug, but die before it does its effects), that has few side effects, that is easy to manufacture... All these properties can be translated into a reward function, except that here, being given a new molecule, the specialists have to go to the laboratory, do clinical trials (which take days or weeks, which are financially expensive ...).\n",
        "\n",
        "An alternative in this case (and that GFlowNets knows very well how to exploit), is to train on the molecules already available (but which are not as satisfactory as we wish), a model able to estimate the reward of new molecules. In the case of covid-19, a dataset can be made of : Moderna, Pfizer-BioNTech, AstraZeneca, Janssen (Johnson & Johnson) ... This reward function (less expensive), can be used to evaluate molecules very quickly.  \n"
      ],
      "metadata": {
        "id": "fLfcDMya97Gw"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "---\n",
        "\n",
        "Note that Reinforcement Learning algorithms can also be used to solve this type of problems, but the maximization of expected return in this algorithms is generally achieved by putting all the probability mass of the state-action policy on the highest-return sequence of actions. GflowNets solves the problem by turning $R$ into a generative policy which samples with a probability proportional to $R$."
      ],
      "metadata": {
        "id": "ZHZxWo3E-p5G"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "---\n",
        "\n",
        "1) **MCMC : Metropolis-Hasting (MH) algorithm**\n",
        "\n",
        "Suppose we want to sample from a distribution $\\pi(x) = f(x) / Z_{\\pi}$, $f(x) > 0$ and $Z_{\\pi}$ unknow.\n",
        "\n",
        "We define a Markov transition matrix $Q$ on $\\mathcal{X}$ : $0 \\le Q \\le 1$ and $\\sum_{x' \\in \\mathcal{X}} Q(x'|x) = 1 \\ \\forall x \\in \\mathcal{X}$.\n",
        "\n",
        "Let $x_0 \\in \\mathcal{X}$ the initial state. We then perform the following two steps repeatedly:\n",
        "$$* \\ x = x_t, \\ generate \\ x' ∼ Q(.|x)$$\n",
        "$$* \\ x_{t+1} = \\left\\{\n",
        "    \\begin{array}{ll}\n",
        "        x' & \\mbox{with probability } \\alpha(x'|x) = min \\bigg\\{ \\frac{f(x') \\ . \\ Q(x|x')}{f(x) \\ . \\ Q(x' | x)}, 1 \\bigg\\} \\\\\n",
        "        x & \\mbox{otherwise.}\n",
        "    \\end{array}\n",
        "\\right.$$\n",
        "\n",
        "Note that $\\alpha$ depends only on $f$, not $\\pi$. So we don't need $\\pi$ in the algorithm, only $f = \\pi Z_{\\pi}$."
      ],
      "metadata": {
        "id": "1IQKYvA1_wBY"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Let $P(x'|x) = \\alpha(x'|x) Q(x'|x)$. \n",
        "\n",
        "We have : \n",
        "\n",
        "$$\\pi(x) P(x'|x)  = \\pi(x) \\ min \\bigg\\{ \\frac{Z_{\\pi} \\pi(x') \\ . \\ Q(x|x')}{Z_{\\pi} \\pi(x) \\ . \\ Q(x' | x)}, 1 \\bigg\\} Q(x'|x) =  min \\bigg\\{ \\pi(x') Q(x|x') , \\pi(x) Q(x'|x) \\bigg\\} = \\pi(x') \\ min \\bigg\\{1,  \\frac{Z_{\\pi} \\pi(x) \\ . \\ Q(x'|x)}{Z_{\\pi} \\pi(x') \\ . \\ Q(x | x')}\\bigg\\} Q(x|x') = \\pi(x') \\alpha(x|x') Q(x|x') = \\pi(x') P(x|x')$$\n",
        "\n",
        "ie $\\forall \\ (x, x') \\in \\mathcal{X}^2, \\pi(x) P(x'|x) = \\pi(x') P(x|x')$ (detailed balance equation), so the  markov chain of transition matrix $P$ is reversible with stationary distribution $\\pi$, since then we have $ \\forall \\ x \\in \\mathcal{X}, \\ \\sum_{x'} \\pi(x')P(x|x') = \\sum_{x'} \\pi(x)P(x'|x) = \\pi(x) \\sum_{x'} P(x'|x) = \\pi(x)$, ie $\\pi P = \\pi$. This condition, although sufficient, is not necessary for $\\pi$ to be a stationary distribution. It is also necessary that the Markov Chain is ergotic (i.e. that there exists $n$ such that $P^n > 0$) to guarantee that $\\pi$ is stationary (in this case, $\\pi$ is the unique stationary distribution of the Markov Chain).\n",
        "\n",
        "Consider our problem above of generating multiples of 27. We can consider an isotropic Gaussian distribution as the proposal distribution, $Q(.|x) ∼ \\mathcal{N} (x, \\sigma)$. In this case, the algorithm will be very sensitive to the value of $\\sigma$ chosen. For example, if $\\sigma$ is too small, the algorithm may turn indefinitely around its starting point (because of the desert that exists between the modes of $R$).\n",
        "\n"
      ],
      "metadata": {
        "id": "6TxNSM3WQ_q2"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "X = list(range(100+1))\n",
        "R = [reward_function(x, a=27, lambda_ = 5) for x in X]\n",
        "\n",
        "figsize=(25,5)\n",
        "fig, ax1 = plt.subplots(1, 1, sharex=False, sharey=False, figsize = figsize)\n",
        "\n",
        "ax1.plot(X, R, label=\"Rewards distribution.\")\n",
        "ax1.set(xlabel='states', ylabel='rewards')\n",
        "ax1.legend()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 351
        },
        "id": "f5XQCNOjp_kO",
        "outputId": "9b66e8c2-f759-410f-a01e-61f5f424f8b7"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<matplotlib.legend.Legend at 0x7f9e61b87990>"
            ]
          },
          "metadata": {},
          "execution_count": 3
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 1800x360 with 1 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "**2. MCMC :  Gibbs sampling and Metropolis-adjusted Langevin**\n",
        "\n",
        "**TODO**\n",
        "\n",
        "See [7] for the moment. I will describe the method and prove that it works later."
      ],
      "metadata": {
        "id": "nboO40HksFO7"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**3. Inverse Transform Sampling, Acceptance-Rejection Method and Important Sampling (wee need $\\pi$)**"
      ],
      "metadata": {
        "id": "eDVfFMjS2gRX"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "\n",
        "\n",
        "If we have a distribution of cumulative function $F : \\mathbb{R} \\to [0, 1]$ (assuming continuous) and $U ∼ \\mathcal{U}([0, 1])$, then for all $x \\in \\mathbb{R}$ : $Pr\\{F^{-1}(U) \\le x\\} = Pr\\{U \\le F(x)\\} = F(x) \\mbox{ since } F(x) \\in [0, 1]$ $\\Rightarrow$ The random variable $F^{-1}(U)$ is distributed as $F$.\n",
        "\n",
        "So to generate a number that follows the probability distribution of cumulative function $F$, we can generate $u ∼ \\mathcal{U}([0, 1])$ (we suppose that we have a random number generator that can do this), then return $F^{-1}(u)$ : this is how Inverse Transform Sampling works. If $F$ is not continuous, the proof must be re-write differently, but the method remains valid."
      ],
      "metadata": {
        "id": "EcbwkXCT2etJ"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "\n",
        "\n",
        "\n",
        "\n",
        "\n",
        "In rejection sampling, another density $q$ is considered from which we can sample directly and easily (e.g., inverse transform\n",
        "method or whatever) under the restriction that $\\pi = p \\le Mq$ where $M > 0$ is an appropriate bound on $\\frac{p}{q}$, $sup_{x} \\frac{p(x)}{q(x)} \\le M$. The easiest way to satisfy this assumption is to make sure that $q$ has heavier tails than $p$, and set $M = sup_{x \\in support(p)} \\frac{p(x)}{q(x)} \\ (\\mbox{assume } < \\infty )$. We must ensure that the support of  $p$ is a subset of the support of $q$.\n",
        "\n",
        "Then we can sample from $q$ directly and then reject the samples in a strategic way to make the resulting non-rejected samples look like they came from $p$."
      ],
      "metadata": {
        "id": "wdHDtQXKqB1M"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "![3.png](data:image/png;base64,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)"
      ],
      "metadata": {
        "id": "KNDVHOIvp0_P"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Figure from [8]"
      ],
      "metadata": {
        "id": "d-aGDfWDrUTZ"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Algorithm :\n",
        "\n",
        "i. Generate $u  ∼ \\mathcal{U}([0, 1])$\n",
        "\n",
        "ii. Generate a candidate $x ∼ q$ from the candidate density\n",
        "\n",
        "iii. $\\left\\{\n",
        "    \\begin{array}{ll}\n",
        "        & \\mbox{Accept the candidate } x \\mbox{ if } u < \\frac{p(x)}{M \\ q(x)} \\\\\n",
        "        & \\mbox{Otherwise, reject }  x \\mbox{ and go back to the beginning (i)}\n",
        "    \\end{array}\n",
        "\\right.$\n",
        "\n"
      ],
      "metadata": {
        "id": "r9_l3Cb8p9gB"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Let $X$ be the random varibal with density function $q$. We want to show that the conditional distribution of $X$ given that $u ∼ \\mathcal{U}([0, 1]) < \\frac{p(X)}{M \\ q(X)}$ is $p$, ie $$Pr\\bigg\\{ X=x \\ | \\ u ∼ \\mathcal{U}([0, 1]) < \\frac{p(X)}{M \\ q(X)} \\bigg\\} = p(x)$$\n",
        "\n",
        "We recall that we need to choose the alternative probability distribution $q$ from which we already have an efficient algorithm for generating from (e.g., inverse transform method or whatever), but also such that the function $q(x)$ is close to $p(x)$. In particular, we assume that the ratio $\\frac{p(x)}{q(x)}$ is bounded by a constant $M > 0$, $sup_{x} \\frac{p(x)}{q(x)} \\le M$ (and in practice we would want $M$ as close to 1 as possible).\n",
        "\n",
        "The number of times $N$ that steps *i* and *ii* need to be called (the number of iterations needed to successfully generate a candidate) is itself a random variable and has a geometric distribution with success probability $$s = Pr\\bigg\\{ u ∼ \\mathcal{U}([0, 1]) < \\frac{p(X)}{M \\ q(X)} \\bigg\\} = \\int_{} F_{\\mathcal{U}([0, 1]) \\ | \\ X = x} \\bigg( \\frac{p(X)}{M \\ q(X)} \\bigg) \\ f_X\\big(x\\big) dx = \\int_{} \\frac{p(x)}{M \\ q(x)} q(x) dx \\mbox{ since } 0 < \\frac{p(x)}{M \\ q(x)} \\le 1$$\n",
        "\n",
        "$$\\Rightarrow s = \\int_{} \\frac{p(x)}{M} dx = \\frac{1}{M} \\mbox{ (p is a density function, hence by definition integrates to 1)}$$ \n",
        "\n",
        "$N ∼ geometric(s) \\Rightarrow Pr \\{N = n\\} = (1 − s)^{n−1} s, \\ n \\ge 1$. Thus on average the number of iterations required is given by $\\mathbb{E}(N) = \\frac{1}{s} = M$, the bounding constant, and we can now indeed see that it is desirable to choose our alternative density $q$ so as to minimize this constant $M=sup_{x} \\frac{p(x)}{q(x)}$. Of course the optimal function would be $q(x) = p(x)$ which is not what we have in mind since the whole point is to choose a different (easy to simulate) alternative from $p$. In short, it is a bit of an art to find an appropriate $q$. In any case, we summarize with : **the expected number of iterations of the algorithm required until an X is successfully generated is exactly the bounding constant $M=sup_{x} \\frac{p(x)}{q(x)}$**\n",
        "\n",
        "Steps *i* ($u ∼ \\mathcal{U}([0, 1])$) and *ii* ($x ∼ q$) being independent, we have :\n",
        "\n",
        "$$Pr\\bigg\\{ X=x \\ | \\ u ∼ \\mathcal{U}([0, 1]) < \\frac{p(X)}{M \\ q(X)} \\bigg\\} = \\frac{Pr\\bigg\\{ X=x, u ∼ \\mathcal{U}([0, 1]) < \\frac{p(x)}{M \\ q(x)} \\bigg\\}}{Pr\\bigg\\{ u ∼ \\mathcal{U}([0, 1]) < \\frac{p(X)}{M \\ q(X)} \\bigg\\}} = M . f_X\\big(x\\big) .  Pr\\bigg\\{u ∼ \\mathcal{U}([0, 1]) < \\frac{p(x)}{M \\ q(x)} \\bigg\\} = M . q(x) . \\frac{p(x)}{M \\ q(x)} = p(x) \\ ▯ $$ \n",
        "\n",
        "\n",
        "\n",
        "\n"
      ],
      "metadata": {
        "id": "XeH_6o3LLFv6"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "\n",
        "\n",
        "---\n",
        "\n",
        "4) GFlowNets\n",
        "\n"
      ],
      "metadata": {
        "id": "f80VsPv50wcZ"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Before giving more details about GflowNets, let's first recall some concepts.\n",
        "\n",
        "A directed graph is a tuple $(\\mathcal{S}, \\mathcal{A})$, where $\\mathcal{S}$  is a set of states, and $\\mathcal{A}$ a subset of $\\mathcal{S} \\times \\mathcal{S}$ representing directed edges. Elements of $\\mathcal{A}$ are denoted $s{\\rightarrow}s'$ and called *edges* or *transitions*. \n",
        "\n",
        "For each element of $s \\in \\mathcal{S}$, we define two sets :  the set of its parents $Par(s) = \\{ s' \\ | \\ s'{\\rightarrow}s \\in \\mathcal{A} \\}$ and the set of its children $Child(s) = \\{ s' \\ | \\ s{\\rightarrow}s' \\in \\mathcal{A} \\}$.\n",
        "\n",
        "A **trajectory** in such a graph is a sequence $\\tau=(s_1, \\dots, s_n)$ of elements of $\\mathcal{S}$ such that every transition $s_t{\\rightarrow}s_{t+1} \\in \\mathcal{A}$. \n",
        "* $s \\in \\tau$ means that $s$ is in the trajectory $\\tau$, i.e., $\\exists \\text{ } t \\in \\{1,\\dots, n\\} \\ / \\ s_t=s$ \n",
        "* $s{\\rightarrow}s' \\in \\tau$ means that $\\exists t \\in \\{1, \\dots, n\\} \\ / \\ s_t=s,s_{t+1}=s'$.\n",
        "\n",
        "A **directed acyclic graph** (DAG) is a directed graph in which there is no trajectory $\\tau=(s_1, \\dots, s_n)$ satisfying $s_n = s_1$, besides trajectories composed of one state only.\n",
        "\n",
        "For the DAGs considered here, we have two special states :\n",
        "* the *source/initial state* $s_0$, $Par(s_0) = \\emptyset$ \n",
        "* the *sink/final state* $s_f$, $Child(s_f) = \\emptyset$ \n",
        "\n",
        "Depending on the problem we can have many of these states. In this case, a special initial (resp. final) state can be created which is connected to all initial states (resp. to which all final states are connected). \n",
        "\n",
        "A trajectory that starts at $s_0$ and ends at $s_f$ is said to be complete. Hereafter, the complete trajectories $\\tau=(s_0, s_1, \\dots, s_n,s_f)$ will be noted simply $\\tau=(s_0, s_1, \\dots, s_n)$.\n"
      ],
      "metadata": {
        "id": "Wsl1qHL1URSA"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Let $\\mathcal{X}$ be the set of final states, and let define on $\\mathcal{X}$ a positive function $R$. The problem we want to solve is the following: how to define on $(\\mathcal{S}, \\mathcal{A})$ a generative model for which the probability $\\pi(x)$ of generating an element $x \\in \\mathcal{X}$ is proportional to $R(x)$, ie $\\pi(x)=\\frac{R(x)}{\\sum_{x \\in \\mathcal{X}}R(x)}$ ?\n",
        "\n",
        "To achieve this, GflowNets define on the set of complete trajectories $\\mathcal{T} = \\{(s_0, \\dots, s_n) | s_n \\in \\mathcal{X} \\}$ a positive function $F$, called flow :\n",
        "* which must respect the following condition, called reward matching condition: $R(x) = \\sum_{\\tau=(s_0, \\dots, s_n) | s_n = x} F(\\tau)$\n",
        "* which must be Markovian, i.e. there must exist distributions $P_F(.|s)$ over the children $Child(s)$ of every non-terminal state $s$, and a constant $Z$, such that for any complete trajectory $\\tau=(s_0, \\dots, s_n)$ we have $P(\\tau) = P_F(\\tau) = P_F(s_n|s_{n-1}) P_F(s_{n-1}|s_{n-2}) \\dots P_F(s_1|s_0) = \\frac{F(\\tau)}{Z}$. \n",
        "\n",
        "$Z$ is called the **total flow**, ie the sum of the flows of all the complete trajectories, $Z = \\sum_{\\tau \\in \\cal T} F(\\tau)$. Note the equivalence between $Z$ and the partition function of a physical system as defined above : as in statistical physics, $Z$ captures how the rewards/probabilities are distributed among the different individual final states (or otherwise among the different complete trajectories).\n",
        "\n",
        "$P_F(s_{t+1}|s_{t})$ is called a **forward policy**, and the authors of GflowNets have shown that by using it to construct a trajectory $\\tau=(s_0, \\dots, s_n)$, the probability that $s_n=x$ is proportional to $R(x)$.\n",
        "\n",
        "We also have a **backwards transition probability** (given a terminal state $x \\in \\mathcal{X}$, $P_B$ can be used to sample a trajectory leading from $s_0$ to $x$). There must exist distributions $P_B(.|s)$ over the parent $Par(s)$ of every non-initial state $s$ such that for any terminal state $x$ we have : $$P(\\tau=(s_0, \\dots, s_n ) \\ | \\ s_n = x) = P_B(s_0|s_1) \\dots P_B(s_{n-2}|s_{n-1}) P_B(s_{n-1}|s_n=x)$$\n",
        "\n",
        "The **flow through a state** (or **state flow**) is the sum of the flows of the complete trajectories passing through that state, $F(s) = \\sum_{\\tau \\in {\\cal T}} 1_{s \\in \\tau} F(\\tau)$.\n",
        "\n",
        "The flow $F(s{\\rightarrow}s')$ through an edge $s{\\rightarrow}s'$ is called an **edge flow**. \n",
        "\n",
        "We then have :\n",
        "* $F(s_0) = F(s_f) = \\sum_{\\tau\\in \\cal T} F(\\tau) = Z$ because $\\forall \\tau \\in \\mathcal{T}$, $s_0 \\in \\tau$ and $s_f \\in \\tau$ \n",
        "* $P_F(s'|s) =\\frac{F(s{\\rightarrow}s')}{F(s)}$ (see the definition of $P_F(\\tau)$ above)\n",
        "* $P_B(s|s') = \\frac{F(s{\\rightarrow}s')}{F(s')}$\n",
        "* In-flow for non-initial states : $F_{in}(s) = \\sum_{s'' \\in Par(s)} F(s''{\\rightarrow}s)$ \n",
        "* Out-flow for non-terminal states : $F_{out}(s) = \\sum_{s' \\in Child(s)} F(s{\\rightarrow}s')$ \n",
        "\n",
        "The objective in the following will be to generate the complete trajectories $\\tau=(s_0, \\dots, s_n = x)$ such that $F(s_t) = F_{in}(s_t) = F_{out}(s_t) \\ \\forall \\ t \\in {1, ...,n}$ and $F_{out}(s_0) = Z$, $F_{out}(s_n) = R(x)$.\n",
        "\n",
        "If this objective is satisfied, we will have $\\pi(x) =\\frac{R(x)}{Z} \\ \\forall \\ x \\in \\ \\mathcal{X}$"
      ],
      "metadata": {
        "id": "Oojm6EhWy3kR"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "---\n",
        "**Flow matching objective**\n",
        "\n",
        "A model $F_{\\theta}(s, s')$ with learnable parameters $\\theta$ approximates the edge flows $F(s{\\rightarrow}s')$. The corresponding forward policy is given by $P_F(s'|s, \\theta) = \\frac{F_{\\theta}(s, s')}{F(s)} = \\frac{F_{\\theta}(s, s')}{ \\sum_{s'' \\in Child(s)} F_{\\theta}(s{\\rightarrow}s'') }$. \n",
        "\n",
        "The parameters are trained to minimize the error in the flow matching constraint for all non-initial nodes $s$ :\n",
        "\n",
        "$$ \n",
        "\\mathcal{L}_{FM} \\big(s\\big) = \\left\\{\n",
        "    \\begin{array}{ll}\n",
        "        \\bigg( log \\frac{ \\sum_{s'' \\rightarrow s} F_{\\theta}(s'', s)}{R(s)} \\bigg)^2 & \\mbox{if } s \\mbox{ terminal} \\\\\n",
        "        \\bigg( log \\frac{ \\sum_{s'' \\rightarrow s} F_{\\theta}(s'', s)}{ \\sum_{s \\rightarrow s'} F_{\\theta}(s, s')} \\bigg)^2 & \\mbox{else.}\n",
        "    \\end{array}\n",
        "\\right. \n",
        "\\\\= \\left\\{\n",
        "    \\begin{array}{ll}\n",
        "        \\bigg[ log \\bigg( \\sum_{s'' \\rightarrow s} F_{\\theta}(s'', s) \\bigg) - log \\bigg( R(s) \\bigg) \\bigg]^2 & \\mbox{if } s \\mbox{ terminal} \\\\\n",
        "        \\bigg[ log \\bigg( \\sum_{s'' \\rightarrow s} F_{\\theta}(s'', s) \\bigg) - log \\bigg( \\sum_{s \\rightarrow s'} F_{\\theta}(s, s') \\bigg) \\bigg]^2 & \\mbox{else.}\n",
        "    \\end{array}\n",
        "\\right. \n",
        "$$\n",
        "\n",
        "From a computational point of view, this loss is difficult to compute when $|Par(s)|$ is large, for many $s$, because it requires, being at a given non-initial state $s$, to enumerate all its parents, to evaluate the model $F_{\\theta}$ on each of the parents, and to consider only the flows leading to $s$. \n",
        "\n",
        "This objective is optimized for nonterminal states $s$ and terminal states $x$ from trajectories sampled from a training policy $\\pi_{\\theta}$, a tempered (higher temperature) version of $P_F(.|s, \\theta)$ which also helps exploration during training.\n",
        "The parameters are updated with stochastic gradient $\\mathbb{E}_{\\tau=(s_0, \\dots, s_n) \\sim \\pi_{\\theta}} \\nabla_{\\theta} \\bigg[ \\sum_{t=1}^n \\mathcal{L}_{FM} \\big(s_t\\big) \\bigg]$\n",
        "\n",
        "One issue from a learning point of view is that the flow will be very large for nodes near the root (early in the trajectory) and tiny for nodes near the leaves (late in the trajectory). In high-dimensional spaces the $F(s, .)$ and $F(s) = \\sum_{.} F(s, .)$ for early states will be exponentially larger than for later states. Since we want $F(s, .)$ to be the output of a neural network, this would lead to serious numerical issues.\n",
        "\n",
        "To avoid this problem, [1] define the flow matching objective on a log-scale, where they match not the incoming and outgoing flows but their logarithms, and they train the predictor to estimate $F_{\\theta}^{log}(s, .) = logF(s, .)$, and exponentiate-sum-log the $F_{\\theta}^{log}$ predictions to compute the loss, yielding the square of a difference of logs:\n",
        "$$\n",
        "\\mathcal{L}_{FM} \\big(s\\big) = \\left\\{\n",
        "    \\begin{array}{ll}\n",
        "        \\bigg[ log \\bigg( \\epsilon + \\sum_{s'' \\rightarrow s} exp \\text{ } F_{\\theta}^{log}(s'', s) \\bigg) - log \\bigg( \\epsilon + R(s) \\bigg) \\bigg]^2 & \\mbox{if } s \\mbox{ terminal} \\\\\n",
        "        \\bigg[ log \\bigg( \\epsilon + \\sum_{s'' \\rightarrow s} exp \\text{ } F_{\\theta} ^{log}(s'', s) \\bigg) - log \\bigg( \\epsilon +  \\sum_{s \\rightarrow s'} exp \\text{ } F_{\\theta}^{log}(s, s') \\bigg) \\bigg]^2 & \\mbox{else.}\n",
        "    \\end{array}\n",
        "\\right. \n",
        "$$\n",
        "\n",
        "The hyper-parameter $\\epsilon$ trades-off how much pressure we put on matching large versus small flows. We can set $\\epsilon = min_{x \\in \\mathcal{X}} R(x)$.\n",
        "\n",
        "We can also use this **Flow Matching** loss [2] :\n",
        "\n",
        "$\\mathcal{L}_{FM} \\big(\\tau = (s_0, ..., s_n = x) \\big) = \\bigg( log \\bigg(  \\frac{R(x)}{F_{\\theta}(x)} \\bigg)  \\bigg)^2$"
      ],
      "metadata": {
        "id": "gmamNNWBcvHM"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "---\n",
        "**Detailed balance objective**\n",
        "\n",
        "$P_F(s'|s) =\\frac{F(s{\\rightarrow}s')}{F(s)}$ and $P_B(s|s') = \\frac{F(s{\\rightarrow}s')}{F(s')}$ $\\Longrightarrow$ $F(s)P_F(s'|s) = F(s')P_B(s|s')$\n",
        "\n",
        "This is called the detailed balance constraint. It is a bit similar to the reversibility condition of Markov chains.\n",
        "\n",
        "A neural network model with parameters $\\theta$ has input $s$ and three kinds of outputs: an estimated state flow $F_{\\theta}(s)$, an estimated distribution over children $P_B(.|s, \\theta)$, and an estimated distribution over parents $P_F(.|s, \\theta)$. The error in the detailed balance constraint is optimized on actions $s{\\rightarrow}s'$ between nodes seen along trajectories.\n",
        "\n",
        "$$ \n",
        "\\mathcal{L}_{DB} \\big(s, s' \\big) = \\left\\{\n",
        "    \\begin{array}{ll}\n",
        "        \\bigg( log \\frac{F_{\\theta}(s) P_{F} (s' | s; \\theta)}{R(s') P_{B} (s | s'; \\theta) } \\bigg)^2 & \\mbox{if } s' \\mbox{ terminal} \\\\\n",
        "        \\bigg( log \\frac{F_{\\theta}(s) P_{F} (s' | s; \\theta) }{F_{\\theta}(s') P_{B} (s | s'; \\theta) } \\bigg)^2 & \\mbox{else.}\n",
        "    \\end{array}\n",
        "\\right. \n",
        "\\\\= \\left\\{\n",
        "    \\begin{array}{ll}\n",
        "        \\bigg[ log(F_{\\theta} (s)) - log\\big( P_{B} (s' | s; \\theta)\\big) + log\\big(P_{F} (s | s'; \\theta)\\big) - log(R(s'))  \\bigg]^2 & \\mbox{if } s' \\mbox{ terminal} \\\\\n",
        "        \\bigg[ log(F_{\\theta} (s)) - log(F_{\\theta} (s')) - log\\big( P_{B} (s' | s; \\theta)\\big) + log\\big(P_{F} (s | s'; \\theta)\\big) \\bigg]^2 & \\mbox{else.}\n",
        "    \\end{array}\n",
        "\\right. \n",
        "$$\n",
        "\n",
        "Similarly to flow matching, the parameters are updated with stochastic gradient\n",
        "$\\mathbb{E}_{\\tau=(s_0, \\dots, s_n) \\sim \\pi_{\\theta}} \\nabla_{\\theta} \\bigg[ \\sum_{t=1}^n \\mathcal{L}_{DB} \\big(s_{t-1}, s_t\\big) \\bigg]$"
      ],
      "metadata": {
        "id": "5Tn7bGwnJ0Av"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "---\n",
        "**Trajectory Balance objective**\n",
        "\n",
        "For any terminal state $x$ we have :\n",
        "\n",
        "$$\\prod_{t=1}^{n} P_B(s_{t-1}|s_t) = P(\\tau=(s_0, \\dots, s_n ) \\ | \\ s_n = x)  = \\frac{P(\\tau=(s_0, \\dots, s_n=x))}{P(s_n=x)} = \\frac{\\prod_{t=1}^{n} P_F(s_t|s_{t-1})}{\\frac{F(x)}{Z}}$$\n",
        "\n",
        "This gives the trajectory balance constraint for any complete trajectory $\\tau = (s_0, ..., s_n = x)$ : $$Z \\prod_{t=1}^{n} P_F(s_t|s_{t-1}) = F(x) \\prod_{t=1}^{n} P_B(s_{t-1}|s_t), \\ Z = F(s_0), \\ F(x) = R(x)$$\n",
        "\n",
        "A model with parameters $\\theta$ outputs estimated forward policy $P_F(.|s, \\theta)$ and backward policy $P_B(.|s, \\theta)$ for states $s$ (just as for detailed balance above), as well as a global scalar $Z_{\\theta}$ estimating $F(s_0)$.\n",
        "\n",
        "For a trajectory $\\tau = (s_0, ..., s_n = x)$, [3] defines the trajectory loss ($F(x)$ is replaced by $R(x)$ because we want the two quantities to be equal) :\n",
        "\n",
        "$$ \n",
        "\\mathcal{L}_{TB} \\big(\\tau = (s_0, ..., s_n = x) \\big) = \\bigg( log \\frac{Z_{\\theta} \\prod_{t=1}^{n} P_{F} (s_t | s_{t-1}; \\theta) }{R(x) \\prod_{t=1}^{n} P_{B} (s_{t-1} | s_t; \\theta) } \\bigg)^2\n",
        "\\\\= \\bigg[ log(Z_{\\theta}) + \\sum_{t=1}^{n} \\bigg( - log\\big( P_{B} (s_{t-1} | s_t; \\theta)\\big) + log\\big(P_{F} (s_t | s_{t-1}; \\theta)\\big) \\bigg) - log \\big(R(x)\\big) \\bigg]^2\n",
        "$$\n",
        "\n",
        "The trajectory loss is updated along trajectories sampled from $\\pi_{\\theta}$, a tempered version of $P_F(.|., \\theta)$, i.e., with stochastic gradient $\\mathbb{E}_{\\tau \\sim \\pi_{\\theta}} \\nabla_{\\theta} \\mathcal{L}_{TB} \\big(\\tau \\big) $"
      ],
      "metadata": {
        "id": "P64Lx-l3J6T9"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "\n",
        "\n",
        "---\n",
        "More details : \n",
        "- https://milayb.notion.site/GFlowNet-Tutorial-919dcf0a0f0c4e978916a2f509938b00\n",
        "- To go in depth: *GflowNets foundations* paper [2] (long, 70 pages) or *Trajectory Balance* paper [3] (very pedagogical paper).\n",
        "- At the time of writing this tutorial, we have these papers on GflowNets : [1], [2], [3], [4], [5] and [6]\n",
        "\n"
      ],
      "metadata": {
        "id": "pwd6Rmdjwml1"
      }
    }
  ]
}