{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "## Relativistic SGHMC \"Relativistic Monte Carlo\"\n", "\n", "In this notebook we reproduce the results of the paper \n", "[Relativistic Monte Carlo](http://proceedings.mlr.press/v54/lu17b/lu17b.pdf#page=7).\n", "\n", "We start by introducing and plotting all relevant log likelihoods of our objective functions." ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": true }, "outputs": [], "source": [ "%matplotlib inline\n", "import sys\n", "import os\n", "sys.path.insert(0, os.path.join(os.path.abspath(\".\"), \"..\", \"..\", \"..\"))\n", "\n", "from pysgmcmc.diagnostics.sample_chains import PYSGMCMCTrace\n", "from pymc3.backends.base import MultiTrace\n", "from pymc3.diagnostics import effective_n as ess\n", "\n", "import matplotlib.pyplot as plt\n", "import tensorflow as tf\n", "import numpy as np\n", "from pysgmcmc.samplers.relativistic_sghmc import RelativisticSGHMCSampler\n", "\n", "from pysgmcmc.diagnostics.objective_functions import (\n", " banana_log_likelihood,\n", " gmm1_log_likelihood, gmm2_log_likelihood,\n", " gmm3_log_likelihood\n", ")\n", "\n", "from collections import namedtuple\n", "\n", "ObjectiveFunction = namedtuple(\n", " \"ObjectiveFunction\", [\"function\", \"dimensionality\"]\n", ")\n", "\n", "objective_functions = (\n", " ObjectiveFunction(\n", " function=banana_log_likelihood, dimensionality=2\n", " ),\n", " ObjectiveFunction(\n", " function=gmm1_log_likelihood, dimensionality=1\n", " ),\n", " ObjectiveFunction(\n", " function=gmm2_log_likelihood, dimensionality=1\n", " ),\n", " ObjectiveFunction(\n", " function=gmm3_log_likelihood, dimensionality=1\n", " ),\n", ")\n", "\n", "\n", "def cost_function(log_likelihood_function):\n", " def wrapped(*args, **kwargs):\n", " return -log_likelihood_function(*args, **kwargs)\n", " wrapped.__name__ = log_likelihood_function.__name__\n", " return wrapped\n", "\n", "# Banana Contour {{{ #\n", "def banana_plot():\n", " #x = np.arange(-30, 30, 0.05)\n", " #y = np.arange(-60, 20, 0.05)\n", " x = np.arange(-25, 25, 0.05)\n", " y = np.arange(-50, 20, 0.05)\n", " xx, yy = np.meshgrid(x, y, sparse=True)\n", " densities = np.asarray([np.exp(banana_log_likelihood((x, y))) for x in xx for y in yy])\n", " f, ax = plt.subplots(1)\n", " xdata = [1, 4, 8]\n", " ydata = [10, 20, 30]\n", " ax.contour(x, y, densities, 1, label=\"Banana\")\n", " ax.plot([], [], label=\"Banana\")\n", " ax.legend()\n", " ax.grid()\n", " #ax.set_ylim(ymin=-10, ymax=40)\n", " #ax.set_xlim(xmin=-6, xmax=7)\n", " ax.set_ylim(ymin=-60, ymax=20)\n", " ax.set_xlim(xmin=-30, xmax=30)\n", " \n", "# }}} Banana Contour #\n", "\n", "# Gaussian Mixture Models {{{ #\n", "\n", "def gmm_plot(gmm_fun, label=None):\n", " xv = np.arange(-10, 10, 0.1)\n", " yv = np.asarray([np.exp(gmm_fun(xi)) for xi in xv])\n", " plt.grid()\n", " plt.plot(xv, yv, label=label)\n", " plt.legend()\n", "\n", "plot_functions = {\n", " \"banana_log_likelihood\": banana_plot,\n", " \"gmm1_log_likelihood\": lambda: gmm_plot(gmm1_log_likelihood, label=\"GMM_1\"),\n", " \"gmm2_log_likelihood\": lambda: gmm_plot(gmm2_log_likelihood, label=\"GMM_2\"),\n", " \"gmm3_log_likelihood\": lambda: gmm_plot(gmm3_log_likelihood, label=\"GMM_3\"),\n", "}\n", "\n", "\n", "def extract_samples(sampler, n_samples=1000, keep_every=10):\n", " from itertools import islice\n", " n_iterations = n_samples * keep_every\n", " return np.asarray(\n", " [sample for sample, _ in\n", " islice(sampler, 0, n_iterations, keep_every)]\n", " )\n", "\n", "def plot_samples(sampler, n_samples=1000, keep_every=10):\n", " samples = extract_samples(\n", " sampler, n_samples=n_samples, keep_every=keep_every\n", " )\n", " plot_functions[sampler.cost_fun.__name__]()\n", " \n", " first_sample = samples[0]\n", " try:\n", " sample_dimensionality, = first_sample.shape\n", " except ValueError:\n", " plt.scatter(samples, np.exp([-sampler.cost_fun(sample) for sample in samples]))\n", " else:\n", " plt.scatter(*[samples[:, i] for i in range(sample_dimensionality)])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Extract samples and plot them\n", "\n", "Below we extract $1000$ samples for each cost function using relativistic sghmc and plot the samples on top of the respective function." ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "scrolled": true }, "outputs": [ { "data": { "image/png": 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EHW8VeOiV3XnpezXVVdx77VRO6n8iWoxxVsgc04IB4jfr+8ZCyRd310zlojOG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\n1oKGnPgvQIlAeYnQrxT6hYR+IRhQJgz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pWR/1nIz23S9mx9Zyikco8m89/bophTRgCqFA8Os2Z2s8PN8mk7gZ4b2M8+nA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\n+XyX5Wwl3ffKS+6gAeiGz3+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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "from pysgmcmc.samplers.relativistic_sghmc import RelativisticSGHMCSampler\n", "from pysgmcmc.stepsize_schedules import ConstantStepsizeSchedule\n", "\n", "for function, dimensionality in objective_functions:\n", " tf.reset_default_graph()\n", " graph = tf.Graph()\n", " \n", " with tf.Session(graph=graph) as session:\n", " if function.__name__ == \"banana_log_likelihood\":\n", " params = [\n", " tf.Variable(0., dtype=tf.float32, name=\"x\"), \n", " tf.Variable(6., dtype=tf.float32, name=\"y\")\n", " ]\n", " else:\n", " params = [tf.Variable(0., dtype=tf.float32, name=\"x\")]\n", " sampler = RelativisticSGHMCSampler(\n", " stepsize_schedule=ConstantStepsizeSchedule(0.1),\n", " params=params, \n", " cost_fun=cost_function(function), \n", " session=session,\n", " dtype=tf.float32\n", " )\n", " session.run(tf.global_variables_initializer())\n", " \n", " plot_samples(sampler)\n", " plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Diagnostics\n", "\n", "Next, we analyze some diagnostics of our relativistic sghmc sampler. Namely, we will study how effective sample sizes (ESS) and mean absolute error (MAE) behave and vary over different values for the stepsize $\\epsilon$.\n" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "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": "3.5.2" } }, "nbformat": 4, "nbformat_minor": 2 }