{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Using Pòlya-Gamma Auxiliary Variables for Binary Classification\n", "\n", "## Overview\n", "\n", "In this notebook, we'll demonstrate how to use Pòlya-Gamma auxiliary variables to do efficient inference for Gaussian Process binary classification as in reference [1]. \n", "We will also use natural gradient descent, as described in more detail in the [Natural gradient descent](./Natural_Gradient_Descent.ipynb) tutorial.\n", "\n", "\n", "[1] Florian Wenzel, Theo Galy-Fajou, Christan Donner, Marius Kloft, Manfred Opper. [Efficient Gaussian process classification using Pòlya-Gamma data augmentation](https://arxiv.org/abs/1802.06383). Proceedings of the AAAI Conference on Artificial Intelligence. 2019.\n", "\n", "## Pòlya-Gamma Augmentation\n", "\n", "When a Gaussian Process prior is paired with a Gaussian likelihood inference can be done exactly with a simple closed form expression.\n", "Unfortunately this attractive feature does not carry over to non-conjugate likelihoods like the Bernoulli likelihood that arises in the context of binary classification with a logistic link function.\n", "Sampling-based stochastic variational inference offers a general strategy for dealing with non-conjugate likelihoods; see the [corresponding tutorial](./Non_Gaussian_Likelihoods.ipynb).\n", "\n", "Another possible strategy is to introduce additional latent variables that restore conjugacy. \n", "This is the strategy we follow here. \n", "In particular we are going to introduce a Pòlya-Gamma auxiliary variable for each data point in our training dataset. \n", "The [Polya-Gamma](https://arxiv.org/abs/1205.0310) distribution $\\rm{PG}$ is a univariate distribution with support on the positive real line. \n", "In our context it is interesting because if $\\omega_i$ is distributed according to $\\rm{PG}(1,0)$ then the logistic likelihood $\\sigma(\\cdot)$ for data point $(x_i, y_i)$ can be represented as\n", "\n", "\\begin{align}\n", "\\sigma(y_i f_i) = \\frac{1}{1 + \\exp(-y_i f_i)} = \\tfrac{1}{2} \\mathbb{E}_{\\omega_i \\sim \\rm{PG}(1,0)} \\left[ \\exp \\left(\\tfrac{1}{2} y_i f_i - \\tfrac{\\omega_i}{2} f_i^2 \\right) \\right]\n", "\\end{align}\n", "\n", "where $y_i \\in \\{-1, 1\\}$ is the binary label of data point $i$\n", "and $f_i$ is the Gaussian Process prior evaluated at input $x_i$. \n", "The crucial point here is that $f_i$ appears quadratically in the exponential within the expectation. \n", "In other words, conditioned on $\\omega_i$, we can integrate out $f_i$ exactly, just as if we were doing regression with a Gaussian likelihood. For more details please see the original reference. \n", "\n", "## Setup" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "import tqdm\n", "import math\n", "import torch\n", "import gpytorch\n", "from matplotlib import pyplot as plt\n", "\n", "# Make plots inline\n", "%matplotlib inline" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For this example notebook, we'll create a simple artificial dataset." ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "import os\n", "from math import floor\n", "\n", "# this is for running the notebook in our testing framework\n", "smoke_test = ('CI' in os.environ)\n", "\n", "N = 100\n", "X = torch.linspace(-1., 1., N)\n", "probs = (torch.sin(X * math.pi).add(1.).div(2.))\n", "y = torch.distributions.Bernoulli(probs=probs).sample()\n", "X = X.unsqueeze(-1)\n", "\n", "train_n = int(floor(0.8 * N))\n", "indices = torch.randperm(N)\n", "train_x = X[indices[:train_n]].contiguous()\n", "train_y = y[indices[:train_n]].contiguous()\n", "\n", "test_x = X[indices[train_n:]].contiguous()\n", "test_y = y[indices[train_n:]].contiguous()\n", "\n", "if torch.cuda.is_available():\n", " train_x, train_y, test_x, test_y = train_x.cuda(), train_y.cuda(), test_x.cuda(), test_y.cuda()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's plot our artificial dataset. \n", "Note that here the binary labels are 0/1-valued; we will need to be careful to translate between this representation and the -1/1 representation that is most natural in the context of Pòlya-Gamma augementation." ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[]" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "plt.plot(train_x.squeeze(-1).cpu(), train_y.cpu(), 'o')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The following steps create the dataloader objects. See the [SVGP regression notebook](./SVGP_Regression_CUDA.ipynb) for details." ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "from torch.utils.data import TensorDataset, DataLoader\n", "\n", "train_dataset = TensorDataset(train_x, train_y)\n", "train_loader = DataLoader(train_dataset, batch_size=100000, shuffle=False)\n", "\n", "test_dataset = TensorDataset(test_x, test_y)\n", "test_loader = DataLoader(test_dataset, batch_size=1024, shuffle=False)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Variational Inference with PG Auxiliaries\n", "\n", "We define a Bernoulli likelihood that leverages Pòlya-Gamma augmentation. \n", "It turns out that we can derive closed form updates for the Pòlya-Gamma auxiliary variables. To deal with the Gaussian Process we introduce inducing points and inducing locations. \n", "In particular we will need to learn a variational covariance matrix and a variational mean vector that control the inducing points. (See the discussion in the [SVGP tutorial](Approximate_GP_Objective_Functions.ipynb) for more details.) \n", "We will use natural gradient updates to deal with these two variational parameters; this will allow us to take large steps, thus yielding fast convergence. " ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [], "source": [ "class PGLikelihood(gpytorch.likelihoods._OneDimensionalLikelihood):\n", " # this method effectively computes the expected log likelihood \n", " # contribution to Eqn (10) in Reference [1].\n", " def expected_log_prob(self, target, input, *args, **kwargs):\n", " mean, variance = input.mean, input.variance\n", " # Compute the expectation E[f_i^2]\n", " raw_second_moment = variance + mean.pow(2)\n", "\n", " # Translate targets to be -1, 1\n", " target = target.to(mean.dtype).mul(2.).sub(1.)\n", "\n", " # We detach the following variable since we do not want\n", " # to differentiate through the closed-form PG update.\n", " c = raw_second_moment.detach().sqrt()\n", " # Compute mean of PG auxiliary variable omega: 0.5 * Expectation[omega]\n", " # See Eqn (11) and Appendix A2 and A3 in Reference [1] for details.\n", " half_omega = 0.25 * torch.tanh(0.5 * c) / c\n", "\n", " # Expected log likelihood\n", " res = 0.5 * target * mean - half_omega * raw_second_moment\n", " # Sum over data points in mini-batch\n", " res = res.sum(dim=-1)\n", "\n", " return res\n", " \n", " # define the likelihood\n", " def forward(self, function_samples):\n", " return torch.distributions.Bernoulli(logits=function_samples)\n", " \n", " # define the marginal likelihood using Gauss Hermite quadrature\n", " def marginal(self, function_dist):\n", " prob_lambda = lambda function_samples: self.forward(function_samples).probs\n", " probs = self.quadrature(prob_lambda, function_dist)\n", " return torch.distributions.Bernoulli(probs=probs)\n", " \n", "\n", "# define the actual GP model (kernels, inducing points, etc.) \n", "class GPModel(gpytorch.models.ApproximateGP):\n", " def __init__(self, inducing_points):\n", " variational_distribution = gpytorch.variational.NaturalVariationalDistribution(inducing_points.size(0))\n", " variational_strategy = gpytorch.variational.VariationalStrategy(\n", " self, inducing_points, variational_distribution, learn_inducing_locations=True\n", " )\n", " super(GPModel, self).__init__(variational_strategy)\n", " self.mean_module = gpytorch.means.ZeroMean()\n", " self.covar_module = gpytorch.kernels.ScaleKernel(gpytorch.kernels.RBFKernel())\n", "\n", " def forward(self, x):\n", " mean_x = self.mean_module(x)\n", " covar_x = self.covar_module(x)\n", " return gpytorch.distributions.MultivariateNormal(mean_x, covar_x)\n", "\n", "# we initialize our model with M = 30 inducing points\n", "M = 30\n", "inducing_points = torch.linspace(-2., 2., M, dtype=train_x.dtype, device=train_x.device).unsqueeze(-1)\n", "model = GPModel(inducing_points=inducing_points)\n", "model.covar_module.base_kernel.initialize(lengthscale=0.2)\n", "likelihood = PGLikelihood()\n", "\n", "if torch.cuda.is_available():\n", " model = model.cuda()\n", " likelihood = likelihood.cuda()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Setup optimizers\n", "\n", "We will use a `NGD` (Natural Gradient Descent) optimizer to deal with the inducing point covariance matrix and corresponding mean vector, while we will use the `Adam` optimizer for all other parameters (the kernel hyperparmaeters as well as the inducing point locations). \n", "Note that we use a pretty large learning rate for the `NGD` optimizer." ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [], "source": [ "variational_ngd_optimizer = gpytorch.optim.NGD(model.variational_parameters(), num_data=train_y.size(0), lr=0.1)\n", "\n", "hyperparameter_optimizer = torch.optim.Adam([\n", " {'params': model.hyperparameters()},\n", " {'params': likelihood.parameters()},\n", "], lr=0.01)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Define training loop" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "application/vnd.jupyter.widget-view+json": { "model_id": "488a3a3d5f704b0f98611a14e3e6143a", "version_major": 2, 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"output_type": "display_data" }, { "data": { "application/vnd.jupyter.widget-view+json": { "model_id": "", "version_major": 2, "version_minor": 0 }, "text/plain": [ "HBox(children=(FloatProgress(value=0.0, description='Minibatch', max=1.0, style=ProgressStyle(description_widt…" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "application/vnd.jupyter.widget-view+json": { "model_id": "", "version_major": 2, "version_minor": 0 }, "text/plain": [ "HBox(children=(FloatProgress(value=0.0, description='Minibatch', max=1.0, style=ProgressStyle(description_widt…" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\n" ] } ], "source": [ "model.train()\n", "likelihood.train()\n", "mll = gpytorch.mlls.VariationalELBO(likelihood, model, num_data=train_y.size(0))\n", "\n", "num_epochs = 1 if smoke_test else 100\n", "epochs_iter = tqdm.notebook.tqdm(range(num_epochs), desc=\"Epoch\")\n", "for i in epochs_iter:\n", " minibatch_iter = tqdm.notebook.tqdm(train_loader, desc=\"Minibatch\", leave=False)\n", " \n", " for x_batch, y_batch in minibatch_iter:\n", " ### Perform NGD step to optimize variational parameters\n", " variational_ngd_optimizer.zero_grad()\n", " hyperparameter_optimizer.zero_grad()\n", " \n", " output = model(x_batch)\n", " loss = -mll(output, y_batch)\n", " minibatch_iter.set_postfix(loss=loss.item())\n", " loss.backward()\n", " variational_ngd_optimizer.step()\n", " hyperparameter_optimizer.step()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Visualization and Evaluation" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[]" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# push training data points through model\n", "train_mean_f = model(train_x).loc.data.cpu()\n", "# plot training data with y being -1/1 valued\n", "plt.plot(train_x.squeeze(-1).cpu(), train_y.mul(2.).sub(1.).cpu(), 'o')\n", "# plot mean gaussian process posterior mean evaluated at training data\n", "plt.plot(train_x.squeeze(-1).cpu(), train_mean_f.cpu(), 'x')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "As expected the Gaussian Process posterior mean (plotted in orange) gives confident predictions in the regions\n", "where the correct label is unambiguous (e.g. for x ~ 0.5) and gives unconfident predictions in regions where\n", "the correct label is ambiguous (e.g. x ~ 0.0).\n", "\n", "We compute the negative log likelihood (NLL) and classification accuracy on the held-out test data." ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Test NLL: 0.3481\n", "Test Acc: 0.9000\n" ] } ], "source": [ "model.eval()\n", "likelihood.eval()\n", "with torch.no_grad():\n", " nlls = -likelihood.log_marginal(test_y, model(test_x))\n", " acc = (likelihood(model(test_x)).probs.gt(0.5) == test_y.bool()).float().mean()\n", "print('Test NLL: {:.4f}'.format(nlls.mean()))\n", "print('Test Acc: {:.4f}'.format(acc.mean()))" ] } ], "metadata": { "anaconda-cloud": {}, "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": "3.8.3" } }, "nbformat": 4, "nbformat_minor": 2 }