tutorials/batch_mode_cross_validation.ipynb

{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "collapsed": true
      },
      "source": [
        "## Application of batch-mode regression to cross-validation\n",
        "\n",
        "botorch provides a helper function `gen_loo_cv_folds` to easily perform leave-one-out (LOO) cross-validation (CV) by taking advantage of batch-mode regression and evaluation in GPyTorch. This tutorial illustrates the process on a noisy sinusoidal function, similar to the example from the batch-mode GP regression [tutorial](https://github.com/cornellius-gp/gpytorch/blob/master/examples/01_Simple_GP_Regression/Simple_Batch_Mode_GP_Regression.ipynb) from GPyTorch:\n",
        "\n",
        "$$y = \\sin(2\\pi x) + \\epsilon, ~\\epsilon \\sim \\mathcal N(0, 0.2).$$\n",
        "\n",
        "Note: this tutorial aims to introduce batch-mode regression and evaluation in GPyTorch with CV as an example application. For alternative, more user-friendly functions to perform CV in Ax, see [ax.modelbridge.cross_validation](https://github.com/facebook/Ax/blob/main/ax/modelbridge/cross_validation.py). However, for larger CV tasks, it may be useful to exploit GPyTorch batch-mode, as shown in this tutorial."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 1,
      "metadata": {
        "collapsed": true
      },
      "outputs": [],
      "source": [
        "import torch\n",
        "import math\n",
        "\n",
        "device = torch.device(\"cpu\")\n",
        "dtype = torch.float\n",
        "torch.manual_seed(3);"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "### Initialize the CV dataset\n",
        "\n",
        "For our training data, we take 20 regularly spaced points on the interval $[0, 1]$ and generate noisy evaluations with an observed noise variance of 0.2. Remember that botorch requires an explicit output dimension."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 2,
      "metadata": {
        "collapsed": true
      },
      "outputs": [],
      "source": [
        "sigma = math.sqrt(0.2)\n",
        "train_X = torch.linspace(0, 1, 20, dtype=dtype, device=device).view(-1, 1)\n",
        "train_Y_noiseless = torch.sin(train_X * (2 * math.pi))\n",
        "train_Y = train_Y_noiseless + sigma * torch.randn_like(train_Y_noiseless)\n",
        "train_Yvar = torch.full_like(train_Y, 0.2)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "The botorch function `gen_loo_cv_folds` takes our observed data `train_X`, `train_Y`, `train_Yvar` as input and returns the LOO CV folds in a `CVFolds` object."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 3,
      "metadata": {
        "collapsed": true
      },
      "outputs": [],
      "source": [
        "from botorch.cross_validation import gen_loo_cv_folds\n",
        "\n",
        "cv_folds = gen_loo_cv_folds(train_X=train_X, train_Y=train_Y, train_Yvar=train_Yvar)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "The `cv_folds` object contains the data, stored as tensors of appropriate batch shape, necessary to perform 20 CVs of 19 training points and 1 test point. For example, we can check that the shapes of the training inputs and training targets are `b x n x d = 20 x 19 x 1` and `b x n x o = 20 x 19 x 1` respectively, where `o` is the number of outputs."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 4,
      "metadata": {},
      "outputs": [
        {
          "data": {
            "text/plain": [
              "(torch.Size([20, 20, 1]), torch.Size([20, 20, 1]))"
            ]
          },
          "execution_count": 4,
          "metadata": {
            "bento_obj_id": "140675275377288"
          },
          "output_type": "execute_result"
        }
      ],
      "source": [
        "cv_folds.train_X.shape, cv_folds.train_Y.shape"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 5,
      "metadata": {},
      "outputs": [
        {
          "data": {
            "text/plain": [
              "(torch.Size([20, 1, 1]), torch.Size([20, 1, 1]))"
            ]
          },
          "execution_count": 5,
          "metadata": {
            "bento_obj_id": "140676807470216"
          },
          "output_type": "execute_result"
        }
      ],
      "source": [
        "cv_folds.test_X.shape, cv_folds.test_Y.shape"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Note that in a situation where the dataset is large, one may not want to perform LOO; in that case, a similar process can be used to perform $k$-fold CV."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "### Perform LOOCV\n",
        "\n",
        "We can use the `batch_cross_validation` function to perform LOOCV using batching (meaning that the `b = 20` sets of training data can be fit as `b = 20` separate GP models with separate hyperparameters in parallel through GPyTorch) and return a CVResult tuple with the batched `GPyTorchPosterior` object over the LOOCV test points and the observed targets. The `batch_cross_validation` requires a model class (`model_cls`) and a marginal log likelihood class (`mll_cls`). Since we have an observed and constant noise level, we will use the FixedNoiseGP as the `model_cls` and an ExactMarginalLogLikelihood as the `mll_cls`."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 6,
      "metadata": {
        "collapsed": true
      },
      "outputs": [],
      "source": [
        "from botorch.cross_validation import batch_cross_validation\n",
        "from botorch.models import FixedNoiseGP\n",
        "from gpytorch.mlls.exact_marginal_log_likelihood import ExactMarginalLogLikelihood\n",
        "\n",
        "# instantiate and fit model\n",
        "cv_results = batch_cross_validation(\n",
        "    model_cls=FixedNoiseGP,\n",
        "    mll_cls=ExactMarginalLogLikelihood,\n",
        "    cv_folds=cv_folds,\n",
        ")"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "#### Compute the cross-validation error and generate plots\n",
        "To compute the cross-validation error, we first evaluate the test points by computing the posterior in batch mode. Next, we compute the squared errors for each test point from the prediction and take an average across all cross-validation folds."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 7,
      "metadata": {},
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "Cross-validation error:  0.062\n"
          ]
        },
        {
          "data": {
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z4HBjTFOXYhEJi3BUuJeWlvLT8pe5/fbb8fl8TJgwgalTp4Y8MYi49YlqDXxQ7uedzrLd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\nRESkEiUHERGpRMlBREQqUXIQEZFK/h+9giAEniipOwAAAABJRU5ErkJggg==\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "bento_obj_id": "140674656098008"
          },
          "output_type": "display_data"
        }
      ],
      "source": [
        "from matplotlib import pyplot as plt\n",
        "%matplotlib inline\n",
        "\n",
        "posterior = cv_results.posterior\n",
        "mean = posterior.mean\n",
        "cv_error = ((cv_folds.test_Y.squeeze() - mean.squeeze()) ** 2).mean()\n",
        "print(f\"Cross-validation error: {cv_error : 4.2}\")\n",
        "\n",
        "# get lower and upper confidence bounds\n",
        "lower, upper = posterior.mvn.confidence_region()\n",
        "\n",
        "# scatterplot of predicted versus test\n",
        "_, axes = plt.subplots(1, 1, figsize=(6, 4))\n",
        "plt.plot([-1.5, 1.5], [-1.5, 1.5], 'k', label=\"true objective\", linewidth=2)\n",
        "\n",
        "axes.set_xlabel(\"Actual\")\n",
        "axes.set_ylabel(\"Predicted\")\n",
        "\n",
        "axes.errorbar(\n",
        "    x=cv_folds.test_Y.numpy().flatten(), \n",
        "    y=mean.numpy().flatten(), \n",
        "    xerr=1.96*sigma,\n",
        "    yerr=((upper-lower)/2).numpy().flatten(),\n",
        "    fmt='*'\n",
        ");"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Finally, we can visualize the fitted models. To do this, we again take advantage of batch-mode evaluation to obtain predictions, including lower and upper confidence regions, from each of the 20 models."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 8,
      "metadata": {
        "collapsed": true
      },
      "outputs": [],
      "source": [
        "model = cv_results.model\n",
        "with torch.no_grad():\n",
        "    # evaluate the models at a series of points for plotting \n",
        "    plot_x = torch.linspace(0, 1, 101).view(1, -1, 1).repeat(cv_folds.train_X.shape[0], 1, 1)\n",
        "    posterior = model.posterior(plot_x)\n",
        "    mean = posterior.mean\n",
        "    \n",
        "    # get lower and upper confidence bounds\n",
        "    lower, upper = posterior.mvn.confidence_region()\n",
        "    plot_x.squeeze_()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "The code snippet below plots the result for the 12th CV fold (by setting `num = 12`), but note that we have computed the results for all folds above (other plots can be obtained by iterating `num` from 1 to 20)."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 9,
      "metadata": {},
      "outputs": [
        {
          "data": {
            "image/png": 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CzJDPBx/tqeRQtTp9iQRlVDTY2VfZpHUY3dLnk7vPB/LfM8jId3Lmj5T/qDU8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\nxVG+OGylxqZOb50am4MPdpb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FUSRu13e/v25fK3kDXAwY1coFP7GSnufsdMVsIEsZeyI7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\n3AVBEMKQSO6CIAhhSCR3QRCEMPT/J+miawYbwJIAAAAASUVORK5CYII=\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "bento_obj_id": "140674628290824"
          },
          "output_type": "display_data"
        }
      ],
      "source": [
        "_, axes = plt.subplots(1, 1, figsize=(6, 4))\n",
        "\n",
        "# plot the 12th CV fold\n",
        "num = 12 \n",
        "\n",
        "# plot the training data in black\n",
        "axes.plot(\n",
        "    cv_folds.train_X[num - 1].detach().numpy(), \n",
        "    cv_folds.train_Y[num - 1].detach().numpy(), \n",
        "    'k*'\n",
        ")\n",
        "\n",
        "# plot the test data in red\n",
        "axes.plot(\n",
        "    cv_folds.test_X[num - 1].detach().numpy(), \n",
        "    cv_folds.test_Y[num - 1].detach().numpy(), \n",
        "    'r*'\n",
        ")\n",
        "\n",
        "# plot posterior means as blue line\n",
        "axes.plot(plot_x[num - 1].numpy(), mean[num-1].numpy(), 'b')\n",
        "\n",
        "# shade between the lower and upper confidence bounds\n",
        "axes.fill_between(\n",
        "    plot_x[num - 1].numpy(), \n",
        "    lower[num - 1].numpy(), \n",
        "    upper[num - 1].numpy(), \n",
        "    alpha=0.5\n",
        ");"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": true
      },
      "outputs": [],
      "source": []
    }
  ],
  "metadata": {
    "kernelspec": {
      "display_name": "python3",
      "language": "python",
      "name": "python3"
    }
  },
  "nbformat": 4,
  "nbformat_minor": 2
}