GPyTorch 1.2 Examples : Exact GP (回帰) – 不確かな入力による GP 回帰 (翻訳/解説)
翻訳 : (株)クラスキャット セールスインフォメーション
作成日時 : 09/05/2020 (1.2)
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Exact GP (回帰) – 不確かな入力による GP 回帰
イントロダクション
このノートブックでは、訓練データの不確かさを扱う一つの方法を実演していきます。次の関数をモデル化する訓練データを集めているとしましょう。
\[
\begin{align}
y &= \sin(2\pi x) + \epsilon \\
\epsilon &\sim \mathcal{N}(0, 0.2)
\end{align}
\]
けれども、今は特徴について少し不確かであると仮定します。特に、総ての x_i 値は点ではなく代わりに分布であると仮定していきます。E.g.
\[
x_i \sim \mathcal{N}(\mu_i, \sigma_i).
\]
不確かな入力を扱うために distributional カーネルを使用する
変分法を使用するのではなく (変分例題の不確かな入力による GP 回帰チュートリアル参照)、入力の不確かさのタイプを明示的に知っているのであれば、それをカーネルに渡すことができます。より具体的には、ガウス入力を仮定した場合、ガウス入力の間の対称的な (= symmetrized) KL divergence を計算します。
import math import torch import tqdm import gpytorch from matplotlib import pyplot as plt %matplotlib inline %load_ext autoreload %autoreload 2
# Training data is 100 points in [0,1] inclusive regularly spaced train_x_mean = torch.linspace(0, 1, 20) # We'll assume the variance shrinks the closer we get to 1 train_x_stdv = torch.linspace(0.03, 0.01, 20) # True function is sin(2*pi*x) with Gaussian noise train_y = torch.sin(train_x_mean * (2 * math.pi)) + torch.randn(train_x_mean.size()) * 0.2
訓練 distributional データを効果的に渡すため、mean と log 分散をスタックする必要があります。
train_x_distributional = torch.stack((train_x_mean, (train_x_stdv**2).log()), dim=1)
f, ax = plt.subplots(1, 1, figsize=(8, 3)) ax.errorbar(train_x_mean, train_y, xerr=(train_x_stdv * 2), fmt="k*", label="Train Data") ax.legend()
多くの設定で標準的であるように、type-II 勾配降下を通して結果としての distributional GP のハイパーパラメータを訓練します。fully Bayesian 推論を行なうこともできるでしょう。
from gpytorch.models import ExactGP
from gpytorch.kernels import GaussianSymmetrizedKLKernel, ScaleKernel
from gpytorch.means import ConstantMean
class ExactGPModel(ExactGP):
def __init__(self, train_x, train_y, likelihood):
super(ExactGPModel, self).__init__(train_x, train_y, likelihood)
self.mean_module = ConstantMean()
self.covar_module = ScaleKernel(GaussianSymmetrizedKLKernel())
def forward(self, x):
mean_x = self.mean_module(x)
covar_x = self.covar_module(x)
return gpytorch.distributions.MultivariateNormal(mean_x, covar_x)
# initialize likelihood and model
likelihood = gpytorch.likelihoods.GaussianLikelihood()
model = ExactGPModel(train_x_distributional, train_y, likelihood)
# this is for running the notebook in our testing framework
import os
smoke_test = ('CI' in os.environ)
training_iter = 2 if smoke_test else 500
# Find optimal model hyperparameters
model.train()
likelihood.train()
# Use the adam optimizer
optimizer = torch.optim.Adam(model.parameters(), lr=0.25) # Includes GaussianLikelihood parameters
# "Loss" for GPs - the marginal log likelihood
mll = gpytorch.mlls.ExactMarginalLogLikelihood(likelihood, model)
for i in range(training_iter):
# Zero gradients from previous iteration
optimizer.zero_grad()
# Output from model
output = model(train_x_distributional)
# Calc loss and backprop gradients
loss = -mll(output, train_y)
loss.backward()
print('Iter %d/%d - Loss: %.3f lengthscale: %.3f noise: %.3f' % (
i + 1, training_iter, loss.item(),
model.covar_module.base_kernel.lengthscale.item(),
model.likelihood.noise.item()
))
optimizer.step()
Iter 1/500 - Loss: 1.274 lengthscale: 0.693 noise: 0.693 Iter 2/500 - Loss: 1.242 lengthscale: 0.826 noise: 0.576 Iter 3/500 - Loss: 1.173 lengthscale: 0.955 noise: 0.475 Iter 4/500 - Loss: 1.141 lengthscale: 1.102 noise: 0.389 Iter 5/500 - Loss: 1.121 lengthscale: 1.266 noise: 0.317 Iter 6/500 - Loss: 1.097 lengthscale: 1.442 noise: 0.261 Iter 7/500 - Loss: 1.093 lengthscale: 1.621 noise: 0.220 Iter 8/500 - Loss: 1.112 lengthscale: 1.801 noise: 0.192 Iter 9/500 - Loss: 1.112 lengthscale: 1.988 noise: 0.177 Iter 10/500 - Loss: 1.095 lengthscale: 2.188 noise: 0.172 Iter 11/500 - Loss: 1.081 lengthscale: 2.401 noise: 0.172 Iter 12/500 - Loss: 1.068 lengthscale: 2.624 noise: 0.177 Iter 13/500 - Loss: 1.049 lengthscale: 2.856 noise: 0.183 Iter 14/500 - Loss: 1.031 lengthscale: 3.095 noise: 0.188 Iter 15/500 - Loss: 1.020 lengthscale: 3.340 noise: 0.191 Iter 16/500 - Loss: 1.014 lengthscale: 3.589 noise: 0.189 Iter 17/500 - Loss: 1.007 lengthscale: 3.842 noise: 0.183 Iter 18/500 - Loss: 0.997 lengthscale: 4.098 noise: 0.173 Iter 19/500 - Loss: 0.983 lengthscale: 4.356 noise: 0.159 Iter 20/500 - Loss: 0.968 lengthscale: 4.616 noise: 0.144 Iter 21/500 - Loss: 0.951 lengthscale: 4.877 noise: 0.128 Iter 22/500 - Loss: 0.933 lengthscale: 5.139 noise: 0.112 Iter 23/500 - Loss: 0.913 lengthscale: 5.402 noise: 0.097 Iter 24/500 - Loss: 0.891 lengthscale: 5.667 noise: 0.083 Iter 25/500 - Loss: 0.870 lengthscale: 5.933 noise: 0.070 Iter 26/500 - Loss: 0.850 lengthscale: 6.202 noise: 0.059 Iter 27/500 - Loss: 0.832 lengthscale: 6.474 noise: 0.050 Iter 28/500 - Loss: 0.814 lengthscale: 6.748 noise: 0.042 Iter 29/500 - Loss: 0.794 lengthscale: 7.025 noise: 0.035 Iter 30/500 - Loss: 0.775 lengthscale: 7.304 noise: 0.029 Iter 31/500 - Loss: 0.756 lengthscale: 7.586 noise: 0.024 Iter 32/500 - Loss: 0.737 lengthscale: 7.868 noise: 0.020 Iter 33/500 - Loss: 0.719 lengthscale: 8.152 noise: 0.017 Iter 34/500 - Loss: 0.701 lengthscale: 8.437 noise: 0.014 Iter 35/500 - Loss: 0.684 lengthscale: 8.722 noise: 0.012 Iter 36/500 - Loss: 0.668 lengthscale: 9.007 noise: 0.010 Iter 37/500 - 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今は、予測をテストします。単純性のために 0.01 の固定分散を仮定します。
# Get into evaluation (predictive posterior) mode
model.eval()
likelihood.eval()
# Test points are regularly spaced along [0,1]
# Make predictions by feeding model through likelihood
with torch.no_grad(), gpytorch.settings.fast_pred_var():
test_x = torch.linspace(0, 1, 51)
test_x_distributional = torch.stack((test_x, (1e-2 * torch.ones_like(test_x)).log()), dim=1)
observed_pred = likelihood(model(test_x_distributional))
with torch.no_grad():
# Initialize plot
f, ax = plt.subplots(1, 1, figsize=(8, 3))
# Get upper and lower confidence bounds
lower, upper = observed_pred.confidence_region()
# Plot training data as black stars
ax.errorbar(train_x_mean.numpy(), train_y.numpy(), xerr=train_x_stdv, fmt='k*')
# Plot predictive means as blue line
ax.plot(test_x.numpy(), observed_pred.mean.numpy(), 'b')
# Shade between the lower and upper confidence bounds
ax.fill_between(test_x.numpy(), lower.numpy(), upper.numpy(), alpha=0.5)
ax.set_ylim([-3, 3])
ax.legend(['Observed Data', 'Mean', 'Confidence'])
最後の説明として、入力として確率分布に渡る任意の距離関数を取る一般的な DistributionalInputKernel クラスを公開することにより distributional カーネル・クラスを拡張することを非常に容易なものとしました。
以上