Show code cell source
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
import matplotlib_inline
matplotlib_inline.backend_inline.set_matplotlib_formats('svg')
import seaborn as sns
sns.set_context("paper")
sns.set_style("ticks")
import torch
import gpytorch
from gpytorch.kernels import RBFKernel, ScaleKernel
class ExactGP(gpytorch.models.ExactGP):
def __init__(self,
train_x,
train_y,
likelihood=gpytorch.likelihoods.GaussianLikelihood(
noise_constraint=gpytorch.constraints.GreaterThan(0.0)
),
mean_module=gpytorch.means.ConstantMean(),
covar_module=ScaleKernel(RBFKernel())
):
super().__init__(train_x, train_y, likelihood)
self.mean_module = mean_module
self.covar_module = covar_module
def forward(self, x):
mean_x = self.mean_module(x)
covar_x = self.covar_module(x)
return gpytorch.distributions.MultivariateNormal(mean_x, covar_x)
def plot_1d_regression(
x_star,
model,
ax=None,
f_true=None,
num_samples=10
):
"""Plot the posterior predictive.
Arguments
x_start -- The test points on which to evaluate.
model -- The trained model.
Keyword Arguments
ax -- An axes object to write on.
f_true -- The true function.
num_samples -- The number of samples.
"""
f_star = model(x_star)
m_star = f_star.mean
v_star = f_star.variance
y_star = model.likelihood(f_star)
yv_star = y_star.variance
f_lower = (
m_star - 2.0 * torch.sqrt(v_star)
)
f_upper = (
m_star + 2.0 * torch.sqrt(v_star)
)
y_lower = m_star - 2.0 * torch.sqrt(yv_star)
y_upper = m_star + 2.0 * torch.sqrt(yv_star)
if ax is None:
fig, ax = plt.subplots()
ax.plot(model.train_inputs[0].flatten().detach(),
model.train_targets.detach(),
'kx',
markersize=10,
markeredgewidth=2,
label='Observations'
)
ax.plot(
x_star,
m_star.detach(),
lw=2,
label='$m_n(x)$',
color=sns.color_palette()[0]
)
ax.fill_between(
x_star.flatten().detach(),
f_lower.flatten().detach(),
f_upper.flatten().detach(),
alpha=0.5,
label='$f(\mathbf{x}^*)$ 95% pred.',
color=sns.color_palette()[0]
)
ax.fill_between(
x_star.detach().flatten(),
y_lower.detach().flatten(),
f_lower.detach().flatten(),
color=sns.color_palette()[1],
alpha=0.5,
label='$y^*$ 95% pred.'
)
ax.fill_between(
x_star.detach().flatten(),
f_upper.detach().flatten(),
y_upper.detach().flatten(),
color=sns.color_palette()[1],
alpha=0.5,
label=None
)
if f_true is not None:
ax.plot(
x_star,
f_true(x_star),
'm-.',
label='True function'
)
if num_samples > 0:
f_post_samples = f_star.sample(
sample_shape=torch.Size([10])
)
ax.plot(
x_star.numpy(),
f_post_samples.T.detach().numpy(),
color="red",
lw=0.5
)
# This is just to add the legend entry
ax.plot(
[],
[],
color="red",
lw=0.5,
label="Posterior samples"
)
ax.set_xlabel('$x$')
ax.set_ylabel('$y$')
plt.legend(loc='best', frameon=False)
sns.despine(trim=True)
return m_star, v_star
def train(model, train_x, train_y, n_iter=10, lr=0.1):
"""Train the model.
Arguments
model -- The model to train.
train_x -- The training inputs.
train_y -- The training labels.
n_iter -- The number of iterations.
"""
model.train()
optimizer = torch.optim.LBFGS(model.parameters(), line_search_fn='strong_wolfe')
likelihood = model.likelihood
mll = gpytorch.mlls.ExactMarginalLogLikelihood(likelihood, model)
def closure():
optimizer.zero_grad()
output = model(train_x)
loss = -mll(output, train_y)
loss.backward()
return loss
for i in range(n_iter):
loss = optimizer.step(closure)
if (i + 1) % 1 == 0:
print(f'Iter {i + 1:3d}/{n_iter} - Loss: {loss.item():.3f}')
model.eval()
def plot_iaf(
x_star,
gpr,
alpha,
alpha_params={},
ax=None,
f_true=None,
iaf_label="Information Acquisition Function"
):
"""Plot the information acquisition function.
Arguments
x_star -- A set of points to plot on.
gpr -- A rained Gaussian process regression
object.
alpha -- The information acquisition function.
This assumed to be a function of the
posterior mean and standard deviation.
Keyword Arguments
ax -- An axes object to plot on.
f_true -- The true function - if available.
alpha_params -- Extra parameters to the information
acquisition function.
ax -- An axes object to plot on.
f_true -- The true function - if available.
iaf_label -- The label for the information acquisition
function. Default is "Information Acquisition".
The evaluation of the information acquisition function
is as follows:
af_values = alpha(mu, sigma, y_max, **alpha_params)
"""
if ax is None:
fig, ax = plt.subplots()
ax.set_title(
", ".join(
f"{n}={k:.2f}"
for n, k in alpha_params.items()
)
)
m, v = plot_1d_regression(
x_star,
gpr,
ax=ax,
f_true=f_true,
num_samples=0
)
sigma = torch.sqrt(v)
af_values = alpha(m, sigma, gpr.train_targets.numpy().max(), **alpha_params)
next_id = torch.argmax(af_values)
next_x = x_star[next_id]
af_max = af_values[next_id]
ax2 = ax.twinx()
ax2.plot(x_star, af_values.detach(), color=sns.color_palette()[1])
ax2.set_ylabel(
iaf_label,
color=sns.color_palette()[1]
)
plt.setp(
ax2.get_yticklabels(),
color=sns.color_palette()[1]
)
ax2.plot(
next_x * np.ones(100),
torch.linspace(0, af_max.item(), 100),
color=sns.color_palette()[1],
linewidth=1
)
def maximize(
f,
model,
X_design,
alpha,
alpha_params={},
max_it=10,
optimize=False,
plot=False,
**kwargs
):
"""Optimize a function using a limited number of evaluations.
Arguments
f -- The function to optimize.
gpr -- A Gaussian process model to use for representing
our state of knowledge.
X_design -- The set of candidate points for identifying the
maximum.
alpha -- The information acquisition function.
This assumed to be a function of the
posterior mean and standard deviation.
Keyword Arguments
alpha_params -- Extra parameters to the information
acquisition function.
max_it -- The maximum number of iterations.
optimize -- Whether or not to optimize the hyper-parameters.
plot -- Determines how often to plot. Make it one
to plot at each iteration. Make it max_it
to plot at the last iteration.
The rest of the keyword arguments are passed to plot_iaf().
"""
af_all = []
for count in range(max_it):
# Predict
f_design = model(X_design)
m = f_design.mean
sigma2 = f_design.variance
sigma = torch.sqrt(sigma2)
# Evaluate information acquisition function
y_train = model.train_targets.numpy()
af_values = alpha(
m,
sigma,
y_train.max(),
**alpha_params
)
# Find best point to include
i = torch.argmax(af_values)
af_all.append(af_values[i])
new_x = X_design[i:(i+1)].float()
new_y = f(new_x)
train_x = torch.cat([model.train_inputs[0], new_x[:, None]])
train_y = torch.cat([model.train_targets, new_y])
model.set_train_data(train_x, train_y, strict=False)
if optimize:
train(model, train_x, train_y, n_iter=100, lr=0.1)
else:
model.train()
model.eval()
# Plot if required
if count % plot == 0:
if "ax" in kwargs:
ax = kwargs[ax]
else:
fig, ax = plt.subplots()
plot_iaf(
X_design,
model,
alpha,
alpha_params=alpha_params,
f_true=f,
ax=ax,
**kwargs
)
ax.set_title(
f"N={count}, " + ax.get_title()
)
return af_all
Show code cell source
!pip install gpytorch
Expected Improvement#
We develop intuition about the expected improvement, the most popular acquisition function for Bayesian optimization. Let’s reintroduce the same running example as the previous hands-on activity.
Show code cell source
def f(x):
"""A function to optimize."""
return -4 * (1. - np.sin(6 * x + 8 * np.exp(6 * x - 7.)))
np.random.seed(12345)
n_init = 3
X = np.random.rand(n_init)
Y = f(X)
plt.plot(X, Y, 'kx', markersize=10, markeredgewidth=2)
plt.xlabel('$x$')
plt.ylabel('$y$')
sns.despine(trim=True);
Just like in the previous hands-on activity, assume that we have made some observations and that we have used them to do Gaussian process regression, resulting in the point-predictive distribution:
\[
p(y|\mathbf{x},\mathcal{D}_{n}) = \mathcal{N}\left(y|m_{n}(\mathbf{x}), \sigma^2_{n}(\mathbf{x})\right),
\]
where \(m_{n}(\mathbf{x})\) and \(\sigma^2_{n}(\mathbf{x})\) are the predictive mean and variance respectively. Here is the code for this:
train_x = torch.from_numpy(X).float()
train_y = torch.from_numpy(Y).float()
model = ExactGP(train_x, train_y)
# It is not a good idea to train the model when we don't have enough data
# So we fix the hyperparameters to something reasonable
model.covar_module.base_kernel.lengthscale = 0.15
model.covar_module.outputscale = 4.0
model.likelihood.noise = 0.0
model.eval()
x = torch.linspace(0, 1, 100)
plot_1d_regression(
x,
model,
f_true=f
);
/Users/ibilion/.pyenv/versions/3.11.6/lib/python3.11/site-packages/linear_operator/utils/cholesky.py:40: NumericalWarning: A not p.d., added jitter of 1.0e-06 to the diagonal
warnings.warn(
/Users/ibilion/.pyenv/versions/3.11.6/lib/python3.11/site-packages/linear_operator/utils/cholesky.py:40: NumericalWarning: A not p.d., added jitter of 1.0e-05 to the diagonal
warnings.warn(
/Users/ibilion/.pyenv/versions/3.11.6/lib/python3.11/site-packages/linear_operator/utils/cholesky.py:40: NumericalWarning: A not p.d., added jitter of 1.0e-04 to the diagonal
warnings.warn(
The expected improvement is more involved, but it serves as a template for deriving more general information acquisition functions. Here is how you think. Consider a hypothetical experiment at \(\mathbf{x}\) and assume that you observed \(y\). How much improvement is that compared to your currently best-observed point \(y_n^*\)? It is:
\[\begin{split}
I_n(\mathbf{x}, y) =
\begin{cases}
0,&\;\text{if}\;y \le y_n^*,\\
y - y_n^*,&\;\text{otherwise}.
\end{cases}
\end{split}\]
But you don’t know what \(y\) is. What do you do now? Well, the only legitimate thing to do is to take the expectation over what you expected \(y\) to be given what you know. So it is:
\[\begin{split}
\begin{align}
\operatorname{EI}_n(\mathbf{x}) &=&
\int_{-\infty}^\infty I_n(\mathbf{x}, y)p(y|\mathbf{x}, \mathcal{D}_n)dy\\
&=& \int_{-\infty}^{y_n^*}0\cdot p(y|\mathbf{x}, \mathcal{D}_n)dy
+ \int_{y_n^*}^{\infty}(y - y_n^*)\cdot p(y|\mathbf{x}, \mathcal{D}_n)dy.
\end{align}
\end{split}\]
You can work this out analytically. You will get:
\[
\operatorname{EI}_n(\mathbf{x}) = \left(m_n(\mathbf{x}) - y_n^*\right)\Phi\left(\frac{m_n(\mathbf{x}) - y_n^*}{\sigma_n(\mathbf{x})}\right)
+ \sigma_n(x)\phi\left(\frac{m_n(\mathbf{x}) - y_n^*}{\sigma_n(\mathbf{x})}\right).
\]
def ei(m, sigma, ymax):
"""Return the expected improvement.
Arguments
m -- The predictive mean at the test points.
sigma -- The predictive standard deviation at
the test points.
ymax -- The maximum observed value (so far).
"""
diff = m - ymax
u = diff / sigma
ei = ( diff * torch.distributions.Normal(0, 1).cdf(u) +
sigma * torch.distributions.Normal(0, 1).log_prob(u).exp()
)
ei[sigma <= 0.] = 0.
return ei
plot_iaf(
x,
model,
ei
)
Bayesian global optimization with the expected improvement#
Let’s now run the Bayesian global optimization algorithm using the expected improvement as the information acquisition function:
train_x = torch.from_numpy(X).float()
train_y = torch.from_numpy(Y).float()
model = ExactGP(train_x, train_y)
# It is not a good idea to train the model when we don't have enough data
# So we fix the hyperparameters to something reasonable
model.covar_module.base_kernel.lengthscale = 0.15
model.covar_module.outputscale = 4.0
model.likelihood.noise = 0.0
model.eval()
# Run the algorithm
X_design = torch.linspace(0, 1, 100)
af_all = maximize(
f,
model,
X_design,
ei,
max_it=5,
plot=1
)
Show code cell output
/Users/ibilion/.pyenv/versions/3.11.6/lib/python3.11/site-packages/gpytorch/distributions/multivariate_normal.py:319: NumericalWarning: Negative variance values detected. This is likely due to numerical instabilities. Rounding negative variances up to 1e-06.
warnings.warn(
Questions#
Rerun the main algorithm for EI by optimizing the hyper-parameters. Hint: Go through the code of
maximize.