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MAKE_BOOK_FIGURES=Trueimport numpy as npimport scipy.stats as stimport matplotlib as mplimport matplotlib.pyplot as plt%matplotlib inlineimport matplotlib_inlinematplotlib_inline.backend_inline.set_matplotlib_formats('svg')import seaborn as snssns.set_context("paper")sns.set_style("ticks")def set_book_style(): plt.style.use('seaborn-v0_8-white') sns.set_style("ticks") sns.set_palette("deep") mpl.rcParams.update({ # Font settings 'font.family': 'serif', # For academic publishing 'font.size': 8, # As requested, 10pt font 'axes.labelsize': 8, 'axes.titlesize': 8, 'xtick.labelsize': 7, # Slightly smaller for better readability 'ytick.labelsize': 7, 'legend.fontsize': 7, # Line and marker settings for consistency 'axes.linewidth': 0.5, 'grid.linewidth': 0.5, 'lines.linewidth': 1.0, 'lines.markersize': 4, # Layout to prevent clipped labels 'figure.constrained_layout.use': True, # Default DPI (will override when saving) 'figure.dpi': 600, 'savefig.dpi': 600, # Despine - remove top and right spines 'axes.spines.top': False, 'axes.spines.right': False, # Remove legend frame 'legend.frameon': False, # Additional trim settings 'figure.autolayout': True, # Alternative to constrained_layout 'savefig.bbox': 'tight', # Trim when saving 'savefig.pad_inches': 0.1 # Small padding to ensure nothing gets cut off })def set_notebook_style(): plt.style.use('seaborn-v0_8-white') sns.set_style("ticks") sns.set_palette("deep") mpl.rcParams.update({ # Font settings - using default sizes 'font.family': 'serif', 'axes.labelsize': 10, 'axes.titlesize': 10, 'xtick.labelsize': 9, 'ytick.labelsize': 9, 'legend.fontsize': 9, # Line and marker settings 'axes.linewidth': 0.5, 'grid.linewidth': 0.5, 'lines.linewidth': 1.0, 'lines.markersize': 4, # Layout settings 'figure.constrained_layout.use': True, # Remove only top and right spines 'axes.spines.top': False, 'axes.spines.right': False, # Remove legend frame 'legend.frameon': False, # Additional settings 'figure.autolayout': True, 'savefig.bbox': 'tight', 'savefig.pad_inches': 0.1 })def save_for_book(fig, filename, is_vector=True, **kwargs): """ Save a figure with book-optimized settings. Parameters: ----------- fig : matplotlib figure The figure to save filename : str Filename without extension is_vector : bool If True, saves as vector at 1000 dpi. If False, saves as raster at 600 dpi. **kwargs : dict Additional kwargs to pass to savefig """ # Set appropriate DPI and format based on figure type if is_vector: dpi = 1000 ext = '.pdf' else: dpi = 600 ext = '.tif' # Save the figure with book settings fig.savefig(f"{filename}{ext}", dpi=dpi, **kwargs)def make_full_width_fig(): return plt.subplots(figsize=(4.7, 2.9), constrained_layout=True)def make_half_width_fig(): return plt.subplots(figsize=(2.35, 1.45), constrained_layout=True)if MAKE_BOOK_FIGURES: set_book_style()else: set_notebook_style()make_full_width_fig = make_full_width_fig if MAKE_BOOK_FIGURES else lambda: plt.subplots()make_half_width_fig = make_half_width_fig if MAKE_BOOK_FIGURES else lambda: plt.subplots()
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!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.
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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
)
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/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.