The following commands will build the slideshow:
!aws s3 cp s3://jupyter/notebooks/talks/DataPhilly\ February\ 2016\ Bayesian\ Optimization\ with\ Gaussian\ Processes.ipynb \
/vagrant/tmp/dataphilly.ipynb
!jupyter nbconvert --to=slides --reveal-prefix=https://cdnjs.cloudflare.com/ajax/libs/reveal.js/3.2.0/ \
--output=/vagrant/tmp/dataphilly /vagrant/tmp/dataphilly.ipynb
download: s3://jupyter/notebooks/talks/DataPhilly February 2016 Bayesian Optimization with Gaussian Processes.ipynb to ../../vagrant/tmp/dataphilly.ipynb [NbConvertApp] Converting notebook /vagrant/tmp/dataphilly.ipynb to slides [NbConvertApp] Writing 4638819 bytes to /vagrant/tmp/dataphilly.slides.html
The built slideshow is available online here.
%matplotlib inline
from __future__ import division
from collections import namedtuple
from warnings import filterwarnings
from IPython.display import display, Image
filterwarnings('ignore', module='matplotlib')
import GPy
from matplotlib import pyplot as plt
import numpy as np
from scipy import stats
import seaborn as sns
warning in stationary: failed to import cython module: falling back to numpy
blue, green, red, purple, gold, teal = sns.color_palette()
np.random.seed(5918927) # from random.org
with open('/vagrant/tmp/dataphilly/ei.gif', 'rb') as gif_data:
gif = Image(gif_data.read())
display(gif)
Minimize a function that is difficult, expensive, or time-consuming to evaluate.
A flexible family of probability distributions over continuous functions
intro_plot_x = np.linspace(0, 1, 100)
intro_plot_X = intro_plot_x[:, np.newaxis]
intro_X = np.array([[0.5]])
intro_Y = np.array([[1]])
rbf_kernel = GPy.kern.RBF(1., lengthscale=0.25)
rbf_model = GPy.models.GPRegression(intro_X, intro_Y, kernel=rbf_kernel)
rbf_samples = rbf_model.posterior_samples_f(intro_plot_X, size=20)
/usr/local/lib/python2.7/dist-packages/GPy/core/gp.py:456: RuntimeWarning:covariance is not positive-semidefinite.
fig, ax = plt.subplots(figsize=(8, 6))
ax.plot(intro_plot_x, rbf_samples, c=blue, alpha=0.75);
fig
$f \sim GP(m, k)$ if for any $\mathbf{x} = (x_1, \ldots, x_n)$, $(f(x_1), \ldots, f(x_n))$ has a multivariate normal distribution
If $f \sim GP(0, k)$, $\mathcal{D} = \{(x_1, y_1), \ldots, (x_n, y_n)\}$, where $y_i = f(x_i) + \varepsilon$, $\varepsilon \sim N(0, \sigma^2)$, $f\ |\ \mathcal{D}$ is also a Gaussian process with
rbf_model.Gaussian_noise.constrain_fixed(0.)
rbf_model.optimize()
rbf_post_mean, _ = rbf_model.predict(intro_plot_X)
rbf_post_low, rbf_post_high = rbf_model.predict_quantiles(intro_plot_X)
rbf_post_samples = rbf_model.posterior_samples_f(intro_plot_X, size=20)
WARNING: reconstraining parameters GP_regression.Gaussian_noise
fig, (l_ax, r_ax) = plt.subplots(ncols=2, sharex=True, sharey=True, figsize=(24, 9))
l_ax.plot(intro_plot_x, rbf_samples, c=blue, alpha=0.75);
r_ax.fill_between(intro_plot_x,
rbf_post_low.ravel(),
rbf_post_high.ravel(),
color='gray', alpha=0.25);
r_ax.plot(intro_plot_x, rbf_post_samples, c=blue, alpha=0.75);
r_ax.scatter(intro_X, intro_Y, s=100, c='k', label='Observations');
r_ax.plot(intro_plot_x, rbf_post_mean, '--', c='k', label='GP mean');
r_ax.set_xlim(-0.01, 1.01);
r_ax.legend();
fig
fig, axes = plt.subplots(nrows=2, ncols=2, sharex=True, figsize=(14, 10.5))
intro_hyperparams = [(1., 1.), (1., 0.25),
(1., 0.1), (1., 0.05)]
for ax, (sigma, l) in zip(axes.ravel(), intro_hyperparams):
rbf_kernel = GPy.kern.RBF(1., variance=sigma**2, lengthscale=l)
rbf_model = GPy.models.GPRegression(intro_X, intro_Y, kernel=rbf_kernel)
rbf_samples = rbf_model.posterior_samples_f(intro_plot_X, size=20)
ax.plot(intro_plot_x, rbf_samples, c=blue, alpha=0.75);
ax.set_title('$\\sigma = {}$, $\\ell = {}$'.format(sigma, l));
fig
fig, axes = plt.subplots(nrows=2, ncols=3, sharex=True, figsize=(14, 10.5))
intro_matern_hyperparams = [(0.5, 1., 0.5), (1.5, 1., 0.5), (2.5, 1., 0.5),
(0.5, 1., 0.5), (1.5, 1., 0.1), (2.5, 1., (0.1))]
matern_kernels = {0.5: GPy.kern.Exponential, 1.5: GPy.kern.Matern32, 2.5: GPy.kern.Matern52}
for ax, (nu, sigma, l) in zip(axes.ravel(), intro_matern_hyperparams):
matern_kernel = matern_kernels[nu](1., variance=sigma**2, lengthscale=l)
matern_model = GPy.models.GPRegression(intro_X, intro_Y, kernel=matern_kernel)
matern_samples = matern_model.posterior_samples_f(intro_plot_X, size=20)
ax.plot(intro_plot_x, matern_samples, c=blue, alpha=0.75);
ax.set_title('$\\nu = {}$, $\\sigma = {}$, $\\ell = {}$'.format(nu, sigma, l));
fig
def f(x):
return np.sin(x) + np.sin(10 / 3 * x)
x_min, x_max = 2.7, 7.5
X0 = np.atleast_2d(7.)
Y0 = f(X0)
fig, ax = plt.subplots(figsize=(8, 6))
plot_x = np.linspace(x_min, x_max, 200)
ax.plot(plot_x, f(plot_x), c='k', label='Actual function');
ax.scatter(X0, Y0, s=100, c=red, zorder=10, label='Current minimum');
ax.set_xlim(x_min, x_max);
ax.legend();
fig
def fit_gp_mle(X, Y):
kernel = GPy.kern.Matern52(1)
model = GPy.models.GPRegression(X, Y, kernel=kernel)
model.Gaussian_noise.constrain_fixed(0)
model.optimize()
return model
model = fit_gp_mle(X0, Y0)
plot_X = plot_x[:, np.newaxis]
mean, _ = model.predict(plot_X)
low, high = model.predict_quantiles(plot_X)
samples = model.posterior_samples_f(plot_X, size=20)
WARNING: reconstraining parameters GP_regression.Gaussian_noise
fig, ax = plt.subplots(figsize=(8, 6))
ax.fill_between(plot_x, low.ravel(), high.ravel(),
color='gray', alpha=0.25);
ax.plot(plot_x, f(plot_x), c='k', label='Actual function');
ax.plot(plot_x, mean, c=blue, label='GP mean');
ax.scatter(X0, Y0, s=100, c=red, zorder=10, label='Current minimum');
ax.set_xlim(x_min, x_max);
ax.legend();
fig
improvement = -np.minimum(0., samples - Y0)
fig, (f_ax, improv_ax) = plt.subplots(ncols=2, sharex=True, figsize=(16, 6))
f_ax.fill_between(plot_x, low.ravel(), high.ravel(),
color='gray', alpha=0.25);
f_ax.plot(plot_x, samples, c=blue, alpha=0.5);
f_ax.plot(plot_x, f(plot_x), c='k', label='Actual function');
f_ax.hlines(Y0, x_min, x_max, red, '--', lw=2, zorder=9);
f_ax.scatter(X0, Y0, s=100, c=red, zorder=10, label='Current minimum');
f_ax.set_xlim(x_min, x_max);
f_ax.legend();
improv_ax.plot(plot_x, improvement, c=blue);
improv_ax.set_ylabel('Sample improvement');
improv_ax.set_ylim(bottom=-0.01);
fig
sample_pi = (improvement > 0).mean(axis=1)
fig, (improv_ax, pi_ax) = plt.subplots(ncols=2, sharex=True, figsize=(16, 6))
improv_ax.plot(plot_x, improvement, c=blue);
improv_ax.set_xlim(x_min, x_max);
improv_ax.set_ylabel('Sample improvement');
improv_ax.set_ylim(bottom=-0.01);
pi_ax.plot(plot_x, sample_pi, c=blue);
pi_ax.set_ylabel('Sample probability of improvement');
fig
def probability_of_improvement(y_min, model, X_grid):
mean, var = model.predict(X_grid)
z = (y_min - mean) / np.sqrt(var)
return stats.norm.cdf(z)
Step = namedtuple('Step', ['X_next', 'acq'])
def do_step(model, X0, Y0, X_grid, acq_func):
acq = acq_func(Y0.min(), model, X_grid)
next_ix = acq.argmax()
X_next = X_grid[next_ix]
return Step(X_next, acq)
def func_plot(model, X_next, f, ax):
X, Y = model.X, model.Y
mean, _ = model.predict(plot_X)
low, high = model.predict_quantiles(plot_X)
ax.fill_between(plot_x, low.ravel(), high.ravel(),
color='gray', alpha=0.25);
ax.plot(plot_x, mean, c=blue, label='GP mean');
ax.plot(plot_x, f(plot_x), c='k', label='Actual function');
ax.scatter(X, Y, s=100, c=blue, zorder=10, label='Current minimum');
ax.scatter(X[Y.argmin()], Y.min(), s=100, c=red, zorder=10, label='Current minimum');
ax.scatter(X_next, f(X_next),
s=100, c=purple, zorder=10, label='Next evaluation point');
ax.set_xlim(x_min, x_max);
ax.legend(loc=2);
def step_plot(model, step, f, axes=None):
if axes is None:
fig, (f_ax, acq_ax) = plt.subplots(ncols=2, sharex=True, figsize=(16, 6))
else:
fig, (f_ax, acq_ax) = plt.gcf(), axes
func_plot(model, step.X_next, f, f_ax)
acq_ax.plot(plot_x, step.acq, c='k', label='Exact');
acq_ax.set_ylim(bottom=0.);
return fig, (f_ax, acq_ax)
pi_step = do_step(model, X0, Y0, plot_X, probability_of_improvement)
fig, (f_ax, pi_ax) = step_plot(model, pi_step, f)
pi_ax.plot(plot_x, sample_pi, label='Sample');
pi_ax.set_ylabel('Probability of improvement');
pi_ax.legend();
fig
sample_ei = improvement.mean(axis=1)
fig, (improv_ax, ei_ax) = plt.subplots(ncols=2, sharex=True, figsize=(16, 6))
improv_ax.plot(plot_x, improvement, c=blue);
improv_ax.set_xlim(x_min, x_max);
improv_ax.set_ylabel('Sample improvement');
improv_ax.set_ylim(bottom=-0.01);
ei_ax.plot(plot_x, sample_ei, c=blue);
ei_ax.set_ylabel('Sample expected improvement');
fig
def expected_improvement_from_params(y_min, mean, var):
z = (y_min - mean) / np.sqrt(var)
return np.sqrt(var) * (z * stats.norm.cdf(z) + stats.norm.pdf(z))
def expected_improvement(y_min, model, X_grid):
mean, var = model.predict(X_grid)
return expected_improvement_from_params(y_min, mean, var)
ei_step = do_step(model, X0, Y0, plot_X, expected_improvement)
fig, (f_ax, ei_ax) = step_plot(model, ei_step, f)
ei_ax.plot(plot_x, sample_ei, label='Sample');
ei_ax.set_ylabel('Expected improvement');
ei_ax.legend();
fig
steps = []
X, Y = X0, Y0
for _ in xrange(15):
step = do_step(model, X, Y, plot_X, expected_improvement)
Y_next = f(step.X_next)
steps.append((model, step))
X = np.concatenate((X, step.X_next[:, np.newaxis]))
Y = np.concatenate((Y, Y_next[:, np.newaxis]))
model = fit_gp_mle(X, Y)
WARNING: reconstraining parameters GP_regression.Gaussian_noise WARNING: reconstraining parameters GP_regression.Gaussian_noise WARNING: reconstraining parameters GP_regression.Gaussian_noise WARNING: reconstraining parameters GP_regression.Gaussian_noise WARNING: reconstraining parameters GP_regression.Gaussian_noise WARNING: reconstraining parameters GP_regression.Gaussian_noise WARNING: reconstraining parameters GP_regression.Gaussian_noise WARNING: reconstraining parameters GP_regression.Gaussian_noise WARNING: reconstraining parameters GP_regression.Gaussian_noise WARNING: reconstraining parameters GP_regression.Gaussian_noise WARNING: reconstraining parameters GP_regression.Gaussian_noise WARNING: reconstraining parameters GP_regression.Gaussian_noise WARNING: reconstraining parameters GP_regression.Gaussian_noise WARNING: reconstraining parameters GP_regression.Gaussian_noise WARNING: reconstraining parameters GP_regression.Gaussian_noise
fig, (f_ax, ei_ax) = step_plot(steps[0][0], steps[0][1], f);
ei_ax.set_ylabel('Expected improvement');
fig
fig, (f_ax, ei_ax) = step_plot(steps[1][0], steps[1][1], f);
ei_ax.set_ylabel('Expected improvement');
fig
fig, (f_ax, ei_ax) = step_plot(steps[2][0], steps[2][1], f);
ei_ax.set_ylabel('Expected improvement');
fig
fig, (f_ax, ei_ax) = step_plot(steps[3][0], steps[3][1], f);
ei_ax.set_ylabel('Expected improvement');
fig
fig, (f_ax, ei_ax) = step_plot(steps[4][0], steps[4][1], f);
ei_ax.set_ylabel('Expected improvement');
fig
fig, (f_ax, ei_ax) = step_plot(steps[5][0], steps[5][1], f);
ei_ax.set_ylabel('Expected improvement');
fig
%mkdir -p /vagrant/tmp/dataphilly/ei/
%rm /vagrant/tmp/dataphilly/ei/*
for i, (model, step) in enumerate(steps):
fig, (_, ei_ax) = step_plot(model, step, f)
ei_ax.set_ylabel('Expected improvement');
fig.savefig('/vagrant/tmp/dataphilly/ei/{:02}.png'.format(i));
!convert -delay 200 -loop 0 /vagrant/tmp/dataphilly/ei/* /vagrant/tmp/dataphilly/ei.gif
with open('/vagrant/tmp/dataphilly/ei.gif', 'rb') as gif_data:
gif = Image(gif_data.read())
display(gif)
fig
ei_min = np.array([model.Y.min() for model, _ in steps])
ei_n_steps = np.arange(ei_min.size) + 1
fig, ax = plt.subplots(figsize=(8, 6))
ax.plot(ei_n_steps, ei_min);
ax.set_xlim(1, ei_n_steps.max());
ax.set_xlabel('Step');
ax.set_ylabel('Minimum function value');
fig
So far, we have chose the covariance hyperparameters $\sigma^2$ and $\ell$ by maximizing the marginal (log) likelihood
$$ \begin{align*} \log \mathbb{P}(\mathbf{y}\ |\ \mathbf{x}, \theta) & \propto -\frac{1}{2} \mathbf{y}^{\intercal} \left(k(\mathbf{x}, \mathbf{x}) + \sigma^2 I\right)^{-1} \mathbf{y} - \frac{1}{2} \log \left|k(\mathbf{x}, \mathbf{x}) + \sigma^2 I\right| \end{align*} $$This can result in overfitting and underestimating the uncertainty in our acquisition function.
def get_model(X, Y):
kernel = GPy.kern.Matern52(1)
kernel.lengthscale.set_prior(GPy.priors.HalfT(1., 4.))
kernel.variance.set_prior(GPy.priors.HalfT(1., 4.))
model = GPy.models.GPRegression(X, Y, kernel)
model.Gaussian_noise.constrain_fixed(0)
return model
def sample_params(X, Y, n_samples):
model = get_model(X, Y)
hmc = GPy.inference.mcmc.HMC(model)
return hmc.sample(n_samples)
def sample_models(X, Y, n_samples):
samples = sample_params(X, Y, n_samples)
sample_models = []
for k_var, k_length in samples:
sample_model = get_model(X, Y)
sample_model.kern.variance.constrain_fixed(k_var)
sample_model.kern.lengthscale.constrain_fixed(k_length)
sample_models.append(sample_model)
return sample_models
def get_expected_improvement_samples(X, Y, X_grid, n_samples):
samples = sample_models(X, Y, n_samples)
n_grid_pts = X_grid.shape[0]
mus = np.empty((n_samples, n_grid_pts))
vars_ = np.empty((n_samples, n_grid_pts))
for i, model in enumerate(samples):
mus[i], vars_[i] = map(np.ravel, model.predict(X_grid))
eis = expected_improvement_from_params(Y.min(), mus, vars_)
return eis
def get_integrated_expected_improvement(X, Y, X_grid, n_samples=100):
eis = get_expected_improvement_samples(X, Y, X_grid, n_samples)
integrated_ei = eis.mean(axis=0)
# sometimes the expected improvement will be nan at a point we have
# already evaluated
integrated_ei[np.isnan(integrated_ei)] = 0
return integrated_ei
%%time
samples = sample_params(X0, Y0, 100)
WARNING: reconstraining parameters Mat52.lengthscale WARNING: reconstraining parameters Mat52.variance WARNING: reconstraining parameters GP_regression.Gaussian_noise CPU times: user 2.63 s, sys: 0 ns, total: 2.63 s Wall time: 2.63 s
grid = sns.jointplot(samples[:, 0], samples[:, 1], stat_func=None)
grid.ax_joint.set_xscale('log');
grid.ax_joint.set_yscale('log');
grid.set_axis_labels(r'$2\ \log\ \sigma$', '$\log\ \ell$');
grid.fig
%%time
ei_samples = get_expected_improvement_samples(X0, Y0, plot_X, 10)
WARNING: reconstraining parameters Mat52.lengthscale WARNING: reconstraining parameters Mat52.variance WARNING: reconstraining parameters GP_regression.Gaussian_noise WARNING: reconstraining parameters Mat52.lengthscale WARNING: reconstraining parameters Mat52.variance WARNING: reconstraining parameters GP_regression.Gaussian_noise WARNING: reconstraining parameters GP_regression.Mat52.variance WARNING: reconstraining parameters GP_regression.Mat52.lengthscale WARNING: reconstraining parameters Mat52.lengthscale WARNING: reconstraining parameters Mat52.variance WARNING: reconstraining parameters GP_regression.Gaussian_noise WARNING: reconstraining parameters GP_regression.Mat52.variance WARNING: reconstraining parameters GP_regression.Mat52.lengthscale WARNING: reconstraining parameters Mat52.lengthscale WARNING: reconstraining parameters Mat52.variance WARNING: reconstraining parameters GP_regression.Gaussian_noise WARNING: reconstraining parameters GP_regression.Mat52.variance WARNING: reconstraining parameters GP_regression.Mat52.lengthscale WARNING: reconstraining parameters Mat52.lengthscale WARNING: reconstraining parameters Mat52.variance WARNING: reconstraining parameters GP_regression.Gaussian_noise WARNING: reconstraining parameters GP_regression.Mat52.variance WARNING: reconstraining parameters GP_regression.Mat52.lengthscale WARNING: reconstraining parameters Mat52.lengthscale WARNING: reconstraining parameters Mat52.variance WARNING: reconstraining parameters GP_regression.Gaussian_noise WARNING: reconstraining parameters GP_regression.Mat52.variance WARNING: reconstraining parameters GP_regression.Mat52.lengthscale WARNING: reconstraining parameters Mat52.lengthscale WARNING: reconstraining parameters Mat52.variance WARNING: reconstraining parameters GP_regression.Gaussian_noise WARNING: reconstraining parameters GP_regression.Mat52.variance WARNING: reconstraining parameters GP_regression.Mat52.lengthscale WARNING: reconstraining parameters Mat52.lengthscale WARNING: reconstraining parameters Mat52.variance WARNING: reconstraining parameters GP_regression.Gaussian_noise WARNING: reconstraining parameters GP_regression.Mat52.variance WARNING: reconstraining parameters GP_regression.Mat52.lengthscale WARNING: reconstraining parameters Mat52.lengthscale WARNING: reconstraining parameters Mat52.variance WARNING: reconstraining parameters GP_regression.Gaussian_noise WARNING: reconstraining parameters GP_regression.Mat52.variance WARNING: reconstraining parameters GP_regression.Mat52.lengthscale WARNING: reconstraining parameters Mat52.lengthscale WARNING: reconstraining parameters Mat52.variance WARNING: reconstraining parameters GP_regression.Gaussian_noise WARNING: reconstraining parameters GP_regression.Mat52.variance WARNING: reconstraining parameters GP_regression.Mat52.lengthscale WARNING: reconstraining parameters Mat52.lengthscale WARNING: reconstraining parameters Mat52.variance WARNING: reconstraining parameters GP_regression.Gaussian_noise WARNING: reconstraining parameters GP_regression.Mat52.variance WARNING: reconstraining parameters GP_regression.Mat52.lengthscale CPU times: user 426 ms, sys: 7.83 ms, total: 434 ms Wall time: 431 ms
model, ei_step = steps[0]
fig, (_, ei_ax) = step_plot(model, ei_step, f)
ei_ax.plot(plot_x, ei_samples.T, c=blue);
ei_ax.plot(plot_x, ei_samples.mean(axis=0).T, '--', c='k', label='Integrated expected improvement');
ei_ax.set_ylim(0, ei_samples.max());
ei_ax.legend(loc=2);
fig
model, ei_step = steps[0]
fig, (_, ei_ax) = step_plot(model, ei_step, f)
ei_ax.plot(plot_x, ei_samples.mean(axis=0), c=blue, label='Integrated expected improvement');
ei_ax.set_ylim(0, ei_samples.mean(axis=0).max());
ei_ax.legend();
def do_int_ei_step(X0, Y0, X_grid, n_samples=100):
ei = get_integrated_expected_improvement(X0, Y0, X_grid, n_samples=n_samples)
next_ix = ei.argmax()
X_next = X_grid[next_ix]
return Step(X_next, ei)
%%time
int_ei_steps = []
X, Y = X0, Y0
model = fit_gp_mle(X0, Y0)
for _ in xrange(15):
step = do_int_ei_step(X, Y, plot_X)
Y_next = f(step.X_next)
int_ei_steps.append((model, step))
X = np.concatenate((X, step.X_next[:, np.newaxis]))
Y = np.concatenate((Y, Y_next[:, np.newaxis]))
model = fit_gp_mle(X, Y)
WARNING: reconstraining parameters GP_regression.Gaussian_noise WARNING: reconstraining parameters Mat52.lengthscale WARNING: reconstraining parameters Mat52.variance WARNING: reconstraining parameters GP_regression.Gaussian_noise WARNING: reconstraining parameters Mat52.lengthscale WARNING: reconstraining parameters Mat52.variance WARNING: reconstraining parameters GP_regression.Gaussian_noise WARNING: reconstraining parameters GP_regression.Mat52.variance WARNING: reconstraining parameters GP_regression.Mat52.lengthscale WARNING: reconstraining parameters Mat52.lengthscale WARNING: reconstraining parameters Mat52.variance WARNING: reconstraining parameters GP_regression.Gaussian_noise WARNING: reconstraining parameters GP_regression.Mat52.variance WARNING: reconstraining parameters GP_regression.Mat52.lengthscale WARNING: reconstraining parameters Mat52.lengthscale WARNING: reconstraining parameters Mat52.variance WARNING: reconstraining parameters GP_regression.Gaussian_noise WARNING: reconstraining parameters GP_regression.Mat52.variance WARNING: reconstraining parameters GP_regression.Mat52.lengthscale WARNING: reconstraining parameters Mat52.lengthscale WARNING: reconstraining parameters Mat52.variance WARNING: reconstraining parameters GP_regression.Gaussian_noise WARNING: reconstraining parameters GP_regression.Mat52.variance WARNING: reconstraining parameters GP_regression.Mat52.lengthscale WARNING: reconstraining parameters Mat52.lengthscale WARNING: reconstraining parameters Mat52.variance WARNING: reconstraining parameters GP_regression.Gaussian_noise WARNING: reconstraining parameters GP_regression.Mat52.variance WARNING: reconstraining parameters GP_regression.Mat52.lengthscale WARNING: reconstraining parameters Mat52.lengthscale WARNING: reconstraining parameters Mat52.variance WARNING: reconstraining parameters GP_regression.Gaussian_noise WARNING: reconstraining parameters GP_regression.Mat52.variance WARNING: reconstraining parameters GP_regression.Mat52.lengthscale WARNING: reconstraining parameters Mat52.lengthscale WARNING: reconstraining parameters Mat52.variance WARNING: reconstraining parameters GP_regression.Gaussian_noise WARNING: reconstraining parameters GP_regression.Mat52.variance WARNING: reconstraining parameters GP_regression.Mat52.lengthscale WARNING: reconstraining parameters Mat52.lengthscale WARNING: reconstraining parameters Mat52.variance WARNING: reconstraining parameters GP_regression.Gaussian_noise WARNING: reconstraining parameters GP_regression.Mat52.variance WARNING: reconstraining parameters GP_regression.Mat52.lengthscale WARNING: reconstraining parameters Mat52.lengthscale WARNING: reconstraining parameters Mat52.variance WARNING: reconstraining parameters GP_regression.Gaussian_noise WARNING: reconstraining parameters GP_regression.Mat52.variance WARNING: reconstraining parameters GP_regression.Mat52.lengthscale WARNING: reconstraining parameters Mat52.lengthscale WARNING: 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WARNING: reconstraining parameters GP_regression.Mat52.variance WARNING: reconstraining parameters GP_regression.Mat52.lengthscale WARNING: reconstraining parameters Mat52.lengthscale WARNING: reconstraining parameters Mat52.variance WARNING: reconstraining parameters GP_regression.Gaussian_noise WARNING: reconstraining parameters GP_regression.Mat52.variance WARNING: reconstraining parameters GP_regression.Mat52.lengthscale WARNING: reconstraining parameters Mat52.lengthscale WARNING: reconstraining parameters Mat52.variance WARNING: reconstraining parameters GP_regression.Gaussian_noise WARNING: reconstraining parameters GP_regression.Mat52.variance WARNING: reconstraining parameters GP_regression.Mat52.lengthscale WARNING: reconstraining parameters Mat52.lengthscale WARNING: reconstraining parameters Mat52.variance WARNING: reconstraining parameters GP_regression.Gaussian_noise WARNING: reconstraining parameters GP_regression.Mat52.variance WARNING: reconstraining parameters 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%mkdir -p /vagrant/tmp/dataphilly/int_ei/
%rm /vagrant/tmp/dataphilly/int_ei/*
for i, (model, step) in enumerate(int_ei_steps):
fig, (_, ei_ax) = step_plot(model, step, f)
ei_ax.set_ylabel('Integrated expected improvement');
fig.savefig('/vagrant/tmp/dataphilly/int_ei/{:02}.png'.format(i));
!convert -delay 200 -loop 0 /vagrant/tmp/dataphilly/int_ei/* /vagrant/tmp/dataphilly/int_ei.gif
with open('/vagrant/tmp/dataphilly/int_ei.gif', 'rb') as gif_data:
gif = Image(gif_data.read())
display(gif)
step_ix = 0
fig, (ei_ax, int_ei_ax) = plt.subplots(ncols=2, sharex=True, sharey=True, figsize=(16, 8))
ei_model, ei_step = steps[step_ix]
func_plot(ei_model, ei_step.X_next, f, ei_ax)
ei_ax.set_title('Expected improvement');
int_ei_model, int_ei_step = int_ei_steps[step_ix]
func_plot(int_ei_model, int_ei_step.X_next, f, int_ei_ax)
int_ei_ax.set_title('Integrated expected improvement');
fig
step_ix = 1
fig, (ei_ax, int_ei_ax) = plt.subplots(ncols=2, sharex=True, sharey=True, figsize=(16, 8))
ei_model, ei_step = steps[step_ix]
func_plot(ei_model, ei_step.X_next, f, ei_ax)
ei_ax.set_title('Expected improvement');
int_ei_model, int_ei_step = int_ei_steps[step_ix]
func_plot(int_ei_model, int_ei_step.X_next, f, int_ei_ax)
int_ei_ax.set_title('Integrated expected improvement');
step_ix = 2
fig, (ei_ax, int_ei_ax) = plt.subplots(ncols=2, sharex=True, sharey=True, figsize=(16, 8))
ei_model, ei_step = steps[step_ix]
func_plot(ei_model, ei_step.X_next, f, ei_ax)
ei_ax.set_title('Expected improvement');
int_ei_model, int_ei_step = int_ei_steps[step_ix]
func_plot(int_ei_model, int_ei_step.X_next, f, int_ei_ax)
int_ei_ax.set_title('Integrated expected improvement');
fig
int_ei_min = np.array([model.Y.min() for model, _ in int_ei_steps])
fig, ax = plt.subplots(figsize=(8, 6))
ax.plot(ei_n_steps, ei_min,
label='Expected improvement');
ax.plot(ei_n_steps, int_ei_min,
label='Integrated expected improvement');
ax.set_xlim(1, ei_n_steps.max());
ax.set_xlabel('Step');
ax.set_ylabel('Minimum function value');
ax.legend();
fig
spearmint
open source Bayesian optimization packagemonetate_img = Image(url='https://tctechcrunch2011.files.wordpress.com/2011/08/040811112223monetate.jpg',
width=600)
monetate_img
arochford@monetate.com