def update_plot(val=None):
sigma = 2**s_sigma.val
lmbda = 2**s_lmbda.val
K = gaussian_kernel(Z, sigma=sigma)
b = _compute_b_sym(Z, K, sigma)
C = _compute_C_sym(Z, K, sigma)
a = score_matching_sym(Z, sigma, lmbda, K, b, C)
J = _objective_sym(Z, sigma, lmbda, a, K, b, C)
J_xval = np.mean(xvalidate(Z, 5, sigma, lmbda, K, num_repetitions=3))
print(a[:5])
kernel = lambda X, Y=None: gaussian_kernel(X, Y, sigma=sigma)
kernel_grad = lambda x, X=None: gaussian_kernel_grad(x, X, sigma)
logq_est = lambda x: log_pdf_estimate(x, a, Z, kernel)
dlogq_est = lambda x: log_pdf_estimate_grad(x, a, Z, kernel_grad)
description = "N=%d, sigma: %.2f, lambda: %.2f, J(a)=%.2f, XJ(a)=%.2f" % \
(N, sigma, lmbda, J, J_xval)
if plot_pdf:
D = evaluate_density_grid(Xs, Ys, logq_est)
description = "log-pdf: " + description
else:
D = evaluate_density_grad_grid(Xs, Ys, dlogq_est)
description = "norm-grad-log-pdf: " + description
ax.clear()
ax.plot(Z[:, 0], Z[:, 1], 'bx')
plot_array(Xs, Ys, D, ax, plot_contour=True)
ax.set_title(description)
fig.canvas.draw_idle()
def callback_sigma_lmbda(model, bounds, info, x, index, ftrue):
global D
fig = pl.figure(1)
fig.clf()
sigma = 2**info[-1]['xbest'][0]
lmbda = 2**info[-1]['xbest'][1]
pl.title("sigma=%.2f, lmbda=%.6f" % (sigma, lmbda))
gamma = 0.5 * (sigma**2)
omega, u = sample_basis(D, m, gamma)
theta = score_matching_sym(Z, lmbda, omega, u)
logq_est = lambda x: np.dot(theta, feature_map_single(x, omega, u))
dlogq_est = lambda x: np.dot(theta, feature_map_grad_single(x, omega, u))
Xs = np.linspace(-3, 3)
Ys = np.linspace(-3, 3)
Q = evaluate_density_grid(Xs, Ys, logq_est)
plot_array(Xs, Ys, Q, pl.gca(), plot_contour=True)
pl.plot(Z[:, 0], Z[:, 1], 'bx')
for ax in fig.axes: # remove tick labels
ax.set_xticklabels([])
ax.set_yticklabels([])
pl.draw()
pl.show(block=False)
def update_plot(val=None):
global omega, u
print("Updating plot")
lmbda = 2**s_lmbda.val
sigma = 2**s_sigma.val
b = compute_b(Z, omega, u)
C = compute_C(Z, omega, u)
theta = score_matching_sym(Z, lmbda, omega, u, b, C)
J = objective(Z, theta, lmbda, omega, u, b, C)
J_xval = np.mean(
xvalidate(Z, lmbda, omega, u, n_folds=5, num_repetitions=3))
logq_est = lambda x: np.dot(theta, feature_map_single(x, omega, u))
dlogq_est = lambda x: np.dot(theta, feature_map_grad_single(x, omega, u))
description = "N=%d, sigma: %.2f, lambda: %.2f, m=%.d, J=%.2f, J_xval=%.2f" % \
(N, sigma, lmbda, m, J, J_xval)
if plot_pdf:
D = evaluate_density_grid(Xs, Ys, logq_est)
description = "log-pdf: " + description
else:
D = evaluate_density_grad_grid(Xs, Ys, dlogq_est)
description = "norm-grad-log-pdf: " + description
ax.clear()
ax.plot(Z[:, 0], Z[:, 1], 'bx')
plot_array(Xs, Ys, D, ax, plot_contour=True)
ax.set_title(description)
fig.canvas.draw_idle()
def plot_true():
G = evaluate_density_grid(Xs, Ys, logq)
G_grad = evaluate_density_grad_grid(Xs, Ys, dlogq)
plt.figure(figsize=(12, 4))
plt.subplot(121)
plot_array(Xs, Ys, G, plot_contour=True)
plt.plot(Z[:, 0], Z[:, 1], 'bx')
plt.title("True log-pdf")
plt.subplot(122)
plot_array(Xs, Ys, G_grad, plot_contour=True)
plt.plot(Z[:, 0], Z[:, 1], 'bx')
plt.title("True gradient norm log-pdf")
def set_up(self):
if self.learn_parameters or self.force_relearn_parameters:
self.sigma, self.lmbda = self.determine_sigma_lmbda()
logger.info("Using sigma=%.2f, lmbda=%.6f" % (self.sigma, self.lmbda))
gamma = 0.5 * (self.sigma**2)
logger.info("Sampling random basis")
omega, u = sample_basis(self.D, self.m, gamma)
logger.info("Estimating density in RKHS, N=%d, m=%d, D=%d" %
(len(self.Z), self.m, self.D))
theta = score_matching_sym(self.Z, self.lmbda, omega, u)
# replace target by kernel estimator to simulate trajectories on
# but keep original target for computing acceptance probability
self.orig_target = self.target
self.target = RandomFeatsEstimator(theta, omega, u)
HMCJob.set_up(self)
# plot density estimate
if self.plot:
import matplotlib.pyplot as plt
from scripts.tools.plotting import evaluate_density_grid, evaluate_gradient_grid, plot_array
Xs = np.linspace(-15, 15)
Ys = np.linspace(-7, 3)
Xs_grad = np.linspace(-40, 40, 40)
Ys_grad = np.linspace(-15, 25, 40)
G = evaluate_density_grid(Xs, Ys, self.target.log_pdf)
G_norm, quiver_U, quiver_V, _, _ = evaluate_gradient_grid(
Xs_grad, Ys_grad, self.target.grad)
plt.subplot(211)
plt.plot(self.Z[:, 0], self.Z[:, 1], 'bx')
plot_array(Xs, Ys, np.exp(G), plot_contour=True)
plt.subplot(212)
plot_array(Xs_grad, Ys_grad, G_norm, plot_contour=True)
plt.quiver(Xs_grad, Ys_grad, quiver_U, quiver_V, color='m')
plt.ioff()
plt.show()
theta = score_matching_sym(Z, lmbda, omega, u)
logq_est = lambda x: log_pdf_estimate(feature_map_single(x, omega, u), theta)
dlogq_est = lambda x: log_pdf_estimate_grad(feature_map_grad_single(x, omega, u),
theta)
# plot density estimate
plt.figure(figsize=(4, 8))
Xs = np.linspace(-15, 15)
Ys = np.linspace(-7, 3)
Xs_grad = np.linspace(-40, 40, 40)
Ys_grad = np.linspace(-15, 25, 40)
G = evaluate_density_grid(Xs, Ys, logq_est)
G_norm, quiver_U, quiver_V, _, _ = evaluate_gradient_grid(Xs_grad, Ys_grad, dlogq_est)
plt.subplot(211)
plt.plot(Z[:, 0], Z[:, 1], 'bx')
plot_array(Xs, Ys, np.exp(G), plot_contour=True)
plt.subplot(212)
plot_array(Xs_grad, Ys_grad, G_norm, plot_contour=True)
plt.quiver(Xs_grad, Ys_grad, quiver_U, quiver_V, color='m')
plt.show()
# plain MCMC parameters
thin = 10
num_warmup = 10
num_iterations = 1000 + num_warmup * thin
q_current = np.array([0., -3.]) + 50
q_current_est = q_current
# hmc parameters
num_steps_min = 10
num_steps_max = 100
step_size_min = 0.05
def callback_lmbda(model, bounds, info, x, index, ftrue):
global D
"""
Plot the current posterior, the index, and the value of the current
recommendation.
"""
xmin, xmax = bounds[0]
xx_ = np.linspace(xmin, xmax, 500) # define grid
xx = xx_[:, None]
# ff = ftrue(xx) # compute true function
acq = index(xx) # compute acquisition
mu, s2 = model.posterior(xx) # compute posterior and
lo = mu - 2 * np.sqrt(s2) # quantiles
hi = mu + 2 * np.sqrt(s2)
# ymin, ymax = ff.min(), ff.max() # get plotting ranges
# ymin -= 0.2 * (ymax - ymin)
# ymax += 0.2 * (ymax - ymin)
kwplot = {'lw': 2, 'alpha': 0.5} # common plotting kwargs
fig = pl.figure(1)
fig.clf()
pl.subplot(221)
# pl.plot(xx, ff, 'k:', **kwplot) # plot true function
pl.plot(xx, mu, 'b-', **kwplot) # plot the posterior and
pl.fill_between(xx_, lo, hi, color='b', alpha=0.1) # uncertainty bands
pl.scatter(
info['x'],
info['y'], # plot data
marker='o',
facecolor='none',
zorder=3)
pl.axvline(x, color='r', **kwplot) # latest selection
pl.axvline(info[-1]['xbest'], color='g',
**kwplot) # current recommendation
# pl.axis((xmin, xmax, ymin, ymax))
pl.ylabel('posterior')
pl.subplot(223)
pl.fill_between(
xx_,
acq.min(),
acq, # plot acquisition
color='r',
alpha=0.1)
pl.axis('tight')
pl.axvline(x, color='r', **kwplot) # plot latest selection
pl.xlabel('input')
pl.ylabel('acquisition')
pl.subplot(224)
pl.plot(info['x'], 'g')
pl.subplot(222)
lmbda = 2**info[-1]['xbest']
gamma = 0.5 * (sigma**2)
omega, u = sample_basis(D, m, gamma)
theta = score_matching_sym(Z, lmbda, omega, u)
logq_est = lambda x: np.dot(theta, feature_map_single(x, omega, u))
dlogq_est = lambda x: np.dot(theta, feature_map_grad_single(x, omega, u))
Xs = np.linspace(-3, 3)
Ys = np.linspace(-3, 3)
Q = evaluate_density_grid(Xs, Ys, logq_est)
plot_array(Xs, Ys, Q, pl.gca(), plot_contour=True)
pl.plot(Z[:, 0], Z[:, 1], 'bx')
for ax in fig.axes: # remove tick labels
ax.set_xticklabels([])
ax.set_yticklabels([])
pl.draw()
pl.show(block=False)
def propose(self, current, current_log_pdf, samples, accepted):
# random variables from a fixed random stream without modifying the current one
rnd_state = np.random.get_state()
np.random.set_state(self.hmc_rnd_state)
if current_log_pdf is None:
current_log_pdf = self.orig_target.log_pdf(current)
# sample momentum and leapfrog parameters
p0 = self.momentum.sample()
num_steps = np.random.randint(self.num_steps_min, self.num_steps_max + 1)
step_size = np.random.rand() * (self.step_size_max - self.step_size_min) + self.step_size_min
# restore random state
self.hmc_rnd_state = np.random.get_state()
np.random.set_state(rnd_state)
logger.debug("Simulating Hamiltonian flow")
Qs, Ps = leapfrog(current, self.target.grad, p0, self.momentum.grad, step_size, num_steps)
q=Qs[-1]
p=Ps[-1]
logger.debug("Momentum start: %s" % str(p0))
logger.debug("Momentum end: %s" % str(p))
# compute acceptance probability, extracting log_pdf of q
p0_log_pdf = self.momentum.log_pdf(p0)
p_log_pdf = self.momentum.log_pdf(p)
# use a function call to be able to overload it for KMC
acc_prob, log_pdf_q = self.accept_prob_log_pdf(current, q, p0_log_pdf, p_log_pdf, current_log_pdf, samples)
if True and (len(samples) % 100) ==0:
logger.debug("Plotting")
import matplotlib.pyplot as plt
res = 50
Xs_q = np.linspace(-4,4, res)
Ys_q = np.linspace(-4,4, res)
# evaluate density and estimate
D1=0
D2=1
def dummy_grad(X_2d):
theta = current.copy()
# theta = np.mean(self.Z, 0)
theta[D1]=X_2d[0]
theta[D2]=X_2d[1]
return self.target.grad(theta)
def dummy(X_2d):
theta = current.copy()
# theta = np.mean(self.Z, 0)
theta[D1]=X_2d[0]
theta[D2]=X_2d[1]
return self.target.log_pdf(theta)
# plt.figure()
# G = evaluate_density_grid(Xs_q, Ys_q, dummy)
# plot_array(Xs_q, Ys_q, G)
# plt.plot(self.Z[:,D1], self.Z[:,D2], '.')
# plt.plot(Qs[:,D1], Qs[:,D2], 'r-')
# plt.plot(samples[:,D1], samples[:,D2], 'm-')
# plt.plot(current[D1], current[D2], 'b*', markersize=15)
# plt.plot(Qs[-1,D1], Qs[-1,D2], 'r*', markersize=15)
plt.figure()
G_norm, U_q, V, X, Y = evaluate_gradient_grid(Xs_q, Ys_q, dummy_grad)
plot_array(Xs_q, Ys_q, G_norm)
plt.plot(self.Z[:,D1], self.Z[:,D2], '.')
plt.plot(Qs[:,D1], Qs[:,D2], 'r-')
plt.plot(samples[:,D1], samples[:,D2], 'm-')
plt.plot(current[D1], current[D2], 'b*', markersize=15)
plt.plot(Qs[-1,D1], Qs[-1,D2], 'r*', markersize=15)
plt.quiver(X, Y, U_q, V, color='m')
# plt.figure()
# plt.plot(Ps[:,D1], Ps[:,D2], 'r-')
# plt.plot(p0[D1], p0[D2], 'b*', markersize=15)
# plt.plot(Ps[-1,D1], Ps[-1,D2], 'r*', markersize=15)
# plt.title('momentum')
acc_probs = np.exp(compute_log_accept_pr(current, p0, Qs, Ps, self.orig_target.log_pdf, self.momentum.log_pdf))
H_ratios = np.exp(compute_log_accept_pr(current, p0, Qs, Ps, self.target.log_pdf, self.momentum.log_pdf))
target_ratio = [np.min([1,np.exp(self.orig_target.log_pdf(x)-current_log_pdf)]) for x in Qs]
momentum_ratio = [np.min([1,np.exp(self.momentum.log_pdf(x)-p0_log_pdf)]) for x in Ps]
target_log_pdf = np.exp(np.array([self.orig_target.log_pdf(x) for x in Qs]))
# #
# plt.figure(figsize=(12,4))
# plt.subplot(151)
# plt.plot(acc_probs)
# plt.plot([0, len(acc_probs)], [acc_probs.mean(), acc_probs.mean()])
# plt.title("acc_probs")
# plt.subplot(152)
# plt.plot(target_ratio)
# plt.title("target_ratio")
# plt.subplot(153)
# plt.plot(momentum_ratio)
# plt.title("momentum_ratio")
# plt.subplot(154)
# plt.plot(H_ratios)
# plt.title("H_ratios")
# plt.subplot(155)
# plt.plot(target_log_pdf)
# plt.title("target_log_pdf")
plt.show()
return q, acc_prob, log_pdf_q
# simulate true and approximate Hamiltonian
Qs, Ps = leapfrog(q0, dlogq, p0, dlogp, step_size, num_steps)
Qs_est, Ps_est = leapfrog(q0, dlogq_est, p0, dlogp, step_size, num_steps)
Hs = compute_hamiltonian(Qs, Ps, logq, logp)
Hs_est = compute_hamiltonian(Qs_est, Ps_est, logq, logp)
# compute acceptance probabilities
log_acc = compute_log_accept_pr(q0, p0, Qs, Ps, logq, logp)
log_acc_est = compute_log_accept_pr(q0, p0, Qs_est, Ps_est, logq, logp)
# normalise Hamiltonians
Hs -= Hs.mean()
Hs_est -= Hs_est.mean()
plt.figure()
plot_array(Xs_q, Ys_q, np.exp(G))
plot_2d_trajectory(Qs)
plt.title("HMC")
plt.gca().xaxis.set_visible(False)
plt.gca().yaxis.set_visible(False)
plt.savefig(fname_base + "_hmc.eps", axis_inches="tight")
plt.figure()
plot_array(Xs_q, Ys_q, np.exp(G_est))
plt.plot(Z[:, 0], Z[:, 1], 'bx')
plot_2d_trajectory(Qs_est)
plt.title("KMC")
plt.gca().xaxis.set_visible(False)
plt.gca().yaxis.set_visible(False)
plt.savefig(fname_base + "_kmc.eps", axis_inches="tight")
# sample data point and predict
f = predict(x, alphas[:i]) * phi_x
# gradient of f at x
f_grad = feature_map_grad_single(x, omega, u) * f
# gradient
grad = 0
for d in range(D):
phi_derivative_d = feature_map_derivative_d(x, omega, u, d)
phi_derivative2_d = feature_map_derivative2_d(x, omega, u, d)
grad += phi_derivative_d * f_grad[d] + phi_derivative2_d
# take gradient step
r = learning_rate(i + 1)
alphas[i] = -r * grad * phi_x
# down-weight past
alphas[:i] *= (1 - r * lmbda)
# visualise log pdf
log_pdf = lambda x: predict(x, alphas)
res = 20
Xs = np.linspace(-3, 3, res)
Ys = np.linspace(-3, 3, res)
# evaluate density and estimate
G = evaluate_density_grid(Xs, Ys, log_pdf)
plot_array(Xs, Ys, np.exp(G))
plt.show()