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deep_gaussian_process_5.py
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deep_gaussian_process_5.py
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from __future__ import absolute_import
from __future__ import print_function
import matplotlib.pyplot as plt
import autograd.numpy as np
import autograd.numpy.random as npr
from autograd import value_and_grad,grad
from scipy.optimize import minimize
from autograd.numpy.linalg import solve
import autograd.scipy.stats.multivariate_normal as mvn
from builtins import range
from autograd.scipy.misc import logsumexp
import itertools
import sys
from optparse import OptionParser
from autograd.util import quick_grad_check
def build_checker_dataset(n_data = 6, noise_std =0.1):
rs = npr.RandomState(0)
inputs = np.array([np.array([x,y]) for x in np.linspace(-1,1,n_data) for y in np.linspace(-1,1,n_data)])
targets = np.sign([np.prod(input) for input in inputs]) + rs.randn(n_data**2)*noise_std
return inputs, targets
def build_step_function_dataset(D=1, n_data=40, noise_std=0.1):
rs = npr.RandomState(0)
inputs = np.linspace(-2, 2, num=n_data)
targets = np.sign(inputs) + rs.randn(n_data) * noise_std
inputs = inputs.reshape((len(inputs), D))
return inputs, targets
def rbf_covariance(cov_params, x, xp):
output_scale = np.exp(cov_params[0])
lengthscales = np.exp(cov_params[1:])
diffs = np.expand_dims(x /lengthscales, 1)\
- np.expand_dims(xp/lengthscales, 0)
return output_scale * np.exp(-0.5 * np.sum(diffs**2, axis=2))
def build_single_gp(cov_func, num_cov_params, num_pseudo_params, input_dimension):
"""Functions that perform Gaussian process regression.
cov_func has signature (cov_params, x, x')"""
def unpack_gp_params(params):
mean = params[0]
noise_scale = np.exp(params[1]) + 0.001
cov_params = params[2:2+num_cov_params]
pseudo_params = params[2+num_cov_params:]
x0, y0 = np.split(pseudo_params,[num_pseudo_params*input_dimension])
x0 = x0.reshape((num_pseudo_params,input_dimension))
gp_details = {'mean': mean, 'noise_scale': noise_scale, 'cov_params': cov_params, 'x0': x0, 'y0': y0}
#x0 = X
#y0 = y
return mean, cov_params, noise_scale, x0, y0
def pack_gp_params(mean, cov_params, noise_scale, x0, y0):
params = np.append(mean,noise_scale)
params = np.concatenate([params,cov_params])
params = np.concatenate([params,np.ndarray.flatten(np.array(x0))])
params = np.concatenate([params,np.ndarray.flatten(np.array(y0))])
return params
def predict(params, xstar):
"""Returns the predictive mean and covariance at locations xstar,
of the latent function value f (without observation noise)."""
mean, cov_params, noise_scale, x0, y0 = unpack_gp_params(params)
cov_f_f = cov_func(cov_params, xstar, xstar)
cov_y_f = cov_func(cov_params, x0, xstar)
cov_y_y = cov_func(cov_params, x0, x0) + noise_scale * np.eye(len(y0))
pred_mean = mean + np.dot(solve(cov_y_y, cov_y_f).T, y0 - mean)
pred_cov = cov_f_f - np.dot(solve(cov_y_y, cov_y_f).T, cov_y_f)
return pred_mean, pred_cov
def predict_with_noise(params, xstar):
pred_mean, pred_cov = predict(params, xstar)
mean, cov_params, noise_scale, x0, y0 = unpack_gp_params(params)
return pred_mean, pred_cov + noise_scale*np.eye(len(xstar))
num_gp_params = 2 + num_cov_params + num_pseudo_params*input_dimension + num_pseudo_params
return num_gp_params, predict, predict_with_noise, unpack_gp_params, pack_gp_params
def build_single_layer(input_dimension, output_dimension, num_pseudo_params, covariance_function, random):
layer_details = [build_single_gp(covariance_function, input_dimension + 1, num_pseudo_params, input_dimension) for i in xrange(output_dimension)]
num_params_each_output, predict_layer_funcs, predict_funcs_with_noise, unpack_gp_params_layer, pack_gp_params_layer = zip(*layer_details)
total_params_layer = sum(num_params_each_output)
def unpack_layer_params(params):
gp_params = np.array_split(params, output_dimension) # assuming all parameters have equal dims, change to what we had below
return gp_params
def sample_from_mvn(mu, sigma):
rs = npr.RandomState(0)
return np.dot(np.linalg.cholesky(sigma+1e-6*np.eye(len(sigma))*np.max(np.diag(sigma))),rs.randn(len(sigma)))+mu if random == 1 else mu
def sample_mean_cov_from_layer(layer_params, xstar, with_noise = False):
predict = predict_funcs_with_noise if with_noise else predict_layer_funcs
gp_params = unpack_layer_params(layer_params)
samples = [predict[i](gp_params[i],xstar) for i in xrange(output_dimension)]
return samples
def sample_values_from_layer(layer_params, xstar, with_noise = False): # should return len(x*)
samples = sample_mean_cov_from_layer(layer_params, xstar, with_noise)
outputs = [sample_from_mvn(mean,cov) for mean,cov in samples]
return np.array(outputs).T
return total_params_layer, sample_mean_cov_from_layer, sample_values_from_layer, predict_layer_funcs, unpack_gp_params_layer, unpack_layer_params, pack_gp_params_layer
def build_deep_gp(dimensions, covariance_function, num_pseudo_params, random):
deep_details = [build_single_layer(dimensions[i],dimensions[i+1],num_pseudo_params,covariance_function,random) for i in xrange(len(dimensions)-1)]
num_params_each_layer, sample_mean_cov_funcs, sample_value_funcs, predict_layer_funcs, unpack_gp_params_all, unpack_layer_params, pack_gp_params_all = zip(*deep_details)
total_params_gp = sum(num_params_each_layer)
def unpack_all_params(all_params):
all_layer_params = np.array_split(all_params,np.cumsum(num_params_each_layer))
return all_layer_params
def sample_mean_cov_from_deep_gp(all_params, xstar, with_noise = False):
xtilde = xstar
all_layer_params = unpack_all_params(all_params)
for layer in xrange(len(dimensions)-2):
layer_params = all_layer_params[layer]
xtilde = sample_value_funcs[layer](layer_params, xtilde, with_noise)
final_layer = len(dimensions)-2
final_layer_params = all_layer_params[final_layer]
final_mean, final_cov = sample_mean_cov_funcs[final_layer](final_layer_params, xtilde, with_noise)[0] # index into 0 because final layer has one unit
#print("Function Means",final_mean)
return final_mean, final_cov
def evaluate_prior(all_params): # clean up code so we don't compute matrices twice
all_layer_params = unpack_all_params(all_params)
log_prior = 0
for layer in xrange(n_layers):
layer_params = all_layer_params[layer]
layer_gp_params = unpack_layer_params[layer](layer_params)
for dim in xrange(dimensions[layer+1]):
gp_params = layer_gp_params[dim]
mean, cov_params, noise_scale, x0, y0 = unpack_gp_params_all[layer][dim](gp_params)
cov_y_y = covariance_function(cov_params,x0,x0) + noise_scale * np.eye(len(y0))
log_prior += mvn.logpdf(y0,np.ones(len(cov_y_y))*mean,cov_y_y+np.eye(len(cov_y_y))*10)
return log_prior
def log_likelihood(all_params):
samples = [sample_mean_cov_from_deep_gp(all_params, X, True) for i in xrange(n_samples)]
return logsumexp(np.array([mvn.logpdf(y,mean,var+1e-6*np.eye(len(var))*np.max(np.diag(var))) for mean,var in samples])) - np.log(n_samples) \
+ evaluate_prior(all_params)
def squared_error(all_params):
samples = np.array([sample_mean_cov_from_deep_gp(all_params, X, False)[0] for i in xrange(n_samples)])
return np.mean((y - np.mean(samples,axis = 0)) ** 2)
return total_params_gp, log_likelihood, sample_mean_cov_from_deep_gp, predict_layer_funcs, squared_error, unpack_gp_params_all, unpack_layer_params, unpack_all_params,\
pack_gp_params_all
if __name__ == '__main__':
random = 0
n_samples = 1
dimensions = [2,1] # Architecture of the GP. Last layer should always be 1
n_data = 20
input_dimension = dimensions[0]
n_layers = len(dimensions)-1
num_pseudo_params = 50
#X, y = build_step_function_dataset(D=input_dimension, n_data=20)
X, y = build_checker_dataset(n_data=16)
total_num_params, log_likelihood, sample_mean_cov_from_deep_gp, predict_layer_funcs, squared_error, unpack_gp_params_all, unpack_layer_params, unpack_all_params, \
pack_gp_params_all = build_deep_gp(dimensions, rbf_covariance, num_pseudo_params, random)
# Set up figure.
if dimensions[0] == 1:
fig = plt.figure(figsize=(12,8), facecolor='white')
ax_first = fig.add_subplot(411, frameon=False)
ax_end_to_end = fig.add_subplot(412, frameon=False)
ax_x_to_h = fig.add_subplot(413, frameon=False)
ax_h_to_y = fig.add_subplot(414, frameon=False)
plt.show(block=False)
else:
fig = plt.figure(figsize=(12,8), facecolor='white')
ax = fig.add_subplot(111, frameon=False)
plt.show(block=False)
def plot_deep_gp_2d(ax,params,plot_xs):
ax.cla()
rs = npr.RandomState(0)
sampled_means_and_covs = [sample_mean_cov_from_deep_gp(params, plot_xs) for i in xrange(n_samples)]
sampled_means, sampled_covs = zip(*sampled_means_and_covs)
avg_pred_mean = np.mean(sampled_means, axis = 0)
avg_pred_cov = np.mean(sampled_covs, axis = 0)
#print("X*",avg_pred_mean)
#rint("X*",plot_xs[0:4])
#sampled_means_and_covs_orig = [sample_mean_cov_from_deep_gp(params, X) for i in xrange(n_samples)]
#sampled_means_orig, sampled_covs_orig = zip(*sampled_means_and_covs_orig)
#avg_pred_mean_orig = np.mean(sampled_means_orig, axis = 0)
#print("Orignal Xs",avg_pred_mean_orig)
X0 = params[5:5+num_pseudo_params*2].reshape(num_pseudo_params,2)
y0 = params[5+num_pseudo_params*2:5+num_pseudo_params*3]
#ax.scatter(X0[:,0],X0[:,1],c = y0)
avg_pred_mean = avg_pred_mean.reshape(40,40)
ax.contourf(np.linspace(-1,1,40),np.linspace(-1,1,40), avg_pred_mean)
ax.scatter(X[:,0],X[:,1],c=y)
ax.set_xticks([])
ax.set_yticks([])
ax.set_title("Full Deep GP")
def plot_deep_gp(ax, params, plot_xs):
ax.cla()
rs = npr.RandomState(0)
sampled_means_and_covs = [sample_mean_cov_from_deep_gp(params, plot_xs) for i in xrange(n_samples)]
sampled_means, sampled_covs = zip(*sampled_means_and_covs)
avg_pred_mean = np.mean(sampled_means, axis = 0)
avg_pred_cov = np.mean(sampled_covs, axis = 0)
marg_std = np.sqrt(np.diag(avg_pred_cov))
if n_samples > 1:
ax.fill(np.concatenate([plot_xs, plot_xs[::-1]]),
np.concatenate([avg_pred_mean - 1.96 * marg_std,
(avg_pred_mean + 1.96 * marg_std)[::-1]]),
alpha=.15, fc='Blue', ec='None')
ax.plot(plot_xs, avg_pred_mean, 'b')
sampled_funcs = np.array([rs.multivariate_normal(mean, cov*(random)) for mean,cov in sampled_means_and_covs])
ax.plot(plot_xs,sampled_funcs.T)
ax.plot(X, y, 'kx')
#ax.set_ylim([-1.5,1.5])
ax.set_xticks([])
ax.set_yticks([])
ax.set_title("Full Deep GP, inputs to outputs")
def plot_single_gp(ax, x0, y0, pred_mean, pred_cov, plot_xs):
ax.cla()
marg_std = np.sqrt(np.diag(pred_cov))
if n_samples > 1:
ax.plot(plot_xs, pred_mean, 'b')
ax.fill(np.concatenate([plot_xs, plot_xs[::-1]]),
np.concatenate([pred_mean - 1.96 * marg_std,
(pred_mean + 1.96 * marg_std)[::-1]]),
alpha=.15, fc='Blue', ec='None')
# Show samples from posterior.
rs = npr.RandomState(0)
sampled_funcs = rs.multivariate_normal(pred_mean, pred_cov*(random), size=n_samples)
ax.plot(plot_xs, sampled_funcs.T)
ax.plot(x0, y0, 'ro')
ax.set_xticks([])
ax.set_yticks([])
def callback(params):
print("Log likelihood {}, Squared Error {}".format(-objective(params),squared_error(params)))
# Show posterior marginals.
if dimensions[0] == 1:
plot_xs = np.reshape(np.linspace(-5, 5, 300), (300,1))
plot_deep_gp(ax_first, init_params, plot_xs)
ax_first.set_title("Initial full predictions")
plot_deep_gp(ax_end_to_end, params, plot_xs)
if dimensions == [1,1]:
ax_end_to_end.plot(params[4:14],params[14:24], 'ro')
elif dimensions == [1,1,1]:
hidden_mean, hidden_cov = predict_layer_funcs[0][0](params[0:24], plot_xs)
plot_single_gp(ax_x_to_h, params[4:14], params[14:24], hidden_mean, hidden_cov, plot_xs)
ax_x_to_h.set_title("Inputs to hiddens, inducing points in red")
y_mean, y_cov = predict_layer_funcs[0][0](params[24:48], plot_xs)
plot_single_gp(ax_h_to_y, params[28:38], params[38:48], y_mean, y_cov, plot_xs)
ax_h_to_y.set_title("Hiddens to outputs, inducing points in red")
plt.draw()
plt.pause(1.0/60.0)
elif dimensions[0] == 2:
plot_xs = np.array([np.array([a,b]) for a in np.linspace(-1,1,40) for b in np.linspace(-1,1,40)])
plot_deep_gp_2d(ax, params, plot_xs)
#all_layer_params = unpack_all_params(init_params)
#for layer in xrange(n_layers):
# layer_params = all_layer_params[layer]
# layer_gp_params = unpack_layer_params[layer](layer_params)
# for dim in xrange(dimensions[layer+1]):
# gp_params = layer_gp_params[dim]
# mean, cov_params, noise_scale, x0, y0 = unpack_gp_params_all[layer][dim](gp_params)
# lengthscales = cov_params[1:]
##ax.scatter(X[:,0],x0[:,0])
plt.draw()
plt.pause(1.0/60.0)
rs = npr.RandomState(1234)
init_params = .1 * rs.randn(total_num_params)
# SMART INITIALIZATION
# Admittedly, this is not a good way to do it
# If you have any tips on how to make this better please let me know
smart_params = np.array([])
all_layer_params = unpack_all_params(init_params)
for layer in xrange(n_layers):
layer_params = all_layer_params[layer]
layer_gp_params = unpack_layer_params[layer](layer_params)
for dim in xrange(dimensions[layer+1]):
gp_params = layer_gp_params[dim]
mean, cov_params, noise_scale, x0, y0 = unpack_gp_params_all[layer][dim](gp_params)
lengthscales = cov_params[1:]
if layer == 0:
pairs = itertools.combinations(X, 2)
dists = np.array([np.abs(p1-p2) for p1,p2 in pairs])
smart_lengthscales = np.array([np.log(np.median(dists[:,i])) for i in xrange(len(lengthscales))])
smart_x0 = np.array(X)[rs.choice(len(X), num_pseudo_params, replace=False),:]
smart_y0 = np.ndarray.flatten(smart_x0)
else:
smart_x0 = x0
smart_y0 = np.ndarray.flatten(x0)
smart_lengthscales = np.array([np.log(1) for i in xrange(len(lengthscales))])
cov_params = np.append(cov_params[0],smart_lengthscales)
params = pack_gp_params_all[layer][dim](mean, cov_params, noise_scale, smart_x0, smart_y0)
smart_params = np.append(smart_params, params)
init_params = smart_params
print("Optimizing covariance parameters...")
objective = lambda params: -log_likelihood(params)
params = minimize(value_and_grad(objective), init_params, jac=True,
method='BFGS', callback=callback)
plt.pause(10.0)