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Added initial version of the refactored EP

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esiivola 2017-06-01 02:19:58 +03:00 • committed by Akash Kumar Dhaka
parent e849a4c62d
commit dfc5bd42dc
1 changed files with 107 additions and 79 deletions
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186
GPy/inference/latent_function_inference/expectation_propagation.py
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@ -1,11 +1,13 @@
# Copyright (c) 2012-2014, GPy authors (see AUTHORS.txt).
# Licensed under the BSD 3-clause license (see LICENSE.txt)
import numpy as np
import itertools
from ...util.linalg import jitchol, DSYR, dtrtrs, dtrtri, pdinv, dpotrs, tdot, symmetrify
from paramz import ObsAr
from . import ExactGaussianInference, VarDTC
from ...util import diag
from .posterior import PosteriorEP as Posterior
from .posterior import MultioutputPosteriorEP as MultioutputPosterior
log_2_pi = np.log(2*np.pi)
@ -36,6 +38,7 @@ class EPBase(object):
def reset(self):
self.old_mutilde, self.old_vtilde = None, None
self.ga_approx_old = None
self._ep_approximation = None
def on_optimization_start(self):
@ -90,41 +93,49 @@ class EP(EPBase, ExactGaussianInference):
# than ObsArrays
Y = Y.values.copy()
#Initial values - Marginal moments
Z_hat = np.empty(num_data,dtype=np.float64)
mu_hat = np.empty(num_data,dtype=np.float64)
sigma2_hat = np.empty(num_data,dtype=np.float64)
tau_cav = np.empty(num_data,dtype=np.float64)
v_cav = np.empty(num_data,dtype=np.float64)
#initial values - Gaussian factors
#Initial values - Posterior distribution parameters: q(f|X,Y) = N(f|mu,Sigma)
if self.old_mutilde is None:
tau_tilde, mu_tilde, v_tilde = np.zeros((3, num_data))
Sigma = K.copy()
diag.add(Sigma, 1e-7)
mu = np.zeros(num_data)
else:
assert self.old_mutilde.size == num_data, "data size mis-match: did you change the data? try resetting!"
mu_tilde, v_tilde = self.old_mutilde, self.old_vtilde
tau_tilde = v_tilde/mu_tilde
mu, Sigma, _ = self._ep_compute_posterior(K, tau_tilde, v_tilde)
diag.add(Sigma, 1e-7)
# TODO: Check the log-marginal under both conditions and choose the best one
#Initial values - Marginal moments, cavity params, gaussian approximation params and posterior params
marg_moments = marginalMoments(num_data)
cav_params = cavityParams(num_data)
ga_approx, post_params = self._init_approximations(K, num_data)
#Approximation
tau_diff = self.epsilon + 1.
v_diff = self.epsilon + 1.
tau_tilde_old = np.nan
v_tilde_old = np.nan
iterations = 0
while ((tau_diff > self.epsilon) or (v_diff > self.epsilon)) and (iterations < self.max_iters):
update_order = np.random.permutation(num_data)
self._update_cavity_params(num_data, cav_params, post_params, marg_moments, ga_approx, likelihood, Y, Y_metadata)
#(re) compute Sigma and mu using full Cholesky decompy
post_params = self._ep_compute_posterior(K, ga_approx.tau, ga_approx.v)
#monitor convergence
if iterations > 0:
tau_diff = np.mean(np.square(ga_approx.tau-self.ga_approx_old.tau))
v_diff = np.mean(np.square(ga_approx.v-self.ga_approx_old.v))
self.ga_approx_old = gaussianApproximation(ga_approx.mu.copy(), ga_approx.v.copy(), ga_approx.tau.copy())
iterations += 1
ga_approx.mu = ga_approx.v/ga_approx.tau
# Z_tilde after removing the terms that can lead to infinite terms due to tau_tilde close to zero.
# This terms cancel with the coreresponding terms in the marginal loglikelihood
log_Z_tilde = self._log_Z_tilde(marg_moments, ga_approx, cav_params)
# - 0.5*np.log(tau_tilde) + 0.5*(v_tilde*v_tilde*1./tau_tilde)
return post_params.mu, post_params.Sigma, ga_approx.mu, ga_approx.tau, log_Z_tilde
def _log_Z_tilde(self, marg_moments, ga_approx, cav_params):
return (np.log(marg_moments.Z_hat) + 0.5*np.log(2*np.pi) + 0.5*np.log(1+ga_approx.tau/cav_params.tau) - 0.5 * ((ga_approx.v)**2 * 1./(cav_params.tau + ga_approx.tau))
+ 0.5*(cav_params.v * ( ( (ga_approx.tau/cav_params.tau) * cav_params.v - 2.0 * ga_approx.v ) * 1./(cav_params.tau + ga_approx.tau))))
def _update_cavity_params(self, num_data, cav_params, post_params, marg_moments, ga_approx, likelihood, Y, Y_metadata, update_order=None):
if update_order is None:
update_order = np.random.permutation(num_data)
for i in update_order:
#Cavity distribution parameters
tau_cav[i] = 1./Sigma[i,i] - self.eta*tau_tilde[i]
v_cav[i] = mu[i]/Sigma[i,i] - self.eta*v_tilde[i]
cav_params.tau[i] = 1./post_params.Sigma[i,i] - self.eta*ga_approx.tau[i]
cav_params.v[i] = post_params.mu[i]/post_params.Sigma[i,i] - self.eta*ga_approx.v[i]
if Y_metadata is not None:
# Pick out the relavent metadata for Yi
Y_metadata_i = {}
@ -133,54 +144,46 @@ class EP(EPBase, ExactGaussianInference):
else:
Y_metadata_i = None
#Marginal moments
Z_hat[i], mu_hat[i], sigma2_hat[i] = likelihood.moments_match_ep(Y[i], tau_cav[i], v_cav[i], Y_metadata_i=Y_metadata_i)
marg_moments.Z_hat[i], marg_moments.mu_hat[i], marg_moments.sigma2_hat[i] = likelihood.moments_match_ep(Y[i], cav_params.tau[i], cav_params.v[i], Y_metadata_i=Y_metadata_i)
#Site parameters update
delta_tau = self.delta/self.eta*(1./sigma2_hat[i] - 1./Sigma[i,i])
delta_v = self.delta/self.eta*(mu_hat[i]/sigma2_hat[i] - mu[i]/Sigma[i,i])
tau_tilde_prev = tau_tilde[i]
tau_tilde[i] += delta_tau
delta_tau = self.delta/self.eta*(1./marg_moments.sigma2_hat[i] - 1./post_params.Sigma[i,i])
delta_v = self.delta/self.eta*(marg_moments.mu_hat[i]/marg_moments.sigma2_hat[i] - post_params.mu[i]/post_params.Sigma[i,i])
tau_tilde_prev = ga_approx.tau[i]
ga_approx.tau[i] += delta_tau
# Enforce positivity of tau_tilde. Even though this is guaranteed for logconcave sites, it is still possible
# to get negative values due to numerical errors. Moreover, the value of tau_tilde should be positive in order to
# update the marginal likelihood without inestability issues.
if tau_tilde[i] < np.finfo(float).eps:
tau_tilde[i] = np.finfo(float).eps
delta_tau = tau_tilde[i] - tau_tilde_prev
v_tilde[i] += delta_v
if ga_approx.tau[i] < np.finfo(float).eps:
ga_approx.tau[i] = np.finfo(float).eps
delta_tau = ga_approx.tau[i] - tau_tilde_prev
ga_approx.v[i] += delta_v
if self.parallel_updates == False:
#Posterior distribution parameters update
ci = delta_tau/(1.+ delta_tau*Sigma[i,i])
DSYR(Sigma, Sigma[:,i].copy(), -ci)
mu = np.dot(Sigma, v_tilde)
#(re) compute Sigma and mu using full Cholesky decompy
mu, Sigma, _ = self._ep_compute_posterior(K, tau_tilde, v_tilde)
#monitor convergence
if iterations > 0:
tau_diff = np.mean(np.square(tau_tilde-tau_tilde_old))
v_diff = np.mean(np.square(v_tilde-v_tilde_old))
tau_tilde_old = tau_tilde.copy()
v_tilde_old = v_tilde.copy()
iterations += 1
mu_tilde = v_tilde/tau_tilde
mu_cav = v_cav/tau_cav
sigma2_sigma2tilde = 1./tau_cav + 1./tau_tilde
# Z_tilde after removing the terms that can lead to infinite terms due to tau_tilde close to zero.
# This terms cancel with the coreresponding terms in the marginal loglikelihood
log_Z_tilde = (np.log(Z_hat) + 0.5*np.log(2*np.pi) + 0.5*np.log(1+tau_tilde/tau_cav)
- 0.5 * ((v_tilde)**2 * 1./(tau_cav + tau_tilde)) + 0.5*(v_cav * ( ( (tau_tilde/tau_cav) * v_cav - 2.0 * v_tilde ) * 1./(tau_cav + tau_tilde))))
# - 0.5*np.log(tau_tilde) + 0.5*(v_tilde*v_tilde*1./tau_tilde)
self.old_mutilde = mu_tilde
self.old_vtilde = v_tilde
return mu, Sigma, mu_tilde, tau_tilde, log_Z_tilde
ci = delta_tau/(1.+ delta_tau*post_params.Sigma[i,i])
DSYR(post_params.Sigma, post_params.Sigma[:,i].copy(), -ci)
post_params.mu = np.dot(post_params.Sigma, ga_approx.v)
def _init_approximations(self, K, num_data):
#initial values - Gaussian factors
#Initial values - Posterior distribution parameters: q(f|X,Y) = N(f|mu,Sigma)
if self.ga_approx_old is None:
mu_tilde, v_tilde, tau_tilde = np.zeros((3, num_data))
ga_approx = gaussianApproximation(mu_tilde, v_tilde, tau_tilde)
Sigma = K.copy()
diag.add(Sigma, 1e-7)
mu = np.zeros(num_data)
post_params = posteriorParams(mu, Sigma)
else:
assert self.ga_approx_old.mu.size == num_data, "data size mis-match: did you change the data? try resetting!"
ga_approx = gaussianApproximation(self.ga_approx_old.mu, self.ga_approx_old.v)
post_params = self._ep_compute_posterior(K, ga_approx.tau, ga_approx.v)
diag.add(post_params.Sigma, 1e-7)
# TODO: Check the log-marginal under both conditions and choose the best one
return (ga_approx, post_params)
def _ep_compute_posterior(self, K, tau_tilde, v_tilde):
num_data = len(tau_tilde)
tau_tilde_root = np.sqrt(tau_tilde)
@ -190,18 +193,18 @@ class EP(EPBase, ExactGaussianInference):
V, _ = dtrtrs(L, Sroot_tilde_K, lower=1)
Sigma = K - np.dot(V.T,V) #K - KS^(1/2)BS^(1/2)K = (K^(-1) + \Sigma^(-1))^(-1)
mu = np.dot(Sigma,v_tilde)
return (mu, Sigma, L)
return posteriorParams(mu, Sigma, L)
def _ep_marginal(self, K, tau_tilde, v_tilde, Z_tilde):
mu, Sigma, L = self._ep_compute_posterior(K, tau_tilde, v_tilde)
post_params = self._ep_compute_posterior(K, tau_tilde, v_tilde)
# Gaussian log marginal excluding terms that can go to infinity due to arbitrarily small tau_tilde.
# These terms cancel out with the terms excluded from Z_tilde
B_logdet = np.sum(2.0*np.log(np.diag(L)))
log_marginal = 0.5*(-len(tau_tilde) * log_2_pi - B_logdet + np.sum(v_tilde * np.dot(Sigma,v_tilde)))
B_logdet = np.sum(2.0*np.log(np.diag(post_params.L)))
log_marginal = 0.5*(-len(tau_tilde) * log_2_pi - B_logdet + np.sum(v_tilde * np.dot(post_params.Sigma,v_tilde)))
log_marginal += Z_tilde
return log_marginal, mu, Sigma, L
return log_marginal, post_params.mu, post_params.Sigma, post_params.L
def _inference(self, K, tau_tilde, v_tilde, likelihood, Z_tilde, Y_metadata=None):
log_marginal, mu, Sigma, L = self._ep_marginal(K, tau_tilde, v_tilde, Z_tilde)
@ -277,8 +280,8 @@ class EPDTC(EPBase, VarDTC):
mu_hat = np.zeros(num_data,dtype=np.float64)
sigma2_hat = np.zeros(num_data,dtype=np.float64)
tau_cav = np.empty(num_data,dtype=np.float64)
v_cav = np.empty(num_data,dtype=np.float64)
tau = np.empty(num_data,dtype=np.float64)
v = np.empty(num_data,dtype=np.float64)
#initial values - Gaussian factors
#Initial values - Posterior distribution parameters: q(f|X,Y) = N(f|mu,Sigma)
@ -315,8 +318,8 @@ class EPDTC(EPBase, VarDTC):
update_order = np.random.permutation(num_data)
for i in update_order:
#Cavity distribution parameters
tau_cav[i] = 1./Sigma_diag[i] - self.eta*tau_tilde[i]
v_cav[i] = mu[i]/Sigma_diag[i] - self.eta*v_tilde[i]
tau[i] = 1./Sigma_diag[i] - self.eta*tau_tilde[i]
v[i] = mu[i]/Sigma_diag[i] - self.eta*v_tilde[i]
if Y_metadata is not None:
# Pick out the relavent metadata for Yi
Y_metadata_i = {}
@ -326,7 +329,7 @@ class EPDTC(EPBase, VarDTC):
Y_metadata_i = None
#Marginal moments
Z_hat[i], mu_hat[i], sigma2_hat[i] = likelihood.moments_match_ep(Y[i], tau_cav[i], v_cav[i], Y_metadata_i=Y_metadata_i)
Z_hat[i], mu_hat[i], sigma2_hat[i] = likelihood.moments_match_ep(Y[i], tau[i], v[i], Y_metadata_i=Y_metadata_i)
#Site parameters update
delta_tau = self.delta/self.eta*(1./sigma2_hat[i] - 1./Sigma_diag[i])
delta_v = self.delta/self.eta*(mu_hat[i]/sigma2_hat[i] - mu[i]/Sigma_diag[i])
@ -365,8 +368,8 @@ class EPDTC(EPBase, VarDTC):
iterations += 1
mu_tilde = v_tilde/tau_tilde
mu_cav = v_cav/tau_cav
sigma2_sigma2tilde = 1./tau_cav + 1./tau_tilde
mu_cav = v/tau
sigma2_sigma2tilde = 1./tau + 1./tau_tilde
log_Z_tilde = (np.log(Z_hat) + 0.5*np.log(2*np.pi) + 0.5*np.log(sigma2_sigma2tilde)
+ 0.5*((mu_cav - mu_tilde)**2) / (sigma2_sigma2tilde))
@ -388,3 +391,28 @@ class EPDTC(EPBase, VarDTC):
Sigma_diag = np.diag(Sigma).copy()
return (mu, Sigma_diag, LLT)
#Four wrapper classes to help modularisation of different EP versions
class marginalMoments(object):
def __init__(self, num_data):
#Initial values - Marginal moments
self.Z_hat = np.empty(num_data,dtype=np.float64)
self.mu_hat = np.empty(num_data,dtype=np.float64)
self.sigma2_hat = np.empty(num_data,dtype=np.float64)
class cavityParams(object):
def __init__(self, num_data):
self.tau = np.empty(num_data,dtype=np.float64)
self.v = np.empty(num_data,dtype=np.float64)
class gaussianApproximation(object):
def __init__(self, mu, v, tau=None):
self.mu = mu
self.v = v
self.tau = mu / v if tau is None else tau
class posteriorParams(object):
def __init__(self, mu=None, Sigma=None, L=None):
self.mu = mu
self.Sigma = Sigma
self.L = L