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Fixing GP_EP
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Ricardo
parent
d286ffe633
commit
b6ffb57263
8 changed files with 638 additions and 5 deletions
314
GPy/inference/EP.py
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314
GPy/inference/EP.py
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import numpy as np
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import random
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import pylab as pb #TODO erase me
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from scipy import stats, linalg
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from .likelihoods import likelihood
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from ..core import model
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from ..util.linalg import pdinv,mdot,jitchol
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from ..util.plot import gpplot
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from .. import kern
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class EP:
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def __init__(self,covariance,likelihood,Kmn=None,Knn_diag=None,epsilon=1e-3,powerep=[1.,1.]):
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"""
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Expectation Propagation
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Arguments
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---------
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X : input observations
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likelihood : Output's likelihood (likelihood class)
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kernel : a GPy kernel (kern class)
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inducing : Either an array specifying the inducing points location or a sacalar defining their number. None value for using a non-sparse model is used.
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powerep : Power-EP parameters (eta,delta) - 2x1 numpy array (floats)
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epsilon : Convergence criterion, maximum squared difference allowed between mean updates to stop iterations (float)
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"""
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self.likelihood = likelihood
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assert covariance.shape[0] == covariance.shape[1]
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if Kmn is not None:
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self.Kmm = covariance
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self.Kmn = Kmn
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self.M = self.Kmn.shape[0]
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self.N = self.Kmn.shape[1]
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assert self.M < self.N, 'The number of inducing inputs must be smaller than the number of observations'
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else:
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self.K = covariance
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self.N = self.K.shape[0]
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if Knn_diag is not None:
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self.Knn_diag = Knn_diag
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assert len(Knn_diag) == self.N, 'Knn_diagonal has size different from N'
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self.epsilon = epsilon
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self.eta, self.delta = powerep
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self.jitter = 1e-12
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"""
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Initial values - Likelihood approximation parameters:
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p(y|f) = t(f|tau_tilde,v_tilde)
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"""
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self.tau_tilde = np.zeros(self.N)
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self.v_tilde = np.zeros(self.N)
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def restart_EP(self):
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"""
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Set the EP approximation to initial state
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"""
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self.tau_tilde = np.zeros(self.N)
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self.v_tilde = np.zeros(self.N)
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self.mu = np.zeros(self.N)
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class Full(EP):
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def fit_EP(self):
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"""
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The expectation-propagation algorithm.
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For nomenclature see Rasmussen & Williams 2006 (pag. 52-60)
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"""
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#Prior distribution parameters: p(f|X) = N(f|0,K)
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#self.K = self.kernel.K(self.X,self.X)
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#Initial values - Posterior distribution parameters: q(f|X,Y) = N(f|mu,Sigma)
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self.mu=np.zeros(self.N)
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self.Sigma=self.K.copy()
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"""
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Initial values - Cavity distribution parameters:
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q_(f|mu_,sigma2_) = Product{q_i(f|mu_i,sigma2_i)}
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sigma_ = 1./tau_
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mu_ = v_/tau_
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"""
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self.tau_ = np.empty(self.N,dtype=float)
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self.v_ = np.empty(self.N,dtype=float)
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#Initial values - Marginal moments
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z = np.empty(self.N,dtype=float)
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self.Z_hat = np.empty(self.N,dtype=float)
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phi = np.empty(self.N,dtype=float)
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mu_hat = np.empty(self.N,dtype=float)
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sigma2_hat = np.empty(self.N,dtype=float)
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self.mu_hat = mu_hat #TODO erase me
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self.sigma2_hat = sigma2_hat #TODO erase me
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#Approximation
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epsilon_np1 = self.epsilon + 1.
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epsilon_np2 = self.epsilon + 1.
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self.iterations = 0
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self.np1 = [self.tau_tilde.copy()]
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self.np2 = [self.v_tilde.copy()]
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while epsilon_np1 > self.epsilon or epsilon_np2 > self.epsilon:
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update_order = np.arange(self.N)
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#random.shuffle(update_order) #TODO uncomment
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for i in update_order:
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#Cavity distribution parameters
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self.tau_[i] = 1./self.Sigma[i,i] - self.eta*self.tau_tilde[i]
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self.v_[i] = self.mu[i]/self.Sigma[i,i] - self.eta*self.v_tilde[i]
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#Marginal moments
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self.Z_hat[i], mu_hat[i], sigma2_hat[i] = self.likelihood.moments_match(i,self.tau_[i],self.v_[i])
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self.mu_hat[i] = mu_hat[i] #TODO erase me
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self.sigma2_hat[i] = sigma2_hat[i] #TODO erase me
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#if i == 3:
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# a = b
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#Site parameters update
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Delta_tau = self.delta/self.eta*(1./sigma2_hat[i] - 1./self.Sigma[i,i])
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Delta_v = self.delta/self.eta*(mu_hat[i]/sigma2_hat[i] - self.mu[i]/self.Sigma[i,i])
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print Delta_tau
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self.tau_tilde[i] = self.tau_tilde[i] + Delta_tau
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self.v_tilde[i] = self.v_tilde[i] + Delta_v
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#Posterior distribution parameters update
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si=self.Sigma[:,i].reshape(self.N,1)
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self.Sigma = self.Sigma - Delta_tau/(1.+ Delta_tau*self.Sigma[i,i])*np.dot(si,si.T)
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self.mu = np.dot(self.Sigma,self.v_tilde)
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self.iterations += 1
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#Sigma recomptutation with Cholesky decompositon
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Sroot_tilde_K = np.sqrt(self.tau_tilde)[:,None]*(self.K)
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B = np.eye(self.N) + np.sqrt(self.tau_tilde)[None,:]*Sroot_tilde_K
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L = jitchol(B)
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V,info = linalg.flapack.dtrtrs(L,Sroot_tilde_K,lower=1)
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self.Sigma = self.K - np.dot(V.T,V)
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self.mu = np.dot(self.Sigma,self.v_tilde)
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epsilon_np1 = sum((self.tau_tilde-self.np1[-1])**2)/self.N
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epsilon_np2 = sum((self.v_tilde-self.np2[-1])**2)/self.N
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self.np1.append(self.tau_tilde.copy())
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self.np2.append(self.v_tilde.copy())
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class DTC(EP):
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def fit_EP(self):
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"""
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The expectation-propagation algorithm with sparse pseudo-input.
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For nomenclature see ... 2013.
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"""
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"""
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Prior approximation parameters:
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q(f|X) = int_{df}{N(f|KfuKuu_invu,diag(Kff-Qff)*N(u|0,Kuu)} = N(f|0,Sigma0)
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Sigma0 = Qnn = Knm*Kmmi*Kmn
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"""
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self.Kmmi, self.Kmm_hld = pdinv(self.Kmm)
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self.KmnKnm = np.dot(self.Kmn, self.Kmn.T)
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self.KmmiKmn = np.dot(self.Kmmi,self.Kmn)
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self.Qnn_diag = np.sum(self.Kmn*self.KmmiKmn,-2)
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self.LLT0 = self.Kmm.copy()
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"""
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Posterior approximation: q(f|y) = N(f| mu, Sigma)
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Sigma = Diag + P*R.T*R*P.T + K
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mu = w + P*gamma
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"""
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self.mu = np.zeros(self.N)
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self.LLT = self.Kmm.copy()
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self.Sigma_diag = self.Qnn_diag.copy()
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"""
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Initial values - Cavity distribution parameters:
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q_(g|mu_,sigma2_) = Product{q_i(g|mu_i,sigma2_i)}
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sigma_ = 1./tau_
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mu_ = v_/tau_
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"""
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self.tau_ = np.empty(self.N,dtype=float)
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self.v_ = np.empty(self.N,dtype=float)
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#Initial values - Marginal moments
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z = np.empty(self.N,dtype=float)
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self.Z_hat = np.empty(self.N,dtype=float)
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phi = np.empty(self.N,dtype=float)
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mu_hat = np.empty(self.N,dtype=float)
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sigma2_hat = np.empty(self.N,dtype=float)
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#Approximation
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epsilon_np1 = 1
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epsilon_np2 = 1
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self.iterations = 0
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self.np1 = [self.tau_tilde.copy()]
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self.np2 = [self.v_tilde.copy()]
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while epsilon_np1 > self.epsilon or epsilon_np2 > self.epsilon:
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update_order = np.arange(self.N)
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random.shuffle(update_order)
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for i in update_order:
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#Cavity distribution parameters
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self.tau_[i] = 1./self.Sigma_diag[i] - self.eta*self.tau_tilde[i]
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self.v_[i] = self.mu[i]/self.Sigma_diag[i] - self.eta*self.v_tilde[i]
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#Marginal moments
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self.Z_hat[i], mu_hat[i], sigma2_hat[i] = self.likelihood.moments_match(i,self.tau_[i],self.v_[i])
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#Site parameters update
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Delta_tau = self.delta/self.eta*(1./sigma2_hat[i] - 1./self.Sigma_diag[i])
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Delta_v = self.delta/self.eta*(mu_hat[i]/sigma2_hat[i] - self.mu[i]/self.Sigma_diag[i])
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self.tau_tilde[i] = self.tau_tilde[i] + Delta_tau
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self.v_tilde[i] = self.v_tilde[i] + Delta_v
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#Posterior distribution parameters update
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self.LLT = self.LLT + np.outer(self.Kmn[:,i],self.Kmn[:,i])*Delta_tau
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L = jitchol(self.LLT)
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V,info = linalg.flapack.dtrtrs(L,self.Kmn,lower=1)
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self.Sigma_diag = np.sum(V*V,-2)
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si = np.sum(V.T*V[:,i],-1)
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self.mu = self.mu + (Delta_v-Delta_tau*self.mu[i])*si
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self.iterations += 1
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#Sigma recomputation with Cholesky decompositon
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self.LLT0 = self.LLT0 + np.dot(self.Kmn*self.tau_tilde[None,:],self.Kmn.T)
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self.L = jitchol(self.LLT)
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V,info = linalg.flapack.dtrtrs(L,self.Kmn,lower=1)
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V2,info = linalg.flapack.dtrtrs(L.T,V,lower=0)
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self.Sigma_diag = np.sum(V*V,-2)
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Knmv_tilde = np.dot(self.Kmn,self.v_tilde)
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self.mu = np.dot(V2.T,Knmv_tilde)
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epsilon_np1 = sum((self.tau_tilde-self.np1[-1])**2)/self.N
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epsilon_np2 = sum((self.v_tilde-self.np2[-1])**2)/self.N
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self.np1.append(self.tau_tilde.copy())
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self.np2.append(self.v_tilde.copy())
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class FITC(EP):
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def fit_EP(self):
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"""
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The expectation-propagation algorithm with sparse pseudo-input.
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For nomenclature see Naish-Guzman and Holden, 2008.
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"""
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"""
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Prior approximation parameters:
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q(f|X) = int_{df}{N(f|KfuKuu_invu,diag(Kff-Qff)*N(u|0,Kuu)} = N(f|0,Sigma0)
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Sigma0 = diag(Knn-Qnn) + Qnn, Qnn = Knm*Kmmi*Kmn
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"""
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self.Kmmi, self.Kmm_hld = pdinv(self.Kmm)
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self.P0 = self.Kmn.T
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self.KmnKnm = np.dot(self.P0.T, self.P0)
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self.KmmiKmn = np.dot(self.Kmmi,self.P0.T)
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self.Qnn_diag = np.sum(self.P0.T*self.KmmiKmn,-2)
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self.Diag0 = self.Knn_diag - self.Qnn_diag
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self.R0 = jitchol(self.Kmmi).T
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"""
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Posterior approximation: q(f|y) = N(f| mu, Sigma)
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Sigma = Diag + P*R.T*R*P.T + K
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mu = w + P*gamma
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"""
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self.w = np.zeros(self.N)
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self.gamma = np.zeros(self.M)
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self.mu = np.zeros(self.N)
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self.P = self.P0.copy()
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self.R = self.R0.copy()
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self.Diag = self.Diag0.copy()
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self.Sigma_diag = self.Knn_diag
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"""
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Initial values - Cavity distribution parameters:
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q_(g|mu_,sigma2_) = Product{q_i(g|mu_i,sigma2_i)}
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sigma_ = 1./tau_
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mu_ = v_/tau_
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"""
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self.tau_ = np.empty(self.N,dtype=float)
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self.v_ = np.empty(self.N,dtype=float)
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#Initial values - Marginal moments
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z = np.empty(self.N,dtype=float)
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self.Z_hat = np.empty(self.N,dtype=float)
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phi = np.empty(self.N,dtype=float)
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mu_hat = np.empty(self.N,dtype=float)
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sigma2_hat = np.empty(self.N,dtype=float)
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#Approximation
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epsilon_np1 = 1
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epsilon_np2 = 1
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self.iterations = 0
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self.np1 = [self.tau_tilde.copy()]
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self.np2 = [self.v_tilde.copy()]
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while epsilon_np1 > self.epsilon or epsilon_np2 > self.epsilon:
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update_order = np.arange(self.N)
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random.shuffle(update_order)
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for i in update_order:
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#Cavity distribution parameters
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self.tau_[i] = 1./self.Sigma_diag[i] - self.eta*self.tau_tilde[i]
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self.v_[i] = self.mu[i]/self.Sigma_diag[i] - self.eta*self.v_tilde[i]
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#Marginal moments
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self.Z_hat[i], mu_hat[i], sigma2_hat[i] = self.likelihood.moments_match(i,self.tau_[i],self.v_[i])
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#Site parameters update
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Delta_tau = self.delta/self.eta*(1./sigma2_hat[i] - 1./self.Sigma_diag[i])
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Delta_v = self.delta/self.eta*(mu_hat[i]/sigma2_hat[i] - self.mu[i]/self.Sigma_diag[i])
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self.tau_tilde[i] = self.tau_tilde[i] + Delta_tau
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self.v_tilde[i] = self.v_tilde[i] + Delta_v
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#Posterior distribution parameters update
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dtd1 = Delta_tau*self.Diag[i] + 1.
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dii = self.Diag[i]
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self.Diag[i] = dii - (Delta_tau * dii**2.)/dtd1
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pi_ = self.P[i,:].reshape(1,self.M)
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self.P[i,:] = pi_ - (Delta_tau*dii)/dtd1 * pi_
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Rp_i = np.dot(self.R,pi_.T)
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RTR = np.dot(self.R.T,np.dot(np.eye(self.M) - Delta_tau/(1.+Delta_tau*self.Sigma_diag[i]) * np.dot(Rp_i,Rp_i.T),self.R))
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self.R = jitchol(RTR).T
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self.w[i] = self.w[i] + (Delta_v - Delta_tau*self.w[i])*dii/dtd1
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self.gamma = self.gamma + (Delta_v - Delta_tau*self.mu[i])*np.dot(RTR,self.P[i,:].T)
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self.RPT = np.dot(self.R,self.P.T)
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self.Sigma_diag = self.Diag + np.sum(self.RPT.T*self.RPT.T,-1)
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self.mu = self.w + np.dot(self.P,self.gamma)
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self.iterations += 1
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#Sigma recomptutation with Cholesky decompositon
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self.Diag = self.Diag0/(1.+ self.Diag0 * self.tau_tilde)
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self.P = (self.Diag / self.Diag0)[:,None] * self.P0
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self.RPT0 = np.dot(self.R0,self.P0.T)
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L = jitchol(np.eye(self.M) + np.dot(self.RPT0,(1./self.Diag0 - self.Diag/(self.Diag0**2))[:,None]*self.RPT0.T))
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self.R,info = linalg.flapack.dtrtrs(L,self.R0,lower=1)
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self.RPT = np.dot(self.R,self.P.T)
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self.Sigma_diag = self.Diag + np.sum(self.RPT.T*self.RPT.T,-1)
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self.w = self.Diag * self.v_tilde
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self.gamma = np.dot(self.R.T, np.dot(self.RPT,self.v_tilde))
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self.mu = self.w + np.dot(self.P,self.gamma)
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epsilon_np1 = sum((self.tau_tilde-self.np1[-1])**2)/self.N
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epsilon_np2 = sum((self.v_tilde-self.np2[-1])**2)/self.N
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self.np1.append(self.tau_tilde.copy())
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self.np2.append(self.v_tilde.copy())
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