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GPy/GPy/inference/latent_function_inference/var_dtc_parallel.py
Zhenwen Dai 216de32c0c fix add kernel and VarDTC_minibatch speed tuning
2014-07-01 18:01:33 +01:00

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# Copyright (c) 2012, GPy authors (see AUTHORS.txt).
# Licensed under the BSD 3-clause license (see LICENSE.txt)
from posterior import Posterior
from ...util.linalg import jitchol, backsub_both_sides, tdot, dtrtrs
from ...util import diag
from ...core.parameterization.variational import VariationalPosterior
import numpy as np
from ...util.misc import param_to_array
from . import LatentFunctionInference
log_2_pi = np.log(2*np.pi)
try:
from mpi4py import MPI
except:
pass
class VarDTC_minibatch(LatentFunctionInference):
"""
An object for inference when the likelihood is Gaussian, but we want to do sparse inference.
The function self.inference returns a Posterior object, which summarizes
the posterior.
For efficiency, we sometimes work with the cholesky of Y*Y.T. To save repeatedly recomputing this, we cache it.
"""
const_jitter = 1e-6
def __init__(self, batchsize=None, limit=1, mpi_comm=None):
self.batchsize = batchsize
self.mpi_comm = mpi_comm
self.limit = limit
# Cache functions
from ...util.caching import Cacher
self.get_trYYT = Cacher(self._get_trYYT, limit)
self.get_YYTfactor = Cacher(self._get_YYTfactor, limit)
self.midRes = {}
self.batch_pos = 0 # the starting position of the current mini-batch
self.Y_speedup = False # Replace Y with the cholesky factor of YY.T, but the posterior inference will be wrong
def __getstate__(self):
# has to be overridden, as Cacher objects cannot be pickled.
return self.batchsize, self.limit, self.Y_speedup
def __setstate__(self, state):
# has to be overridden, as Cacher objects cannot be pickled.
self.batchsize, self.limit, self.Y_speedup = state
self.mpi_comm = None
self.midRes = {}
self.batch_pos = 0
from ...util.caching import Cacher
self.get_trYYT = Cacher(self._get_trYYT, self.limit)
self.get_YYTfactor = Cacher(self._get_YYTfactor, self.limit)
def set_limit(self, limit):
self.get_trYYT.limit = limit
self.get_YYTfactor.limit = limit
def _get_trYYT(self, Y):
return param_to_array(np.sum(np.square(Y)))
def _get_YYTfactor(self, Y):
"""
find a matrix L which satisfies LLT = YYT.
Note that L may have fewer columns than Y.
"""
N, D = Y.shape
if (N>=D):
return param_to_array(Y)
else:
return jitchol(tdot(Y))
def gatherPsiStat(self, kern, X, Z, Y, beta, uncertain_inputs):
het_noise = beta.size > 1
trYYT = self.get_trYYT(Y)
if self.Y_speedup and not het_noise:
Y = self.get_YYTfactor(Y)
num_inducing = Z.shape[0]
num_data, output_dim = Y.shape
if self.batchsize == None:
self.batchsize = num_data
psi2_full = np.zeros((num_inducing,num_inducing))
psi1Y_full = np.zeros((output_dim,num_inducing)) # DxM
psi0_full = 0.
YRY_full = 0.
for n_start in xrange(0,num_data,self.batchsize):
n_end = min(self.batchsize+n_start, num_data)
if (n_end-n_start)==num_data:
Y_slice = Y
X_slice = X
else:
Y_slice = Y[n_start:n_end]
X_slice = X[n_start:n_end]
if het_noise:
b = beta[n_start]
YRY_full += np.inner(Y_slice, Y_slice)*b
else:
b = beta
if uncertain_inputs:
psi0 = kern.psi0(Z, X_slice)
psi1 = kern.psi1(Z, X_slice)
psi2_full += kern.psi2(Z, X_slice)*b
else:
psi0 = kern.Kdiag(X_slice)
psi1 = kern.K(X_slice, Z)
psi2_full += np.dot(psi1.T,psi1)*b
psi0_full += psi0.sum()*b
psi1Y_full += np.dot(Y_slice.T,psi1)*b # DxM
if not het_noise:
YRY_full = trYYT*beta
if self.mpi_comm != None:
psi0_all = np.array(psi0_full)
psi1Y_all = psi1Y_full.copy()
psi2_all = psi2_full.copy()
YRY_all = np.array(YRY_full)
self.mpi_comm.Allreduce([psi0_full, MPI.DOUBLE], [psi0_all, MPI.DOUBLE])
self.mpi_comm.Allreduce([psi1Y_full, MPI.DOUBLE], [psi1Y_all, MPI.DOUBLE])
self.mpi_comm.Allreduce([psi2_full, MPI.DOUBLE], [psi2_all, MPI.DOUBLE])
self.mpi_comm.Allreduce([YRY_full, MPI.DOUBLE], [YRY_all, MPI.DOUBLE])
return psi0_all, psi1Y_all, psi2_all, YRY_all
return psi0_full, psi1Y_full, psi2_full, YRY_full
def inference_likelihood(self, kern, X, Z, likelihood, Y):
"""
The first phase of inference:
Compute: log-likelihood, dL_dKmm
Cached intermediate results: Kmm, KmmInv,
"""
num_data, output_dim = Y.shape
if self.mpi_comm != None:
num_data_all = np.array(num_data,dtype=np.int32)
self.mpi_comm.Allreduce([np.int32(num_data), MPI.INT], [num_data_all, MPI.INT])
num_data = num_data_all
if isinstance(X, VariationalPosterior):
uncertain_inputs = True
else:
uncertain_inputs = False
#see whether we've got a different noise variance for each datum
beta = 1./np.fmax(likelihood.variance, 1e-6)
het_noise = beta.size > 1
if het_noise:
self.batchsize = 1
psi0_full, psi1Y_full, psi2_full, YRY_full = self.gatherPsiStat(kern, X, Z, Y, beta, uncertain_inputs)
#======================================================================
# Compute Common Components
#======================================================================
Kmm = kern.K(Z).copy()
diag.add(Kmm, self.const_jitter)
Lm = jitchol(Kmm)
Lambda = Kmm+psi2_full
LL = jitchol(Lambda)
b,_ = dtrtrs(LL, psi1Y_full.T)
bbt = np.square(b).sum()
v,_ = dtrtrs(LL.T,b,lower=False)
vvt = np.einsum('md,od->mo',v,v)
LmInvPsi2LmInvT = backsub_both_sides(Lm,psi2_full,transpose='right')
Psi2LLInvT = dtrtrs(LL,psi2_full)[0].T
LmInvPsi2LLInvT= dtrtrs(Lm,Psi2LLInvT)[0]
KmmInvPsi2LLInvT = dtrtrs(Lm,LmInvPsi2LLInvT,trans=True)[0]
KmmInvPsi2P = dtrtrs(LL,KmmInvPsi2LLInvT.T, trans=True)[0].T
dL_dpsi2R = (output_dim*KmmInvPsi2P - vvt)/2. # dL_dpsi2 with R inside psi2
# Cache intermediate results
self.midRes['dL_dpsi2R'] = dL_dpsi2R
self.midRes['v'] = v
#======================================================================
# Compute log-likelihood
#======================================================================
if het_noise:
logL_R = -np.log(beta).sum()
else:
logL_R = -num_data*np.log(beta)
logL = -(output_dim*(num_data*log_2_pi+logL_R+psi0_full-np.trace(LmInvPsi2LmInvT))+YRY_full-bbt)/2.-output_dim*(-np.log(np.diag(Lm)).sum()+np.log(np.diag(LL)).sum())
#======================================================================
# Compute dL_dKmm
#======================================================================
dL_dKmm = -(output_dim*np.einsum('md,od->mo',KmmInvPsi2LLInvT,KmmInvPsi2LLInvT) + vvt)/2.
#======================================================================
# Compute the Posterior distribution of inducing points p(u|Y)
#======================================================================
if not self.Y_speedup or het_noise:
post = Posterior(woodbury_inv=KmmInvPsi2P, woodbury_vector=v, K=Kmm, mean=None, cov=None, K_chol=Lm)
else:
post = None
#======================================================================
# Compute dL_dthetaL for uncertian input and non-heter noise
#======================================================================
if not het_noise:
dL_dthetaL = (YRY_full*beta + beta*output_dim*psi0_full - num_data*output_dim*beta)/2. - beta*(dL_dpsi2R*psi2_full).sum() - beta*(v.T*psi1Y_full).sum()
self.midRes['dL_dthetaL'] = dL_dthetaL
return logL, dL_dKmm, post
def inference_minibatch(self, kern, X, Z, likelihood, Y):
"""
The second phase of inference: Computing the derivatives over a minibatch of Y
Compute: dL_dpsi0, dL_dpsi1, dL_dpsi2, dL_dthetaL
return a flag showing whether it reached the end of Y (isEnd)
"""
num_data, output_dim = Y.shape
if isinstance(X, VariationalPosterior):
uncertain_inputs = True
else:
uncertain_inputs = False
#see whether we've got a different noise variance for each datum
beta = 1./np.fmax(likelihood.variance, 1e-6)
het_noise = beta.size > 1
# VVT_factor is a matrix such that tdot(VVT_factor) = VVT...this is for efficiency!
#self.YYTfactor = beta*self.get_YYTfactor(Y)
if self.Y_speedup and not het_noise:
YYT_factor = self.get_YYTfactor(Y)
else:
YYT_factor = Y
n_start = self.batch_pos
n_end = min(self.batchsize+n_start, num_data)
if n_end==num_data:
isEnd = True
self.batch_pos = 0
else:
isEnd = False
self.batch_pos = n_end
Y_slice = YYT_factor[n_start:n_end]
X_slice = X[n_start:n_end]
if not uncertain_inputs:
psi0 = kern.Kdiag(X_slice)
psi1 = kern.K(X_slice, Z)
psi2 = None
betapsi1 = np.einsum('n,nm->nm',beta,psi1)
elif het_noise:
psi0 = kern.psi0(Z, X_slice)
psi1 = kern.psi1(Z, X_slice)
psi2 = kern.psi2(Z, X_slice)
betapsi1 = np.einsum('n,nm->nm',beta,psi1)
if het_noise:
beta = beta[n_start] # assuming batchsize==1
betaY = beta*Y_slice
#======================================================================
# Load Intermediate Results
#======================================================================
dL_dpsi2R = self.midRes['dL_dpsi2R']
v = self.midRes['v']
#======================================================================
# Compute dL_dpsi
#======================================================================
dL_dpsi0 = -output_dim * (beta * np.ones((n_end-n_start,)))/2.
dL_dpsi1 = np.dot(betaY,v.T)
if uncertain_inputs:
dL_dpsi2 = beta* dL_dpsi2R
else:
dL_dpsi1 += np.dot(betapsi1,dL_dpsi2R)*2.
dL_dpsi2 = None
#======================================================================
# Compute dL_dthetaL
#======================================================================
if het_noise:
if uncertain_inputs:
psiR = np.einsum('mo,mo->',dL_dpsi2R,psi2)
else:
psiR = np.einsum('nm,no,mo->',psi1,psi1,dL_dpsi2R)
dL_dthetaL = ((np.square(betaY)).sum(axis=-1) + np.square(beta)*(output_dim*psi0)-output_dim*beta)/2. - np.square(beta)*psiR- (betaY*np.dot(betapsi1,v)).sum(axis=-1)
else:
if isEnd:
dL_dthetaL = self.midRes['dL_dthetaL']
else:
dL_dthetaL = 0.
if uncertain_inputs:
grad_dict = {'dL_dpsi0':dL_dpsi0,
'dL_dpsi1':dL_dpsi1,
'dL_dpsi2':dL_dpsi2,
'dL_dthetaL':dL_dthetaL}
else:
grad_dict = {'dL_dKdiag':dL_dpsi0,
'dL_dKnm':dL_dpsi1,
'dL_dthetaL':dL_dthetaL}
return isEnd, (n_start,n_end), grad_dict
def update_gradients(model, mpi_comm=None):
if mpi_comm == None:
Y = model.Y
X = model.X
else:
Y = model.Y_local
X = model.X[model.N_range[0]:model.N_range[1]]
model._log_marginal_likelihood, dL_dKmm, model.posterior = model.inference_method.inference_likelihood(model.kern, X, model.Z, model.likelihood, Y)
het_noise = model.likelihood.variance.size > 1
if het_noise:
dL_dthetaL = np.empty((model.Y.shape[0],))
else:
dL_dthetaL = np.float64(0.)
kern_grad = model.kern.gradient.copy()
kern_grad[:] = 0.
model.Z.gradient = 0.
isEnd = False
while not isEnd:
isEnd, n_range, grad_dict = model.inference_method.inference_minibatch(model.kern, X, model.Z, model.likelihood, Y)
if isinstance(model.X, VariationalPosterior):
if (n_range[1]-n_range[0])==X.shape[0]:
X_slice = X
elif mpi_comm ==None:
X_slice = model.X[n_range[0]:n_range[1]]
else:
X_slice = model.X[model.N_range[0]+n_range[0]:model.N_range[0]+n_range[1]]
#gradients w.r.t. kernel
model.kern.update_gradients_expectations(variational_posterior=X_slice, Z=model.Z, dL_dpsi0=grad_dict['dL_dpsi0'], dL_dpsi1=grad_dict['dL_dpsi1'], dL_dpsi2=grad_dict['dL_dpsi2'])
kern_grad += model.kern.gradient
#gradients w.r.t. Z
model.Z.gradient += model.kern.gradients_Z_expectations(
dL_dpsi0=grad_dict['dL_dpsi0'], dL_dpsi1=grad_dict['dL_dpsi1'], dL_dpsi2=grad_dict['dL_dpsi2'], Z=model.Z, variational_posterior=X_slice)
#gradients w.r.t. posterior parameters of X
X_grad = model.kern.gradients_qX_expectations(variational_posterior=X_slice, Z=model.Z, dL_dpsi0=grad_dict['dL_dpsi0'], dL_dpsi1=grad_dict['dL_dpsi1'], dL_dpsi2=grad_dict['dL_dpsi2'])
model.set_X_gradients(X_slice, X_grad)
if het_noise:
dL_dthetaL[n_range[0]:n_range[1]] = grad_dict['dL_dthetaL']
else:
dL_dthetaL += grad_dict['dL_dthetaL']
# Gather the gradients from multiple MPI nodes
if mpi_comm != None:
if het_noise:
assert False, "Not implemented!"
kern_grad_all = kern_grad.copy()
Z_grad_all = model.Z.gradient.copy()
mpi_comm.Allreduce([kern_grad, MPI.DOUBLE], [kern_grad_all, MPI.DOUBLE])
mpi_comm.Allreduce([model.Z.gradient, MPI.DOUBLE], [Z_grad_all, MPI.DOUBLE])
kern_grad = kern_grad_all
model.Z.gradient = Z_grad_all
#gradients w.r.t. kernel
model.kern.update_gradients_full(dL_dKmm, model.Z, None)
model.kern.gradient += kern_grad
#gradients w.r.t. Z
model.Z.gradient += model.kern.gradients_X(dL_dKmm, model.Z)
# Update Log-likelihood
KL_div = model.variational_prior.KL_divergence(X)
# update for the KL divergence
model.variational_prior.update_gradients_KL(X)
if mpi_comm != None:
KL_div_all = np.array(KL_div)
mpi_comm.Allreduce([np.float64(KL_div), MPI.DOUBLE], [KL_div_all, MPI.DOUBLE])
KL_div = KL_div_all
[mpi_comm.Allgatherv([pp.copy(), MPI.DOUBLE], [pa, (model.N_list*pa.shape[-1], None), MPI.DOUBLE]) for pp,pa in zip(model.get_X_gradients(X),model.get_X_gradients(model.X))]
from ...models import SSGPLVM
if isinstance(model, SSGPLVM):
grad_pi = np.array(model.variational_prior.pi.gradient)
mpi_comm.Allreduce([grad_pi.copy(), MPI.DOUBLE], [model.variational_prior.pi.gradient, MPI.DOUBLE])
model._log_marginal_likelihood -= KL_div
# dL_dthetaL
model.likelihood.update_gradients(dL_dthetaL)
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