def _LMLgrad_s(self,hyperparams,debugging=False):
"""
evaluate gradients with respect to covariance matrix Sigma
"""
try:
KV = self.get_covariances(hyperparams, debugging=debugging)
except LA.LinAlgError:
LG.error('linalg exception in _LMLgrad_x_sigma')
return {'X_s':SP.zeros(hyperparams['X_s'].shape)}
Si = 1./KV['Stilde_os']
SS = SP.dot(unravel(Si,self.n,self.t).T,KV['Stilde_o'])
USU = SP.dot(KV['USi_c'],KV['Utilde_s'])
Yhat = unravel(Si * ravel(KV['UYtildeU_os']),self.n,self.t)
RV = {}
if 'X_s' in hyperparams:
USUY = SP.dot(USU,Yhat.T)
USUYSYUSU = SP.dot(USUY,(KV['Stilde_o']*USUY).T)
LMLgrad = SP.zeros((self.t,self.covar_s.n_dimensions))
LMLgrad_det = SP.zeros((self.t,self.covar_s.n_dimensions))
LMLgrad_quad = SP.zeros((self.t,self.covar_s.n_dimensions))
for d in xrange(self.covar_s.n_dimensions):
Kd_grad = self.covar_s.Kgrad_x(hyperparams['covar_s'],d)
UsU = SP.dot(Kd_grad.T,USU)*USU
LMLgrad_det[:,d] = SP.dot(UsU,SS.T)
# calculate gradient of squared form
LMLgrad_quad[:,d] = -(USUYSYUSU*Kd_grad).sum(0)
LMLgrad = LMLgrad_det + LMLgrad_quad
RV['X_s'] = LMLgrad
if debugging:
_LMLgrad = SP.zeros((self.t,self.covar_s.n_dimensions))
for t in xrange(self.t):
for d in xrange(self.covar_s.n_dimensions):
Kgrad_x = self.covar_s.Kgrad_x(hyperparams['covar_s'],d,t)
Kgrad_x = SP.kron(Kgrad_x,KV['K_o'])
_LMLgrad[t,d] = 0.5*(KV['W']*Kgrad_x).sum()
assert SP.allclose(LMLgrad,_LMLgrad,rtol=1E-3,atol=1E-2), 'ouch, something is wrong: %.2f'%LA.norm(LMLgrad-_LMLgrad)
if 'covar_s' in hyperparams:
theta = SP.zeros(len(hyperparams['covar_s']))
for i in range(len(theta)):
Kgrad_s = self.covar_s.Kgrad_theta(hyperparams['covar_s'],i)
UdKU = SP.dot(USU.T, SP.dot(Kgrad_s, USU))
SYUdKU = SP.dot(UdKU,KV['Stilde_o'] * Yhat.T)
LMLgrad_det = SP.sum(Si*SP.kron(SP.diag(UdKU),KV['Stilde_o']))
LMLgrad_quad = -(Yhat.T*SYUdKU).sum()
LMLgrad = 0.5*(LMLgrad_det + LMLgrad_quad)
theta[i] = LMLgrad
if debugging:
Kd = SP.kron(Kgrad_s, KV['K_o'])
_LMLgrad = 0.5 * (KV['W']*Kd).sum()
assert SP.allclose(LMLgrad,_LMLgrad,rtol=1E-3,atol=1E-2), 'ouch, something is wrong: %.2f'%LA.norm(LMLgrad-_LMLgrad)
RV['covar_s'] = theta
return RV
def predict(self, hyperparams, Xstar_r=None, debugging=False):
"""
predict over new training points
"""
self._update_inputs(hyperparams)
KV = self.get_covariances(hyperparams, debugging=debugging)
if Xstar_r != None:
self.covar_r.Xcross = Xstar_r
Kstar_r = self.covar_r.Kcross(hyperparams['covar_r'])
Kstar_c = self.covar_c.K(
hyperparams['covar_c']) # kernel over tasks is fixed!
S = SP.kron(KV['Stilde_c'], KV['Stilde_r']) + 1
USUc = SP.dot(SP.sqrt(1. / KV['S_s']) * KV['U_s'], KV['Utilde_c'])
USUr = SP.dot(SP.sqrt(1. / KV['S_o']) * KV['U_o'], KV['Utilde_r'])
KinvY = SP.dot(
USUr,
SP.dot(unravel(ravel(KV['UYtildeU_rc']) * 1. / S, self.n, self.t),
USUc.T))
Ystar = SP.dot(Kstar_r.T, SP.dot(KinvY, Kstar_c))
Ystar = unravel(Ystar, self.covar_r.n_cross, self.t)
if debugging:
Kstar = SP.kron(Kstar_c, Kstar_r)
Ynaive = SP.dot(Kstar.T, KV['alpha'])
Ynaive = unravel(Ynaive, self.covar_r.n_cross, self.t)
assert SP.allclose(Ystar,
Ynaive), 'ouch, prediction does not work out'
return Ystar
def predict(self,
hyperparams,
Xstar_r,
compute_cov=False,
debugging=False):
"""
predict over new training points
"""
self._update_inputs(hyperparams)
KV = self.get_covariances(hyperparams, debugging=debugging)
self.covar_r.Xcross = Xstar_r
Kstar_r = self.covar_r.Kcross(hyperparams['covar_r'])
Kstar_c = self.covar_c.K(
hyperparams['covar_c']) # kernel over tasks is fixed!
S = SP.kron(KV['Stilde_c'], KV['Stilde_r']) + 1
USUc = SP.dot(SP.sqrt(1. / KV['S_s']) * KV['U_s'], KV['Utilde_c'])
#USUc1 = SP.dot(KV['Utilde_c'].T, SP.dot(SP.diag(SP.sqrt(1./KV['S_s'])),KV['U_s'].T))
USUr = SP.dot(SP.sqrt(1. / KV['S_o']) * KV['U_o'], KV['Utilde_r'])
#USUr = SP.dot(SP.dot(KV['U_o'], SP.diag(SP.sqrt(1./KV['S_o']))),KV['Utilde_r'])
KinvY = SP.dot(
USUr,
SP.dot(unravel(ravel(KV['UYtildeU_rc']) * 1. / S, self.n, self.t),
USUc.T))
#KinvY = SP.dot(USUr, SP.dot(unravel(ravel(KV['UYtildeU_rc']) * 1./S, self.n, self.t), USUc))
Ystar = SP.dot(Kstar_r.T, SP.dot(KinvY, Kstar_c))
Ystar = unravel(Ystar, self.covar_r.n_cross, self.t)
if debugging:
Kstar = SP.kron(Kstar_c, Kstar_r)
Ynaive = SP.dot(Kstar.T, KV['alpha'])
Ynaive = unravel(Ynaive, self.covar_r.n_cross, self.t)
assert SP.allclose(Ystar,
Ynaive), 'ouch, prediction does not work out'
Ystar_covar = []
if compute_cov:
R_star_star = SP.exp(2 * hyperparams['covar_r']) * fast_dot(
Xstar_r, Xstar_r.T)
R_tr_star = Kstar_r
C = Kstar_c
# temp1 = fast_kron(fast_dot(fast_dot(fast_dot(C, KV['U_s']), np.diag(SP.sqrt(1./KV['S_s']))),KV['Utilde_c']),
# fast_dot(fast_dot(fast_dot(R_tr_star.T, KV['U_o']), np.diag(SP.sqrt(1./KV['S_o']))),KV['Utilde_r']))
# Ystar_covar1 = SP.diag(fast_kron(C, R_star_star) - fast_dot(fast_dot(temp1,np.diag(1./S )),temp1.T))
temp = fast_kron(
fast_dot(fast_dot(C, KV['U_s'] * SP.sqrt(1. / KV['S_s'])),
KV['Utilde_c']),
fast_dot(
fast_dot(R_tr_star.T, KV['U_o'] * SP.sqrt(1. / KV['S_o'])),
KV['Utilde_r']))
Ystar_covar = SP.diag(fast_kron(C, R_star_star)) - SP.sum(
(1. / S * temp).T * temp.T, axis=0)
Ystar_covar = unravel(Ystar_covar, Xstar_r.shape[0], self.t)
return Ystar, Ystar_covar
def _LMLgrad_x_omega(self, hyperparams, debugging=False):
try:
KV = self.get_covariances(hyperparams)
except LA.LinAlgError:
LG.error('linalg exception in _LMLgrad_x_omega')
return {'X_o': SP.zeros(hyperparams['X_o'].shape)}
LMLgrad = SP.zeros((self.n, self.covar_o.n_dimensions))
for d in xrange(self.covar_o.n_dimensions):
Kgrad_x = self.covar_o.Kgrad_x(hyperparams['covar_o'], d)
Kgrad_x = SP.tile(Kgrad_x, self.n)
Kgrad_x = SP.kron(KV['K_s'], Kgrad_x)
LMLgrad[:, d] = unravel(SP.sum(KV['W'] * Kgrad_x, axis=0), self.n,
self.t).sum(1)
if debugging:
_LMLgrad = SP.zeros((self.n, self.covar_o.n_dimensions))
for n in xrange(self.n):
for d in xrange(self.covar_o.n_dimensions):
Kgrad_x = self.covar_o.Kgrad_x(hyperparams['covar_o'], d,
n)
Kgrad_x = SP.kron(KV['K_s'], Kgrad_x)
_LMLgrad[n, d] = 0.5 * (KV['W'] * Kgrad_x).sum()
assert SP.allclose(LMLgrad, _LMLgrad), 'ouch,something is wrong'
return {'X_o': LMLgrad}
def predict(self, hyperparams, Xstar_r=None, **kwargs):
"""
predict on Xstar
"""
if Xstar_r != None:
self.covar_r.Xcross = Xstar_r
self._update_inputs(hyperparams)
KV = self.get_covariances(hyperparams)
Kstar_r = self.covar_r.Kcross(hyperparams['covar_r'])
Kstar_c = self.covar_c.K(
hyperparams['covar_c']) # kernel over tasks is fixed!
Kstar = SP.kron(Kstar_c, Kstar_r)
Ystar = SP.dot(Kstar.T, KV['alpha'])
Ystar = unravel(Ystar, self.covar_r.n_cross, self.t)
return Ystar
def predict(self,hyperparams,Xstar_r=None,debugging=False):
"""
predict over new training points
"""
self._update_inputs(hyperparams)
KV = self.get_covariances(hyperparams,debugging=debugging)
if Xstar_r!=None:
self.covar_r.Xcross = Xstar_r
Kstar_r = self.covar_r.Kcross(hyperparams['covar_r'])
Kstar_c = self.covar_c.K(hyperparams['covar_c']) # kernel over tasks is fixed!
S = SP.kron(KV['Stilde_c'],KV['Stilde_r'])+1
USUc = SP.dot(SP.sqrt(1./KV['S_s']) * KV['U_s'],KV['Utilde_c'])
USUr = SP.dot(SP.sqrt(1./KV['S_o']) * KV['U_o'],KV['Utilde_r'])
KinvY = SP.dot(USUr,SP.dot(unravel(ravel(KV['UYtildeU_rc']) * 1./S,self.n,self.t),USUc.T))
Ystar = SP.dot(Kstar_r.T,SP.dot(KinvY,Kstar_c))
Ystar = unravel(Ystar,self.covar_r.n_cross,self.t)
if debugging:
Kstar = SP.kron(Kstar_c,Kstar_r)
Ynaive = SP.dot(Kstar.T,KV['alpha'])
Ynaive = unravel(Ynaive,self.covar_r.n_cross,self.t)
assert SP.allclose(Ystar,Ynaive), 'ouch, prediction does not work out'
return Ystar
def _LMLgrad_o(self, hyperparams, debugging=False):
"""
evaluates the gradient with respect to the covariance matrix Omega
"""
try:
KV = self.get_covariances(hyperparams, debugging=debugging)
except LA.LinAlgError:
LG.error('linalg exception in _LMLgrad_x_omega')
return {'X_o': SP.zeros(hyperparams['X_o'].shape)}
Si = 1. / KV['Stilde_os']
SS = SP.dot(unravel(Si, self.n, self.t), KV['Stilde_s'])
USU = SP.dot(KV['USi_r'], KV['Utilde_o'])
Yhat = unravel(Si * ravel(KV['UYtildeU_os']), self.n, self.t)
RV = {}
if 'X_o' in hyperparams:
USUY = SP.dot(USU, Yhat)
USUYSYUSU = SP.dot(USUY, (KV['Stilde_s'] * USUY).T)
LMLgrad = SP.zeros((self.n, self.covar_o.n_dimensions))
LMLgrad_det = SP.zeros((self.n, self.covar_o.n_dimensions))
LMLgrad_quad = SP.zeros((self.n, self.covar_o.n_dimensions))
for d in xrange(self.covar_o.n_dimensions):
Kd_grad = self.covar_o.Kgrad_x(hyperparams['covar_o'], d)
# calculate gradient of logdet
UoU = SP.dot(Kd_grad.T, USU) * USU
LMLgrad_det[:, d] = SP.dot(UoU, SS.T)
# calculate gradient of squared form
LMLgrad_quad[:, d] = -(USUYSYUSU * Kd_grad).sum(0)
LMLgrad = LMLgrad_det + LMLgrad_quad
RV['X_o'] = LMLgrad
if debugging:
_LMLgrad = SP.zeros((self.n, self.covar_o.n_dimensions))
for n in xrange(self.n):
for d in xrange(self.covar_o.n_dimensions):
Kgrad_x = self.covar_o.Kgrad_x(hyperparams['covar_o'],
d, n)
Kgrad_x = SP.kron(KV['K_s'], Kgrad_x)
_LMLgrad[n, d] = 0.5 * (KV['W'] * Kgrad_x).sum()
assert SP.allclose(
LMLgrad, _LMLgrad, rtol=1E-3, atol=1E-2
), 'ouch, something is wrong: %.2f' % LA.norm(LMLgrad - _LMLgrad)
if 'covar_o' in hyperparams:
theta = SP.zeros(len(hyperparams['covar_o']))
for i in range(len(theta)):
Kgrad_o = self.covar_o.Kgrad_theta(hyperparams['covar_o'], i)
UdKU = SP.dot(USU.T, SP.dot(Kgrad_o, USU))
SYUdKU = SP.dot(UdKU, Yhat * KV['Stilde_s'])
LMLgrad_det = SP.sum(Si *
SP.kron(KV['Stilde_s'], SP.diag(UdKU)))
LMLgrad_quad = -(Yhat * SYUdKU).sum()
LMLgrad = 0.5 * (LMLgrad_det + LMLgrad_quad)
theta[i] = LMLgrad
if debugging:
Kd = SP.kron(KV['K_s'], Kgrad_o)
_LMLgrad = 0.5 * (KV['W'] * Kd).sum()
assert SP.allclose(
LMLgrad, _LMLgrad, rtol=1E-2,
atol=1E-2), 'ouch, something is wrong: %.2f' % LA.norm(
LMLgrad - _LMLgrad)
RV['covar_o'] = theta
return RV