def grad_optimize_ei(self, cand, comp, vals, durs, compute_grad=True):
# Here we have to compute the gradients for ei per second
# This means deriving through the two kernels, the one for predicting
# time and the one predicting ei
best = np.min(vals)
cand = np.reshape(cand, (-1, comp.shape[1]))
# First we make predictions for the durations
# Compute covariances
comp_time_cov = self.cov(self.time_amp2, self.time_ls, comp)
cand_time_cross = self.cov(self.time_amp2, self.time_ls,comp,cand)
# Cholesky decompositions
obsv_time_cov = comp_time_cov + self.time_noise*np.eye(comp.shape[0])
obsv_time_chol = spla.cholesky( obsv_time_cov, lower=True )
# Linear systems
t_alpha = spla.cho_solve((obsv_time_chol, True), durs - self.time_mean)
# Predict marginal mean times and (possibly) variances
func_time_m = np.dot(cand_time_cross.T, t_alpha) + self.time_mean
# We don't really need the time variances now
#func_time_v = self.time_amp2*(1+1e-6) - np.sum(t_beta**2, axis=0)
# Bring time out of the log domain
func_time_m = np.exp(func_time_m)
# Compute derivative of cross-distances.
grad_cross_r = gp.grad_dist2(self.time_ls, comp, cand)
# Apply covariance function
cov_grad_func = getattr(gp, 'grad_' + self.cov_func.__name__)
cand_cross_grad = cov_grad_func(self.time_ls, comp, cand)
grad_cross_t = np.squeeze(cand_cross_grad)
# Now compute the gradients w.r.t. ei
# The primary covariances for prediction.
comp_cov = self.cov(self.amp2, self.ls, comp)
cand_cross = self.cov(self.amp2, self.ls, comp, cand)
# Compute the required Cholesky.
obsv_cov = comp_cov + self.noise*np.eye(comp.shape[0])
obsv_chol = spla.cholesky( obsv_cov, lower=True )
cand_cross_grad = cov_grad_func(self.ls, comp, cand)
# Predictive things.
# Solve the linear systems.
alpha = spla.cho_solve((obsv_chol, True), vals - self.mean)
beta = spla.solve_triangular(obsv_chol, cand_cross, lower=True)
# Predict the marginal means and variances at candidates.
func_m = np.dot(cand_cross.T, alpha) + self.mean
func_v = self.amp2*(1+1e-6) - np.sum(beta**2, axis=0)
# Expected improvement
func_s = np.sqrt(func_v)
u = (best - func_m) / func_s
ncdf = sps.norm.cdf(u)
npdf = sps.norm.pdf(u)
ei = func_s*(u*ncdf + npdf)
ei_per_s = -np.sum(ei/func_time_m)
if not compute_grad:
return ei
grad_time_xp_m = np.dot(t_alpha.transpose(),grad_cross_t)
# Gradients of ei w.r.t. mean and variance
g_ei_m = -ncdf
g_ei_s2 = 0.5*npdf / func_s
# Apply covariance function
grad_cross = np.squeeze(cand_cross_grad)
grad_xp_m = np.dot(alpha.transpose(),grad_cross)
grad_xp_v = np.dot(-2*spla.cho_solve((obsv_chol, True),
cand_cross).transpose(),grad_cross)
grad_xp = 0.5*self.amp2*(grad_xp_m*g_ei_m + grad_xp_v*g_ei_s2)
grad_time_xp_m = 0.5*self.time_amp2*grad_time_xp_m*func_time_m
grad_xp = (func_time_m*grad_xp - ei*grad_time_xp_m)/(func_time_m**2)
return ei_per_s, grad_xp.flatten()
def grad_optimize_ei(self, cand, comp, vals, durs, compute_grad=True):
# Here we have to compute the gradients for ei per second
# This means deriving through the two kernels, the one for predicting
# time and the one predicting ei
best = np.min(vals)
cand = np.reshape(cand, (-1, comp.shape[1]))
# First we make predictions for the durations
# Compute covariances
comp_time_cov = self.cov(self.time_amp2, self.time_ls, comp)
cand_time_cross = self.cov(self.time_amp2, self.time_ls,comp,cand)
# Cholesky decompositions
obsv_time_cov = comp_time_cov + self.time_noise*np.eye(comp.shape[0])
obsv_time_chol = spla.cholesky( obsv_time_cov, lower=True )
# Linear systems
t_alpha = spla.cho_solve((obsv_time_chol, True), durs - self.time_mean)
# Predict marginal mean times and (possibly) variances
func_time_m = np.dot(cand_time_cross.T, t_alpha) + self.time_mean
# We don't really need the time variances now
#func_time_v = self.time_amp2*(1+1e-6) - np.sum(t_beta**2, axis=0)
# Bring time out of the log domain
func_time_m = np.exp(func_time_m)
# Compute derivative of cross-distances.
grad_cross_r = gp.grad_dist2(self.time_ls, comp, cand)
# Apply covariance function
cov_grad_func = getattr(gp, 'grad_' + self.cov_func.__name__)
cand_cross_grad = cov_grad_func(self.time_ls, comp, cand)
grad_cross_t = np.squeeze(cand_cross_grad)
# Now compute the gradients w.r.t. ei
# The primary covariances for prediction.
comp_cov = self.cov(self.amp2, self.ls, comp)
cand_cross = self.cov(self.amp2, self.ls, comp, cand)
# Compute the required Cholesky.
obsv_cov = comp_cov + self.noise*np.eye(comp.shape[0])
obsv_chol = spla.cholesky( obsv_cov, lower=True )
cand_cross_grad = cov_grad_func(self.ls, comp, cand)
# Predictive things.
# Solve the linear systems.
alpha = spla.cho_solve((obsv_chol, True), vals - self.mean)
beta = spla.solve_triangular(obsv_chol, cand_cross, lower=True)
# Predict the marginal means and variances at candidates.
func_m = np.dot(cand_cross.T, alpha) + self.mean
func_v = self.amp2*(1+1e-6) - np.sum(beta**2, axis=0)
# Expected improvement
func_s = np.sqrt(func_v)
u = (best - func_m) / func_s
ncdf = sps.norm.cdf(u)
npdf = sps.norm.pdf(u)
ei = func_s*(u*ncdf + npdf)
ei_per_s = -np.sum(ei/func_time_m)
if not compute_grad:
return ei
grad_time_xp_m = np.dot(t_alpha.transpose(),grad_cross_t)
# Gradients of ei w.r.t. mean and variance
g_ei_m = -ncdf
g_ei_s2 = 0.5*npdf / func_s
# Apply covariance function
grad_cross = np.squeeze(cand_cross_grad)
grad_xp_m = np.dot(alpha.transpose(),grad_cross)
grad_xp_v = np.dot(-2*spla.cho_solve((obsv_chol, True),
cand_cross).transpose(),grad_cross)
grad_xp = 0.5*self.amp2*(grad_xp_m*g_ei_m + grad_xp_v*g_ei_s2)
grad_time_xp_m = 0.5*self.time_amp2*grad_time_xp_m*func_time_m
grad_xp = (func_time_m*grad_xp - ei*grad_time_xp_m)/(func_time_m**2)
return ei_per_s, grad_xp.flatten()