def gradient_and_hessian_of_Phi(x): H = zeros((2,2)) # Hessian g = zeros(2) # gradient V = zeros((2,1)) F = zeros((2,2)) Fdot = zeros((2,2)) D = 1 keep = D+1 for n in range(2): # 1: hos_forward, propagate two directions V[n,0] = 1. (y,W) = adolc.hos_forward(1,x,V,keep) V[n,0] = 0. F[0,:] = y[:2] F[1,:] = y[2:] Fdot[0,:] = W[:2,0] Fdot[1,:] = W[2:,0] # 2: matrix forward cg.forward([Mtc(F,Fdot)]) # 3: matrix reverse Phibar = array([[1.]]) Phibardot = array([[0.]]) cg.reverse([Mtc(Phibar, Phibardot)]) # 4: hov_reverse U = zeros((1,4,2)) U[0,:,0] = cg.independentFunctionList[0].xbar.X.flatten() U[0,:,1] = cg.independentFunctionList[0].xbar.Xdot.flatten() res = adolc.hov_ti_reverse(1,U)[0].copy() g[:] = res[0,:,0] H[n,:] = res[0,:,1] return (g,H)
def gradient_of_E_PHI(v,DM): """ computes the gradient of the expectation of PHI, i.e. grad( E[PHI] ), where E[PHI] is approximated by the method of moments up to moment degree DM. This gradient is then used in the steepest descent optimization below""" # multi indices for the full tensor of order DM-1 to compute d^(DM-1) PHI # the remaining order to compute d^DM PHI is achieved by a reverse sweep later Jq = generate_multi_indices(Nq,DM) NJq = shape(Jq)[0] # since Nq is usually much larger than Np # we want to be able to get one higher order derivative # w.r.t. q by reverse mode # that means we have to evaluate the mixed partial derivatives # d^2 F/dpdq in the forward mode # since UTPS cannot do those partial derivatives straightforward # we need to use interpolation J = zeros((DM+1,1,Nm,Np)) # Taylor series of degree DM has DM+1 taylor coefficients (i.e. count also the zero-derivative) qbar_rays = zeros((NJq,Nq,DM+1)) for nq in range(NJq): # 1: evaluation of J # here, we have to use nested interpolation # we need to compute # d^d F(x + s_1 z_1 + s_2 z_2) | # ---------------------------- | # d z_1 d^(d-1) z_2 |_z=0 # for s_1 = [1,0,0] resp. s_1 = [0,1,0] (one direction for each parameter p) # we use here formula (13) of the paper "Evaluating higher derivative tensors by forward propagation of univariate Taylor series" for np in range(Np): # each column of J for dm in range(DM+1): I = array([1,dm]) K = zeros((dm+1,2),dtype=int) K[:,0] = 1 K[:,1] = list(range(dm+1)) V = zeros((Nv,dm+1)) for k in K: s1 = zeros(Nv) s1[np] = k[0] s2 = zeros(Nv) s2[Np:] = k[1]*Jq[nq,:] V[:,0] = s1+s2 tmp = adolc.hos_forward(1,v,V,0)[1] J[dm,0,:,np] += (-1)**multi_index_abs( I - k) * multi_index_binomial(I,k) * tmp[:,dm] scale_factor = array([1./prod(list(range(1,d+1))) for d in range(DM+1)]) for dm in range(DM+1): J[dm,:,:,:] *= scale_factor[dm] # 2: forward evaluation of Phi Jtc=Mtc(J[:,:,:,:]) cg.forward([Jtc]) # 3: reverse evaluation of Phi Phibar = zeros((DM+1,1,1,1)) Phibar[0,0,0,0]=1. cg.reverse([Mtc(Phibar)]) Jbar = FJ.xbar.TC[:,0,:,:] #print Jbar #print shape(Jbar) ## 4: reverse evaluation of J vbar = zeros(Nv) dJ = zeros((Nq,DM+1)) # taylor coefficients for one direction in q for np in range(Np): # each column of J for dm in range(DM+1): I = array([1,dm]) K = zeros((dm+1,2),dtype=int) K[:,0] = 1 K[:,1] = list(range(dm+1)) V = zeros((Nv,dm+1)) for k in K: s1 = zeros(Nv) s1[np] = k[0] s2 = zeros(Nv) s2[Np:] = k[1]*Jq[nq,:] V[:,0] = s1+s2 keep = dm + 2 adolc.hos_forward(1,v,V,keep)[1] # U is a (Q,M,D) array # Jbar is a (D,M,Np) array U = zeros((1,Nm,keep)) for d in range(dm+1): U[0,:,d] = Jbar[d,:,np] #print 'U=',U Z = adolc.hov_ti_reverse(1,U)[0] #print 'Z=',Z #J[dm,0,:,np] += (-1)**multi_index_abs( I - k) * multi_index_binomial(I,k) * tmp[:,dm] #print shape(Z[0,:,:]) #exit() #print Z[0,:,dm+1] tmp = 1./prod(list(range(1,dm+2))) * (-1)**multi_index_abs( I - k) * multi_index_binomial(I,k) * Z[0,Np:,dm+1] dJ[:,dm] += tmp #vbar += tmp qbar_rays[nq,:,:] = dJ # use interpolation to build all PHI^(dm) 0<= dm <= DM derivative_tensor_list = [] for dm in range(DM+1): derivative_tensor = numpy.zeros([Nq for i in range(dm+1)]) I = generate_multi_indices(Nq,dm) d = sum(I[0,:]) NI = shape(I)[0] NJq = shape(Jq)[0] for ni in range(NI): for nj in range(NJq): derivative_tensor[tuple([0 for i in range(dm+1)])] += gamma(I[ni,:], Jq[nj,:]) * qbar_rays[nj,:,d] derivative_tensor_list.append(derivative_tensor) print(derivative_tensor_list) #print qbar_rays vbar[Np:] = derivative_tensor_list[0] #vbar[Np:] = dJ[:,0] return vbar
def gradient_of_E_PHI(v, DM):
""" computes the gradient of the expectation of PHI, i.e. grad( E[PHI] ),
where E[PHI] is approximated by the method of moments up to moment degree DM.
This gradient is then used in the steepest descent optimization below"""
# multi indices for the full tensor of order DM-1 to compute d^(DM-1) PHI
# the remaining order to compute d^DM PHI is achieved by a reverse sweep later
Jq = generate_multi_indices(Nq, DM)
NJq = shape(Jq)[0]
# since Nq is usually much larger than Np
# we want to be able to get one higher order derivative
# w.r.t. q by reverse mode
# that means we have to evaluate the mixed partial derivatives
# d^2 F/dpdq in the forward mode
# since UTPS cannot do those partial derivatives straightforward
# we need to use interpolation
J = zeros(
(DM + 1, 1, Nm, Np)
) # Taylor series of degree DM has DM+1 taylor coefficients (i.e. count also the zero-derivative)
qbar_rays = zeros((NJq, Nq, DM + 1))
for nq in range(NJq):
# 1: evaluation of J
# here, we have to use nested interpolation
# we need to compute
# d^d F(x + s_1 z_1 + s_2 z_2) |
# ---------------------------- |
# d z_1 d^(d-1) z_2 |_z=0
# for s_1 = [1,0,0] resp. s_1 = [0,1,0] (one direction for each parameter p)
# we use here formula (13) of the paper "Evaluating higher derivative tensors by forward propagation of univariate Taylor series"
for np in range(Np): # each column of J
for dm in range(DM + 1):
I = array([1, dm])
K = zeros((dm + 1, 2), dtype=int)
K[:, 0] = 1
K[:, 1] = list(range(dm + 1))
V = zeros((Nv, dm + 1))
for k in K:
s1 = zeros(Nv)
s1[np] = k[0]
s2 = zeros(Nv)
s2[Np:] = k[1] * Jq[nq, :]
V[:, 0] = s1 + s2
tmp = adolc.hos_forward(1, v, V, 0)[1]
J[dm, 0, :, np] += (-1)**multi_index_abs(
I - k) * multi_index_binomial(I, k) * tmp[:, dm]
scale_factor = array(
[1. / prod(list(range(1, d + 1))) for d in range(DM + 1)])
for dm in range(DM + 1):
J[dm, :, :, :] *= scale_factor[dm]
# 2: forward evaluation of Phi
Jtc = Mtc(J[:, :, :, :])
cg.forward([Jtc])
# 3: reverse evaluation of Phi
Phibar = zeros((DM + 1, 1, 1, 1))
Phibar[0, 0, 0, 0] = 1.
cg.reverse([Mtc(Phibar)])
Jbar = FJ.xbar.TC[:, 0, :, :]
#print Jbar
#print shape(Jbar)
## 4: reverse evaluation of J
vbar = zeros(Nv)
dJ = zeros(
(Nq, DM + 1)) # taylor coefficients for one direction in q
for np in range(Np): # each column of J
for dm in range(DM + 1):
I = array([1, dm])
K = zeros((dm + 1, 2), dtype=int)
K[:, 0] = 1
K[:, 1] = list(range(dm + 1))
V = zeros((Nv, dm + 1))
for k in K:
s1 = zeros(Nv)
s1[np] = k[0]
s2 = zeros(Nv)
s2[Np:] = k[1] * Jq[nq, :]
V[:, 0] = s1 + s2
keep = dm + 2
adolc.hos_forward(1, v, V, keep)[1]
# U is a (Q,M,D) array
# Jbar is a (D,M,Np) array
U = zeros((1, Nm, keep))
for d in range(dm + 1):
U[0, :, d] = Jbar[d, :, np]
#print 'U=',U
Z = adolc.hov_ti_reverse(1, U)[0]
#print 'Z=',Z
#J[dm,0,:,np] += (-1)**multi_index_abs( I - k) * multi_index_binomial(I,k) * tmp[:,dm]
#print shape(Z[0,:,:])
#exit()
#print Z[0,:,dm+1]
tmp = 1. / prod(list(range(1, dm + 2))) * (
-1)**multi_index_abs(I - k) * multi_index_binomial(
I, k) * Z[0, Np:, dm + 1]
dJ[:, dm] += tmp
#vbar += tmp
qbar_rays[nq, :, :] = dJ
# use interpolation to build all PHI^(dm) 0<= dm <= DM
derivative_tensor_list = []
for dm in range(DM + 1):
derivative_tensor = numpy.zeros([Nq for i in range(dm + 1)])
I = generate_multi_indices(Nq, dm)
d = sum(I[0, :])
NI = shape(I)[0]
NJq = shape(Jq)[0]
for ni in range(NI):
for nj in range(NJq):
derivative_tensor[tuple([
0 for i in range(dm + 1)
])] += gamma(I[ni, :], Jq[nj, :]) * qbar_rays[nj, :, d]
derivative_tensor_list.append(derivative_tensor)
print(derivative_tensor_list)
#print qbar_rays
vbar[Np:] = derivative_tensor_list[0]
#vbar[Np:] = dJ[:,0]
return vbar
# time ALGOPY hov_reverse
ybar = UTPM(numpy.random.rand(D, P))
start_time = time.time()
for rep in range(reps):
cg.pullback([ybar])
end_time = time.time()
time_hov_reverse_algopy = end_time - start_time
# time PYADOLC hov_reverse
W = numpy.random.rand(1, 1, D)
start_time = time.time()
V = numpy.random.rand(N, P, D - 1)
for rep in range(reps):
for p in range(P):
adolc.hos_forward(1, x, V[:, p, :], keep=D)
adolc.hov_ti_reverse(1, W)
end_time = time.time()
time_hov_reverse_adolc = end_time - start_time
# time PYADOLC.cgraph hov_reverse
W = numpy.random.rand(1, 1, P, D)
for rep in range(reps):
ap.reverse([W])
end_time = time.time()
time_hov_reverse_cgraph = end_time - start_time
print "----------------"
print time_trace_algopy
print time_trace_adolc
print time_trace_cgraph
print "----------------"
# time ALGOPY hov_reverse
ybar = UTPM(numpy.random.rand(D, P))
start_time = time.time()
for rep in range(reps):
cg.pullback([ybar])
end_time = time.time()
time_hov_reverse_algopy = end_time - start_time
# time PYADOLC hov_reverse
W = numpy.random.rand(1, 1, D)
start_time = time.time()
V = numpy.random.rand(N, P, D - 1)
for rep in range(reps):
for p in range(P):
adolc.hos_forward(1, x, V[:, p, :], keep=D)
adolc.hov_ti_reverse(1, W)
end_time = time.time()
time_hov_reverse_adolc = end_time - start_time
# time PYADOLC.cgraph hov_reverse
W = numpy.random.rand(1, 1, P, D)
for rep in range(reps):
ap.reverse([W])
end_time = time.time()
time_hov_reverse_cgraph = end_time - start_time
print('----------------')
print(time_trace_algopy)
print(time_trace_adolc)
print(time_trace_cgraph)
print('----------------')