def Q_solve(block_dim, A, D, F, interactive=False, disp=True,
verbose=False, debug=False, Rs=[10, 100, 1000], N_iter=400,
gamma=.5, tol=1e-1, min_step_size=1e-6,
methods=['frank_wolfe']):
"""
Solves Q optimization.
min_Q -log det R + Tr(RF)
-------------------
|D-ADA.T I |
X = | I R |
| D cI |
| cI R |
-------------------
X is PSD
"""
dim = 4*block_dim
search_tol = 1.
# Copy over initial data
D = np.copy(D)
F = np.copy(F)
c = np.sqrt(1/gamma)
# Numerical stability
scale = 1./np.linalg.norm(D,2)
# Rescaling
D *= scale
# Scale down objective matrices
scale_factor = np.linalg.norm(F, 2)
if scale_factor < 1e-6:
# F can be zero if there are no observations for this state
return np.eye(block_dim)
# Improving conditioning
delta=1e-2
D = D + delta*np.eye(block_dim)
Dinv = np.linalg.inv(D)
D_ADA_T = D - np.dot(A, np.dot(D, A.T)) + delta*np.eye(block_dim)
# Compute trace upper bound
R = (2*np.trace(D) + 2*(1./gamma)*np.trace(Dinv))
Rs = [R]
As, bs, Cs, ds, Fs, gradFs, Gs, gradGs = \
Q_constraints(block_dim, A, F, D, c)
(D_ADA_T_cds, I_1_cds, I_2_cds, R_1_cds,
D_cds, c_I_1_cds, c_I_2_cds, R_2_cds) = \
Q_coords(block_dim)
# Construct init matrix
X_init = np.zeros((dim, dim))
set_entries(X_init, D_ADA_T_cds, D_ADA_T)
set_entries(X_init, I_1_cds, np.eye(block_dim))
set_entries(X_init, I_2_cds, np.eye(block_dim))
Qinv_init_1 = np.linalg.inv(D_ADA_T)
set_entries(X_init, R_1_cds, Qinv_init_1)
set_entries(X_init, D_cds, D)
set_entries(X_init, c_I_1_cds, c*np.eye(block_dim))
set_entries(X_init, c_I_2_cds, c*np.eye(block_dim))
Qinv_init_2 = np.linalg.inv((1./c)**2 * D)
set_entries(X_init, R_2_cds, Qinv_init_2)
X_init = X_init + (1e-4)*np.eye(dim)
if min(np.linalg.eigh(X_init)[0]) < 0:
print("Q_SOLVE INIT FAILED!")
X_init == None
else:
print("Q_SOLVE SUCCESS!")
g = GeneralSolver()
def obj(X):
return (1./scale_factor) * log_det_tr(X, F)
def grad_obj(X):
return (1./scale_factor) * grad_log_det_tr(X, F)
g.save_constraints(dim, obj, grad_obj, As, bs, Cs, ds,
Fs, gradFs, Gs, gradGs)
(U, X, succeed) = g.solve(N_iter, tol, search_tol,
interactive=interactive, disp=disp, verbose=verbose,
debug=debug, Rs=Rs, min_step_size=min_step_size,
methods=methods, X_init=X_init)
if succeed:
R_1 = scale*get_entries(X, R_1_cds)
R_2 = scale*get_entries(X, R_2_cds)
R_avg = (R_1 + R_2) / 2.
# Ensure stability
R_avg = R_avg + (1e-3) * np.eye(block_dim)
Q = np.linalg.inv(R_avg)
# Unscale answer
Q *= (1./scale)
if disp:
print("Q:\n", Q)
return Q
def A_solve(block_dim, B, C, D, E, Q, mu, interactive=False,
disp=True, verbose=False, debug=False, Rs=[10, 100, 1000],
N_iter=400, tol=1e-1, min_step_size=1e-6,
methods=['frank_wolfe']):
"""
Solves A optimization.
min_A Tr [ Q^{-1} ([C - B] A.T + A [C - B].T + A E A.T]
--------------------
| D-Q A |
X = | A.T D^{-1} |
| I A |
| A.T I |
--------------------
A mu == 0
X is PSD
"""
dim = 4*block_dim
search_tol = 1.
# Copy in inputs
B = np.copy(B)
C = np.copy(C)
D = np.copy(D)
E = np.copy(E)
Q = np.copy(Q)
mu = np.copy(mu)
# Scale down objective matrices
scale_factor = (max(np.linalg.norm(C-B, 2), np.linalg.norm(E,2)))
if scale_factor < 1e-6 or np.linalg.norm(D, 2) < 1e-6:
# If A has no observations, not much we can say
return .5*np.eye(block_dim)
C = C/scale_factor
B = B/scale_factor
E = E/scale_factor
# Numerical stability
scale = 1./np.linalg.norm(D, 2)
# Rescaling
D *= scale
Q *= scale
# Improving conditioning
delta=1e-2
D = D + delta*np.eye(block_dim)
Q = Q + delta*np.eye(block_dim)
Dinv = np.linalg.inv(D)
# Compute post-scaled inverses
Dinv = np.linalg.inv(D)
Qinv = np.linalg.inv(Q)
# Compute trace upper bound
R = np.abs(np.trace(D)) + np.abs(np.trace(Dinv)) + 2 * block_dim
Rs = [R]
As, bs, Cs, ds, Fs, gradFs, Gs, gradGs = \
A_constraints(block_dim, D, Dinv, Q, mu)
(D_Q_cds, Dinv_cds, I_1_cds, I_2_cds,
A_1_cds, A_T_1_cds, A_2_cds, A_T_2_cds) = A_coords(block_dim)
# Construct init matrix
upper_norm = np.linalg.norm(D-Q, 2)
lower_norm = np.linalg.norm(D, 2)
const = np.sqrt(upper_norm/lower_norm)
factor = .95
for i in range(10):
X_init = np.zeros((dim, dim))
set_entries(X_init, D_Q_cds, D-Q)
set_entries(X_init, A_1_cds, const*np.eye(block_dim))
set_entries(X_init, A_T_1_cds, const*np.eye(block_dim))
set_entries(X_init, Dinv_cds, Dinv)
set_entries(X_init, I_1_cds, np.eye(block_dim))
set_entries(X_init, A_2_cds, const*np.eye(block_dim))
set_entries(X_init, A_T_2_cds, const*np.eye(block_dim))
set_entries(X_init, I_2_cds, np.eye(block_dim))
X_init = X_init + (1e-2)*np.eye(dim)
if min(np.linalg.eigh(X_init)[0]) < 0:
X_init = None
const = const * factor
else:
print("A_SOLVE SUCCESS AT %d" % i)
print("const: ", const)
break
if X_init == None:
print("A_SOLVE INIT FAILED!")
def obj(X):
return A_dynamics(X, block_dim, C, B, E, Qinv)
def grad_obj(X):
return grad_A_dynamics(X, block_dim, C, B, E, Qinv)
g = GeneralSolver()
g.save_constraints(dim, obj, grad_obj, As, bs, Cs, ds,
Fs, gradFs, Gs, gradGs)
(U, X, succeed) = g.solve(N_iter, tol, search_tol,
interactive=interactive, disp=disp, verbose=verbose,
debug=debug, Rs=Rs, min_step_size=min_step_size,
methods=methods, X_init=X_init)
if succeed:
A_1 = get_entries(X, A_1_cds)
A_T_1 = get_entries(X, A_T_1_cds)
A_2 = get_entries(X, A_2_cds)
A_T_2 = get_entries(X, A_T_2_cds)
A = (A_1 + A_T_1 + A_2 + A_T_2) / 4.
if disp:
print("A:\n", A)
return A