def test_add_centered_entries():
history = LeastSquaresHistory()
history.add_entries(np.ones((2, 2)), np.ones((2, 4)))
center_info = {
"x": history.get_xs(index=-1),
"residuals": history.get_residuals(index=-1),
"radius": 0.5,
}
history.add_centered_entries(
xs=np.ones(2), residuals=np.ones(4) * 2, center_info=center_info
)
xs, residuals, critvals = history.get_entries(index=-1)
aaae(xs, np.array([1.5, 1.5]))
aaae(residuals, np.array([3, 3, 3, 3]))
assert critvals == 36
assert history.get_n_fun() == 3
def test_add_entries_initialized_extension_needed():
history = LeastSquaresHistory()
history.add_entries(np.ones((4, 3)), np.zeros((4, 5)))
history.xs = history.xs[:5]
history.residuals = history.residuals[:5]
history.critvals = history.critvals[:5]
history.add_entries(np.arange(12).reshape(4, 3), np.arange(20).reshape(4, 5))
assert len(history.xs) == 10
assert len(history.residuals) == 10
assert len(history.critvals) == 10
xs, residuals, _ = history.get_entries(index=-1)
xs_sinlge = history.get_xs(index=-1)
residuals_sinlge = history.get_residuals(index=-1)
for entry in xs, xs_sinlge, residuals, residuals_sinlge:
assert isinstance(entry, np.ndarray)
assert history.get_n_fun() == 8
def test_add_entries_not_initialized(entries, is_center):
history = LeastSquaresHistory()
if is_center:
c_info = {"x": np.zeros(3), "residuals": np.zeros(5), "radius": 1}
history.add_centered_entries(*entries, c_info)
else:
history.add_entries(*entries)
xs, residuals, critvals = history.get_entries()
xs_sinlge = history.get_xs()
residuals_sinlge = history.get_residuals()
critvals_sinlge = history.get_critvals()
for entry in xs, residuals, critvals:
assert isinstance(entry, np.ndarray)
aaae(xs, np.arange(3).reshape(1, 3))
aaae(xs_sinlge, np.arange(3).reshape(1, 3))
aaae(residuals, np.arange(5).reshape(1, 5))
aaae(residuals_sinlge, np.arange(5).reshape(1, 5))
aaae(critvals, np.array([30.0]))
aaae(critvals_sinlge, np.array([30.0]))
def test_add_entries_initialized_with_space(entries, is_center):
history = LeastSquaresHistory()
history.add_entries(np.ones((4, 3)), np.zeros((4, 5)))
if is_center:
c_info = {"x": np.zeros(3), "residuals": np.zeros(5), "radius": 1}
history.add_centered_entries(*entries, c_info)
else:
history.add_entries(*entries)
xs, residuals, critvals = history.get_entries(index=-1)
xs_sinlge = history.get_xs(index=-1)
residuals_sinlge = history.get_residuals(index=-1)
critvals_sinlge = history.get_critvals(index=-1)
for entry in xs, residuals:
assert isinstance(entry, np.ndarray)
aaae(xs, np.arange(3))
aaae(xs_sinlge, np.arange(3))
aaae(residuals, np.arange(5))
aaae(residuals_sinlge, np.arange(5))
assert critvals == 30
assert critvals_sinlge == 30
def internal_solve_pounders(
criterion,
x0,
lower_bounds,
upper_bounds,
gtol,
maxiter,
delta,
delta_min,
delta_max,
gamma0,
gamma1,
theta1,
theta2,
eta0,
eta1,
c1,
c2,
solver_sub,
ftol_sub,
xtol_sub,
gtol_sub,
batch_evaluator,
n_cores,
):
"""Minimize the criterion function using POUNDERS.
Args:
criterion_and_derivative (callable): Function that returns criterion
and derivative as a tuple.
x0 (np.ndarray): Initial guess of the parameter vector. Starting points.
lower_bounds (np.ndarray): Lower bounds.
Must have same length as the initial guess of the
parameter vector. Equal to -1 if not provided by the user.
upper_bounds (np.ndarray): Upper bounds.
Must have same length as the initial guess of the
parameter vector. Equal to 1 if not provided by the user.
gtol (float): Convergence criterion of the absolute gradient norm.
Default is 1e-4.
maxiter (int): Maximum number of iterations. If reached, terminate.
delta (float): Delta, initial trust-region radius.
delta_min (float): Minimal trust-region radius.
delta_max (float): Maximal trust-region radius.
gamma0 (float): Shrinking factor of the trust-region radius in case the
solution vector of the suproblem is not accepted, but the model is fully
linar (i.e. "valid").
gamma1 (float): Expansion factor of the trust-region radius in case the
solution vector of the suproblem is accepted.
theta1 (float): Threshold for adding the current x candidate
to the model. Function argument to find_affine_points().
theta2 (float): Threshold for adding the current x candidate
to the model. Argument to get_interpolation_matrices_residual_model().
eta0 (float): First threshold for accepting the solution vector of the
subproblem as the best x candidate.
eta1 (float): Second threshold for accepting the solution vector of the
subproblem as the best x candidate.
c1 (float): Treshold for accepting the norm of our current x candidate.
Equal to sqrt(n) by default. Argument to find_affine_points() in case
the input array *model_improving_points* is zero.
c2 (int)): Treshold for accepting the norm of our current x candidate.
Equal to 10 by default. Argument to find_affine_points() in case
the input array *model_improving_points* is not zero.
solver_sub (str): Bound-constraint minimizer for the subproblem.
Currently, three solvers from the scipy library are supported.
- "trust-constr" (default)
- "L-BFGS-B"
- "SLSQP"
ftol_sub (float): Tolerance for f, the criterion function value.
Stopping criterion for the subproblem.
xtol_sub (float): Tolerance for solution vector x.
Stopping criterion for the subproblem.
gtol_sub (float): Tolerance for the absolute gradient norm.
Stopping criterion for the subproblem.
batch_evaluator (str or callable): Name of a pre-implemented batch evaluator
(currently 'joblib' and 'pathos_mp') or callable with the same interface
as the estimagic batch_evaluators.
n_cores (int): Number of processes used to parallelize the function
evaluations. Default is 1.
Returns:
(dict) Result dictionary containing:
- solution_x (np.ndarray): Solution vector of shape (n,).
- solution_criterion (np.ndarray): Values of the criterion function at the
solution vector. Shape (n_obs,).
- history_x (np.ndarray): Entire history of x. Shape (history.get_n_fun(), n).
- history_criterion (np.ndarray): Entire history of the criterion function
evaluations. Shape (history.get_n_fun(), n_obs)
- n_iterations (int): Number of iterations the algorithm ran before finding a
solution vector or reaching maxiter.
- message (str): Message to the user. Currently it says: "Under development."
"""
history = LeastSquaresHistory()
n = x0.shape[0]
n_maxinterp = 2 * n + 1
model_indices = np.zeros(n_maxinterp, dtype=int)
last_n_modelpoints = 0
niter = 0
if lower_bounds is not None and upper_bounds is not None:
if np.max(x0 + delta - upper_bounds) > 1e-10:
raise ValueError("Starting points + delta > upper bounds.")
xs = [x0]
for i in range(n):
x1 = np.copy(x0)
x1[i] += delta
xs.append(x1)
residuals = batch_evaluator(criterion, arguments=xs, n_cores=n_cores)
history.add_entries(xs, residuals)
accepted_index = history.get_best_index()
# Center around new trust-region and normalize to [-1, 1]
indices_not_min = [i for i in range(n + 1) if i != accepted_index]
center_info = {
"x": history.get_best_x(),
"residuals": history.get_best_residuals(),
"radius": delta,
}
x_candidate, residuals_candidate, _ = history.get_centered_entries(
center_info=center_info,
index=indices_not_min,
)
initial_residual_model = {"intercepts": history.get_best_residuals()}
residual_model = update_initial_residual_model(initial_residual_model,
x_candidate,
residuals_candidate)
main_model = update_main_from_residual_model(
residual_model, multiply_square_terms_with_residuals=False)
x_accepted = history.get_best_x()
gradient_norm = np.linalg.norm(main_model["linear_terms"])
gradient_norm *= delta
valid = True
reason = True
n_modelpoints = n + 1
last_model_indices = np.zeros(n_maxinterp, dtype=int)
while reason is True:
niter += 1
# Solve the subproblem min{Q(s): ||s|| <= 1.0}
result_sub = solve_subproblem(
solution=x_accepted,
delta=delta,
main_model=main_model,
ftol=ftol_sub,
xtol=xtol_sub,
gtol=gtol_sub,
solver=solver_sub,
lower_bounds=lower_bounds,
upper_bounds=upper_bounds,
)
qmin = -result_sub.fun
x_candidate = x_accepted + result_sub.x * delta
residuals_candidate = criterion(x_candidate)
history.add_entries(x_candidate, residuals_candidate)
rho = (history.get_critvals(accepted_index) -
history.get_critvals(-1)) / qmin
if (rho >= eta1) or (rho > eta0 and valid is True):
residual_model["intercepts"] = history.get_residuals(
index=accepted_index)
center_info = {"x": history.get_best_x(), "radius": delta}
x_candidate = history.get_centered_xs(center_info, index=-1)
residual_model = update_residual_model_with_new_accepted_x(
residual_model=residual_model, x_candidate=x_candidate)
main_model = update_main_model_with_new_accepted_x(
main_model=main_model, x_candidate=x_candidate)
x_accepted = history.get_best_x()
accepted_index = history.get_best_index()
# The model is deemend "not valid" if it has less than n model points.
# Otherwise, if the model has n points it is considered "valid" or
# "fully linear".
# Note: valid is True in first iteration
if valid is False:
(
model_improving_points,
model_indices,
n_modelpoints,
project_x_onto_null,
) = find_affine_points(
history=history,
x_accepted=x_accepted,
model_improving_points=np.zeros((n, n)),
project_x_onto_null=False,
delta=delta,
theta1=theta1,
c=c1,
model_indices=model_indices,
n_modelpoints=0,
)
if n_modelpoints < n:
history, model_indices = add_points_to_make_main_model_fully_linear(
history=history,
main_model=main_model,
model_improving_points=model_improving_points,
model_indices=model_indices,
x_accepted=x_accepted,
n_modelpoints=n_modelpoints,
delta=delta,
criterion=criterion,
lower_bounds=lower_bounds,
upper_bounds=upper_bounds,
batch_evaluator=batch_evaluator,
n_cores=n_cores,
)
n_modelpoints = n
# Update the trust region radius
delta_old = delta
norm_x_sub = np.sqrt(np.sum(result_sub.x**2))
if rho >= eta1 and norm_x_sub > 0.5 * delta:
delta = min(delta * gamma1, delta_max)
elif valid is True:
delta = max(delta * gamma0, delta_min)
# Compute the next interpolation set
(
model_improving_points,
model_indices,
n_modelpoints,
project_x_onto_null,
) = find_affine_points(
history=history,
x_accepted=x_accepted,
model_improving_points=np.zeros((n, n)),
project_x_onto_null=False,
delta=delta,
theta1=theta1,
c=c1,
model_indices=model_indices,
n_modelpoints=0,
)
if n_modelpoints == n:
valid = True
else:
valid = False
(
model_improving_points,
model_indices,
n_modelpoints,
project_x_onto_null,
) = find_affine_points(
history=history,
x_accepted=x_accepted,
model_improving_points=model_improving_points,
project_x_onto_null=project_x_onto_null,
delta=delta,
theta1=theta1,
c=c2,
model_indices=model_indices,
n_modelpoints=n_modelpoints,
)
if n_modelpoints < n:
# Model not valid. Add geometry points
(
history,
model_indices,
n_modelpoints,
) = add_points_to_make_main_model_fully_linear(
history=history,
main_model=main_model,
model_improving_points=model_improving_points,
model_indices=model_indices,
x_accepted=x_accepted,
n_modelpoints=n_modelpoints,
delta=delta,
criterion=criterion,
lower_bounds=lower_bounds,
upper_bounds=upper_bounds,
batch_evaluator=batch_evaluator,
n_cores=n_cores,
)
model_indices[1:n_modelpoints + 1] = model_indices[:n_modelpoints]
n_modelpoints += 1
model_indices[0] = accepted_index
(
x_sample_monomial_basis,
monomial_basis,
basis_null_space,
lower_triangular,
n_modelpoints,
) = get_interpolation_matrices_residual_model(
history=history,
x_accepted=x_accepted,
model_indices=model_indices,
delta=delta,
c2=c2,
theta2=theta2,
n_maxinterp=n_maxinterp,
n_modelpoints=n_modelpoints,
)
center_info = {"x": x_accepted, "radius": delta_old}
interpolation_set = history.get_centered_xs(
center_info, index=model_indices[:n_modelpoints])
f_interpolated = interpolate_f(
history=history,
interpolation_set=interpolation_set,
residual_model=residual_model,
model_indices=model_indices,
n_modelpoints=n_modelpoints,
n_maxinterp=n_maxinterp,
)
coefficients_residual_model = get_coefficients_residual_model(
x_sample_monomial_basis=x_sample_monomial_basis,
monomial_basis=monomial_basis,
basis_null_space=basis_null_space,
lower_triangular=lower_triangular,
f_interpolated=f_interpolated,
n_modelpoints=n_modelpoints,
)
residual_model["intercepts"] = history.get_residuals(
index=accepted_index)
residual_model = update_residual_model(
residual_model=residual_model,
coefficients_to_add=coefficients_residual_model,
delta=delta,
delta_old=delta_old,
)
main_model = update_main_from_residual_model(residual_model)
gradient_norm = np.linalg.norm(main_model["linear_terms"])
gradient_norm *= delta
if gradient_norm < gtol:
reason = False
if niter > maxiter:
reason = False
# Test for repeated model
if n_modelpoints == last_n_modelpoints:
same = True
else:
same = False
for i in range(n_modelpoints):
if same:
if model_indices[i] == last_model_indices[i]:
same = True
else:
same = False
last_model_indices[i] = model_indices[i]
last_n_modelpoints = n_modelpoints
if (same is True) and (delta == delta_old):
# Identical model used in successive iterations
reason = False
result_dict = {
"solution_x": history.get_best_x(),
"solution_criterion": history.get_best_residuals(),
"history_x": history.get_xs(),
"history_criterion": history.get_residuals(),
"n_iterations": niter,
"message": "Under development.",
}
return result_dict