|
"""Trust Region Reflective algorithm for least-squares optimization.
The algorithm is based on ideas from paper [STIR]_. The main idea is to account for presence of the bounds by appropriate scaling of the variables (or equivalently changing a trust-region shape). Let's introduce a vector v:
| ub[i] - x[i], if g[i] < 0 and ub[i] < np.inf v[i] = | x[i] - lb[i], if g[i] > 0 and lb[i] > -np.inf | 1, otherwise
where g is the gradient of a cost function and lb, ub are the bounds. Its components are distances to the bounds at which the anti-gradient points (if this distance is finite). Define a scaling matrix D = diag(v**0.5). First-order optimality conditions can be stated as
D^2 g(x) = 0.
Meaning that components of the gradient should be zero for strictly interior variables, and components must point inside the feasible region for variables on the bound.
Now consider this system of equations as a new optimization problem. If the point x is strictly interior (not on the bound) then the left-hand side is differentiable and the Newton step for it satisfies
(D^2 H + diag(g) Jv) p = -D^2 g
where H is the Hessian matrix (or its J^T J approximation in least squares), Jv is the Jacobian matrix of v with components -1, 1 or 0, such that all elements of matrix C = diag(g) Jv are non-negative. Introduce the change of the variables x = D x_h (_h would be "hat" in LaTeX). In the new variables we have a Newton step satisfying
B_h p_h = -g_h,
where B_h = D H D + C, g_h = D g. In least squares B_h = J_h^T J_h, where J_h = J D. Note that J_h and g_h are proper Jacobian and gradient with respect to "hat" variables. To guarantee global convergence we formulate a trust-region problem based on the Newton step in the new variables:
0.5 * p_h^T B_h p + g_h^T p_h -> min, ||p_h|| <= Delta
In the original space B = H + D^{-1} C D^{-1}, and the equivalent trust-region problem is
0.5 * p^T B p + g^T p -> min, ||D^{-1} p|| <= Delta
Here the meaning of the matrix D becomes more clear: it alters the shape of a trust-region, such that large steps towards the bounds are not allowed. In the implementation the trust-region problem is solved in "hat" space, but handling of the bounds is done in the original space (see below and read the code).
The introduction of the matrix D doesn't allow to ignore bounds, the algorithm must keep iterates strictly feasible (to satisfy aforementioned differentiability), the parameter theta controls step back from the boundary (see the code for details).
The algorithm does another important trick. If the trust-region solution doesn't fit into the bounds, then a reflected (from a firstly encountered bound) search direction is considered. For motivation and analysis refer to [STIR]_ paper (and other papers of the authors). In practice it doesn't need a lot of justifications, the algorithm simply chooses the best step among three: a constrained trust-region step, a reflected step and a constrained Cauchy step (a minimizer along -g_h in "hat" space, or -D^2 g in the original space).
Another feature is that a trust-region radius control strategy is modified to account for appearance of the diagonal C matrix (called diag_h in the code).
Note, that all described peculiarities are completely gone as we consider problems without bounds (the algorithm becomes a standard trust-region type algorithm very similar to ones implemented in MINPACK).
The implementation supports two methods of solving the trust-region problem. The first, called 'exact', applies SVD on Jacobian and then solves the problem very accurately using the algorithm described in [JJMore]_. It is not applicable to large problem. The second, called 'lsmr', uses the 2-D subspace approach (sometimes called "indefinite dogleg"), where the problem is solved in a subspace spanned by the gradient and the approximate Gauss-Newton step found by ``scipy.sparse.linalg.lsmr``. A 2-D trust-region problem is reformulated as a 4-th order algebraic equation and solved very accurately by ``numpy.roots``. The subspace approach allows to solve very large problems (up to couple of millions of residuals on a regular PC), provided the Jacobian matrix is sufficiently sparse.
References ---------- .. [STIR] Branch, M.A., T.F. Coleman, and Y. Li, "A Subspace, Interior, and Conjugate Gradient Method for Large-Scale Bound-Constrained Minimization Problems," SIAM Journal on Scientific Computing, Vol. 21, Number 1, pp 1-23, 1999. .. [JJMore] More, J. J., "The Levenberg-Marquardt Algorithm: Implementation and Theory," Numerical Analysis, ed. G. A. Watson, Lecture """
step_size_to_bound, find_active_constraints, in_bounds, make_strictly_feasible, intersect_trust_region, solve_lsq_trust_region, solve_trust_region_2d, minimize_quadratic_1d, build_quadratic_1d, evaluate_quadratic, right_multiplied_operator, regularized_lsq_operator, CL_scaling_vector, compute_grad, compute_jac_scale, check_termination, update_tr_radius, scale_for_robust_loss_function, print_header_nonlinear, print_iteration_nonlinear)
loss_function, tr_solver, tr_options, verbose): # For efficiency it makes sense to run the simplified version of the # algorithm when no bounds are imposed. We decided to write the two # separate functions. It violates DRY principle, but the individual # functions are kept the most readable. if np.all(lb == -np.inf) and np.all(ub == np.inf): return trf_no_bounds( fun, jac, x0, f0, J0, ftol, xtol, gtol, max_nfev, x_scale, loss_function, tr_solver, tr_options, verbose) else: return trf_bounds( fun, jac, x0, f0, J0, lb, ub, ftol, xtol, gtol, max_nfev, x_scale, loss_function, tr_solver, tr_options, verbose)
"""Select the best step according to Trust Region Reflective algorithm.""" if in_bounds(x + p, lb, ub): p_value = evaluate_quadratic(J_h, g_h, p_h, diag=diag_h) return p, p_h, -p_value
p_stride, hits = step_size_to_bound(x, p, lb, ub)
# Compute the reflected direction. r_h = np.copy(p_h) r_h[hits.astype(bool)] *= -1 r = d * r_h
# Restrict trust-region step, such that it hits the bound. p *= p_stride p_h *= p_stride x_on_bound = x + p
# Reflected direction will cross first either feasible region or trust # region boundary. _, to_tr = intersect_trust_region(p_h, r_h, Delta) to_bound, _ = step_size_to_bound(x_on_bound, r, lb, ub)
# Find lower and upper bounds on a step size along the reflected # direction, considering the strict feasibility requirement. There is no # single correct way to do that, the chosen approach seems to work best # on test problems. r_stride = min(to_bound, to_tr) if r_stride > 0: r_stride_l = (1 - theta) * p_stride / r_stride if r_stride == to_bound: r_stride_u = theta * to_bound else: r_stride_u = to_tr else: r_stride_l = 0 r_stride_u = -1
# Check if reflection step is available. if r_stride_l <= r_stride_u: a, b, c = build_quadratic_1d(J_h, g_h, r_h, s0=p_h, diag=diag_h) r_stride, r_value = minimize_quadratic_1d( a, b, r_stride_l, r_stride_u, c=c) r_h *= r_stride r_h += p_h r = r_h * d else: r_value = np.inf
# Now correct p_h to make it strictly interior. p *= theta p_h *= theta p_value = evaluate_quadratic(J_h, g_h, p_h, diag=diag_h)
ag_h = -g_h ag = d * ag_h
to_tr = Delta / norm(ag_h) to_bound, _ = step_size_to_bound(x, ag, lb, ub) if to_bound < to_tr: ag_stride = theta * to_bound else: ag_stride = to_tr
a, b = build_quadratic_1d(J_h, g_h, ag_h, diag=diag_h) ag_stride, ag_value = minimize_quadratic_1d(a, b, 0, ag_stride) ag_h *= ag_stride ag *= ag_stride
if p_value < r_value and p_value < ag_value: return p, p_h, -p_value elif r_value < p_value and r_value < ag_value: return r, r_h, -r_value else: return ag, ag_h, -ag_value
x_scale, loss_function, tr_solver, tr_options, verbose): x = x0.copy()
f = f0 f_true = f.copy() nfev = 1
J = J0 njev = 1 m, n = J.shape
if loss_function is not None: rho = loss_function(f) cost = 0.5 * np.sum(rho[0]) J, f = scale_for_robust_loss_function(J, f, rho) else: cost = 0.5 * np.dot(f, f)
g = compute_grad(J, f)
jac_scale = isinstance(x_scale, string_types) and x_scale == 'jac' if jac_scale: scale, scale_inv = compute_jac_scale(J) else: scale, scale_inv = x_scale, 1 / x_scale
v, dv = CL_scaling_vector(x, g, lb, ub) v[dv != 0] *= scale_inv[dv != 0] Delta = norm(x0 * scale_inv / v**0.5) if Delta == 0: Delta = 1.0
g_norm = norm(g * v, ord=np.inf)
f_augmented = np.zeros((m + n)) if tr_solver == 'exact': J_augmented = np.empty((m + n, n)) elif tr_solver == 'lsmr': reg_term = 0.0 regularize = tr_options.pop('regularize', True)
if max_nfev is None: max_nfev = x0.size * 100
alpha = 0.0 # "Levenberg-Marquardt" parameter
termination_status = None iteration = 0 step_norm = None actual_reduction = None
if verbose == 2: print_header_nonlinear()
while True: v, dv = CL_scaling_vector(x, g, lb, ub)
g_norm = norm(g * v, ord=np.inf) if g_norm < gtol: termination_status = 1
if verbose == 2: print_iteration_nonlinear(iteration, nfev, cost, actual_reduction, step_norm, g_norm)
if termination_status is not None or nfev == max_nfev: break
# Now compute variables in "hat" space. Here we also account for # scaling introduced by `x_scale` parameter. This part is a bit tricky, # you have to write down the formulas and see how the trust-region # problem is formulated when the two types of scaling are applied. # The idea is that first we apply `x_scale` and then apply Coleman-Li # approach in the new variables.
# v is recomputed in the variables after applying `x_scale`, note that # components which were identically 1 not affected. v[dv != 0] *= scale_inv[dv != 0]
# Here we apply two types of scaling. d = v**0.5 * scale
# C = diag(g * scale) Jv diag_h = g * dv * scale
# After all this were done, we continue normally.
# "hat" gradient. g_h = d * g
f_augmented[:m] = f if tr_solver == 'exact': J_augmented[:m] = J * d J_h = J_augmented[:m] # Memory view. J_augmented[m:] = np.diag(diag_h**0.5) U, s, V = svd(J_augmented, full_matrices=False) V = V.T uf = U.T.dot(f_augmented) elif tr_solver == 'lsmr': J_h = right_multiplied_operator(J, d)
if regularize: a, b = build_quadratic_1d(J_h, g_h, -g_h, diag=diag_h) to_tr = Delta / norm(g_h) ag_value = minimize_quadratic_1d(a, b, 0, to_tr)[1] reg_term = -ag_value / Delta**2
lsmr_op = regularized_lsq_operator(J_h, (diag_h + reg_term)**0.5) gn_h = lsmr(lsmr_op, f_augmented, **tr_options)[0] S = np.vstack((g_h, gn_h)).T S, _ = qr(S, mode='economic') JS = J_h.dot(S) # LinearOperator does dot too. B_S = np.dot(JS.T, JS) + np.dot(S.T * diag_h, S) g_S = S.T.dot(g_h)
# theta controls step back step ratio from the bounds. theta = max(0.995, 1 - g_norm)
actual_reduction = -1 while actual_reduction <= 0 and nfev < max_nfev: if tr_solver == 'exact': p_h, alpha, n_iter = solve_lsq_trust_region( n, m, uf, s, V, Delta, initial_alpha=alpha) elif tr_solver == 'lsmr': p_S, _ = solve_trust_region_2d(B_S, g_S, Delta) p_h = S.dot(p_S)
p = d * p_h # Trust-region solution in the original space. step, step_h, predicted_reduction = select_step( x, J_h, diag_h, g_h, p, p_h, d, Delta, lb, ub, theta)
x_new = make_strictly_feasible(x + step, lb, ub, rstep=0) f_new = fun(x_new) nfev += 1
step_h_norm = norm(step_h)
if not np.all(np.isfinite(f_new)): Delta = 0.25 * step_h_norm continue
# Usual trust-region step quality estimation. if loss_function is not None: cost_new = loss_function(f_new, cost_only=True) else: cost_new = 0.5 * np.dot(f_new, f_new) actual_reduction = cost - cost_new # Correction term is specific to the algorithm, # vanishes in unbounded case. correction = 0.5 * np.dot(step_h * diag_h, step_h)
Delta_new, ratio = update_tr_radius( Delta, actual_reduction - correction, predicted_reduction, step_h_norm, step_h_norm > 0.95 * Delta ) alpha *= Delta / Delta_new Delta = Delta_new
step_norm = norm(step) termination_status = check_termination( actual_reduction, cost, step_norm, norm(x), ratio, ftol, xtol)
if termination_status is not None: break
if actual_reduction > 0: x = x_new
f = f_new f_true = f.copy()
cost = cost_new
J = jac(x, f) njev += 1
if loss_function is not None: rho = loss_function(f) J, f = scale_for_robust_loss_function(J, f, rho)
g = compute_grad(J, f)
if jac_scale: scale, scale_inv = compute_jac_scale(J, scale_inv) else: step_norm = 0 actual_reduction = 0
iteration += 1
if termination_status is None: termination_status = 0
active_mask = find_active_constraints(x, lb, ub, rtol=xtol) return OptimizeResult( x=x, cost=cost, fun=f_true, jac=J, grad=g, optimality=g_norm, active_mask=active_mask, nfev=nfev, njev=njev, status=termination_status)
x_scale, loss_function, tr_solver, tr_options, verbose): x = x0.copy()
f = f0 f_true = f.copy() nfev = 1
J = J0 njev = 1 m, n = J.shape
if loss_function is not None: rho = loss_function(f) cost = 0.5 * np.sum(rho[0]) J, f = scale_for_robust_loss_function(J, f, rho) else: cost = 0.5 * np.dot(f, f)
g = compute_grad(J, f)
jac_scale = isinstance(x_scale, string_types) and x_scale == 'jac' if jac_scale: scale, scale_inv = compute_jac_scale(J) else: scale, scale_inv = x_scale, 1 / x_scale
Delta = norm(x0 * scale_inv) if Delta == 0: Delta = 1.0
if tr_solver == 'lsmr': reg_term = 0 damp = tr_options.pop('damp', 0.0) regularize = tr_options.pop('regularize', True)
if max_nfev is None: max_nfev = x0.size * 100
alpha = 0.0 # "Levenberg-Marquardt" parameter
termination_status = None iteration = 0 step_norm = None actual_reduction = None
if verbose == 2: print_header_nonlinear()
while True: g_norm = norm(g, ord=np.inf) if g_norm < gtol: termination_status = 1
if verbose == 2: print_iteration_nonlinear(iteration, nfev, cost, actual_reduction, step_norm, g_norm)
if termination_status is not None or nfev == max_nfev: break
d = scale g_h = d * g
if tr_solver == 'exact': J_h = J * d U, s, V = svd(J_h, full_matrices=False) V = V.T uf = U.T.dot(f) elif tr_solver == 'lsmr': J_h = right_multiplied_operator(J, d)
if regularize: a, b = build_quadratic_1d(J_h, g_h, -g_h) to_tr = Delta / norm(g_h) ag_value = minimize_quadratic_1d(a, b, 0, to_tr)[1] reg_term = -ag_value / Delta**2
damp_full = (damp**2 + reg_term)**0.5 gn_h = lsmr(J_h, f, damp=damp_full, **tr_options)[0] S = np.vstack((g_h, gn_h)).T S, _ = qr(S, mode='economic') JS = J_h.dot(S) B_S = np.dot(JS.T, JS) g_S = S.T.dot(g_h)
actual_reduction = -1 while actual_reduction <= 0 and nfev < max_nfev: if tr_solver == 'exact': step_h, alpha, n_iter = solve_lsq_trust_region( n, m, uf, s, V, Delta, initial_alpha=alpha) elif tr_solver == 'lsmr': p_S, _ = solve_trust_region_2d(B_S, g_S, Delta) step_h = S.dot(p_S)
predicted_reduction = -evaluate_quadratic(J_h, g_h, step_h) step = d * step_h x_new = x + step f_new = fun(x_new) nfev += 1
step_h_norm = norm(step_h)
if not np.all(np.isfinite(f_new)): Delta = 0.25 * step_h_norm continue
# Usual trust-region step quality estimation. if loss_function is not None: cost_new = loss_function(f_new, cost_only=True) else: cost_new = 0.5 * np.dot(f_new, f_new) actual_reduction = cost - cost_new
Delta_new, ratio = update_tr_radius( Delta, actual_reduction, predicted_reduction, step_h_norm, step_h_norm > 0.95 * Delta) alpha *= Delta / Delta_new Delta = Delta_new
step_norm = norm(step) termination_status = check_termination( actual_reduction, cost, step_norm, norm(x), ratio, ftol, xtol)
if termination_status is not None: break
if actual_reduction > 0: x = x_new
f = f_new f_true = f.copy()
cost = cost_new
J = jac(x, f) njev += 1
if loss_function is not None: rho = loss_function(f) J, f = scale_for_robust_loss_function(J, f, rho)
g = compute_grad(J, f)
if jac_scale: scale, scale_inv = compute_jac_scale(J, scale_inv) else: step_norm = 0 actual_reduction = 0
iteration += 1
if termination_status is None: termination_status = 0
active_mask = np.zeros_like(x) return OptimizeResult( x=x, cost=cost, fun=f_true, jac=J, grad=g, optimality=g_norm, active_mask=active_mask, nfev=nfev, njev=njev, status=termination_status) |