# This program is free software: you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by # the Free Software Foundation, either version 3 of the License, or # (at your option) any later version. # # This program is distributed in the hope that it will be useful, # but WITHOUT ANY WARRANTY; without even the implied warranty of # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the # GNU General Public License for more details. # # You should have received a copy of the GNU General Public License # along with this program. If not, see . # # Copyright(C) 2013-2017 Max-Planck-Society # # NIFTy is being developed at the Max-Planck-Institut fuer Astrophysik # and financially supported by the Studienstiftung des deutschen Volkes. import numpy as np from .line_search import LineSearch class LineSearchStrongWolfe(LineSearch): """Class for finding a step size that satisfies the strong Wolfe conditions. Algorithm contains two stages. It begins whit a trial step length and keeps increasing it until it finds an acceptable step length or an interval. If it does not satisfy the Wolfe conditions, it performs the Zoom algorithm (second stage). By interpolating it decreases the size of the interval until an acceptable step length is found. Parameters ---------- c1 : float Parameter for Armijo condition rule. (Default: 1e-4) c2 : float Parameter for curvature condition rule. (Default: 0.9) max_step_size : float Maximum step allowed in to be made in the descent direction. (Default: 50) max_iterations : integer Maximum number of iterations performed by the line search algorithm. (Default: 10) max_zoom_iterations : integer Maximum number of iterations performed by the zoom algorithm. (Default: 10) Attributes ---------- c1 : float Parameter for Armijo condition rule. c2 : float Parameter for curvature condition rule. max_step_size : float Maximum step allowed in to be made in the descent direction. max_iterations : integer Maximum number of iterations performed by the line search algorithm. max_zoom_iterations : integer Maximum number of iterations performed by the zoom algorithm. """ def __init__(self, c1=1e-4, c2=0.9, max_step_size=50, max_iterations=10, max_zoom_iterations=10): super(LineSearchStrongWolfe, self).__init__() self.c1 = np.float(c1) self.c2 = np.float(c2) self.max_step_size = max_step_size self.max_iterations = int(max_iterations) self.max_zoom_iterations = int(max_zoom_iterations) def perform_line_search(self, energy, pk, f_k_minus_1=None): """Performs the first stage of the algorithm. It starts with a trial step size and it keeps increasing it until it satisfy the strong Wolf conditions. It also performs the descent and returns the optimal step length and the new enrgy. Parameters ---------- energy : Energy object Energy object from which we will calculate the energy and the gradient at a specific point. pk : Field Unit vector pointing into the search direction. f_k_minus_1 : float Value of the fuction (which is being minimized) at the k-1 iteration of the line search procedure. (Default: None) Returns ------- alpha_star : float The optimal step length in the descent direction. phi_star : float Value of the energy after the performed descent. energy_star : Energy object The new Energy object on the new position. """ self._set_line_energy(energy, pk, f_k_minus_1=f_k_minus_1) c1 = self.c1 c2 = self.c2 max_step_size = self.max_step_size max_iterations = self.max_iterations # initialize the zero phis old_phi_0 = self.f_k_minus_1 energy_0 = self.line_energy.at(0) phi_0 = energy_0.value phiprime_0 = energy_0.gradient if phiprime_0 == 0: self.logger.warn("Flat gradient in search direction.") return 0., 0. # set alphas alpha0 = 0. if self.preferred_initial_step_size is not None: alpha1 = self.preferred_initial_step_size elif old_phi_0 is not None and phiprime_0 != 0: alpha1 = min(1.0, 1.01*2*(phi_0 - old_phi_0)/phiprime_0) if alpha1 < 0: alpha1 = 1.0 else: alpha1 = 1.0 # give the alpha0 phis the right init value phi_alpha0 = phi_0 phiprime_alpha0 = phiprime_0 # start the minimization loop for i in xrange(max_iterations): energy_alpha1 = self.line_energy.at(alpha1) phi_alpha1 = energy_alpha1.value if alpha1 == 0: self.logger.warn("Increment size became 0.") alpha_star = 0. phi_star = phi_0 energy_star = energy_0 break if (phi_alpha1 > phi_0 + c1*alpha1*phiprime_0) or \ ((phi_alpha1 >= phi_alpha0) and (i > 1)): (alpha_star, phi_star, energy_star) = self._zoom( alpha0, alpha1, phi_0, phiprime_0, phi_alpha0, phiprime_alpha0, phi_alpha1, c1, c2) break phiprime_alpha1 = energy_alpha1.gradient if abs(phiprime_alpha1) <= -c2*phiprime_0: alpha_star = alpha1 phi_star = phi_alpha1 energy_star = energy_alpha1 break if phiprime_alpha1 >= 0: (alpha_star, phi_star, energy_star) = self._zoom( alpha1, alpha0, phi_0, phiprime_0, phi_alpha1, phiprime_alpha1, phi_alpha0, c1, c2) break # update alphas alpha0, alpha1 = alpha1, min(2*alpha1, max_step_size) phi_alpha0 = phi_alpha1 phiprime_alpha0 = phiprime_alpha1 # phi_alpha1 = self._phi(alpha1) else: # max_iterations was reached alpha_star = alpha1 phi_star = phi_alpha1 energy_star = energy_alpha1 self.logger.error("The line search algorithm did not converge.") # extract the full energy from the line_energy energy_star = energy_star.energy direction_length = pk.norm() step_length = alpha_star * direction_length return step_length, phi_star, energy_star def _zoom(self, alpha_lo, alpha_hi, phi_0, phiprime_0, phi_lo, phiprime_lo, phi_hi, c1, c2): """Performs the second stage of the line search algorithm. If the first stage was not successful then the Zoom algorithm tries to find a suitable step length by using bisection, quadratic, cubic interpolation. Parameters ---------- alpha_lo : float The lower boundary for the step length interval. alph_hi : float The upper boundary for the step length interval. phi_0 : float Value of the energy at the starting point of the line search algorithm. phiprime_0 : Field Gradient at the starting point of the line search algorithm. phi_lo : float Value of the energy if we perform a step of length alpha_lo in descent direction. phiprime_lo : Field Gradient at the nwe position if we perform a step of length alpha_lo in descent direction. phi_hi : float Value of the energy if we perform a step of length alpha_hi in descent direction. c1 : float Parameter for Armijo condition rule. c2 : float Parameter for curvature condition rule. Returns ------- alpha_star : float The optimal step length in the descent direction. phi_star : float Value of the energy after the performed descent. energy_star : Energy object The new Energy object on the new position. """ max_iterations = self.max_zoom_iterations # define the cubic and quadratic interpolant checks cubic_delta = 0.2 # cubic quad_delta = 0.1 # quadratic # initialize the most recent versions (j-1) of phi and alpha alpha_recent = 0 phi_recent = phi_0 for i in xrange(max_iterations): delta_alpha = alpha_hi - alpha_lo if delta_alpha < 0: a, b = alpha_hi, alpha_lo else: a, b = alpha_lo, alpha_hi # Try cubic interpolation if i > 0: cubic_check = cubic_delta * delta_alpha alpha_j = self._cubicmin(alpha_lo, phi_lo, phiprime_lo, alpha_hi, phi_hi, alpha_recent, phi_recent) # If cubic was not successful or not available, try quadratic if (i == 0) or (alpha_j is None) or (alpha_j > b - cubic_check) or\ (alpha_j < a + cubic_check): quad_check = quad_delta * delta_alpha alpha_j = self._quadmin(alpha_lo, phi_lo, phiprime_lo, alpha_hi, phi_hi) # If quadratic was not successfull, try bisection if (alpha_j is None) or (alpha_j > b - quad_check) or \ (alpha_j < a + quad_check): alpha_j = alpha_lo + 0.5*delta_alpha # Check if the current value of alpha_j is already sufficient energy_alphaj = self.line_energy.at(alpha_j) phi_alphaj = energy_alphaj.value # If the first Wolfe condition is not met replace alpha_hi # by alpha_j if (phi_alphaj > phi_0 + c1*alpha_j*phiprime_0) or\ (phi_alphaj >= phi_lo): alpha_recent, phi_recent = alpha_hi, phi_hi alpha_hi, phi_hi = alpha_j, phi_alphaj else: phiprime_alphaj = energy_alphaj.gradient # If the second Wolfe condition is met, return the result if abs(phiprime_alphaj) <= -c2*phiprime_0: alpha_star = alpha_j phi_star = phi_alphaj energy_star = energy_alphaj break # If not, check the sign of the slope if phiprime_alphaj*delta_alpha >= 0: alpha_recent, phi_recent = alpha_hi, phi_hi alpha_hi, phi_hi = alpha_lo, phi_lo else: alpha_recent, phi_recent = alpha_lo, phi_lo # Replace alpha_lo by alpha_j (alpha_lo, phi_lo, phiprime_lo) = (alpha_j, phi_alphaj, phiprime_alphaj) else: alpha_star, phi_star, energy_star = \ alpha_j, phi_alphaj, energy_alphaj self.logger.error("The line search algorithm (zoom) did not " "converge.") return alpha_star, phi_star, energy_star def _cubicmin(self, a, fa, fpa, b, fb, c, fc): """Estimating the minimum with cubic interpolation. Finds the minimizer for a cubic polynomial that goes through the points ( a,f(a) ), ( b,f(b) ), and ( c,f(c) ) with derivative at point a of fpa. f(x) = A *(x-a)^3 + B*(x-a)^2 + C*(x-a) + D If no minimizer can be found return None Parameters ---------- a : float Selected point. fa : float Value of polynomial at point a. fpa : Field Derivative at point a. b : float Selected point. fb : float Value of polynomial at point b. c : float Selected point. fc : float Value of polynomial at point c. Returns ------- xmin : float Position of the approximated minimum. """ with np.errstate(divide='raise', over='raise', invalid='raise'): try: C = fpa db = b - a dc = c - a denom = db * db * dc * dc * (db - dc) d1 = np.empty((2, 2)) d1[0, 0] = dc * dc d1[0, 1] = -(db*db) d1[1, 0] = -(dc*dc*dc) d1[1, 1] = db*db*db [A, B] = np.dot(d1, np.asarray([fb - fa - C * db, fc - fa - C * dc]).flatten()) A /= denom B /= denom radical = B * B - 3 * A * C xmin = a + (-B + np.sqrt(radical)) / (3 * A) except ArithmeticError: return None if not np.isfinite(xmin): return None return xmin def _quadmin(self, a, fa, fpa, b, fb): """Estimating the minimum with quadratic interpolation. Finds the minimizer for a quadratic polynomial that goes through the points ( a,f(a) ), ( b,f(b) ) with derivative at point a of fpa. f(x) = B*(x-a)^2 + C*(x-a) + D Parameters ---------- a : float Selected point. fa : float Value of polynomial at point a. fpa : Field Derivative at point a. b : float Selected point. fb : float Value of polynomial at point b. Returns ------- xmin : float Position of the approximated minimum. """ # f(x) = B*(x-a)^2 + C*(x-a) + D with np.errstate(divide='raise', over='raise', invalid='raise'): try: D = fa C = fpa db = b - a * 1.0 B = (fb - D - C * db) / (db * db) xmin = a - C / (2.0 * B) except ArithmeticError: return None if not np.isfinite(xmin): return None return xmin