line_search_strong_wolfe.py

# This program is free software: you can redistribute it and/or modify
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# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
# GNU General Public License for more details.
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# along with this program.  If not, see <http://www.gnu.org/licenses/>.
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# 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=1000000000, max_iterations=100,
                 max_zoom_iterations=30):

        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)
        max_step_size = self.max_step_size
        max_iterations = self.max_iterations

        # initialize the zero phis
        old_phi_0 = self.f_k_minus_1
        le_0 = self.line_energy.at(0)
        phi_0 = le_0.value
        phiprime_0 = le_0.directional_derivative
        if phiprime_0 >= 0:
            self.logger.error("Input direction must be a descent direction")
            raise RuntimeError

        # 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):
            le_alpha1 = self.line_energy.at(alpha1)
            phi_alpha1 = le_alpha1.value
            if alpha1 == 0:
                self.logger.warn("Increment size became 0.")
                alpha_star = 0.
                phi_star = phi_0
                le_star = le_0
                break

            if (phi_alpha1 > phi_0 + self.c1*alpha1*phiprime_0) or \
               ((phi_alpha1 >= phi_alpha0) and (i > 0)):
                (alpha_star, phi_star, le_star) = self._zoom(
                                                    alpha0, alpha1,
                                                    phi_0, phiprime_0,
                                                    phi_alpha0,
                                                    phiprime_alpha0,
                                                    phi_alpha1)
                break

            phiprime_alpha1 = le_alpha1.directional_derivative
            if abs(phiprime_alpha1) <= -self.c2*phiprime_0:
                alpha_star = alpha1
                phi_star = phi_alpha1
                le_star = le_alpha1
                break

            if phiprime_alpha1 >= 0:
                (alpha_star, phi_star, le_star) = self._zoom(
                                                    alpha1, alpha0,
                                                    phi_0, phiprime_0,
                                                    phi_alpha1,
                                                    phiprime_alpha1,
                                                    phi_alpha0)
                break

            # update alphas
            alpha0, alpha1 = alpha1, min(2*alpha1, max_step_size)
            if alpha1 == max_step_size:
                print "reached max step size, bailing out"
                alpha_star = alpha1
                phi_star = phi_alpha1
                le_star = le_alpha1
                break
            phi_alpha0 = phi_alpha1
            phiprime_alpha0 = phiprime_alpha1

        else:
            # max_iterations was reached
            alpha_star = alpha1
            phi_star = phi_alpha1
            le_star = le_alpha1
            self.logger.error("The line search algorithm did not converge.")

        # extract the full energy from the line_energy
        energy_star = le_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):
        """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
            A boundary for the step length interval.
            Fulfills Wolfe condition 1.
        alpha_hi : float
            The other 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.

        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
        phiprime_alphaj = 0.
        alpha_recent = None
        phi_recent = None

        assert phi_lo <= phi_0 + self.c1*alpha_lo*phiprime_0
        assert phiprime_lo*(alpha_hi-alpha_lo) < 0.
        for i in xrange(max_iterations):
            # assert phi_lo <= phi_0 + self.c1*alpha_lo*phiprime_0
            # assert phiprime_lo*(alpha_hi-alpha_lo)<0.
            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 successful, 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
            le_alphaj = self.line_energy.at(alpha_j)
            phi_alphaj = le_alphaj.value

            # If the first Wolfe condition is not met replace alpha_hi
            # by alpha_j
            if (phi_alphaj > phi_0 + self.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 = le_alphaj.directional_derivative
                # If the second Wolfe condition is met, return the result
                if abs(phiprime_alphaj) <= -self.c2*phiprime_0:
                    alpha_star = alpha_j
                    phi_star = phi_alphaj
                    le_star = le_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, le_star = \
                alpha_j, phi_alphaj, le_alphaj
            self.logger.error("The line search algorithm (zoom) did not "
                              "converge.")

        return alpha_star, phi_star, le_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