path: root/ot/optim.py

# -*- coding: utf-8 -*-
"""
Generic solvers for regularized OT
"""

# Author: Remi Flamary <remi.flamary@unice.fr>
#         Titouan Vayer <titouan.vayer@irisa.fr>
#
# License: MIT License

import numpy as np
from scipy.optimize.linesearch import scalar_search_armijo
from .lp import emd
from .bregman import sinkhorn

# The corresponding scipy function does not work for matrices


def line_search_armijo(f, xk, pk, gfk, old_fval,
                       args=(), c1=1e-4, alpha0=0.99):
    """
    Armijo linesearch function that works with matrices

    find an approximate minimum of f(xk+alpha*pk) that satifies the
    armijo conditions.

    Parameters
    ----------
    f : callable
        loss function
    xk : ndarray
        initial position
    pk : ndarray
        descent direction
    gfk : ndarray
        gradient of f at xk
    old_fval : float
        loss value at xk
    args : tuple, optional
        arguments given to f
    c1 : float, optional
        c1 const in armijo rule (>0)
    alpha0 : float, optional
        initial step (>0)

    Returns
    -------
    alpha : float
        step that satisfy armijo conditions
    fc : int
        nb of function call
    fa : float
        loss value at step alpha

    """
    xk = np.atleast_1d(xk)
    fc = [0]

    def phi(alpha1):
        fc[0] += 1
        return f(xk + alpha1 * pk, *args)

    if old_fval is None:
        phi0 = phi(0.)
    else:
        phi0 = old_fval

    derphi0 = np.sum(pk * gfk)  # Quickfix for matrices
    alpha, phi1 = scalar_search_armijo(
        phi, phi0, derphi0, c1=c1, alpha0=alpha0)

    return alpha, fc[0], phi1


def solve_linesearch(cost, G, deltaG, Mi, f_val,
                     armijo=True, C1=None, C2=None, reg=None, Gc=None, constC=None, M=None):
    """
    Solve the linesearch in the FW iterations
    Parameters
    ----------
    cost : method
        Cost in the FW for the linesearch
    G : ndarray, shape(ns,nt)
        The transport map at a given iteration of the FW
    deltaG : ndarray (ns,nt)
        Difference between the optimal map found by linearization in the FW algorithm and the value at a given iteration
    Mi : ndarray (ns,nt)
        Cost matrix of the linearized transport problem. Corresponds to the gradient of the cost
    f_val :  float
        Value of the cost at G
    armijo : bool, optional
            If True the steps of the line-search is found via an armijo research. Else closed form is used.
            If there is convergence issues use False.
    C1 : ndarray (ns,ns), optional
        Structure matrix in the source domain. Only used and necessary when armijo=False
    C2 : ndarray (nt,nt), optional
        Structure matrix in the target domain. Only used and necessary when armijo=False
    reg : float, optional
          Regularization parameter. Only used and necessary when armijo=False
    Gc : ndarray (ns,nt)
        Optimal map found by linearization in the FW algorithm. Only used and necessary when armijo=False
    constC : ndarray (ns,nt)
             Constant for the gromov cost. See [24]. Only used and necessary when armijo=False
    M : ndarray (ns,nt), optional
        Cost matrix between the features. Only used and necessary when armijo=False
    Returns
    -------
    alpha : float
            The optimal step size of the FW
    fc : int
         nb of function call. Useless here
    f_val :  float
             The value of the cost for the next iteration
    References
    ----------
    .. [24] Vayer Titouan, Chapel Laetitia, Flamary R{\'e}mi, Tavenard Romain
          and Courty Nicolas
        "Optimal Transport for structured data with application on graphs"
        International Conference on Machine Learning (ICML). 2019.
    """
    if armijo:
        alpha, fc, f_val = line_search_armijo(cost, G, deltaG, Mi, f_val)
    else:  # requires symetric matrices
        dot1 = np.dot(C1, deltaG)
        dot12 = dot1.dot(C2)
        a = -2 * reg * np.sum(dot12 * deltaG)
        b = np.sum((M + reg * constC) * deltaG) - 2 * reg * (np.sum(dot12 * G) + np.sum(np.dot(C1, G).dot(C2) * deltaG))
        c = cost(G)

        alpha = solve_1d_linesearch_quad(a, b, c)
        fc = None
        f_val = cost(G + alpha * deltaG)

    return alpha, fc, f_val


def cg(a, b, M, reg, f, df, G0=None, numItermax=200, numItermaxEmd=100000,
       stopThr=1e-9, stopThr2=1e-9, verbose=False, log=False, **kwargs):
    """
    Solve the general regularized OT problem with conditional gradient

        The function solves the following optimization problem:

    .. math::
        \gamma = arg\min_\gamma <\gamma,M>_F + reg*f(\gamma)

        s.t. \gamma 1 = a

             \gamma^T 1= b

             \gamma\geq 0
    where :

    - M is the (ns,nt) metric cost matrix
    - :math:`f` is the regularization term ( and df is its gradient)
    - a and b are source and target weights (sum to 1)

    The algorithm used for solving the problem is conditional gradient as discussed in  [1]_


    Parameters
    ----------
    a : ndarray, shape (ns,)
        samples weights in the source domain
    b : ndarray, shape (nt,)
        samples in the target domain
    M : ndarray, shape (ns, nt)
        loss matrix
    reg : float
        Regularization term >0
    G0 :  ndarray, shape (ns,nt), optional
        initial guess (default is indep joint density)
    numItermax : int, optional
        Max number of iterations
    numItermaxEmd : int, optional
        Max number of iterations for emd
    stopThr : float, optional
        Stop threshol on the relative variation (>0)
    stopThr2 : float, optional
        Stop threshol on the absolute variation (>0)
    verbose : bool, optional
        Print information along iterations
    log : bool, optional
        record log if True
    **kwargs : dict
             Parameters for linesearch

    Returns
    -------
    gamma : (ns x nt) ndarray
        Optimal transportation matrix for the given parameters
    log : dict
        log dictionary return only if log==True in parameters


    References
    ----------

    .. [1] Ferradans, S., Papadakis, N., Peyré, G., & Aujol, J. F. (2014). Regularized discrete optimal transport. SIAM Journal on Imaging Sciences, 7(3), 1853-1882.

    See Also
    --------
    ot.lp.emd : Unregularized optimal ransport
    ot.bregman.sinkhorn : Entropic regularized optimal transport

    """

    loop = 1

    if log:
        log = {'loss': []}

    if G0 is None:
        G = np.outer(a, b)
    else:
        G = G0

    def cost(G):
        return np.sum(M * G) + reg * f(G)

    f_val = cost(G)
    if log:
        log['loss'].append(f_val)

    it = 0

    if verbose:
        print('{:5s}|{:12s}|{:8s}|{:8s}'.format(
            'It.', 'Loss', 'Relative loss', 'Absolute loss') + '\n' + '-' * 48)
        print('{:5d}|{:8e}|{:8e}|{:8e}'.format(it, f_val, 0, 0))

    while loop:

        it += 1
        old_fval = f_val

        # problem linearization
        Mi = M + reg * df(G)
        # set M positive
        Mi += Mi.min()

        # solve linear program
        Gc = emd(a, b, Mi, numItermax=numItermaxEmd)

        deltaG = Gc - G

        # line search
        alpha, fc, f_val = solve_linesearch(cost, G, deltaG, Mi, f_val, reg=reg, M=M, Gc=Gc, **kwargs)

        G = G + alpha * deltaG

        # test convergence
        if it >= numItermax:
            loop = 0

        abs_delta_fval = abs(f_val - old_fval)
        relative_delta_fval = abs_delta_fval / abs(f_val)
        if relative_delta_fval < stopThr or abs_delta_fval < stopThr2:
            loop = 0

        if log:
            log['loss'].append(f_val)

        if verbose:
            if it % 20 == 0:
                print('{:5s}|{:12s}|{:8s}|{:8s}'.format(
                    'It.', 'Loss', 'Relative loss', 'Absolute loss') + '\n' + '-' * 48)
            print('{:5d}|{:8e}|{:8e}|{:8e}'.format(it, f_val, relative_delta_fval, abs_delta_fval))

    if log:
        return G, log
    else:
        return G


def gcg(a, b, M, reg1, reg2, f, df, G0=None, numItermax=10,
        numInnerItermax=200, stopThr=1e-9, stopThr2=1e-9, verbose=False, log=False):
    """
    Solve the general regularized OT problem with the generalized conditional gradient

        The function solves the following optimization problem:

    .. math::
        \gamma = arg\min_\gamma <\gamma,M>_F + reg1\cdot\Omega(\gamma) + reg2\cdot f(\gamma)

        s.t. \gamma 1 = a

             \gamma^T 1= b

             \gamma\geq 0
    where :

    - M is the (ns,nt) metric cost matrix
    - :math:`\Omega` is the entropic regularization term :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})`
    - :math:`f` is the regularization term ( and df is its gradient)
    - a and b are source and target weights (sum to 1)

    The algorithm used for solving the problem is the generalized conditional gradient as discussed in  [5,7]_


    Parameters
    ----------
    a : ndarray, shape (ns,)
        samples weights in the source domain
    b : ndarrayv (nt,)
        samples in the target domain
    M : ndarray, shape (ns, nt)
        loss matrix
    reg1 : float
        Entropic Regularization term >0
    reg2 : float
        Second Regularization term >0
    G0 : ndarray, shape (ns, nt), optional
        initial guess (default is indep joint density)
    numItermax : int, optional
        Max number of iterations
    numInnerItermax : int, optional
        Max number of iterations of Sinkhorn
    stopThr : float, optional
        Stop threshol on the relative variation (>0)
    stopThr2 : float, optional
        Stop threshol on the absolute variation (>0)
    verbose : bool, optional
        Print information along iterations
    log : bool, optional
        record log if True

    Returns
    -------
    gamma : ndarray, shape (ns, nt)
        Optimal transportation matrix for the given parameters
    log : dict
        log dictionary return only if log==True in parameters

    References
    ----------
    .. [5] N. Courty; R. Flamary; D. Tuia; A. Rakotomamonjy, "Optimal Transport for Domain Adaptation," in IEEE Transactions on Pattern Analysis and Machine Intelligence , vol.PP, no.99, pp.1-1
    .. [7] Rakotomamonjy, A., Flamary, R., & Courty, N. (2015). Generalized conditional gradient: analysis of convergence and applications. arXiv preprint arXiv:1510.06567.

    See Also
    --------
    ot.optim.cg : conditional gradient

    """

    loop = 1

    if log:
        log = {'loss': []}

    if G0 is None:
        G = np.outer(a, b)
    else:
        G = G0

    def cost(G):
        return np.sum(M * G) + reg1 * np.sum(G * np.log(G)) + reg2 * f(G)

    f_val = cost(G)
    if log:
        log['loss'].append(f_val)

    it = 0

    if verbose:
        print('{:5s}|{:12s}|{:8s}|{:8s}'.format(
            'It.', 'Loss', 'Relative loss', 'Absolute loss') + '\n' + '-' * 48)
        print('{:5d}|{:8e}|{:8e}|{:8e}'.format(it, f_val, 0, 0))

    while loop:

        it += 1
        old_fval = f_val

        # problem linearization
        Mi = M + reg2 * df(G)

        # solve linear program with Sinkhorn
        # Gc = sinkhorn_stabilized(a,b, Mi, reg1, numItermax = numInnerItermax)
        Gc = sinkhorn(a, b, Mi, reg1, numItermax=numInnerItermax)

        deltaG = Gc - G

        # line search
        dcost = Mi + reg1 * (1 + np.log(G))  # ??
        alpha, fc, f_val = line_search_armijo(cost, G, deltaG, dcost, f_val)

        G = G + alpha * deltaG

        # test convergence
        if it >= numItermax:
            loop = 0

        abs_delta_fval = abs(f_val - old_fval)
        relative_delta_fval = abs_delta_fval / abs(f_val)

        if relative_delta_fval < stopThr or abs_delta_fval < stopThr2:
            loop = 0

        if log:
            log['loss'].append(f_val)

        if verbose:
            if it % 20 == 0:
                print('{:5s}|{:12s}|{:8s}|{:8s}'.format(
                    'It.', 'Loss', 'Relative loss', 'Absolute loss') + '\n' + '-' * 48)
            print('{:5d}|{:8e}|{:8e}|{:8e}'.format(it, f_val, relative_delta_fval, abs_delta_fval))

    if log:
        return G, log
    else:
        return G


def solve_1d_linesearch_quad(a, b, c):
    """
    For any convex or non-convex 1d quadratic function f, solve on [0,1] the following problem:
    .. math::
        \argmin f(x)=a*x^{2}+b*x+c

    Parameters
    ----------
    a,b,c : float
        The coefficients of the quadratic function

    Returns
    -------
    x : float
        The optimal value which leads to the minimal cost
    """
    f0 = c
    df0 = b
    f1 = a + f0 + df0

    if a > 0:  # convex
        minimum = min(1, max(0, np.divide(-b, 2.0 * a)))
        return minimum
    else:  # non convex
        if f0 > f1:
            return 1
        else:
            return 0