<SYSTEM_TASK:>
Solve the following problem using Python, implementing the functions described below, one line at a time
<END_TASK>
<USER_TASK:>
Description:
 def gcg(a, b, M, reg1, reg2, f, df, G0=None, numItermax=10, numInnerItermax=200, stopThr=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 : np.ndarray (ns,) samples weights in the source domain b : np.ndarray (nt,) samples in the target domain M : np.ndarray (ns,nt) loss matrix reg1 : float Entropic Regularization term >0 reg2 : float Second Regularization term >0 G0 : np.ndarray (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 error (>0) verbose : bool, optional Print information along iterations log : bool, optional record log if True 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 .. [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}'.format(
            'It.', 'Loss', 'Delta loss') + '\n' + '-' * 32)
        print('{:5d}|{:8e}|{:8e}'.format(it, f_val, 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

        delta_fval = (f_val - old_fval) / abs(f_val)
        if abs(delta_fval) < stopThr:
            loop = 0

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

        if verbose:
            if it % 20 == 0:
                print('{:5s}|{:12s}|{:8s}'.format(
                    'It.', 'Loss', 'Delta loss') + '\n' + '-' * 32)
            print('{:5d}|{:8e}|{:8e}'.format(it, f_val, delta_fval))

    if log:
        return G, log
    else:
        return G