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diff --git a/ot/stochastic.py b/ot/stochastic.py new file mode 100644 index 0000000..85c4230 --- /dev/null +++ b/ot/stochastic.py @@ -0,0 +1,736 @@ +# Author: Kilian Fatras <kilian.fatras@gmail.com> +# +# License: MIT License + +import numpy as np + + +############################################################################## +# Optimization toolbox for SEMI - DUAL problems +############################################################################## + + +def coordinate_grad_semi_dual(b, M, reg, beta, i): + ''' + Compute the coordinate gradient update for regularized discrete distributions for (i, :) + + The function computes the gradient of the semi dual problem: + + .. math:: + \max_v \sum_i (\sum_j v_j * b_j - reg * log(\sum_j exp((v_j - M_{i,j})/reg) * b_j)) * a_i + + Where : + + - M is the (ns,nt) metric cost matrix + - v is a dual variable in R^J + - reg is the regularization term + - a and b are source and target weights (sum to 1) + + The algorithm used for solving the problem is the ASGD & SAG algorithms + as proposed in [18]_ [alg.1 & alg.2] + + + Parameters + ---------- + + b : np.ndarray(nt,) + target measure + M : np.ndarray(ns, nt) + cost matrix + reg : float nu + Regularization term > 0 + v : np.ndarray(nt,) + dual variable + i : number int + picked number i + + Returns + ------- + + coordinate gradient : np.ndarray(nt,) + + Examples + -------- + + >>> n_source = 7 + >>> n_target = 4 + >>> reg = 1 + >>> numItermax = 300000 + >>> a = ot.utils.unif(n_source) + >>> b = ot.utils.unif(n_target) + >>> rng = np.random.RandomState(0) + >>> X_source = rng.randn(n_source, 2) + >>> Y_target = rng.randn(n_target, 2) + >>> M = ot.dist(X_source, Y_target) + >>> method = "ASGD" + >>> asgd_pi = stochastic.solve_semi_dual_entropic(a, b, M, reg, + method, numItermax) + >>> print(asgd_pi) + + References + ---------- + + [Genevay et al., 2016] : + Stochastic Optimization for Large-scale Optimal Transport, + Advances in Neural Information Processing Systems (2016), + arXiv preprint arxiv:1605.08527. + + ''' + + r = M[i, :] - beta + exp_beta = np.exp(-r / reg) * b + khi = exp_beta / (np.sum(exp_beta)) + return b - khi + + +def sag_entropic_transport(a, b, M, reg, numItermax=10000, lr=None): + ''' + Compute the SAG algorithm to solve the regularized discrete measures + optimal transport max problem + + The function solves the following optimization problem: + + .. math:: + \gamma = arg\min_\gamma <\gamma,M>_F + reg\cdot\Omega(\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 with :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})` + - a and b are source and target weights (sum to 1) + + The algorithm used for solving the problem is the SAG algorithm + as proposed in [18]_ [alg.1] + + + Parameters + ---------- + + a : np.ndarray(ns,), + source measure + b : np.ndarray(nt,), + target measure + M : np.ndarray(ns, nt), + cost matrix + reg : float number, + Regularization term > 0 + numItermax : int number + number of iteration + lr : float number + learning rate + + Returns + ------- + + v : np.ndarray(nt,) + dual variable + + Examples + -------- + + >>> n_source = 7 + >>> n_target = 4 + >>> reg = 1 + >>> numItermax = 300000 + >>> a = ot.utils.unif(n_source) + >>> b = ot.utils.unif(n_target) + >>> rng = np.random.RandomState(0) + >>> X_source = rng.randn(n_source, 2) + >>> Y_target = rng.randn(n_target, 2) + >>> M = ot.dist(X_source, Y_target) + >>> method = "ASGD" + >>> asgd_pi = stochastic.solve_semi_dual_entropic(a, b, M, reg, + method, numItermax) + >>> print(asgd_pi) + + References + ---------- + + [Genevay et al., 2016] : + Stochastic Optimization for Large-scale Optimal Transport, + Advances in Neural Information Processing Systems (2016), + arXiv preprint arxiv:1605.08527. + ''' + + if lr is None: + lr = 1. / max(a / reg) + n_source = np.shape(M)[0] + n_target = np.shape(M)[1] + cur_beta = np.zeros(n_target) + stored_gradient = np.zeros((n_source, n_target)) + sum_stored_gradient = np.zeros(n_target) + for _ in range(numItermax): + i = np.random.randint(n_source) + cur_coord_grad = a[i] * coordinate_grad_semi_dual(b, M, reg, + cur_beta, i) + sum_stored_gradient += (cur_coord_grad - stored_gradient[i]) + stored_gradient[i] = cur_coord_grad + cur_beta += lr * (1. / n_source) * sum_stored_gradient + return cur_beta + + +def averaged_sgd_entropic_transport(a, b, M, reg, numItermax=300000, lr=None): + ''' + Compute the ASGD algorithm to solve the regularized semi continous measures optimal transport max problem + + The function solves the following optimization problem: + + .. math:: + \gamma = arg\min_\gamma <\gamma,M>_F + reg\cdot\Omega(\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 with :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})` + - a and b are source and target weights (sum to 1) + + The algorithm used for solving the problem is the ASGD algorithm + as proposed in [18]_ [alg.2] + + + Parameters + ---------- + + b : np.ndarray(nt,) + target measure + M : np.ndarray(ns, nt) + cost matrix + reg : float number + Regularization term > 0 + numItermax : int number + number of iteration + lr : float number + learning rate + + + Returns + ------- + + ave_v : np.ndarray(nt,) + dual variable + + Examples + -------- + + >>> n_source = 7 + >>> n_target = 4 + >>> reg = 1 + >>> numItermax = 300000 + >>> a = ot.utils.unif(n_source) + >>> b = ot.utils.unif(n_target) + >>> rng = np.random.RandomState(0) + >>> X_source = rng.randn(n_source, 2) + >>> Y_target = rng.randn(n_target, 2) + >>> M = ot.dist(X_source, Y_target) + >>> method = "ASGD" + >>> asgd_pi = stochastic.solve_semi_dual_entropic(a, b, M, reg, + method, numItermax) + >>> print(asgd_pi) + + References + ---------- + + [Genevay et al., 2016] : + Stochastic Optimization for Large-scale Optimal Transport, + Advances in Neural Information Processing Systems (2016), + arXiv preprint arxiv:1605.08527. + ''' + + if lr is None: + lr = 1. / max(a / reg) + n_source = np.shape(M)[0] + n_target = np.shape(M)[1] + cur_beta = np.zeros(n_target) + ave_beta = np.zeros(n_target) + for cur_iter in range(numItermax): + k = cur_iter + 1 + i = np.random.randint(n_source) + cur_coord_grad = coordinate_grad_semi_dual(b, M, reg, cur_beta, i) + cur_beta += (lr / np.sqrt(k)) * cur_coord_grad + ave_beta = (1. / k) * cur_beta + (1 - 1. / k) * ave_beta + return ave_beta + + +def c_transform_entropic(b, M, reg, beta): + ''' + The goal is to recover u from the c-transform. + + The function computes the c_transform of a dual variable from the other + dual variable: + + .. math:: + u = v^{c,reg} = -reg \sum_j exp((v - M)/reg) b_j + + Where : + + - M is the (ns,nt) metric cost matrix + - u, v are dual variables in R^IxR^J + - reg is the regularization term + + It is used to recover an optimal u from optimal v solving the semi dual + problem, see Proposition 2.1 of [18]_ + + + Parameters + ---------- + + b : np.ndarray(nt,) + target measure + M : np.ndarray(ns, nt) + cost matrix + reg : float + regularization term > 0 + v : np.ndarray(nt,) + dual variable + + Returns + ------- + + u : np.ndarray(ns,) + dual variable + + Examples + -------- + + >>> n_source = 7 + >>> n_target = 4 + >>> reg = 1 + >>> numItermax = 300000 + >>> a = ot.utils.unif(n_source) + >>> b = ot.utils.unif(n_target) + >>> rng = np.random.RandomState(0) + >>> X_source = rng.randn(n_source, 2) + >>> Y_target = rng.randn(n_target, 2) + >>> M = ot.dist(X_source, Y_target) + >>> method = "ASGD" + >>> asgd_pi = stochastic.solve_semi_dual_entropic(a, b, M, reg, + method, numItermax) + >>> print(asgd_pi) + + References + ---------- + + [Genevay et al., 2016] : + Stochastic Optimization for Large-scale Optimal Transport, + Advances in Neural Information Processing Systems (2016), + arXiv preprint arxiv:1605.08527. + ''' + + n_source = np.shape(M)[0] + alpha = np.zeros(n_source) + for i in range(n_source): + r = M[i, :] - beta + min_r = np.min(r) + exp_beta = np.exp(-(r - min_r) / reg) * b + alpha[i] = min_r - reg * np.log(np.sum(exp_beta)) + return alpha + + +def solve_semi_dual_entropic(a, b, M, reg, method, numItermax=10000, lr=None, + log=False): + ''' + Compute the transportation matrix to solve the regularized discrete + measures optimal transport max problem + + The function solves the following optimization problem: + + .. math:: + \gamma = arg\min_\gamma <\gamma,M>_F + reg\cdot\Omega(\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 with :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})` + - a and b are source and target weights (sum to 1) + The algorithm used for solving the problem is the SAG or ASGD algorithms + as proposed in [18]_ + + + Parameters + ---------- + + a : np.ndarray(ns,) + source measure + b : np.ndarray(nt,) + target measure + M : np.ndarray(ns, nt) + cost matrix + reg : float number + Regularization term > 0 + methode : str + used method (SAG or ASGD) + numItermax : int number + number of iteration + lr : float number + learning rate + n_source : int number + size of the source measure + n_target : int number + size of the target measure + log : bool, optional + record log if True + + Returns + ------- + + pi : np.ndarray(ns, nt) + transportation matrix + log : dict + log dictionary return only if log==True in parameters + + Examples + -------- + + >>> n_source = 7 + >>> n_target = 4 + >>> reg = 1 + >>> numItermax = 300000 + >>> a = ot.utils.unif(n_source) + >>> b = ot.utils.unif(n_target) + >>> rng = np.random.RandomState(0) + >>> X_source = rng.randn(n_source, 2) + >>> Y_target = rng.randn(n_target, 2) + >>> M = ot.dist(X_source, Y_target) + >>> method = "ASGD" + >>> asgd_pi = stochastic.solve_semi_dual_entropic(a, b, M, reg, + method, numItermax) + >>> print(asgd_pi) + + References + ---------- + + [Genevay et al., 2016] : + Stochastic Optimization for Large-scale Optimal Transport, + Advances in Neural Information Processing Systems (2016), + arXiv preprint arxiv:1605.08527. + ''' + + if method.lower() == "sag": + opt_beta = sag_entropic_transport(a, b, M, reg, numItermax, lr) + elif method.lower() == "asgd": + opt_beta = averaged_sgd_entropic_transport(a, b, M, reg, numItermax, lr) + else: + print("Please, select your method between SAG and ASGD") + return None + + opt_alpha = c_transform_entropic(b, M, reg, opt_beta) + pi = (np.exp((opt_alpha[:, None] + opt_beta[None, :] - M[:, :]) / reg) * + a[:, None] * b[None, :]) + + if log: + log = {} + log['alpha'] = opt_alpha + log['beta'] = opt_beta + return pi, log + else: + return pi + + +############################################################################## +# Optimization toolbox for DUAL problems +############################################################################## + + +def batch_grad_dual(a, b, M, reg, alpha, beta, batch_size, batch_alpha, + batch_beta): + ''' + Computes the partial gradient of the dual optimal transport problem. + + For each (i,j) in a batch of coordinates, the partial gradients are : + + .. math:: + \partial_{u_i} F = u_i * b_s/l_{v} - \sum_{j \in B_v} exp((u_i + v_j - M_{i,j})/reg) * a_i * b_j + + \partial_{v_j} F = v_j * b_s/l_{u} - \sum_{i \in B_u} exp((u_i + v_j - M_{i,j})/reg) * a_i * b_j + + Where : + + - M is the (ns,nt) metric cost matrix + - u, v are dual variables in R^ixR^J + - reg is the regularization term + - :math:`B_u` and :math:`B_v` are lists of index + - :math:`b_s` is the size of the batchs :math:`B_u` and :math:`B_v` + - :math:`l_u` and :math:`l_v` are the lenghts of :math:`B_u` and :math:`B_v` + - a and b are source and target weights (sum to 1) + + + The algorithm used for solving the dual problem is the SGD algorithm + as proposed in [19]_ [alg.1] + + + Parameters + ---------- + + a : np.ndarray(ns,) + source measure + b : np.ndarray(nt,) + target measure + M : np.ndarray(ns, nt) + cost matrix + reg : float number + Regularization term > 0 + alpha : np.ndarray(ns,) + dual variable + beta : np.ndarray(nt,) + dual variable + batch_size : int number + size of the batch + batch_alpha : np.ndarray(bs,) + batch of index of alpha + batch_beta : np.ndarray(bs,) + batch of index of beta + + Returns + ------- + + grad : np.ndarray(ns,) + partial grad F + + Examples + -------- + + >>> n_source = 7 + >>> n_target = 4 + >>> reg = 1 + >>> numItermax = 20000 + >>> lr = 0.1 + >>> batch_size = 3 + >>> log = True + >>> a = ot.utils.unif(n_source) + >>> b = ot.utils.unif(n_target) + >>> rng = np.random.RandomState(0) + >>> X_source = rng.randn(n_source, 2) + >>> Y_target = rng.randn(n_target, 2) + >>> M = ot.dist(X_source, Y_target) + >>> sgd_dual_pi, log = stochastic.solve_dual_entropic(a, b, M, reg, + batch_size, + numItermax, lr, log) + >>> print(log['alpha'], log['beta']) + >>> print(sgd_dual_pi) + + References + ---------- + + [Seguy et al., 2018] : + International Conference on Learning Representation (2018), + arXiv preprint arxiv:1711.02283. + ''' + + G = - (np.exp((alpha[batch_alpha, None] + beta[None, batch_beta] - + M[batch_alpha, :][:, batch_beta]) / reg) * + a[batch_alpha, None] * b[None, batch_beta]) + grad_beta = np.zeros(np.shape(M)[1]) + grad_alpha = np.zeros(np.shape(M)[0]) + grad_beta[batch_beta] = (b[batch_beta] * len(batch_alpha) / np.shape(M)[0] + + G.sum(0)) + grad_alpha[batch_alpha] = (a[batch_alpha] * len(batch_beta) + / np.shape(M)[1] + G.sum(1)) + + return grad_alpha, grad_beta + + +def sgd_entropic_regularization(a, b, M, reg, batch_size, numItermax, lr): + ''' + Compute the sgd algorithm to solve the regularized discrete measures + optimal transport dual problem + + The function solves the following optimization problem: + + .. math:: + \gamma = arg\min_\gamma <\gamma,M>_F + reg\cdot\Omega(\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 with :math:`\Omega(\gamma)=\sum_{i,j} \gamma_{i,j}\log(\gamma_{i,j})` + - a and b are source and target weights (sum to 1) + + Parameters + ---------- + + a : np.ndarray(ns,) + source measure + b : np.ndarray(nt,) + target measure + M : np.ndarray(ns, nt) + cost matrix + reg : float number + Regularization term > 0 + batch_size : int number + size of the batch + numItermax : int number + number of iteration + lr : float number + learning rate + + Returns + ------- + + alpha : np.ndarray(ns,) + dual variable + beta : np.ndarray(nt,) + dual variable + + Examples + -------- + + >>> n_source = 7 + >>> n_target = 4 + >>> reg = 1 + >>> numItermax = 20000 + >>> lr = 0.1 + >>> batch_size = 3 + >>> log = True + >>> a = ot.utils.unif(n_source) + >>> b = ot.utils.unif(n_target) + >>> rng = np.random.RandomState(0) + >>> X_source = rng.randn(n_source, 2) + >>> Y_target = rng.randn(n_target, 2) + >>> M = ot.dist(X_source, Y_target) + >>> sgd_dual_pi, log = stochastic.solve_dual_entropic(a, b, M, reg, + batch_size, + numItermax, lr, log) + >>> print(log['alpha'], log['beta']) + >>> print(sgd_dual_pi) + + References + ---------- + + [Seguy et al., 2018] : + International Conference on Learning Representation (2018), + arXiv preprint arxiv:1711.02283. + ''' + + n_source = np.shape(M)[0] + n_target = np.shape(M)[1] + cur_alpha = np.zeros(n_source) + cur_beta = np.zeros(n_target) + for cur_iter in range(numItermax): + k = np.sqrt(cur_iter + 1) + batch_alpha = np.random.choice(n_source, batch_size, replace=False) + batch_beta = np.random.choice(n_target, batch_size, replace=False) + update_alpha, update_beta = batch_grad_dual(a, b, M, reg, cur_alpha, + cur_beta, batch_size, + batch_alpha, batch_beta) + cur_alpha[batch_alpha] += (lr / k) * update_alpha[batch_alpha] + cur_beta[batch_beta] += (lr / k) * update_beta[batch_beta] + + return cur_alpha, cur_beta + + +def solve_dual_entropic(a, b, M, reg, batch_size, numItermax=10000, lr=1, + log=False): + ''' + Compute the transportation matrix to solve the regularized discrete measures + optimal transport dual problem + + The function solves the following optimization problem: + + .. math:: + \gamma = arg\min_\gamma <\gamma,M>_F + reg\cdot\Omega(\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})` + - a and b are source and target weights (sum to 1) + + Parameters + ---------- + + a : np.ndarray(ns,) + source measure + b : np.ndarray(nt,) + target measure + M : np.ndarray(ns, nt) + cost matrix + reg : float number + Regularization term > 0 + batch_size : int number + size of the batch + numItermax : int number + number of iteration + lr : float number + learning rate + log : bool, optional + record log if True + + Returns + ------- + + pi : np.ndarray(ns, nt) + transportation matrix + log : dict + log dictionary return only if log==True in parameters + + Examples + -------- + + >>> n_source = 7 + >>> n_target = 4 + >>> reg = 1 + >>> numItermax = 20000 + >>> lr = 0.1 + >>> batch_size = 3 + >>> log = True + >>> a = ot.utils.unif(n_source) + >>> b = ot.utils.unif(n_target) + >>> rng = np.random.RandomState(0) + >>> X_source = rng.randn(n_source, 2) + >>> Y_target = rng.randn(n_target, 2) + >>> M = ot.dist(X_source, Y_target) + >>> sgd_dual_pi, log = stochastic.solve_dual_entropic(a, b, M, reg, + batch_size, + numItermax, lr, log) + >>> print(log['alpha'], log['beta']) + >>> print(sgd_dual_pi) + + References + ---------- + + [Seguy et al., 2018] : + International Conference on Learning Representation (2018), + arXiv preprint arxiv:1711.02283. + ''' + + opt_alpha, opt_beta = sgd_entropic_regularization(a, b, M, reg, batch_size, + numItermax, lr) + pi = (np.exp((opt_alpha[:, None] + opt_beta[None, :] - M[:, :]) / reg) * + a[:, None] * b[None, :]) + if log: + log = {} + log['alpha'] = opt_alpha + log['beta'] = opt_beta + return pi, log + else: + return pi |