"""Solvers for Ridge and LogisticRegression using SAG algorithm""" # Authors: Tom Dupre la Tour # # License: BSD 3 clause import warnings import numpy as np from .base import make_dataset from .sag_fast import sag from ..exceptions import ConvergenceWarning from ..utils import check_array from ..utils.extmath import row_norms def get_auto_step_size(max_squared_sum, alpha_scaled, loss, fit_intercept, n_samples=None, is_saga=False): """Compute automatic step size for SAG solver The step size is set to 1 / (alpha_scaled + L + fit_intercept) where L is the max sum of squares for over all samples. Parameters ---------- max_squared_sum : float Maximum squared sum of X over samples. alpha_scaled : float Constant that multiplies the regularization term, scaled by 1. / n_samples, the number of samples. loss : string, in {"log", "squared"} The loss function used in SAG solver. fit_intercept : bool Specifies if a constant (a.k.a. bias or intercept) will be added to the decision function. n_samples : int, optional Number of rows in X. Useful if is_saga=True. is_saga : boolean, optional Whether to return step size for the SAGA algorithm or the SAG algorithm. Returns ------- step_size : float Step size used in SAG solver. References ---------- Schmidt, M., Roux, N. L., & Bach, F. (2013). Minimizing finite sums with the stochastic average gradient https://hal.inria.fr/hal-00860051/document Defazio, A., Bach F. & Lacoste-Julien S. (2014). SAGA: A Fast Incremental Gradient Method With Support for Non-Strongly Convex Composite Objectives https://arxiv.org/abs/1407.0202 """ if loss in ('log', 'multinomial'): L = (0.25 * (max_squared_sum + int(fit_intercept)) + alpha_scaled) elif loss == 'squared': # inverse Lipschitz constant for squared loss L = max_squared_sum + int(fit_intercept) + alpha_scaled else: raise ValueError("Unknown loss function for SAG solver, got %s " "instead of 'log' or 'squared'" % loss) if is_saga: # SAGA theoretical step size is 1/3L or 1 / (2 * (L + mu n)) # See Defazio et al. 2014 mun = min(2 * n_samples * alpha_scaled, L) step = 1. / (2 * L + mun) else: # SAG theoretical step size is 1/16L but it is recommended to use 1 / L # see http://www.birs.ca//workshops//2014/14w5003/files/schmidt.pdf, # slide 65 step = 1. / L return step def sag_solver(X, y, sample_weight=None, loss='log', alpha=1., beta=0., max_iter=1000, tol=0.001, verbose=0, random_state=None, check_input=True, max_squared_sum=None, warm_start_mem=None, is_saga=False): """SAG solver for Ridge and LogisticRegression SAG stands for Stochastic Average Gradient: the gradient of the loss is estimated each sample at a time and the model is updated along the way with a constant learning rate. IMPORTANT NOTE: 'sag' solver converges faster on columns that are on the same scale. You can normalize the data by using sklearn.preprocessing.StandardScaler on your data before passing it to the fit method. This implementation works with data represented as dense numpy arrays or sparse scipy arrays of floating point values for the features. It will fit the data according to squared loss or log loss. The regularizer is a penalty added to the loss function that shrinks model parameters towards the zero vector using the squared euclidean norm L2. .. versionadded:: 0.17 Parameters ---------- X : {array-like, sparse matrix}, shape (n_samples, n_features) Training data y : numpy array, shape (n_samples,) Target values. With loss='multinomial', y must be label encoded (see preprocessing.LabelEncoder). sample_weight : array-like, shape (n_samples,), optional Weights applied to individual samples (1. for unweighted). loss : 'log' | 'squared' | 'multinomial' Loss function that will be optimized: -'log' is the binary logistic loss, as used in LogisticRegression. -'squared' is the squared loss, as used in Ridge. -'multinomial' is the multinomial logistic loss, as used in LogisticRegression. .. versionadded:: 0.18 *loss='multinomial'* alpha : float, optional Constant that multiplies the regularization term. Defaults to 1. max_iter : int, optional The max number of passes over the training data if the stopping criteria is not reached. Defaults to 1000. tol : double, optional The stopping criteria for the weights. The iterations will stop when max(change in weights) / max(weights) < tol. Defaults to .001 verbose : integer, optional The verbosity level. random_state : int, RandomState instance or None, optional, default None The seed of the pseudo random number generator to use when shuffling the data. If int, random_state is the seed used by the random number generator; If RandomState instance, random_state is the random number generator; If None, the random number generator is the RandomState instance used by `np.random`. check_input : bool, default True If False, the input arrays X and y will not be checked. max_squared_sum : float, default None Maximum squared sum of X over samples. If None, it will be computed, going through all the samples. The value should be precomputed to speed up cross validation. warm_start_mem : dict, optional The initialization parameters used for warm starting. Warm starting is currently used in LogisticRegression but not in Ridge. It contains: - 'coef': the weight vector, with the intercept in last line if the intercept is fitted. - 'gradient_memory': the scalar gradient for all seen samples. - 'sum_gradient': the sum of gradient over all seen samples, for each feature. - 'intercept_sum_gradient': the sum of gradient over all seen samples, for the intercept. - 'seen': array of boolean describing the seen samples. - 'num_seen': the number of seen samples. is_saga : boolean, optional Whether to use the SAGA algorithm or the SAG algorithm. SAGA behaves better in the first epochs, and allow for l1 regularisation. Returns ------- coef_ : array, shape (n_features) Weight vector. n_iter_ : int The number of full pass on all samples. warm_start_mem : dict Contains a 'coef' key with the fitted result, and possibly the fitted intercept at the end of the array. Contains also other keys used for warm starting. Examples -------- >>> import numpy as np >>> from sklearn import linear_model >>> n_samples, n_features = 10, 5 >>> np.random.seed(0) >>> X = np.random.randn(n_samples, n_features) >>> y = np.random.randn(n_samples) >>> clf = linear_model.Ridge(solver='sag') >>> clf.fit(X, y) ... #doctest: +NORMALIZE_WHITESPACE Ridge(alpha=1.0, copy_X=True, fit_intercept=True, max_iter=None, normalize=False, random_state=None, solver='sag', tol=0.001) >>> X = np.array([[-1, -1], [-2, -1], [1, 1], [2, 1]]) >>> y = np.array([1, 1, 2, 2]) >>> clf = linear_model.LogisticRegression(solver='sag') >>> clf.fit(X, y) ... #doctest: +NORMALIZE_WHITESPACE LogisticRegression(C=1.0, class_weight=None, dual=False, fit_intercept=True, intercept_scaling=1, max_iter=100, multi_class='ovr', n_jobs=1, penalty='l2', random_state=None, solver='sag', tol=0.0001, verbose=0, warm_start=False) References ---------- Schmidt, M., Roux, N. L., & Bach, F. (2013). Minimizing finite sums with the stochastic average gradient https://hal.inria.fr/hal-00860051/document Defazio, A., Bach F. & Lacoste-Julien S. (2014). SAGA: A Fast Incremental Gradient Method With Support for Non-Strongly Convex Composite Objectives https://arxiv.org/abs/1407.0202 See also -------- Ridge, SGDRegressor, ElasticNet, Lasso, SVR, and LogisticRegression, SGDClassifier, LinearSVC, Perceptron """ if warm_start_mem is None: warm_start_mem = {} # Ridge default max_iter is None if max_iter is None: max_iter = 1000 if check_input: X = check_array(X, dtype=np.float64, accept_sparse='csr', order='C') y = check_array(y, dtype=np.float64, ensure_2d=False, order='C') n_samples, n_features = X.shape[0], X.shape[1] # As in SGD, the alpha is scaled by n_samples. alpha_scaled = float(alpha) / n_samples beta_scaled = float(beta) / n_samples # if loss == 'multinomial', y should be label encoded. n_classes = int(y.max()) + 1 if loss == 'multinomial' else 1 # initialization if sample_weight is None: sample_weight = np.ones(n_samples, dtype=np.float64, order='C') if 'coef' in warm_start_mem.keys(): coef_init = warm_start_mem['coef'] else: # assume fit_intercept is False coef_init = np.zeros((n_features, n_classes), dtype=np.float64, order='C') # coef_init contains possibly the intercept_init at the end. # Note that Ridge centers the data before fitting, so fit_intercept=False. fit_intercept = coef_init.shape[0] == (n_features + 1) if fit_intercept: intercept_init = coef_init[-1, :] coef_init = coef_init[:-1, :] else: intercept_init = np.zeros(n_classes, dtype=np.float64) if 'intercept_sum_gradient' in warm_start_mem.keys(): intercept_sum_gradient = warm_start_mem['intercept_sum_gradient'] else: intercept_sum_gradient = np.zeros(n_classes, dtype=np.float64) if 'gradient_memory' in warm_start_mem.keys(): gradient_memory_init = warm_start_mem['gradient_memory'] else: gradient_memory_init = np.zeros((n_samples, n_classes), dtype=np.float64, order='C') if 'sum_gradient' in warm_start_mem.keys(): sum_gradient_init = warm_start_mem['sum_gradient'] else: sum_gradient_init = np.zeros((n_features, n_classes), dtype=np.float64, order='C') if 'seen' in warm_start_mem.keys(): seen_init = warm_start_mem['seen'] else: seen_init = np.zeros(n_samples, dtype=np.int32, order='C') if 'num_seen' in warm_start_mem.keys(): num_seen_init = warm_start_mem['num_seen'] else: num_seen_init = 0 dataset, intercept_decay = make_dataset(X, y, sample_weight, random_state) if max_squared_sum is None: max_squared_sum = row_norms(X, squared=True).max() step_size = get_auto_step_size(max_squared_sum, alpha_scaled, loss, fit_intercept, n_samples=n_samples, is_saga=is_saga) if step_size * alpha_scaled == 1: raise ZeroDivisionError("Current sag implementation does not handle " "the case step_size * alpha_scaled == 1") num_seen, n_iter_ = sag(dataset, coef_init, intercept_init, n_samples, n_features, n_classes, tol, max_iter, loss, step_size, alpha_scaled, beta_scaled, sum_gradient_init, gradient_memory_init, seen_init, num_seen_init, fit_intercept, intercept_sum_gradient, intercept_decay, is_saga, verbose) if n_iter_ == max_iter: warnings.warn("The max_iter was reached which means " "the coef_ did not converge", ConvergenceWarning) if fit_intercept: coef_init = np.vstack((coef_init, intercept_init)) warm_start_mem = {'coef': coef_init, 'sum_gradient': sum_gradient_init, 'intercept_sum_gradient': intercept_sum_gradient, 'gradient_memory': gradient_memory_init, 'seen': seen_init, 'num_seen': num_seen} if loss == 'multinomial': coef_ = coef_init.T else: coef_ = coef_init[:, 0] return coef_, n_iter_, warm_start_mem