414 lines
16 KiB
Python
414 lines
16 KiB
Python
"""
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Loss functions for linear models with raw_prediction = X @ coef
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"""
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import numpy as np
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from scipy import sparse
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from ..utils.extmath import squared_norm
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class LinearModelLoss:
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"""General class for loss functions with raw_prediction = X @ coef + intercept.
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Note that raw_prediction is also known as linear predictor.
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The loss is the sum of per sample losses and includes a term for L2
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regularization::
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loss = sum_i s_i loss(y_i, X_i @ coef + intercept)
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+ 1/2 * l2_reg_strength * ||coef||_2^2
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with sample weights s_i=1 if sample_weight=None.
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Gradient and hessian, for simplicity without intercept, are::
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gradient = X.T @ loss.gradient + l2_reg_strength * coef
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hessian = X.T @ diag(loss.hessian) @ X + l2_reg_strength * identity
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Conventions:
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if fit_intercept:
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n_dof = n_features + 1
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else:
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n_dof = n_features
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if base_loss.is_multiclass:
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coef.shape = (n_classes, n_dof) or ravelled (n_classes * n_dof,)
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else:
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coef.shape = (n_dof,)
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The intercept term is at the end of the coef array:
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if base_loss.is_multiclass:
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if coef.shape (n_classes, n_dof):
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intercept = coef[:, -1]
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if coef.shape (n_classes * n_dof,)
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intercept = coef[n_features::n_dof] = coef[(n_dof-1)::n_dof]
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intercept.shape = (n_classes,)
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else:
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intercept = coef[-1]
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Note: If coef has shape (n_classes * n_dof,), the 2d-array can be reconstructed as
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coef.reshape((n_classes, -1), order="F")
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The option order="F" makes coef[:, i] contiguous. This, in turn, makes the
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coefficients without intercept, coef[:, :-1], contiguous and speeds up
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matrix-vector computations.
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Note: If the average loss per sample is wanted instead of the sum of the loss per
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sample, one can simply use a rescaled sample_weight such that
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sum(sample_weight) = 1.
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Parameters
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----------
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base_loss : instance of class BaseLoss from sklearn._loss.
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fit_intercept : bool
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"""
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def __init__(self, base_loss, fit_intercept):
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self.base_loss = base_loss
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self.fit_intercept = fit_intercept
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def _w_intercept_raw(self, coef, X):
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"""Helper function to get coefficients, intercept and raw_prediction.
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Parameters
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----------
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coef : ndarray of shape (n_dof,), (n_classes, n_dof) or (n_classes * n_dof,)
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Coefficients of a linear model.
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If shape (n_classes * n_dof,), the classes of one feature are contiguous,
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i.e. one reconstructs the 2d-array via
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coef.reshape((n_classes, -1), order="F").
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X : {array-like, sparse matrix} of shape (n_samples, n_features)
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Training data.
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Returns
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-------
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weights : ndarray of shape (n_features,) or (n_classes, n_features)
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Coefficients without intercept term.
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intercept : float or ndarray of shape (n_classes,)
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Intercept terms.
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raw_prediction : ndarray of shape (n_samples,) or \
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(n_samples, n_classes)
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"""
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if not self.base_loss.is_multiclass:
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if self.fit_intercept:
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intercept = coef[-1]
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weights = coef[:-1]
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else:
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intercept = 0.0
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weights = coef
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raw_prediction = X @ weights + intercept
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else:
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# reshape to (n_classes, n_dof)
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if coef.ndim == 1:
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weights = coef.reshape((self.base_loss.n_classes, -1), order="F")
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else:
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weights = coef
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if self.fit_intercept:
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intercept = weights[:, -1]
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weights = weights[:, :-1]
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else:
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intercept = 0.0
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raw_prediction = X @ weights.T + intercept # ndarray, likely C-contiguous
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return weights, intercept, raw_prediction
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def loss(self, coef, X, y, sample_weight=None, l2_reg_strength=0.0, n_threads=1):
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"""Compute the loss as sum over point-wise losses.
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Parameters
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----------
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coef : ndarray of shape (n_dof,), (n_classes, n_dof) or (n_classes * n_dof,)
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Coefficients of a linear model.
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If shape (n_classes * n_dof,), the classes of one feature are contiguous,
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i.e. one reconstructs the 2d-array via
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coef.reshape((n_classes, -1), order="F").
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X : {array-like, sparse matrix} of shape (n_samples, n_features)
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Training data.
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y : contiguous array of shape (n_samples,)
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Observed, true target values.
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sample_weight : None or contiguous array of shape (n_samples,), default=None
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Sample weights.
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l2_reg_strength : float, default=0.0
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L2 regularization strength
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n_threads : int, default=1
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Number of OpenMP threads to use.
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Returns
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-------
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loss : float
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Sum of losses per sample plus penalty.
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"""
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weights, intercept, raw_prediction = self._w_intercept_raw(coef, X)
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loss = self.base_loss.loss(
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y_true=y,
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raw_prediction=raw_prediction,
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sample_weight=sample_weight,
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n_threads=n_threads,
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)
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loss = loss.sum()
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norm2_w = weights @ weights if weights.ndim == 1 else squared_norm(weights)
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return loss + 0.5 * l2_reg_strength * norm2_w
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def loss_gradient(
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self, coef, X, y, sample_weight=None, l2_reg_strength=0.0, n_threads=1
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):
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"""Computes the sum of loss and gradient w.r.t. coef.
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Parameters
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----------
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coef : ndarray of shape (n_dof,), (n_classes, n_dof) or (n_classes * n_dof,)
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Coefficients of a linear model.
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If shape (n_classes * n_dof,), the classes of one feature are contiguous,
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i.e. one reconstructs the 2d-array via
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coef.reshape((n_classes, -1), order="F").
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X : {array-like, sparse matrix} of shape (n_samples, n_features)
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Training data.
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y : contiguous array of shape (n_samples,)
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Observed, true target values.
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sample_weight : None or contiguous array of shape (n_samples,), default=None
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Sample weights.
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l2_reg_strength : float, default=0.0
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L2 regularization strength
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n_threads : int, default=1
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Number of OpenMP threads to use.
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Returns
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-------
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loss : float
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Sum of losses per sample plus penalty.
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gradient : ndarray of shape coef.shape
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The gradient of the loss.
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"""
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n_features, n_classes = X.shape[1], self.base_loss.n_classes
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n_dof = n_features + int(self.fit_intercept)
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weights, intercept, raw_prediction = self._w_intercept_raw(coef, X)
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loss, grad_per_sample = self.base_loss.loss_gradient(
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y_true=y,
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raw_prediction=raw_prediction,
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sample_weight=sample_weight,
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n_threads=n_threads,
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)
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loss = loss.sum()
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if not self.base_loss.is_multiclass:
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loss += 0.5 * l2_reg_strength * (weights @ weights)
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grad = np.empty_like(coef, dtype=weights.dtype)
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grad[:n_features] = X.T @ grad_per_sample + l2_reg_strength * weights
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if self.fit_intercept:
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grad[-1] = grad_per_sample.sum()
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else:
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loss += 0.5 * l2_reg_strength * squared_norm(weights)
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grad = np.empty((n_classes, n_dof), dtype=weights.dtype, order="F")
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# grad_per_sample.shape = (n_samples, n_classes)
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grad[:, :n_features] = grad_per_sample.T @ X + l2_reg_strength * weights
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if self.fit_intercept:
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grad[:, -1] = grad_per_sample.sum(axis=0)
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if coef.ndim == 1:
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grad = grad.ravel(order="F")
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return loss, grad
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def gradient(
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self, coef, X, y, sample_weight=None, l2_reg_strength=0.0, n_threads=1
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):
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"""Computes the gradient w.r.t. coef.
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Parameters
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----------
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coef : ndarray of shape (n_dof,), (n_classes, n_dof) or (n_classes * n_dof,)
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Coefficients of a linear model.
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If shape (n_classes * n_dof,), the classes of one feature are contiguous,
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i.e. one reconstructs the 2d-array via
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coef.reshape((n_classes, -1), order="F").
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X : {array-like, sparse matrix} of shape (n_samples, n_features)
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Training data.
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y : contiguous array of shape (n_samples,)
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Observed, true target values.
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sample_weight : None or contiguous array of shape (n_samples,), default=None
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Sample weights.
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l2_reg_strength : float, default=0.0
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L2 regularization strength
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n_threads : int, default=1
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Number of OpenMP threads to use.
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Returns
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-------
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gradient : ndarray of shape coef.shape
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The gradient of the loss.
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"""
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n_features, n_classes = X.shape[1], self.base_loss.n_classes
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n_dof = n_features + int(self.fit_intercept)
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weights, intercept, raw_prediction = self._w_intercept_raw(coef, X)
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grad_per_sample = self.base_loss.gradient(
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y_true=y,
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raw_prediction=raw_prediction,
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sample_weight=sample_weight,
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n_threads=n_threads,
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)
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if not self.base_loss.is_multiclass:
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grad = np.empty_like(coef, dtype=weights.dtype)
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grad[:n_features] = X.T @ grad_per_sample + l2_reg_strength * weights
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if self.fit_intercept:
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grad[-1] = grad_per_sample.sum()
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return grad
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else:
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grad = np.empty((n_classes, n_dof), dtype=weights.dtype, order="F")
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# gradient.shape = (n_samples, n_classes)
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grad[:, :n_features] = grad_per_sample.T @ X + l2_reg_strength * weights
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if self.fit_intercept:
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grad[:, -1] = grad_per_sample.sum(axis=0)
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if coef.ndim == 1:
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return grad.ravel(order="F")
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else:
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return grad
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def gradient_hessian_product(
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self, coef, X, y, sample_weight=None, l2_reg_strength=0.0, n_threads=1
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):
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"""Computes gradient and hessp (hessian product function) w.r.t. coef.
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Parameters
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----------
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coef : ndarray of shape (n_dof,), (n_classes, n_dof) or (n_classes * n_dof,)
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Coefficients of a linear model.
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If shape (n_classes * n_dof,), the classes of one feature are contiguous,
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i.e. one reconstructs the 2d-array via
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coef.reshape((n_classes, -1), order="F").
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X : {array-like, sparse matrix} of shape (n_samples, n_features)
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Training data.
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y : contiguous array of shape (n_samples,)
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Observed, true target values.
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sample_weight : None or contiguous array of shape (n_samples,), default=None
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Sample weights.
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l2_reg_strength : float, default=0.0
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L2 regularization strength
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n_threads : int, default=1
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Number of OpenMP threads to use.
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Returns
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-------
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gradient : ndarray of shape coef.shape
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The gradient of the loss.
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hessp : callable
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Function that takes in a vector input of shape of gradient and
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and returns matrix-vector product with hessian.
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"""
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(n_samples, n_features), n_classes = X.shape, self.base_loss.n_classes
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n_dof = n_features + int(self.fit_intercept)
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weights, intercept, raw_prediction = self._w_intercept_raw(coef, X)
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if not self.base_loss.is_multiclass:
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gradient, hessian = self.base_loss.gradient_hessian(
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y_true=y,
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raw_prediction=raw_prediction,
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sample_weight=sample_weight,
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n_threads=n_threads,
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)
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grad = np.empty_like(coef, dtype=weights.dtype)
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grad[:n_features] = X.T @ gradient + l2_reg_strength * weights
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if self.fit_intercept:
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grad[-1] = gradient.sum()
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# Precompute as much as possible: hX, hX_sum and hessian_sum
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hessian_sum = hessian.sum()
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if sparse.issparse(X):
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hX = sparse.dia_matrix((hessian, 0), shape=(n_samples, n_samples)) @ X
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else:
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hX = hessian[:, np.newaxis] * X
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if self.fit_intercept:
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# Calculate the double derivative with respect to intercept.
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# Note: In case hX is sparse, hX.sum is a matrix object.
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hX_sum = np.squeeze(np.asarray(hX.sum(axis=0)))
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# With intercept included and l2_reg_strength = 0, hessp returns
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# res = (X, 1)' @ diag(h) @ (X, 1) @ s
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# = (X, 1)' @ (hX @ s[:n_features], sum(h) * s[-1])
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# res[:n_features] = X' @ hX @ s[:n_features] + sum(h) * s[-1]
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# res[-1] = 1' @ hX @ s[:n_features] + sum(h) * s[-1]
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def hessp(s):
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ret = np.empty_like(s)
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if sparse.issparse(X):
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ret[:n_features] = X.T @ (hX @ s[:n_features])
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else:
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ret[:n_features] = np.linalg.multi_dot([X.T, hX, s[:n_features]])
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ret[:n_features] += l2_reg_strength * s[:n_features]
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if self.fit_intercept:
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ret[:n_features] += s[-1] * hX_sum
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ret[-1] = hX_sum @ s[:n_features] + hessian_sum * s[-1]
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return ret
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else:
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# Here we may safely assume HalfMultinomialLoss aka categorical
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# cross-entropy.
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# HalfMultinomialLoss computes only the diagonal part of the hessian, i.e.
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# diagonal in the classes. Here, we want the matrix-vector product of the
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# full hessian. Therefore, we call gradient_proba.
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gradient, proba = self.base_loss.gradient_proba(
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y_true=y,
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raw_prediction=raw_prediction,
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sample_weight=sample_weight,
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n_threads=n_threads,
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)
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grad = np.empty((n_classes, n_dof), dtype=weights.dtype, order="F")
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grad[:, :n_features] = gradient.T @ X + l2_reg_strength * weights
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if self.fit_intercept:
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grad[:, -1] = gradient.sum(axis=0)
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# Full hessian-vector product, i.e. not only the diagonal part of the
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# hessian. Derivation with some index battle for input vector s:
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# - sample index i
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# - feature indices j, m
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# - class indices k, l
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# - 1_{k=l} is one if k=l else 0
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# - p_i_k is the (predicted) probability that sample i belongs to class k
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# for all i: sum_k p_i_k = 1
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# - s_l_m is input vector for class l and feature m
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# - X' = X transposed
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#
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# Note: Hessian with dropping most indices is just:
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# X' @ p_k (1(k=l) - p_l) @ X
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#
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# result_{k j} = sum_{i, l, m} Hessian_{i, k j, m l} * s_l_m
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# = sum_{i, l, m} (X')_{ji} * p_i_k * (1_{k=l} - p_i_l)
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# * X_{im} s_l_m
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# = sum_{i, m} (X')_{ji} * p_i_k
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# * (X_{im} * s_k_m - sum_l p_i_l * X_{im} * s_l_m)
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#
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# See also https://github.com/scikit-learn/scikit-learn/pull/3646#discussion_r17461411 # noqa
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def hessp(s):
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s = s.reshape((n_classes, -1), order="F") # shape = (n_classes, n_dof)
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if self.fit_intercept:
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s_intercept = s[:, -1]
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s = s[:, :-1] # shape = (n_classes, n_features)
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else:
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s_intercept = 0
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tmp = X @ s.T + s_intercept # X_{im} * s_k_m
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tmp += (-proba * tmp).sum(axis=1)[:, np.newaxis] # - sum_l ..
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tmp *= proba # * p_i_k
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if sample_weight is not None:
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tmp *= sample_weight[:, np.newaxis]
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# hess_prod = empty_like(grad), but we ravel grad below and this
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# function is run after that.
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hess_prod = np.empty((n_classes, n_dof), dtype=weights.dtype, order="F")
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hess_prod[:, :n_features] = tmp.T @ X + l2_reg_strength * s
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if self.fit_intercept:
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hess_prod[:, -1] = tmp.sum(axis=0)
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if coef.ndim == 1:
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return hess_prod.ravel(order="F")
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else:
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return hess_prod
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if coef.ndim == 1:
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return grad.ravel(order="F"), hessp
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return grad, hessp
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