658 lines
25 KiB
Python
658 lines
25 KiB
Python
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"""
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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 init_zero_coef(self, X, dtype=None):
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"""Allocate coef of correct shape with zeros.
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Parameters:
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-----------
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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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dtype : data-type, default=None
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Overrides the data type of coef. With dtype=None, coef will have the same
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dtype as X.
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Returns
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-------
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coef : ndarray of shape (n_dof,) or (n_classes, n_dof)
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Coefficients of a linear model.
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"""
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n_features = X.shape[1]
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n_classes = self.base_loss.n_classes
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if self.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 self.base_loss.is_multiclass:
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coef = np.zeros_like(X, shape=(n_classes, n_dof), dtype=dtype, order="F")
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else:
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coef = np.zeros_like(X, shape=n_dof, dtype=dtype)
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return coef
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def weight_intercept(self, coef):
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"""Helper function to get coefficients and intercept.
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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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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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"""
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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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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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return weights, intercept
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def weight_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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weights, intercept = self.weight_intercept(coef)
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if not self.base_loss.is_multiclass:
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raw_prediction = X @ weights + intercept
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else:
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# weights has shape (n_classes, n_dof)
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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 l2_penalty(self, weights, l2_reg_strength):
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"""Compute L2 penalty term l2_reg_strength/2 *||w||_2^2."""
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norm2_w = weights @ weights if weights.ndim == 1 else squared_norm(weights)
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return 0.5 * l2_reg_strength * norm2_w
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def loss(
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self,
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coef,
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X,
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y,
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sample_weight=None,
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l2_reg_strength=0.0,
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n_threads=1,
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raw_prediction=None,
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):
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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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raw_prediction : C-contiguous array of shape (n_samples,) or array of \
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shape (n_samples, n_classes)
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Raw prediction values (in link space). If provided, these are used. If
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None, then raw_prediction = X @ coef + intercept is calculated.
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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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if raw_prediction is None:
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weights, intercept, raw_prediction = self.weight_intercept_raw(coef, X)
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else:
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weights, intercept = self.weight_intercept(coef)
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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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return loss + self.l2_penalty(weights, l2_reg_strength)
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def loss_gradient(
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self,
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coef,
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X,
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y,
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sample_weight=None,
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l2_reg_strength=0.0,
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n_threads=1,
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raw_prediction=None,
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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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raw_prediction : C-contiguous array of shape (n_samples,) or array of \
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shape (n_samples, n_classes)
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Raw prediction values (in link space). If provided, these are used. If
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None, then raw_prediction = X @ coef + intercept is calculated.
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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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if raw_prediction is None:
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weights, intercept, raw_prediction = self.weight_intercept_raw(coef, X)
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else:
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weights, intercept = self.weight_intercept(coef)
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loss, grad_pointwise = 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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loss += self.l2_penalty(weights, l2_reg_strength)
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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_pointwise + l2_reg_strength * weights
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if self.fit_intercept:
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grad[-1] = grad_pointwise.sum()
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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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# grad_pointwise.shape = (n_samples, n_classes)
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grad[:, :n_features] = grad_pointwise.T @ X + l2_reg_strength * weights
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if self.fit_intercept:
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grad[:, -1] = grad_pointwise.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,
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coef,
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X,
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y,
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sample_weight=None,
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l2_reg_strength=0.0,
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n_threads=1,
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raw_prediction=None,
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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,
|
||
|
|
i.e. one reconstructs the 2d-array via
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||
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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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raw_prediction : C-contiguous array of shape (n_samples,) or array of \
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shape (n_samples, n_classes)
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Raw prediction values (in link space). If provided, these are used. If
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None, then raw_prediction = X @ coef + intercept is calculated.
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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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if raw_prediction is None:
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weights, intercept, raw_prediction = self.weight_intercept_raw(coef, X)
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else:
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weights, intercept = self.weight_intercept(coef)
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grad_pointwise = 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_pointwise + l2_reg_strength * weights
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if self.fit_intercept:
|
||
|
|
grad[-1] = grad_pointwise.sum()
|
||
|
|
return grad
|
||
|
|
else:
|
||
|
|
grad = np.empty((n_classes, n_dof), dtype=weights.dtype, order="F")
|
||
|
|
# gradient.shape = (n_samples, n_classes)
|
||
|
|
grad[:, :n_features] = grad_pointwise.T @ X + l2_reg_strength * weights
|
||
|
|
if self.fit_intercept:
|
||
|
|
grad[:, -1] = grad_pointwise.sum(axis=0)
|
||
|
|
if coef.ndim == 1:
|
||
|
|
return grad.ravel(order="F")
|
||
|
|
else:
|
||
|
|
return grad
|
||
|
|
|
||
|
|
def gradient_hessian(
|
||
|
|
self,
|
||
|
|
coef,
|
||
|
|
X,
|
||
|
|
y,
|
||
|
|
sample_weight=None,
|
||
|
|
l2_reg_strength=0.0,
|
||
|
|
n_threads=1,
|
||
|
|
gradient_out=None,
|
||
|
|
hessian_out=None,
|
||
|
|
raw_prediction=None,
|
||
|
|
):
|
||
|
|
"""Computes gradient and hessian w.r.t. coef.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
coef : ndarray of shape (n_dof,), (n_classes, n_dof) or (n_classes * n_dof,)
|
||
|
|
Coefficients of a linear model.
|
||
|
|
If shape (n_classes * n_dof,), the classes of one feature are contiguous,
|
||
|
|
i.e. one reconstructs the 2d-array via
|
||
|
|
coef.reshape((n_classes, -1), order="F").
|
||
|
|
X : {array-like, sparse matrix} of shape (n_samples, n_features)
|
||
|
|
Training data.
|
||
|
|
y : contiguous array of shape (n_samples,)
|
||
|
|
Observed, true target values.
|
||
|
|
sample_weight : None or contiguous array of shape (n_samples,), default=None
|
||
|
|
Sample weights.
|
||
|
|
l2_reg_strength : float, default=0.0
|
||
|
|
L2 regularization strength
|
||
|
|
n_threads : int, default=1
|
||
|
|
Number of OpenMP threads to use.
|
||
|
|
gradient_out : None or ndarray of shape coef.shape
|
||
|
|
A location into which the gradient is stored. If None, a new array
|
||
|
|
might be created.
|
||
|
|
hessian_out : None or ndarray
|
||
|
|
A location into which the hessian is stored. If None, a new array
|
||
|
|
might be created.
|
||
|
|
raw_prediction : C-contiguous array of shape (n_samples,) or array of \
|
||
|
|
shape (n_samples, n_classes)
|
||
|
|
Raw prediction values (in link space). If provided, these are used. If
|
||
|
|
None, then raw_prediction = X @ coef + intercept is calculated.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
gradient : ndarray of shape coef.shape
|
||
|
|
The gradient of the loss.
|
||
|
|
|
||
|
|
hessian : ndarray
|
||
|
|
Hessian matrix.
|
||
|
|
|
||
|
|
hessian_warning : bool
|
||
|
|
True if pointwise hessian has more than half of its elements non-positive.
|
||
|
|
"""
|
||
|
|
n_samples, n_features = X.shape
|
||
|
|
n_dof = n_features + int(self.fit_intercept)
|
||
|
|
|
||
|
|
if raw_prediction is None:
|
||
|
|
weights, intercept, raw_prediction = self.weight_intercept_raw(coef, X)
|
||
|
|
else:
|
||
|
|
weights, intercept = self.weight_intercept(coef)
|
||
|
|
|
||
|
|
grad_pointwise, hess_pointwise = self.base_loss.gradient_hessian(
|
||
|
|
y_true=y,
|
||
|
|
raw_prediction=raw_prediction,
|
||
|
|
sample_weight=sample_weight,
|
||
|
|
n_threads=n_threads,
|
||
|
|
)
|
||
|
|
|
||
|
|
# For non-canonical link functions and far away from the optimum, the pointwise
|
||
|
|
# hessian can be negative. We take care that 75% ot the hessian entries are
|
||
|
|
# positive.
|
||
|
|
hessian_warning = np.mean(hess_pointwise <= 0) > 0.25
|
||
|
|
hess_pointwise = np.abs(hess_pointwise)
|
||
|
|
|
||
|
|
if not self.base_loss.is_multiclass:
|
||
|
|
# gradient
|
||
|
|
if gradient_out is None:
|
||
|
|
grad = np.empty_like(coef, dtype=weights.dtype)
|
||
|
|
else:
|
||
|
|
grad = gradient_out
|
||
|
|
grad[:n_features] = X.T @ grad_pointwise + l2_reg_strength * weights
|
||
|
|
if self.fit_intercept:
|
||
|
|
grad[-1] = grad_pointwise.sum()
|
||
|
|
|
||
|
|
# hessian
|
||
|
|
if hessian_out is None:
|
||
|
|
hess = np.empty(shape=(n_dof, n_dof), dtype=weights.dtype)
|
||
|
|
else:
|
||
|
|
hess = hessian_out
|
||
|
|
|
||
|
|
if hessian_warning:
|
||
|
|
# Exit early without computing the hessian.
|
||
|
|
return grad, hess, hessian_warning
|
||
|
|
|
||
|
|
# TODO: This "sandwich product", X' diag(W) X, is the main computational
|
||
|
|
# bottleneck for solvers. A dedicated Cython routine might improve it
|
||
|
|
# exploiting the symmetry (as opposed to, e.g., BLAS gemm).
|
||
|
|
if sparse.issparse(X):
|
||
|
|
hess[:n_features, :n_features] = (
|
||
|
|
X.T
|
||
|
|
@ sparse.dia_matrix(
|
||
|
|
(hess_pointwise, 0), shape=(n_samples, n_samples)
|
||
|
|
)
|
||
|
|
@ X
|
||
|
|
).toarray()
|
||
|
|
else:
|
||
|
|
# np.einsum may use less memory but the following, using BLAS matrix
|
||
|
|
# multiplication (gemm), is by far faster.
|
||
|
|
WX = hess_pointwise[:, None] * X
|
||
|
|
hess[:n_features, :n_features] = np.dot(X.T, WX)
|
||
|
|
|
||
|
|
if l2_reg_strength > 0:
|
||
|
|
# The L2 penalty enters the Hessian on the diagonal only. To add those
|
||
|
|
# terms, we use a flattened view on the array.
|
||
|
|
hess.reshape(-1)[
|
||
|
|
: (n_features * n_dof) : (n_dof + 1)
|
||
|
|
] += l2_reg_strength
|
||
|
|
|
||
|
|
if self.fit_intercept:
|
||
|
|
# With intercept included as added column to X, the hessian becomes
|
||
|
|
# hess = (X, 1)' @ diag(h) @ (X, 1)
|
||
|
|
# = (X' @ diag(h) @ X, X' @ h)
|
||
|
|
# ( h @ X, sum(h))
|
||
|
|
# The left upper part has already been filled, it remains to compute
|
||
|
|
# the last row and the last column.
|
||
|
|
Xh = X.T @ hess_pointwise
|
||
|
|
hess[:-1, -1] = Xh
|
||
|
|
hess[-1, :-1] = Xh
|
||
|
|
hess[-1, -1] = hess_pointwise.sum()
|
||
|
|
else:
|
||
|
|
# Here we may safely assume HalfMultinomialLoss aka categorical
|
||
|
|
# cross-entropy.
|
||
|
|
raise NotImplementedError
|
||
|
|
|
||
|
|
return grad, hess, hessian_warning
|
||
|
|
|
||
|
|
def gradient_hessian_product(
|
||
|
|
self, coef, X, y, sample_weight=None, l2_reg_strength=0.0, n_threads=1
|
||
|
|
):
|
||
|
|
"""Computes gradient and hessp (hessian product function) w.r.t. coef.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
coef : ndarray of shape (n_dof,), (n_classes, n_dof) or (n_classes * n_dof,)
|
||
|
|
Coefficients of a linear model.
|
||
|
|
If shape (n_classes * n_dof,), the classes of one feature are contiguous,
|
||
|
|
i.e. one reconstructs the 2d-array via
|
||
|
|
coef.reshape((n_classes, -1), order="F").
|
||
|
|
X : {array-like, sparse matrix} of shape (n_samples, n_features)
|
||
|
|
Training data.
|
||
|
|
y : contiguous array of shape (n_samples,)
|
||
|
|
Observed, true target values.
|
||
|
|
sample_weight : None or contiguous array of shape (n_samples,), default=None
|
||
|
|
Sample weights.
|
||
|
|
l2_reg_strength : float, default=0.0
|
||
|
|
L2 regularization strength
|
||
|
|
n_threads : int, default=1
|
||
|
|
Number of OpenMP threads to use.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
gradient : ndarray of shape coef.shape
|
||
|
|
The gradient of the loss.
|
||
|
|
|
||
|
|
hessp : callable
|
||
|
|
Function that takes in a vector input of shape of gradient and
|
||
|
|
and returns matrix-vector product with hessian.
|
||
|
|
"""
|
||
|
|
(n_samples, n_features), n_classes = X.shape, self.base_loss.n_classes
|
||
|
|
n_dof = n_features + int(self.fit_intercept)
|
||
|
|
weights, intercept, raw_prediction = self.weight_intercept_raw(coef, X)
|
||
|
|
|
||
|
|
if not self.base_loss.is_multiclass:
|
||
|
|
grad_pointwise, hess_pointwise = self.base_loss.gradient_hessian(
|
||
|
|
y_true=y,
|
||
|
|
raw_prediction=raw_prediction,
|
||
|
|
sample_weight=sample_weight,
|
||
|
|
n_threads=n_threads,
|
||
|
|
)
|
||
|
|
grad = np.empty_like(coef, dtype=weights.dtype)
|
||
|
|
grad[:n_features] = X.T @ grad_pointwise + l2_reg_strength * weights
|
||
|
|
if self.fit_intercept:
|
||
|
|
grad[-1] = grad_pointwise.sum()
|
||
|
|
|
||
|
|
# Precompute as much as possible: hX, hX_sum and hessian_sum
|
||
|
|
hessian_sum = hess_pointwise.sum()
|
||
|
|
if sparse.issparse(X):
|
||
|
|
hX = (
|
||
|
|
sparse.dia_matrix((hess_pointwise, 0), shape=(n_samples, n_samples))
|
||
|
|
@ X
|
||
|
|
)
|
||
|
|
else:
|
||
|
|
hX = hess_pointwise[:, np.newaxis] * X
|
||
|
|
|
||
|
|
if self.fit_intercept:
|
||
|
|
# Calculate the double derivative with respect to intercept.
|
||
|
|
# Note: In case hX is sparse, hX.sum is a matrix object.
|
||
|
|
hX_sum = np.squeeze(np.asarray(hX.sum(axis=0)))
|
||
|
|
# prevent squeezing to zero-dim array if n_features == 1
|
||
|
|
hX_sum = np.atleast_1d(hX_sum)
|
||
|
|
|
||
|
|
# With intercept included and l2_reg_strength = 0, hessp returns
|
||
|
|
# res = (X, 1)' @ diag(h) @ (X, 1) @ s
|
||
|
|
# = (X, 1)' @ (hX @ s[:n_features], sum(h) * s[-1])
|
||
|
|
# res[:n_features] = X' @ hX @ s[:n_features] + sum(h) * s[-1]
|
||
|
|
# res[-1] = 1' @ hX @ s[:n_features] + sum(h) * s[-1]
|
||
|
|
def hessp(s):
|
||
|
|
ret = np.empty_like(s)
|
||
|
|
if sparse.issparse(X):
|
||
|
|
ret[:n_features] = X.T @ (hX @ s[:n_features])
|
||
|
|
else:
|
||
|
|
ret[:n_features] = np.linalg.multi_dot([X.T, hX, s[:n_features]])
|
||
|
|
ret[:n_features] += l2_reg_strength * s[:n_features]
|
||
|
|
|
||
|
|
if self.fit_intercept:
|
||
|
|
ret[:n_features] += s[-1] * hX_sum
|
||
|
|
ret[-1] = hX_sum @ s[:n_features] + hessian_sum * s[-1]
|
||
|
|
return ret
|
||
|
|
|
||
|
|
else:
|
||
|
|
# Here we may safely assume HalfMultinomialLoss aka categorical
|
||
|
|
# cross-entropy.
|
||
|
|
# HalfMultinomialLoss computes only the diagonal part of the hessian, i.e.
|
||
|
|
# diagonal in the classes. Here, we want the matrix-vector product of the
|
||
|
|
# full hessian. Therefore, we call gradient_proba.
|
||
|
|
grad_pointwise, proba = self.base_loss.gradient_proba(
|
||
|
|
y_true=y,
|
||
|
|
raw_prediction=raw_prediction,
|
||
|
|
sample_weight=sample_weight,
|
||
|
|
n_threads=n_threads,
|
||
|
|
)
|
||
|
|
grad = np.empty((n_classes, n_dof), dtype=weights.dtype, order="F")
|
||
|
|
grad[:, :n_features] = grad_pointwise.T @ X + l2_reg_strength * weights
|
||
|
|
if self.fit_intercept:
|
||
|
|
grad[:, -1] = grad_pointwise.sum(axis=0)
|
||
|
|
|
||
|
|
# Full hessian-vector product, i.e. not only the diagonal part of the
|
||
|
|
# hessian. Derivation with some index battle for input vector s:
|
||
|
|
# - sample index i
|
||
|
|
# - feature indices j, m
|
||
|
|
# - class indices k, l
|
||
|
|
# - 1_{k=l} is one if k=l else 0
|
||
|
|
# - p_i_k is the (predicted) probability that sample i belongs to class k
|
||
|
|
# for all i: sum_k p_i_k = 1
|
||
|
|
# - s_l_m is input vector for class l and feature m
|
||
|
|
# - X' = X transposed
|
||
|
|
#
|
||
|
|
# Note: Hessian with dropping most indices is just:
|
||
|
|
# X' @ p_k (1(k=l) - p_l) @ X
|
||
|
|
#
|
||
|
|
# result_{k j} = sum_{i, l, m} Hessian_{i, k j, m l} * s_l_m
|
||
|
|
# = sum_{i, l, m} (X')_{ji} * p_i_k * (1_{k=l} - p_i_l)
|
||
|
|
# * X_{im} s_l_m
|
||
|
|
# = sum_{i, m} (X')_{ji} * p_i_k
|
||
|
|
# * (X_{im} * s_k_m - sum_l p_i_l * X_{im} * s_l_m)
|
||
|
|
#
|
||
|
|
# See also https://github.com/scikit-learn/scikit-learn/pull/3646#discussion_r17461411 # noqa
|
||
|
|
def hessp(s):
|
||
|
|
s = s.reshape((n_classes, -1), order="F") # shape = (n_classes, n_dof)
|
||
|
|
if self.fit_intercept:
|
||
|
|
s_intercept = s[:, -1]
|
||
|
|
s = s[:, :-1] # shape = (n_classes, n_features)
|
||
|
|
else:
|
||
|
|
s_intercept = 0
|
||
|
|
tmp = X @ s.T + s_intercept # X_{im} * s_k_m
|
||
|
|
tmp += (-proba * tmp).sum(axis=1)[:, np.newaxis] # - sum_l ..
|
||
|
|
tmp *= proba # * p_i_k
|
||
|
|
if sample_weight is not None:
|
||
|
|
tmp *= sample_weight[:, np.newaxis]
|
||
|
|
# hess_prod = empty_like(grad), but we ravel grad below and this
|
||
|
|
# function is run after that.
|
||
|
|
hess_prod = np.empty((n_classes, n_dof), dtype=weights.dtype, order="F")
|
||
|
|
hess_prod[:, :n_features] = tmp.T @ X + l2_reg_strength * s
|
||
|
|
if self.fit_intercept:
|
||
|
|
hess_prod[:, -1] = tmp.sum(axis=0)
|
||
|
|
if coef.ndim == 1:
|
||
|
|
return hess_prod.ravel(order="F")
|
||
|
|
else:
|
||
|
|
return hess_prod
|
||
|
|
|
||
|
|
if coef.ndim == 1:
|
||
|
|
return grad.ravel(order="F"), hessp
|
||
|
|
|
||
|
|
return grad, hessp
|