431 lines
16 KiB
Python
431 lines
16 KiB
Python
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"""Hessian update strategies for quasi-Newton optimization methods."""
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from __future__ import division, print_function, absolute_import
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import numpy as np
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from numpy.linalg import norm
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from scipy.linalg import get_blas_funcs
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from warnings import warn
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__all__ = ['HessianUpdateStrategy', 'BFGS', 'SR1']
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class HessianUpdateStrategy(object):
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"""Interface for implementing Hessian update strategies.
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Many optimization methods make use of Hessian (or inverse Hessian)
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approximations, such as the quasi-Newton methods BFGS, SR1, L-BFGS.
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Some of these approximations, however, do not actually need to store
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the entire matrix or can compute the internal matrix product with a
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given vector in a very efficiently manner. This class serves as an
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abstract interface between the optimization algorithm and the
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quasi-Newton update strategies, giving freedom of implementation
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to store and update the internal matrix as efficiently as possible.
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Different choices of initialization and update procedure will result
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in different quasi-Newton strategies.
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Four methods should be implemented in derived classes: ``initialize``,
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``update``, ``dot`` and ``get_matrix``.
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Notes
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-----
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Any instance of a class that implements this interface,
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can be accepted by the method ``minimize`` and used by
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the compatible solvers to approximate the Hessian (or
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inverse Hessian) used by the optimization algorithms.
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"""
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def initialize(self, n, approx_type):
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"""Initialize internal matrix.
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Allocate internal memory for storing and updating
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the Hessian or its inverse.
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Parameters
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----------
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n : int
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Problem dimension.
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approx_type : {'hess', 'inv_hess'}
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Selects either the Hessian or the inverse Hessian.
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When set to 'hess' the Hessian will be stored and updated.
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When set to 'inv_hess' its inverse will be used instead.
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"""
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raise NotImplementedError("The method ``initialize(n, approx_type)``"
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" is not implemented.")
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def update(self, delta_x, delta_grad):
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"""Update internal matrix.
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Update Hessian matrix or its inverse (depending on how 'approx_type'
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is defined) using information about the last evaluated points.
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Parameters
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----------
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delta_x : ndarray
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The difference between two points the gradient
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function have been evaluated at: ``delta_x = x2 - x1``.
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delta_grad : ndarray
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The difference between the gradients:
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``delta_grad = grad(x2) - grad(x1)``.
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"""
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raise NotImplementedError("The method ``update(delta_x, delta_grad)``"
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" is not implemented.")
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def dot(self, p):
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"""Compute the product of the internal matrix with the given vector.
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Parameters
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----------
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p : array_like
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1-d array representing a vector.
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Returns
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-------
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Hp : array
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1-d represents the result of multiplying the approximation matrix
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by vector p.
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"""
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raise NotImplementedError("The method ``dot(p)``"
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" is not implemented.")
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def get_matrix(self):
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"""Return current internal matrix.
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Returns
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-------
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H : ndarray, shape (n, n)
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Dense matrix containing either the Hessian
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or its inverse (depending on how 'approx_type'
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is defined).
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"""
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raise NotImplementedError("The method ``get_matrix(p)``"
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" is not implemented.")
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class FullHessianUpdateStrategy(HessianUpdateStrategy):
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"""Hessian update strategy with full dimensional internal representation.
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"""
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_syr = get_blas_funcs('syr', dtype='d') # Symmetric rank 1 update
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_syr2 = get_blas_funcs('syr2', dtype='d') # Symmetric rank 2 update
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# Symmetric matrix-vector product
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_symv = get_blas_funcs('symv', dtype='d')
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def __init__(self, init_scale='auto'):
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self.init_scale = init_scale
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# Until initialize is called we can't really use the class,
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# so it makes sense to set everything to None.
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self.first_iteration = None
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self.approx_type = None
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self.B = None
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self.H = None
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def initialize(self, n, approx_type):
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"""Initialize internal matrix.
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Allocate internal memory for storing and updating
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the Hessian or its inverse.
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Parameters
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----------
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n : int
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Problem dimension.
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approx_type : {'hess', 'inv_hess'}
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Selects either the Hessian or the inverse Hessian.
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When set to 'hess' the Hessian will be stored and updated.
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When set to 'inv_hess' its inverse will be used instead.
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"""
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self.first_iteration = True
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self.n = n
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self.approx_type = approx_type
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if approx_type not in ('hess', 'inv_hess'):
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raise ValueError("`approx_type` must be 'hess' or 'inv_hess'.")
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# Create matrix
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if self.approx_type == 'hess':
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self.B = np.eye(n, dtype=float)
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else:
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self.H = np.eye(n, dtype=float)
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def _auto_scale(self, delta_x, delta_grad):
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# Heuristic to scale matrix at first iteration.
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# Described in Nocedal and Wright "Numerical Optimization"
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# p.143 formula (6.20).
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s_norm2 = np.dot(delta_x, delta_x)
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y_norm2 = np.dot(delta_grad, delta_grad)
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ys = np.abs(np.dot(delta_grad, delta_x))
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if ys == 0.0 or y_norm2 == 0 or s_norm2 == 0:
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return 1
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if self.approx_type == 'hess':
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return y_norm2 / ys
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else:
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return ys / y_norm2
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def _update_implementation(self, delta_x, delta_grad):
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raise NotImplementedError("The method ``_update_implementation``"
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" is not implemented.")
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def update(self, delta_x, delta_grad):
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"""Update internal matrix.
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Update Hessian matrix or its inverse (depending on how 'approx_type'
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is defined) using information about the last evaluated points.
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Parameters
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----------
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delta_x : ndarray
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The difference between two points the gradient
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function have been evaluated at: ``delta_x = x2 - x1``.
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delta_grad : ndarray
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The difference between the gradients:
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``delta_grad = grad(x2) - grad(x1)``.
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"""
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if np.all(delta_x == 0.0):
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return
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if np.all(delta_grad == 0.0):
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warn('delta_grad == 0.0. Check if the approximated '
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'function is linear. If the function is linear '
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'better results can be obtained by defining the '
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'Hessian as zero instead of using quasi-Newton '
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'approximations.', UserWarning)
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return
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if self.first_iteration:
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# Get user specific scale
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if self.init_scale == "auto":
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scale = self._auto_scale(delta_x, delta_grad)
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else:
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scale = float(self.init_scale)
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# Scale initial matrix with ``scale * np.eye(n)``
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if self.approx_type == 'hess':
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self.B *= scale
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else:
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self.H *= scale
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self.first_iteration = False
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self._update_implementation(delta_x, delta_grad)
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def dot(self, p):
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"""Compute the product of the internal matrix with the given vector.
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Parameters
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----------
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p : array_like
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1-d array representing a vector.
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Returns
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-------
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Hp : array
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1-d represents the result of multiplying the approximation matrix
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by vector p.
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"""
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if self.approx_type == 'hess':
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return self._symv(1, self.B, p)
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else:
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return self._symv(1, self.H, p)
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def get_matrix(self):
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"""Return the current internal matrix.
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Returns
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-------
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M : ndarray, shape (n, n)
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Dense matrix containing either the Hessian or its inverse
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(depending on how `approx_type` was defined).
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"""
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if self.approx_type == 'hess':
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M = np.copy(self.B)
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else:
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M = np.copy(self.H)
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li = np.tril_indices_from(M, k=-1)
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M[li] = M.T[li]
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return M
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class BFGS(FullHessianUpdateStrategy):
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"""Broyden-Fletcher-Goldfarb-Shanno (BFGS) Hessian update strategy.
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Parameters
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----------
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exception_strategy : {'skip_update', 'damp_update'}, optional
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Define how to proceed when the curvature condition is violated.
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Set it to 'skip_update' to just skip the update. Or, alternatively,
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set it to 'damp_update' to interpolate between the actual BFGS
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result and the unmodified matrix. Both exceptions strategies
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are explained in [1]_, p.536-537.
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min_curvature : float
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This number, scaled by a normalization factor, defines the
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minimum curvature ``dot(delta_grad, delta_x)`` allowed to go
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unaffected by the exception strategy. By default is equal to
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1e-8 when ``exception_strategy = 'skip_update'`` and equal
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to 0.2 when ``exception_strategy = 'damp_update'``.
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init_scale : {float, 'auto'}
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Matrix scale at first iteration. At the first
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iteration the Hessian matrix or its inverse will be initialized
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with ``init_scale*np.eye(n)``, where ``n`` is the problem dimension.
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Set it to 'auto' in order to use an automatic heuristic for choosing
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the initial scale. The heuristic is described in [1]_, p.143.
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By default uses 'auto'.
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Notes
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-----
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The update is based on the description in [1]_, p.140.
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References
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----------
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.. [1] Nocedal, Jorge, and Stephen J. Wright. "Numerical optimization"
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Second Edition (2006).
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"""
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def __init__(self, exception_strategy='skip_update', min_curvature=None,
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init_scale='auto'):
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if exception_strategy == 'skip_update':
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if min_curvature is not None:
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self.min_curvature = min_curvature
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else:
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self.min_curvature = 1e-8
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elif exception_strategy == 'damp_update':
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if min_curvature is not None:
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self.min_curvature = min_curvature
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else:
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self.min_curvature = 0.2
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else:
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raise ValueError("`exception_strategy` must be 'skip_update' "
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"or 'damp_update'.")
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super(BFGS, self).__init__(init_scale)
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self.exception_strategy = exception_strategy
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def _update_inverse_hessian(self, ys, Hy, yHy, s):
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"""Update the inverse Hessian matrix.
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BFGS update using the formula:
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``H <- H + ((H*y).T*y + s.T*y)/(s.T*y)^2 * (s*s.T)
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- 1/(s.T*y) * ((H*y)*s.T + s*(H*y).T)``
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where ``s = delta_x`` and ``y = delta_grad``. This formula is
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equivalent to (6.17) in [1]_ written in a more efficient way
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for implementation.
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References
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----------
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.. [1] Nocedal, Jorge, and Stephen J. Wright. "Numerical optimization"
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Second Edition (2006).
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"""
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self.H = self._syr2(-1.0 / ys, s, Hy, a=self.H)
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self.H = self._syr((ys+yHy)/ys**2, s, a=self.H)
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def _update_hessian(self, ys, Bs, sBs, y):
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"""Update the Hessian matrix.
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BFGS update using the formula:
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``B <- B - (B*s)*(B*s).T/s.T*(B*s) + y*y^T/s.T*y``
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where ``s`` is short for ``delta_x`` and ``y`` is short
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for ``delta_grad``. Formula (6.19) in [1]_.
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References
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----------
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.. [1] Nocedal, Jorge, and Stephen J. Wright. "Numerical optimization"
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Second Edition (2006).
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"""
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self.B = self._syr(1.0 / ys, y, a=self.B)
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self.B = self._syr(-1.0 / sBs, Bs, a=self.B)
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def _update_implementation(self, delta_x, delta_grad):
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# Auxiliary variables w and z
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if self.approx_type == 'hess':
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w = delta_x
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z = delta_grad
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else:
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w = delta_grad
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z = delta_x
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# Do some common operations
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wz = np.dot(w, z)
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Mw = self.dot(w)
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wMw = Mw.dot(w)
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# Guarantee that wMw > 0 by reinitializing matrix.
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# While this is always true in exact arithmetics,
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# indefinite matrix may appear due to roundoff errors.
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if wMw <= 0.0:
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scale = self._auto_scale(delta_x, delta_grad)
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# Reinitialize matrix
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if self.approx_type == 'hess':
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self.B = scale * np.eye(self.n, dtype=float)
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else:
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self.H = scale * np.eye(self.n, dtype=float)
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# Do common operations for new matrix
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Mw = self.dot(w)
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wMw = Mw.dot(w)
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# Check if curvature condition is violated
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if wz <= self.min_curvature * wMw:
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# If the option 'skip_update' is set
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# we just skip the update when the condion
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# is violated.
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if self.exception_strategy == 'skip_update':
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return
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# If the option 'damp_update' is set we
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# interpolate between the actual BFGS
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# result and the unmodified matrix.
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elif self.exception_strategy == 'damp_update':
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update_factor = (1-self.min_curvature) / (1 - wz/wMw)
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z = update_factor*z + (1-update_factor)*Mw
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wz = np.dot(w, z)
|
||
|
|
# Update matrix
|
||
|
|
if self.approx_type == 'hess':
|
||
|
|
self._update_hessian(wz, Mw, wMw, z)
|
||
|
|
else:
|
||
|
|
self._update_inverse_hessian(wz, Mw, wMw, z)
|
||
|
|
|
||
|
|
|
||
|
|
class SR1(FullHessianUpdateStrategy):
|
||
|
|
"""Symmetric-rank-1 Hessian update strategy.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
min_denominator : float
|
||
|
|
This number, scaled by a normalization factor,
|
||
|
|
defines the minimum denominator magnitude allowed
|
||
|
|
in the update. When the condition is violated we skip
|
||
|
|
the update. By default uses ``1e-8``.
|
||
|
|
init_scale : {float, 'auto'}, optional
|
||
|
|
Matrix scale at first iteration. At the first
|
||
|
|
iteration the Hessian matrix or its inverse will be initialized
|
||
|
|
with ``init_scale*np.eye(n)``, where ``n`` is the problem dimension.
|
||
|
|
Set it to 'auto' in order to use an automatic heuristic for choosing
|
||
|
|
the initial scale. The heuristic is described in [1]_, p.143.
|
||
|
|
By default uses 'auto'.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
The update is based on the description in [1]_, p.144-146.
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
.. [1] Nocedal, Jorge, and Stephen J. Wright. "Numerical optimization"
|
||
|
|
Second Edition (2006).
|
||
|
|
"""
|
||
|
|
|
||
|
|
def __init__(self, min_denominator=1e-8, init_scale='auto'):
|
||
|
|
self.min_denominator = min_denominator
|
||
|
|
super(SR1, self).__init__(init_scale)
|
||
|
|
|
||
|
|
def _update_implementation(self, delta_x, delta_grad):
|
||
|
|
# Auxiliary variables w and z
|
||
|
|
if self.approx_type == 'hess':
|
||
|
|
w = delta_x
|
||
|
|
z = delta_grad
|
||
|
|
else:
|
||
|
|
w = delta_grad
|
||
|
|
z = delta_x
|
||
|
|
# Do some common operations
|
||
|
|
Mw = self.dot(w)
|
||
|
|
z_minus_Mw = z - Mw
|
||
|
|
denominator = np.dot(w, z_minus_Mw)
|
||
|
|
# If the denominator is too small
|
||
|
|
# we just skip the update.
|
||
|
|
if np.abs(denominator) <= self.min_denominator*norm(w)*norm(z_minus_Mw):
|
||
|
|
return
|
||
|
|
# Update matrix
|
||
|
|
if self.approx_type == 'hess':
|
||
|
|
self.B = self._syr(1/denominator, z_minus_Mw, a=self.B)
|
||
|
|
else:
|
||
|
|
self.H = self._syr(1/denominator, z_minus_Mw, a=self.H)
|