Source code for picos.expressions.exp_entropy

# coding: utf-8

# ------------------------------------------------------------------------------
# Copyright (C) 2019 Maximilian Stahlberg
# Based on the original picos.expressions module by Guillaume Sagnol.
#
# This file is part of PICOS.
#
# PICOS is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# PICOS is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program.  If not, see <http://www.gnu.org/licenses/>.
# ------------------------------------------------------------------------------

"""Implements :class:`Entropy` and :class:`NegativeEntropy`."""

import math
import operator
from collections import namedtuple

import cvxopt
import numpy

from .. import glyphs
from ..apidoc import api_end, api_start
from ..caching import cached_selfinverse_unary_operator, cached_unary_operator
from ..constraints import KullbackLeiblerConstraint
from .data import convert_and_refine_arguments, convert_operands, cvx2np
from .exp_affine import AffineExpression
from .expression import Expression, refine_operands, validate_prediction

_API_START = api_start(globals())
# -------------------------------


[docs]class Entropy(Expression): r"""Entropy or negative relative entropy of an affine expression. Negative relative entropy is also known as the perspective of the logarithm. :Definition: Let :math:`x` be an :math:`n`-dimensional real affine expression. 1. If no additional expression :math:`y` is given, this is the entropy .. math:: -\sum_{i = 1}^n \operatorname{vec}(x)_i \log(\operatorname{vec}(x)_i). 2. If an additional affine expression :math:`y` of same shape as :math:`x` is given, this is the negative relative entropy (or logarithmic perspective) .. math:: -\sum_{i = 1}^n \operatorname{vec}(x)_i \log\left( \frac{\operatorname{vec}(x)_i}{\operatorname{vec}(y)_i} \right) &= -\sum_{i = 1}^n \operatorname{vec}(x)_i \left[ \log(\operatorname{vec}(x)_i) - \log(\operatorname{vec}(y)_i) \right] \\ &= \sum_{i = 1}^n \operatorname{vec}(x)_i \left[ \log(\operatorname{vec}(y)_i) - \log(\operatorname{vec}(x)_i \right] \\ &= \sum_{i = 1}^n \operatorname{vec}(x)_i \log\left( \frac{\operatorname{vec}(y)_i}{\operatorname{vec}(x)_i} \right). .. warning:: When you pose a lower bound on this expression, then PICOS enforces :math:`x \geq 0` through an auxiliary constraint during solution search. When an additional expression :math:`y` is given, PICOS enforces :math:`y \geq 0` as well. """ # -------------------------------------------------------------------------- # Initialization and factory methods. # --------------------------------------------------------------------------
[docs] @convert_and_refine_arguments("x", "y", allowNone=True) def __init__(self, x, y=None): """Construct an :class:`Entropy`. :param x: The affine expression :math:`x`. :type x: ~picos.expressions.AffineExpression :param y: An additional affine expression :math:`y`. If necessary, PICOS will attempt to reshape or broadcast it to the shape of :math:`x`. :type y: ~picos.expressions.AffineExpression """ if not isinstance(x, AffineExpression): raise TypeError("Can only take the elementwise logarithm of a real " "affine expression, not of {}.".format(type(x).__name__)) if y is not None: if not isinstance(y, AffineExpression): raise TypeError("The additional parameter y must be a real " "affine expression, not {}.".format(type(x).__name__)) elif x.shape != y.shape: y = y.reshaped_or_broadcasted(x.shape) if y.is1: y = None self._x = x self._y = y if y is None: typeStr = "Entropy" if len(x) == 1: symbStr = glyphs.neg(glyphs.mul(x.string, glyphs.log(x.string))) else: symbStr = glyphs.neg( glyphs.dotp(x.string, glyphs.log(x.string))) else: typeStr = "Logarithmic Perspective" if len(x) == 1: symbStr = glyphs.mul( x.string, glyphs.log(glyphs.div(y.string, x.string))) else: symbStr = glyphs.dotp(x.string, glyphs.sub(glyphs.log(y.string), glyphs.log(x.string))) Expression.__init__(self, typeStr, symbStr)
# -------------------------------------------------------------------------- # Abstract method implementations and method overridings, except _predict. # -------------------------------------------------------------------------- def _get_refined(self): if self._x.constant and (self._y is None or self._y.constant): return AffineExpression.from_constant(self.value, 1, self._symbStr) else: return self Subtype = namedtuple("Subtype", ("argdim", "y")) def _get_subtype(self): return self.Subtype(len(self._x), self._y is not None) def _get_value(self): x = numpy.ravel(cvx2np(self._x._get_value())) if self._y is None: s = -numpy.dot(x, numpy.log(x)) else: y = numpy.ravel(cvx2np(self._y._get_value())) s = numpy.dot(x, numpy.log(y / x)) return cvxopt.matrix(s) @cached_unary_operator def _get_variables(self): if self._y is None: return self._x.variables else: return self._x.variables.union(self._y.variables) def _is_convex(self): return False def _is_concave(self): return True def _replace_variables(self, var_map): return self.__class__(self._x._replace_variables(var_map), None if self._y is None else self._y._replace_variables(var_map)) # -------------------------------------------------------------------------- # Python special method implementations, except constraint-creating ones. # -------------------------------------------------------------------------- @cached_selfinverse_unary_operator def __neg__(self): return NegativeEntropy(self._x, self._y) @convert_operands(scalarRHS=True) @refine_operands() def __add__(self, other): if isinstance(other, AffineExpression): other = Entropy(other, other * math.e) if isinstance(other, Entropy): if self._y is None and other._y is None: entropy = Entropy(self._x.vec // other._x.vec) elif self._y is not None and other._y is None: one = AffineExpression.from_constant(1.0, (other.n, 1)) entropy = Entropy( self._x.vec // other._x.vec, self._y.vec // one) elif self._y is None and other._y is not None: one = AffineExpression.from_constant(1.0, (self.n, 1)) entropy = Entropy( self._x.vec // other._x.vec, one // other._y.vec) else: entropy = Entropy( self._x.vec // other._x.vec, self._y.vec // other._y.vec) entropy._symbStr = glyphs.clever_add(self.string, other.string) return entropy else: return NotImplemented @convert_operands(scalarRHS=True) @refine_operands() def __sub__(self, other): if isinstance(other, AffineExpression): other = NegativeEntropy(other, other / math.e) if isinstance(other, NegativeEntropy): return self + (-other) else: return NotImplemented @convert_operands(scalarRHS=True) @refine_operands() def __radd__(self, other): if isinstance(other, (AffineExpression, Entropy)): entropy = self.__add__(other) # NOTE: __add__ always creates a fresh expression. entropy._symbStr = glyphs.clever_add(other.string, self.string) return entropy else: return NotImplemented # -------------------------------------------------------------------------- # Methods and properties that return expressions. # -------------------------------------------------------------------------- @property def x(self): """The expression :math:`x`.""" return self._x @property def y(self): """The additional expression :math:`y`, or :obj:`None`.""" return self._y # -------------------------------------------------------------------------- # Methods and properties that describe the expression. # -------------------------------------------------------------------------- @property def n(self): """Length of :attr:`x`.""" return len(self._x) # -------------------------------------------------------------------------- # Constraint-creating operators, and _predict. # -------------------------------------------------------------------------- @classmethod def _predict(cls, subtype, relation, other): assert isinstance(subtype, cls.Subtype) if relation == operator.__ge__: if issubclass(other.clstype, AffineExpression) \ and other.subtype.dim == 1: return KullbackLeiblerConstraint.make_type( argdim=subtype.argdim) return NotImplemented @convert_operands(scalarRHS=True) @validate_prediction @refine_operands() def __ge__(self, other): if isinstance(other, AffineExpression): return KullbackLeiblerConstraint(-self, -other) else: return NotImplemented
[docs]class NegativeEntropy(Expression): r"""Negative or relative entropy of an affine expression. Relative entropy is also known as the Kullback-Leibler divergence. :Definition: Let :math:`x` be an :math:`n`-dimensional real affine expression. 1. If no additional expression :math:`y` is given, this is the negative entropy .. math:: \sum_{i = 1}^n \operatorname{vec}(x)_i \log(\operatorname{vec}(x)_i). 2. If an additional affine expression :math:`y` of same shape as :math:`x` is given, this is the relative entropy (or Kullback-Leibler divergence) .. math:: \sum_{i = 1}^n \operatorname{vec}(x)_i \log\left( \frac{\operatorname{vec}(x)_i}{\operatorname{vec}(y)_i} \right) = \sum_{i = 1}^n \operatorname{vec}(x)_i \left[ \log(\operatorname{vec}(x)_i) - \log(\operatorname{vec}(y)_i) \right]. .. warning:: When you pose an upper bound on this expression, then PICOS enforces :math:`x \geq 0` through an auxiliary constraint during solution search. When an additional expression :math:`y` is given, PICOS enforces :math:`y \geq 0` as well. """ # -------------------------------------------------------------------------- # Initialization and factory methods. # --------------------------------------------------------------------------
[docs] @convert_and_refine_arguments("x", "y", allowNone=True) def __init__(self, x, y=None): """Construct a :class:`NegativeEntropy`. :param x: The affine expression :math:`x`. :type x: ~picos.expressions.AffineExpression :param y: An additional affine expression :math:`y`. If necessary, PICOS will attempt to reshape or broadcast it to the shape of :math:`x`. :type y: ~picos.expressions.AffineExpression """ if not isinstance(x, AffineExpression): raise TypeError("Can only take the elementwise logarithm of a real " "affine expression, not of {}.".format(type(x).__name__)) if y is not None: if not isinstance(y, AffineExpression): raise TypeError("The additional parameter y must be a real " "affine expression, not {}.".format(type(x).__name__)) elif x.shape != y.shape: y = y.reshaped_or_broadcasted(x.shape) if y.is1: y = None self._x = x self._y = y if y is None: typeStr = "Negative Entropy" if len(x) == 1: symbStr = glyphs.mul(x.string, glyphs.log(x.string)) else: symbStr = glyphs.dotp(x.string, glyphs.log(x.string)) else: typeStr = "Relative Entropy" if len(x) == 1: symbStr = glyphs.mul( x.string, glyphs.log(glyphs.div(x.string, y.string))) else: symbStr = glyphs.dotp(x.string, glyphs.sub(glyphs.log(x.string), glyphs.log(y.string))) Expression.__init__(self, typeStr, symbStr)
# -------------------------------------------------------------------------- # Abstract method implementations and method overridings, except _predict. # -------------------------------------------------------------------------- def _get_refined(self): if self._x.constant and (self._y is None or self._y.constant): return AffineExpression.from_constant(self.value, 1, self._symbStr) else: return self Subtype = namedtuple("Subtype", ("argdim", "y")) def _get_subtype(self): return self.Subtype(len(self._x), self._y is not None) def _get_value(self): x = numpy.ravel(cvx2np(self._x._get_value())) if self._y is None: s = numpy.dot(x, numpy.log(x)) else: y = numpy.ravel(cvx2np(self._y._get_value())) s = numpy.dot(x, numpy.log(x / y)) return cvxopt.matrix(s) @cached_unary_operator def _get_variables(self): if self._y is None: return self._x.variables else: return self._x.variables.union(self._y.variables) def _is_convex(self): return True def _is_concave(self): return False def _replace_variables(self, var_map): return self.__class__(self._x._replace_variables(var_map), None if self._y is None else self._y._replace_variables(var_map)) # -------------------------------------------------------------------------- # Python special method implementations, except constraint-creating ones. # -------------------------------------------------------------------------- @cached_selfinverse_unary_operator def __neg__(self): return Entropy(self._x, self._y) @convert_operands(scalarRHS=True) @refine_operands() def __add__(self, other): if isinstance(other, AffineExpression): other = NegativeEntropy(other, other / math.e) if isinstance(other, NegativeEntropy): if self._y is None and other._y is None: negent = NegativeEntropy(self._x.vec // other._x.vec) elif self._y is not None and other._y is None: one = AffineExpression.from_constant(1.0, (other.n, 1)) negent = NegativeEntropy( self._x.vec // other._x.vec, self._y.vec // one) elif self._y is None and other._y is not None: one = AffineExpression.from_constant(1.0, (self.n, 1)) negent = NegativeEntropy( self._x.vec // other._x.vec, one // other._y.vec) else: negent = NegativeEntropy( self._x.vec // other._x.vec, self._y.vec // other._y.vec) negent._symbStr = glyphs.clever_add(self.string, other.string) return negent else: return NotImplemented @convert_operands(scalarRHS=True) @refine_operands() def __radd__(self, other): if isinstance(other, (AffineExpression, NegativeEntropy)): negent = self.__add__(other) # NOTE: __add__ always creates a fresh expression. negent._symbStr = glyphs.clever_add(other.string, self.string) return negent else: return NotImplemented @convert_operands(scalarRHS=True) @refine_operands() def __sub__(self, other): if isinstance(other, AffineExpression): other = Entropy(other, other * math.e) if isinstance(other, Entropy): return self + (-other) else: return NotImplemented # -------------------------------------------------------------------------- # Methods and properties that return expressions. # -------------------------------------------------------------------------- @property def x(self): """The expression :math:`x`.""" return self._x @property def y(self): """The additional expression :math:`y`, or :obj:`None`.""" return self._y # -------------------------------------------------------------------------- # Methods and properties that describe the expression. # -------------------------------------------------------------------------- @property def n(self): """Length of :attr:`x`.""" return len(self._x) # -------------------------------------------------------------------------- # Constraint-creating operators, and _predict. # -------------------------------------------------------------------------- @classmethod def _predict(cls, subtype, relation, other): assert isinstance(subtype, cls.Subtype) if relation == operator.__le__: if issubclass(other.clstype, AffineExpression) \ and other.subtype.dim == 1: return KullbackLeiblerConstraint.make_type( argdim=subtype.argdim) return NotImplemented @convert_operands(scalarRHS=True) @validate_prediction @refine_operands() def __le__(self, other): if isinstance(other, AffineExpression): return KullbackLeiblerConstraint(self, other) else: return NotImplemented
# -------------------------------------- __all__ = api_end(_API_START, globals())