X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=dunshire%2Fgames.py;h=75f5329bb973e05d9070609fcb77e5eeba80fdee;hb=a35db50836050e28ee4e06a12caeaa30ebbb4b11;hp=f9877b334cfa3bcf2d60c6269ecbf4340de24eb1;hpb=f5b5ef66e41ae0538eb32e4b8420c36a23b95361;p=dunshire.git diff --git a/dunshire/games.py b/dunshire/games.py index f9877b3..75f5329 100644 --- a/dunshire/games.py +++ b/dunshire/games.py @@ -5,13 +5,14 @@ This module contains the main :class:`SymmetricLinearGame` class that knows how to solve a linear game. """ from cvxopt import matrix, printing, solvers -from .cones import CartesianProduct, IceCream, NonnegativeOrthant +from .cones import CartesianProduct from .errors import GameUnsolvableException, PoorScalingException from .matrices import (append_col, append_row, condition_number, identity, inner_product, norm, specnorm) -from . import options +from .options import ABS_TOL, FLOAT_FORMAT, DEBUG_FLOAT_FORMAT + +printing.options['dformat'] = FLOAT_FORMAT -printing.options['dformat'] = options.FLOAT_FORMAT class Solution: """ @@ -219,8 +220,7 @@ class SymmetricLinearGame: [ 1], e2 = [ 1] [ 2] - [ 3], - Condition((L, K, e1, e2)) = 31.834... + [ 3] Lists can (and probably should) be used for every argument:: @@ -238,8 +238,7 @@ class SymmetricLinearGame: e1 = [ 1] [ 1], e2 = [ 1] - [ 1], - Condition((L, K, e1, e2)) = 1.707... + [ 1] The points ``e1`` and ``e2`` can also be passed as some other enumerable type (of the correct length) without much harm, since @@ -261,8 +260,7 @@ class SymmetricLinearGame: e1 = [ 1] [ 1], e2 = [ 1] - [ 1], - Condition((L, K, e1, e2)) = 1.707... + [ 1] However, ``L`` will always be intepreted as a list of rows, even if it is passed as a :class:`cvxopt.base.matrix` which is @@ -283,8 +281,7 @@ class SymmetricLinearGame: e1 = [ 1] [ 1], e2 = [ 1] - [ 1], - Condition((L, K, e1, e2)) = 6.073... + [ 1] >>> L = cvxopt.matrix(L) >>> print(L) [ 1 3] @@ -299,8 +296,7 @@ class SymmetricLinearGame: e1 = [ 1] [ 1], e2 = [ 1] - [ 1], - Condition((L, K, e1, e2)) = 6.073... + [ 1] """ def __init__(self, L, K, e1, e2): @@ -322,6 +318,8 @@ class SymmetricLinearGame: if not self._e2 in K: raise ValueError('the point e2 must lie in the interior of K') + # Initial value of cached method. + self._L_specnorm_value = None def __str__(self): @@ -332,8 +330,7 @@ class SymmetricLinearGame: ' L = {:s},\n' \ ' K = {!s},\n' \ ' e1 = {:s},\n' \ - ' e2 = {:s},\n' \ - ' Condition((L, K, e1, e2)) = {:f}.' + ' e2 = {:s}' indented_L = '\n '.join(str(self.L()).splitlines()) indented_e1 = '\n '.join(str(self.e1()).splitlines()) indented_e2 = '\n '.join(str(self.e2()).splitlines()) @@ -341,8 +338,7 @@ class SymmetricLinearGame: return tpl.format(indented_L, str(self.K()), indented_e1, - indented_e2, - self.condition()) + indented_e2) def L(self): @@ -617,7 +613,7 @@ class SymmetricLinearGame: - def _G(self): + def G(self): r""" Return the matrix ``G`` used in our CVXOPT construction. @@ -644,7 +640,7 @@ class SymmetricLinearGame: >>> e1 = [1,2,3] >>> e2 = [1,1,1] >>> SLG = SymmetricLinearGame(L, K, e1, e2) - >>> print(SLG._G()) + >>> print(SLG.G()) [ 0.0000000 -1.0000000 0.0000000 0.0000000] [ 0.0000000 0.0000000 -1.0000000 0.0000000] [ 0.0000000 0.0000000 0.0000000 -1.0000000] @@ -659,7 +655,7 @@ class SymmetricLinearGame: append_col(self.e1(), -self.L())) - def _c(self): + def c(self): """ Return the vector ``c`` used in our CVXOPT construction. @@ -686,7 +682,7 @@ class SymmetricLinearGame: >>> e1 = [1,2,3] >>> e2 = [1,1,1] >>> SLG = SymmetricLinearGame(L, K, e1, e2) - >>> print(SLG._c()) + >>> print(SLG.c()) [-1.0000000] [ 0.0000000] [ 0.0000000] @@ -727,8 +723,8 @@ class SymmetricLinearGame: """ return CartesianProduct(self._K, self._K) - def _h(self): - """ + def h(self): + r""" Return the ``h`` vector used in our CVXOPT construction. The ``h`` vector appears on the right-hand side of :math:`Gx + s @@ -754,7 +750,7 @@ class SymmetricLinearGame: >>> e1 = [1,2,3] >>> e2 = [1,1,1] >>> SLG = SymmetricLinearGame(L, K, e1, e2) - >>> print(SLG._h()) + >>> print(SLG.h()) [0.0000000] [0.0000000] [0.0000000] @@ -770,7 +766,7 @@ class SymmetricLinearGame: @staticmethod def b(): - """ + r""" Return the ``b`` vector used in our CVXOPT construction. The vector ``b`` appears on the right-hand side of :math:`Ax = @@ -820,27 +816,10 @@ class SymmetricLinearGame: :meth:`L` is satisfied. """ p = self.e2() / (norm(self.e2()) ** 2) - - # Compute the distance from p to the outside of K. - if isinstance(self.K(), NonnegativeOrthant): - # How far is it to a wall? - dist = min(list(self.e1())) - elif isinstance(self.K(), IceCream): - # How far is it to the boundary of the ball that defines - # the ice-cream cone at a given height? Now draw a - # 45-45-90 triangle and the shortest distance to the - # outside of the cone should be 1/sqrt(2) of that. - # It works in R^2, so it works everywhere, right? - # We use "2" because it's better numerically than sqrt(2). - height = self.e1()[0] - radius = norm(self.e1()[1:]) - dist = (height - radius) / 2 - else: - raise NotImplementedError - - nu = - specnorm(self.L())/(dist*norm(self.e2())) - x = matrix([nu,p], (self.dimension() + 1, 1)) - s = - self._G()*x + dist = self.K().ball_radius(self.e1()) + nu = - self._L_specnorm()/(dist*norm(self.e2())) + x = matrix([nu, p], (self.dimension() + 1, 1)) + s = - self.G()*x return {'x': x, 's': s} @@ -850,33 +829,96 @@ class SymmetricLinearGame: Return a feasible starting point for player two. """ q = self.e1() / (norm(self.e1()) ** 2) - - # Compute the distance from p to the outside of K. - if isinstance(self.K(), NonnegativeOrthant): - # How far is it to a wall? - dist = min(list(self.e2())) - elif isinstance(self.K(), IceCream): - # How far is it to the boundary of the ball that defines - # the ice-cream cone at a given height? Now draw a - # 45-45-90 triangle and the shortest distance to the - # outside of the cone should be 1/sqrt(2) of that. - # It works in R^2, so it works everywhere, right? - # We use "2" because it's better numerically than sqrt(2). - height = self.e2()[0] - radius = norm(self.e2()[1:]) - dist = (height - radius) / 2 - else: - raise NotImplementedError - - omega = specnorm(self.L())/(dist*norm(self.e1())) + dist = self.K().ball_radius(self.e2()) + omega = self._L_specnorm()/(dist*norm(self.e1())) y = matrix([omega]) z2 = q z1 = y*self.e2() - self.L().trans()*z2 - z = matrix([z1,z2], (self.dimension()*2, 1)) + z = matrix([z1, z2], (self.dimension()*2, 1)) return {'y': y, 'z': z} + def _L_specnorm(self): + """ + Compute the spectral norm of :meth:`L` and cache it. + + The spectral norm of the matrix :meth:`L` is used in a few + places. Since it can be expensive to compute, we want to cache + its value. That is not possible in :func:`specnorm`, which lies + outside of a class, so this is the place to do it. + + Returns + ------- + + float + A nonnegative real number; the largest singular value of + the matrix :meth:`L`. + + """ + if self._L_specnorm_value is None: + self._L_specnorm_value = specnorm(self.L()) + return self._L_specnorm_value + + + def tolerance_scale(self, solution): + r""" + Return a scaling factor that should be applied to ``ABS_TOL`` + for this game. + + When performing certain comparisons, the default tolernace + ``ABS_TOL`` may not be appropriate. For example, if we expect + ``x`` and ``y`` to be within ``ABS_TOL`` of each other, than the + inner product of ``L*x`` and ``y`` can be as far apart as the + spectral norm of ``L`` times the sum of the norms of ``x`` and + ``y``. Such a comparison is made in :meth:`solution`, and in + many of our unit tests. + + The returned scaling factor found from the inner product mentioned + above is + + .. math:: + + \left\lVert L \right\rVert_{2} + \left( \left\lVert \bar{x} \right\rVert + + \left\lVert \bar{y} \right\rVert + \right), + + where :math:`\bar{x}` and :math:`\bar{y}` are optimal solutions + for players one and two respectively. This scaling factor is not + formally justified, but attempting anything smaller leads to + test failures. + + .. warning:: + + Optimal solutions are not unique, so the scaling factor + obtained from ``solution`` may not work when comparing other + solutions. + + Parameters + ---------- + + solution : Solution + A solution of this game, used to obtain the norms of the + optimal strategies. + + Returns + ------- + + float + A scaling factor to be multiplied by ``ABS_TOL`` when + making comparisons involving solutions of this game. + + """ + norm_p1_opt = norm(solution.player1_optimal()) + norm_p2_opt = norm(solution.player2_optimal()) + scale = self._L_specnorm()*(norm_p1_opt + norm_p2_opt) + + # Don't return anything smaller than 1... we can't go below + # out "minimum tolerance." + return max(1, scale) + + def solution(self): """ Solve this linear game and return a :class:`Solution`. @@ -991,23 +1033,62 @@ class SymmetricLinearGame: [2.506...] [0.000...] + This is another one that was difficult numerically, and caused + trouble even after we fixed the first two:: + + >>> from dunshire import * + >>> L = [[57.22233908627052301199, 41.70631373437460354126], + ... [83.04512571985074487202, 57.82581810406928468637]] + >>> K = NonnegativeOrthant(2) + >>> e1 = [7.31887017043399268346, 0.89744171905822367474] + >>> e2 = [0.11099824781179848388, 6.12564670639315345113] + >>> SLG = SymmetricLinearGame(L,K,e1,e2) + >>> print(SLG.solution()) + Game value: 70.437... + Player 1 optimal: + [9.009...] + [0.000...] + Player 2 optimal: + [0.136...] + [0.000...] + + And finally, here's one that returns an "optimal" solution, but + whose primal/dual objective function values are far apart:: + + >>> from dunshire import * + >>> L = [[ 6.49260076597376212248, -0.60528030227678542019], + ... [ 2.59896077096751731972, -0.97685530240286766457]] + >>> K = IceCream(2) + >>> e1 = [1, 0.43749513972645248661] + >>> e2 = [1, 0.46008379832200291260] + >>> SLG = SymmetricLinearGame(L, K, e1, e2) + >>> print(SLG.solution()) + Game value: 11.596... + Player 1 optimal: + [ 1.852...] + [-1.852...] + Player 2 optimal: + [ 1.777...] + [-1.777...] + """ try: opts = {'show_progress': False} - soln_dict = solvers.conelp(self._c(), - self._G(), - self._h(), + soln_dict = solvers.conelp(self.c(), + self.G(), + self.h(), self.C().cvxopt_dims(), self.A(), self.b(), primalstart=self.player1_start(), + dualstart=self.player2_start(), options=opts) except ValueError as error: if str(error) == 'math domain error': # Oops, CVXOPT tried to take the square root of a # negative number. Report some details about the game # rather than just the underlying CVXOPT crash. - printing.options['dformat'] = options.DEBUG_FLOAT_FORMAT + printing.options['dformat'] = DEBUG_FLOAT_FORMAT raise PoorScalingException(self) else: raise error @@ -1032,9 +1113,23 @@ class SymmetricLinearGame: # that CVXOPT is convinced the problem is infeasible (and that # cannot happen). if soln_dict['status'] in ['primal infeasible', 'dual infeasible']: - printing.options['dformat'] = options.DEBUG_FLOAT_FORMAT + printing.options['dformat'] = DEBUG_FLOAT_FORMAT raise GameUnsolvableException(self, soln_dict) + # For the game value, we could use any of: + # + # * p1_value + # * p2_value + # * (p1_value + p2_value)/2 + # * the game payoff + # + # We want the game value to be the payoff, however, so it + # makes the most sense to just use that, even if it means we + # can't test the fact that p1_value/p2_value are close to the + # payoff. + payoff = self.payoff(p1_optimal, p2_optimal) + soln = Solution(payoff, p1_optimal, p2_optimal) + # The "optimal" and "unknown" results, we actually treat the # same. Even if CVXOPT bails out due to numerical difficulty, # it will have some candidate points in mind. If those @@ -1045,29 +1140,18 @@ class SymmetricLinearGame: # close enough (one could be low by ABS_TOL, the other high by # it) because otherwise CVXOPT might return "unknown" and give # us two points in the cone that are nowhere near optimal. - if abs(p1_value - p2_value) > 2*options.ABS_TOL: - printing.options['dformat'] = options.DEBUG_FLOAT_FORMAT + # + if abs(p1_value - p2_value) > self.tolerance_scale(soln)*ABS_TOL: + printing.options['dformat'] = DEBUG_FLOAT_FORMAT raise GameUnsolvableException(self, soln_dict) # And we also check that the points it gave us belong to the # cone, just in case... if (p1_optimal not in self._K) or (p2_optimal not in self._K): - printing.options['dformat'] = options.DEBUG_FLOAT_FORMAT + printing.options['dformat'] = DEBUG_FLOAT_FORMAT raise GameUnsolvableException(self, soln_dict) - # For the game value, we could use any of: - # - # * p1_value - # * p2_value - # * (p1_value + p2_value)/2 - # * the game payoff - # - # We want the game value to be the payoff, however, so it - # makes the most sense to just use that, even if it means we - # can't test the fact that p1_value/p2_value are close to the - # payoff. - payoff = self.payoff(p1_optimal, p2_optimal) - return Solution(payoff, p1_optimal, p2_optimal) + return soln def condition(self): @@ -1098,13 +1182,11 @@ class SymmetricLinearGame: >>> e1 = [1] >>> e2 = e1 >>> SLG = SymmetricLinearGame(L, K, e1, e2) - >>> actual = SLG.condition() - >>> expected = 1.8090169943749477 - >>> abs(actual - expected) < options.ABS_TOL - True + >>> SLG.condition() + 1.809... """ - return (condition_number(self._G()) + condition_number(self.A()))/2 + return (condition_number(self.G()) + condition_number(self.A()))/2 def dual(self): @@ -1136,8 +1218,7 @@ class SymmetricLinearGame: [ 3], e2 = [ 1] [ 1] - [ 1], - Condition((L, K, e1, e2)) = 44.476... + [ 1] """ # We pass ``self.L()`` right back into the constructor, because