def solve(self, sample, c_value, kernel):
r"""
Solve the variable-quality SVM classification optimization problem
corresponding to the supplied sample, according to specified value for
tradeoff constant `C` and kernel `k`.
INPUT:
- ``sample`` -- list or tuple of ``AccuracyExample`` instances whose
labels are all set either to `1` or `-1`.
- ``c_value`` -- float or None (the former choice selects the
soft-margin version of the algorithm) value for the tradeoff constant
`C`.
- ``kernel`` -- ``Kernel`` instance defining the kernel to be used.
OUTPUT:
list of float values -- optimal values for the optimization problem.
EXAMPLES:
Consider the following representation of the AND binary function, and a
default instantiation for ``CVXOPTVQClassificationSolver``:
::
>>> from yaplf.data import LabeledExample, AccuracyExample
>>> and_sample = [AccuracyExample(LabeledExample((1, 1), 1), 0),
... AccuracyExample(LabeledExample((0, 0), -1), 0),
... AccuracyExample(LabeledExample((0, 1), -1), 1),
... AccuracyExample(LabeledExample((1, 0), -1), 0)]
>>> from yaplf.algorithms.svm.solvers \
... import CVXOPTVQClassificationSolver
>>> s = CVXOPTVQClassificationSolver()
Once the solver instance is available, it is possible to invoke its
``solve``function, specifying a labeled sample such as ``and_sample``,
a positive value for the constant `c_value` and a kernel instance in
order to get the solution of the corresponding SV classification
optimization problem:
::
>>> from yaplf.models.kernel import LinearKernel
>>> s.solve(and_sample, 2, LinearKernel())
[2, 0.0, 2, 0.0]
The value for `c_value` can be set to ``None``, in order to build and
solve the original optimization problem rather than the soft-margin
formulation; analogously, a different kernel can be used as argument to
the solver:
::
>>> from yaplf.models.kernel import PolynomialKernel
>>> s.solve(and_sample, None, PolynomialKernel(3))
[0.15135135150351597, 0.0, 0.097297297016552056,
0.054054053943170456]
Note however that this class should never be used directly. It is
automatically used by ``SVMVQClassificationAlgorithm``.
AUTHORS:
- Dario Malchiodi (2010-04-12)
"""
# cvxopt solves the problem
# min 1/2 x' Q x + p' x
# subject to G x >= h and A x = b
# dict below is mapped to the above symbols as follows:
# problem["obj_quad"] -> Q
# problem["obj_lin"] -> p
# problem["ineq_coeff"] -> G
# problem["ineq_const"] -> h
# problem["eq_coeff"] -> A
# problem["eq_const"] -> b
solvers.options['show_progress'] = self.verbose
solvers.options['maxiters'] = self.max_iterations
solvers.options['solver'] = self.solver
# coercion to float in the following assignment is required
# in order to work with sage notebook
num_examples = len(sample)
problem = {}
problem["obj_quad"] = cvxopt_matrix([[ \
float(elem_i.example.label * elem_j.example.label * \
(kernel.compute(elem_i.example.pattern, elem_j.example.pattern) - \
elem_i.example.label * elem_j.example.label * (elem_i.accuracy + \
elem_j.accuracy))) for elem_i in sample] for elem_j in sample])
problem["obj_lin"] = cvxopt_matrix([-1.0 for i in range(num_examples)])
if c_value is None:
problem["ineq_coeff"] = cvxopt_matrix([
[float(-1.0 * kronecker_delta(i, j))
for i in range(num_examples)] +
[float(-1.0 * sample[j].accuracy)]
for j in range(num_examples)])
problem["ineq_const"] = cvxopt_matrix(
[float(0.0)] * num_examples + [float(1.0 - self.epsilon)])
else:
problem["ineq_coeff"] = cvxopt_matrix([
[float(-1.0 * kronecker_delta(i, j))
for i in range(num_examples)] +
[float(kronecker_delta(i, j))
for i in range(num_examples)] +
[float(-1.0 * sample[j].accuracy)]
for j in range(num_examples)])
problem["ineq_const"] = cvxopt_matrix([float(0.0)] * num_examples +
[float(c_value)] * num_examples + [float(1.0 - self.epsilon)])
problem["eq_coeff"] = cvxopt_matrix([float(elem.example.label)
for elem in sample], (1, num_examples))
problem["eq_const"] = cvxopt_matrix(0.0)
sol = solvers.qp(problem["obj_quad"], problem["obj_lin"],
problem["ineq_coeff"], problem["ineq_const"],
problem["eq_coeff"], problem["eq_const"])
return [chop(x, right=c_value) for x in list(sol['x'])]
def solve(self, sample, c=float("inf"), kernel=LinearKernel()):
r"""
Solve the SVM classification optimization problem corresponding
to the supplied sample, according to specified value for the tradeoff
constant `C`.
INPUT:
- ``sample`` -- list or tuple of ``LabeledExample`` instances whose
labels are all set either to `1` or `-1`.
- ``c`` -- float or ``float('inf')`` (the former choice selects the
soft-margin version of the algorithm) value for the tradeoff constant
`C`.
- ``kernel`` -- ``Kernel`` instance defining the kernel to be used
(default: ``LinearKernel()``, accounting for a linear kernel).
OUTPUT:
list of float values -- optimal values for the optimization problem.
EXAMPLES:
Consider the following representation of the AND binary function, and a
default instantiation for ``NEOSClassificationSolver``:
::
>>> from yaplf.data import LabeledExample
>>> and_sample = [LabeledExample((1, 1), 1),
... LabeledExample((0, 0), -1), LabeledExample((0, 1), -1),
... LabeledExample((1, 0), -1)]
>>> from yaplf.algorithms.svm.classification.solvers \
... import NEOSClassificationSolver
>>> s = NEOSClassificationSolver()
Once the solver instance is available, it is possible to invoke its
``solve``function, specifying a labeled sample such as ``and_sample``,
a positive value for the constant `C` and a kernel instance in order to
get the solution of the corresponding SV classification optimization
problem:
::
>>> from yaplf.models.kernel import LinearKernel
>>> s.solve(and_sample, 2, LinearKernel())
[2, 0, 1.0, 1.0]
The value for `C` can be set to ``float('inf')``, in order to build
and solve the original optimization problem rather than the
soft-margin formulation:
::
>>> s.solve(and_sample, float('inf'), LinearKernel())
[4.0, 0, 2.0, 2.0]
Note however that this class should never be used directly. It is
automatically used by ``SVMClassificationAlgorithm``.
AUTHORS:
- Dario Malchiodi (2011-02-05)
"""
neos = xmlrpclib.Server("http://%s:%d" % ("www.neos-server.org", 3332))
num_examples = len(sample)
input_dimension = len(sample[0].pattern)
constraint = " <= " + str(c) if c != float("inf") else ""
kernel_description = AMPLKernelFactory(kernel).get_kernel_description()
# that is, something like sum{k in 1..n}(x[i,k]*x[j,k])
pattern_description = ["param x:\t"]
label_description = ["param y:=\n"]
for component_index in range(input_dimension):
pattern_description.append(str(component_index + 1))
pattern_description.append("\t")
pattern_description.append(":=\n")
example_number = 1
for example in sample:
pattern_description.append(str(example_number))
for component in example.pattern:
pattern_description.append("\t")
pattern_description.append(str(component))
label_description.append(str(example_number))
label_description.append("\t")
label_description.append(str(sample[example_number - 1].label))
example_number = example_number + 1
if example_number > len(sample):
pattern_description.append(";")
label_description.append(";")
pattern_description.append("\n")
label_description.append("\n")
xml = """
<document>
<category>nco</category>
<solver>SNOPT</solver>
<inputMethod>AMPL</inputMethod>
<model><![CDATA[
param m integer > 0 default %d; # number of sample points
param n integer > 0 default %d; # sample space dimension
param x {1..m,1..n}; # sample points
param y {1..m}; # sample labels
param dot{i in 1..m,j in 1..m}:=%s;
var alpha{1..m} >=0%s;
maximize quadratic_form:
sum{i in 1..m} alpha[i]
-1/2*sum{i in 1..m,j in 1..m}alpha[i]*alpha[j]*y[i]*y[j]*dot[i,j];
subject to linear_constraint:
sum{i in 1..m} alpha[i]*y[i]=0;
]]></model>
<data><![CDATA[
data;
%s
%s
]]></data>
<commands><![CDATA[
option solver snopt;
solve;
printf: "(";
printf {i in 1..m-1}:"%%f,",alpha[i];
printf: "%%f)",alpha[m];
]]></commands>
</document>
""" % (
num_examples,
input_dimension,
kernel_description,
constraint,
"".join(pattern_description),
"".join(label_description),
)
(job_number, password) = neos.submitJob(xml)
if self.verbose:
print xml
print "job number: %s" % job_number
offset = 0
status = ""
while status != "Done":
(msg, offset) = neos.getIntermediateResults(job_number, password, offset)
if self.verbose:
print msg.data
status = neos.getJobStatus(job_number, password)
msg = neos.getFinalResults(job_number, password).data
if self.verbose:
print msg
begin = 0
while msg[begin] != "(":
begin = begin + 1
end = len(msg) - 1
while msg[end] != ")":
end = end - 1
return [chop(alpha, right=c) for alpha in eval(msg[begin : end + 1])]
def solve(self, sample, c=float('inf'), kernel=LinearKernel(),
tolerance=1e-6):
r"""
Solve the SVM classification optimization problem corresponding
to the supplied sample, according to specified value for the tradeoff
constant `C`.
INPUT:
- ``sample`` -- list or tuple of ``LabeledExample`` instances whose
labels are all set either to `1` or `-1`.
- ``c`` -- float value for the tradeoff constant `C`.
``float('inf')`` selects the soft-margin version of the algorithm)
- ``kernel`` -- ``Kernel`` instance defining the kernel to be used.
- ``tolerance`` -- tolerance to be used when clipping values to the
extremes of an interval.
OUTPUT:
list of float values -- optimal values for the optimization problem.
EXAMPLES:
Consider the following representation of the AND binary function, and a
default instantiation for ``GurobiClassificationSolver``:
::
>>> from yaplf.data import LabeledExample
>>> and_sample = [LabeledExample((1, 1), 1),
... LabeledExample((0, 0), -1), LabeledExample((0, 1), -1),
... LabeledExample((1, 0), -1)]
>>> from yaplf.algorithms.svm.classification.solvers \
... import GurobiClassificationSolver
>>> s = GurobiClassificationSolver()
Once the solver instance is available, it is possible to invoke its
``solve`` function, specifying a labeled sample such as ``and_sample``,
a positive value for the constant `C` and a kernel instance in order to
get the solution of the corresponding SV classification optimization
problem:
::
>>> from yaplf.models.kernel import LinearKernel
>>> s.solve(and_sample, 2, LinearKernel())
[2, 0, 0.999999999992222, 0.999999999992222]
The value for `C` can be set to ``float('inf')`` (which is also its
default value), in order to build and solve the original optimization
problem rather than the soft-margin formulation:
::
>>> s.solve(and_sample, float('inf'), LinearKernel())
[4.00000000000204, 0, 1.999999999976717, 1.99999999997672]
Note however that this class should never be used directly. It is
automatically used by ``SVMClassificationAlgorithm``.
AUTHORS:
- Dario Malchiodi (2014-03-03)
"""
m = len(sample)
patterns = [e.example.pattern for e in sample]
labels = [e.example.label for e in sample]
accuracy = [e.accuracy for e in sample]
model = gurobipy.Model('classify')
for i in range(m):
if c == float('inf'):
model.addVar(name='alpha_%d' % i, lb=0,
vtype=gurobipy.GRB.CONTINUOUS)
else:
model.addVar(name='alpha_%d' % i, lb=0, ub=c,
vtype=gurobipy.GRB.CONTINUOUS)
model.update()
alphas = model.getVars()
obj = gurobipy.QuadExpr() + sum(alphas)
map(lambda (i, j):
obj.add(alphas[i] * alphas[j] * labels[i] * labels[j],
-0.5*(kernel.compute(patterns[i], patterns[j]) - labels[i] * labels[j] * (accuracy[i]+accuracy[j]))),
[(i, j) for i in xrange(m) for j in xrange(m)])
model.setObjective(obj, gurobipy.GRB.MAXIMIZE)
constEqual = gurobipy.LinExpr()
# map(lambda x: constEqual.add(x, 1.0),
# [a*l for a, l in zip(alphas, labels)])
# model.addConstr(constEqual, gurobipy.GRB.EQUAL, 0)
constGreater = gurobipy.LinExpr(1 + tolerance)
for addend in [a*l for a, l in zip(alphas, labels)]:
constEqual.add(addend, 1.0)
constGreater.add(addend, 1.0)
model.addConstr(constEqual, gurobipy.GRB.EQUAL, 0)
model.addConstr(constGreater, gurobipy.GRB.GREATER_EQUAL, tolerance)
if not self.verbose:
model.setParam('OutputFlag', False)
model.optimize()
alphas_opt = [chop(a.x, right=c, tolerance=tolerance) for a in alphas]
return alphas_opt
def solve(self, sample, c=float("inf"), kernel=LinearKernel()):
r"""
Solve the SVM classification optimization problem corresponding
to the supplied sample, according to specified value for the tradeoff
constant `C`.
INPUT:
- ``sample`` -- list or tuple of ``LabeledExample`` instances whose
labels are all set either to `1` or `-1`.
- ``c`` -- float or ``float('inf')`` (the former choice selects the
soft-margin version of the algorithm) value for the tradeoff constant
`C`.
- ``kernel`` -- ``Kernel`` instance defining the kernel to be used
(default value: ``LinearKernel()``, using a linear kernel)
OUTPUT:
list of float values -- optimal values for the optimization problem.
EXAMPLES:
Consider the following representation of the AND binary function, and a
default instantiation for ``CVXOPTClassificationSolver``:
::
>>> from yaplf.data import LabeledExample
>>> and_sample = [LabeledExample((1, 1), 1),
... LabeledExample((0, 0), -1), LabeledExample((0, 1), -1),
... LabeledExample((1, 0), -1)]
>>> from yaplf.algorithms.svm.classification.solvers \
... import CVXOPTClassificationSolver
>>> s = CVXOPTClassificationSolver()
Once the solver instance is available, it is possible to invoke its
``solve``function, specifying a labeled sample such as ``and_sample``,
a positive value for the constant `C` and a kernel instance in order to
get the solution of the corresponding SV classification optimization
problem:
::
>>> from yaplf.models.kernel import LinearKernel
>>> s.solve(and_sample, 2, LinearKernel())
[2, 0, 0.9999998669645057, 0.9999998669645057]
The value for `C` can be set to ``float('inf')``, in order to build
and solve the original optimization problem rather than the
soft-margin formulation:
::
>>> s.solve(and_sample, float('inf'), LinearKernel())
[4.000001003300218, 0, 2.000000364577095, 2.000000364577095]
Note however that this class should never be used directly. It is
automatically used by ``SVMClassificationAlgorithm``.
AUTHORS:
- Dario Malchiodi (2010-02-22)
"""
solvers.options["show_progress"] = self.verbose
solvers.options["maxiters"] = self.max_iterations
solvers.options["solver"] = self.solver
# cvxopt solves the problem
# min 1/2 x' Q x + p' x
# subject to G x >= h and A x = b
# dict below is mapped to the above symbols as follows:
# problem["obj_quad"] -> Q
# problem["obj_lin"] -> p
# problem["ineq_coeff"] -> G
# problem["ineq_const"] -> h
# problem["eq_coeff"] -> A
# problem["eq_const"] -> b
num_examples = len(sample)
problem = {}
problem["obj_quad"] = cvxopt_matrix(
[
[elem_i.label * elem_j.label * kernel.compute(elem_i.pattern, elem_j.pattern) for elem_i in sample]
for elem_j in sample
]
)
problem["obj_lin"] = cvxopt_matrix([-1.0] * num_examples)
if c == float("inf"):
problem["ineq_coeff"] = cvxopt_matrix(-1.0 * eye(num_examples))
problem["ineq_const"] = cvxopt_matrix([0.0] * num_examples)
else:
problem["ineq_coeff"] = cvxopt_matrix(
[
[-1.0 * kronecker_delta(i, j) for i in range(num_examples)]
+ [kronecker_delta(i, j) for i in range(num_examples)]
for j in range(num_examples)
]
)
problem["ineq_const"] = cvxopt_matrix([float(0.0)] * num_examples + [float(c)] * num_examples)
# coercion to float in the following assignment is required
# in order to work with sage notebooks
problem["eq_coeff"] = cvxopt_matrix([float(elem.label) for elem in sample], (1, num_examples))
problem["eq_const"] = cvxopt_matrix(0.0)
# was
# sol = solvers.qp(quad_coeff, lin_coeff, ineq_coeff, ineq_const, \
# eq_coeff, eq_const)
sol = solvers.qp(
problem["obj_quad"],
problem["obj_lin"],
problem["ineq_coeff"],
problem["ineq_const"],
problem["eq_coeff"],
problem["eq_const"],
)
if sol["status"] != "optimal":
raise ValueError("cvxopt returned status " + sol["status"])
# was
# alpha = map(lambda x: chop(x, right = c), list(sol['x']))
alpha = [chop(x, right=c) for x in list(sol["x"])]
return alpha