def solve_sqrt_lasso_skinny(X,
Y,
weights=None,
initial=None,
quadratic=None,
solve_args={}):
"""
Solve the square-root LASSO optimization problem:
$$
\text{minimize}_{\beta} \|y-X\beta\|_2 + D |\beta|,
$$
where $D$ is the diagonal matrix with weights on its diagonal.
Parameters
----------
y : np.float((n,))
The target, in the model $y = X\beta$
X : np.float((n, p))
The data, in the model $y = X\beta$
weights : np.float
Coefficients of the L-1 penalty in
optimization problem, note that different
coordinates can have different coefficients.
initial : np.float(p)
Initial point for optimization.
solve_args : dict
Arguments passed to regreg solver.
quadratic : `regreg.identity_quadratic`
A quadratic term added to objective function.
"""
X = rr.astransform(X)
n, p = X.output_shape[0], X.input_shape[0]
if weights is None:
lam = choose_lambda(X)
weights = lam * np.ones((p, ))
weight_dict = dict(zip(np.arange(p), 2 * weights))
penalty = rr.mixed_lasso(range(p) + [rr.NONNEGATIVE],
lagrange=1.,
weights=weight_dict)
loss = sqlasso_objective_skinny(X, Y)
problem = rr.simple_problem(loss, penalty)
problem.coefs[-1] = np.linalg.norm(Y)
if initial is not None:
problem.coefs[:-1] = initial
soln = problem.solve(quadratic, **solve_args)
_loss = sqlasso_objective(X, Y)
return soln[:-1], _loss
def test_path_group_lasso():
'''
this test looks at the paths of three different parameterizations
of the same problem
'''
n = 100
X = np.random.standard_normal((n, 10))
U = np.random.standard_normal((n, 2))
Y = np.random.standard_normal(100)
betaX = np.array([3, 4, 5, 0, 0] + [0] * 5)
betaU = np.array([10, -5])
Y += (np.dot(X, betaX) + np.dot(U, betaU)) * 5
Xn = rr.normalize(np.hstack([np.ones((100, 1)), X]),
inplace=True,
center=True,
scale=True,
intercept_column=0).normalized_array()
lasso = mixed_lasso.mixed_lasso_path.gaussian(Xn[:, 1:],
Y,
penalty_structure=[0] * 7 +
[1] * 3,
nstep=10)
sol = lasso.main(inner_tol=1.e-12, verbose=True)
beta = np.array(sol['beta'].todense())
sols = []
sols_sep = []
for l in sol['lagrange']:
loss = rr.glm.gaussian(Xn, Y)
penalty = rr.mixed_lasso([mixed_lasso.UNPENALIZED] + [0] * 7 + [1] * 3,
lagrange=l) # matrix contains an intercept...
problem = rr.simple_problem(loss, penalty)
sols.append(problem.solve(tol=1.e-12).copy())
sep = rr.separable((11, ), [
rr.l2norm((7, ),
np.sqrt(7) * l),
rr.l2norm((3, ),
np.sqrt(3) * l)
], [np.arange(1, 8), np.arange(8, 11)])
sep_problem = rr.simple_problem(loss, sep)
sols_sep.append(sep_problem.solve(tol=1.e-12).copy())
sols = np.array(sols).T
sols_sep = np.array(sols_sep).T
nt.assert_true(
np.linalg.norm(beta - sols) / (1 + np.linalg.norm(beta)) <= 1.e-4)
nt.assert_true(
np.linalg.norm(beta - sols_sep) / (1 + np.linalg.norm(beta)) <= 1.e-4)
def solve_sqrt_lasso_skinny(X, Y, weights=None, initial=None, quadratic=None, solve_args={}):
"""
Solve the square-root LASSO optimization problem:
$$
\text{minimize}_{\beta} \|y-X\beta\|_2 + D |\beta|,
$$
where $D$ is the diagonal matrix with weights on its diagonal.
Parameters
----------
y : np.float((n,))
The target, in the model $y = X\beta$
X : np.float((n, p))
The data, in the model $y = X\beta$
weights : np.float
Coefficients of the L-1 penalty in
optimization problem, note that different
coordinates can have different coefficients.
initial : np.float(p)
Initial point for optimization.
solve_args : dict
Arguments passed to regreg solver.
quadratic : `regreg.identity_quadratic`
A quadratic term added to objective function.
"""
n, p = X.shape
if weights is None:
lam = choose_lambda(X)
weights = lam * np.ones((p,))
weight_dict = dict(zip(np.arange(p),
2 * weights))
penalty = rr.mixed_lasso(range(p) + [rr.NONNEGATIVE], lagrange=1.,
weights=weight_dict)
loss = sqlasso_objective_skinny(X, Y)
problem = rr.simple_problem(loss, penalty)
problem.coefs[-1] = np.linalg.norm(Y)
if initial is not None:
problem.coefs[:-1] = initial
soln = problem.solve(quadratic, **solve_args)
_loss = sqlasso_objective(X, Y)
return soln[:-1], _loss
def test_path_group_lasso():
"""
this test looks at the paths of three different parameterizations
of the same problem
"""
n = 100
X = np.random.standard_normal((n, 10))
U = np.random.standard_normal((n, 2))
Y = np.random.standard_normal(100)
betaX = np.array([3, 4, 5, 0, 0] + [0] * 5)
betaU = np.array([10, -5])
Y += (np.dot(X, betaX) + np.dot(U, betaU)) * 5
Xn = rr.normalize(
np.hstack([np.ones((100, 1)), X]), inplace=True, center=True, scale=True, intercept_column=0
).normalized_array()
lasso = rr.lasso.squared_error(Xn[:, 1:], Y, penalty_structure=[0] * 7 + [1] * 3, nstep=10)
sol = lasso.main(inner_tol=1.0e-12, verbose=True)
beta = np.array(sol["beta"].todense())
sols = []
sols_sep = []
for l in sol["lagrange"]:
loss = rr.squared_error(Xn, Y, coef=1.0 / n)
penalty = rr.mixed_lasso([rr.UNPENALIZED] + [0] * 7 + [1] * 3, lagrange=l) # matrix contains an intercept...
problem = rr.simple_problem(loss, penalty)
sols.append(problem.solve(tol=1.0e-12).copy())
sep = rr.separable(
(11,),
[rr.l2norm((7,), np.sqrt(7) * l), rr.l2norm((3,), np.sqrt(3) * l)],
[np.arange(1, 8), np.arange(8, 11)],
)
sep_problem = rr.simple_problem(loss, sep)
sols_sep.append(sep_problem.solve(tol=1.0e-12).copy())
sols = np.array(sols).T
sols_sep = np.array(sols_sep).T
nt.assert_true(np.linalg.norm(beta - sols) / (1 + np.linalg.norm(beta)) <= 1.0e-4)
nt.assert_true(np.linalg.norm(beta - sols_sep) / (1 + np.linalg.norm(beta)) <= 1.0e-4)
