import warnings
from typing import Tuple
try:
import cvxpy as cp
cvxpy_flag = True
except ImportError:
cvxpy_flag = False
pass
import numpy as np
from scipy.linalg import cho_factor
from sklearn.exceptions import ConvergenceWarning
from ..utils import get_regularization
from ..utils import reorder_constraints
from .sr3 import SR3
[docs]class ConstrainedSR3(SR3):
"""
Sparse relaxed regularized regression with linear (in)equality constraints.
Attempts to minimize the objective function
.. math::
0.5\\|y-Xw\\|^2_2 + \\lambda R(u)
+ (0.5 / \\nu)\\|w-u\\|^2_2
.. math::
\\text{subject to } Cw = d
over u and w, where :math:`R(u)` is a regularization function, C is a
constraint matrix, and d is a vector of values. See the following
reference for more details:
Champion, Kathleen, et al. "A unified sparse optimization framework
to learn parsimonious physics-informed models from data."
IEEE Access 8 (2020): 169259-169271.
Zheng, Peng, et al. "A unified framework for sparse relaxed
regularized regression: Sr3." IEEE Access 7 (2018): 1404-1423.
Parameters
----------
threshold : float, optional (default 0.1)
Determines the strength of the regularization. When the
regularization function R is the l0 norm, the regularization
is equivalent to performing hard thresholding, and lambda
is chosen to threshold at the value given by this parameter.
This is equivalent to choosing lambda = threshold^2 / (2 * nu).
nu : float, optional (default 1)
Determines the level of relaxation. Decreasing nu encourages
w and v to be close, whereas increasing nu allows the
regularized coefficients v to be farther from w.
tol : float, optional (default 1e-5)
Tolerance used for determining convergence of the optimization
algorithm.
thresholder : string, optional (default 'l0')
Regularization function to use. Currently implemented options
are 'l0' (l0 norm), 'l1' (l1 norm), 'l2' (l2 norm), 'cad' (clipped
absolute deviation), 'weighted_l0' (weighted l0 norm),
'weighted_l1' (weighted l1 norm), and 'weighted_l2' (weighted l2 norm).
max_iter : int, optional (default 30)
Maximum iterations of the optimization algorithm.
constraint_lhs : numpy ndarray, optional (default None)
Shape should be (n_constraints, n_features * n_targets),
The left hand side matrix C of Cw <= d.
There should be one row per constraint.
constraint_rhs : numpy ndarray, shape (n_constraints,), optional (default None)
The right hand side vector d of Cw <= d.
constraint_order : string, optional (default "target")
The format in which the constraints ``constraint_lhs`` were passed.
Must be one of "target" or "feature".
"target" indicates that the constraints are grouped by target:
i.e. the first ``n_features`` columns
correspond to constraint coefficients on the library features
for the first target (variable), the next ``n_features`` columns to
the library features for the second target (variable), and so on.
"feature" indicates that the constraints are grouped by library
feature: the first ``n_targets`` columns correspond to the first
library feature, the next ``n_targets`` columns to the second library
feature, and so on.
normalize_columns : boolean, optional (default False)
Normalize the columns of x (the SINDy library terms) before regression
by dividing by the L2-norm. Note that the 'normalize' option in sklearn
is deprecated in sklearn versions >= 1.0 and will be removed. Note that
this parameter is incompatible with the constraints!
initial_guess : np.ndarray, optional (default None)
Shape should be (n_features) or (n_targets, n_features).
Initial guess for coefficients ``coef_``, (v in the mathematical equations)
If None, least-squares is used to obtain an initial guess.
thresholds : np.ndarray, shape (n_targets, n_features), optional (default None)
Array of thresholds for each library function coefficient.
Each row corresponds to a measurement variable and each column
to a function from the feature library.
Recall that SINDy seeks a matrix :math:`\\Xi` such that
:math:`\\dot{X} \\approx \\Theta(X)\\Xi`.
``thresholds[i, j]`` should specify the threshold to be used for the
(j + 1, i + 1) entry of :math:`\\Xi`. That is to say it should give the
threshold to be used for the (j + 1)st library function in the equation
for the (i + 1)st measurement variable.
inequality_constraints : bool, optional (default False)
If True, CVXPY methods are used to solve the problem.
verbose : bool, optional (default False)
If True, prints out the different error terms every
max_iter / 10 iterations.
verbose_cvxpy : bool, optional (default False)
Boolean flag which is passed to CVXPY solve function to indicate if
output should be verbose or not. Only relevant for optimizers that
use the CVXPY package in some capabity.
unbias: bool (default False)
See base class for definition. Most options are incompatible
with unbiasing.
Attributes
----------
coef_ : array, shape (n_features,) or (n_targets, n_features)
Regularized weight vector(s). This is the v in the objective
function.
coef_full_ : array, shape (n_features,) or (n_targets, n_features)
Weight vector(s) that are not subjected to the regularization.
This is the w in the objective function.
history_ : list
History of sparse coefficients. ``history_[k]`` contains the
sparse coefficients (v in the optimization objective function)
at iteration k.
objective_history_ : list
History of the value of the objective at each step. Note that
the trapping SINDy problem is nonconvex, meaning that this value
may increase and decrease as the algorithm works.
"""
def __init__(
self,
threshold=0.1,
nu=1.0,
tol=1e-5,
thresholder="l0",
max_iter=30,
trimming_fraction=0.0,
trimming_step_size=1.0,
constraint_lhs=None,
constraint_rhs=None,
constraint_order="target",
normalize_columns=False,
copy_X=True,
initial_guess=None,
thresholds=None,
equality_constraints=False,
inequality_constraints=False,
constraint_separation_index=0,
verbose=False,
verbose_cvxpy=False,
unbias=False,
):
super().__init__(
threshold=threshold,
nu=nu,
tol=tol,
thresholder=thresholder,
thresholds=thresholds,
trimming_fraction=trimming_fraction,
trimming_step_size=trimming_step_size,
max_iter=max_iter,
initial_guess=initial_guess,
copy_X=copy_X,
normalize_columns=normalize_columns,
verbose=verbose,
unbias=unbias,
)
self.verbose_cvxpy = verbose_cvxpy
self.reg = get_regularization(thresholder)
self.constraint_lhs = constraint_lhs
self.constraint_rhs = constraint_rhs
self.constraint_order = constraint_order
self.use_constraints = (constraint_lhs is not None) and (
constraint_rhs is not None
)
if (
self.use_constraints
and not equality_constraints
and not inequality_constraints
):
warnings.warn(
"constraint_lhs and constraint_rhs passed to the optimizer, "
" but user did not specify if the constraints were equality or"
" inequality constraints. Assuming equality constraints."
)
self.equality_constraints = True
if self.use_constraints:
if constraint_order not in ("feature", "target"):
raise ValueError(
"constraint_order must be either 'feature' or 'target'"
)
if unbias:
raise ValueError(
"Constraints are incompatible with an unbiasing step. Set"
" unbias=False"
)
if inequality_constraints and not cvxpy_flag:
raise ValueError(
"Cannot use inequality constraints without cvxpy installed."
)
if inequality_constraints and not self.use_constraints:
raise ValueError(
"Use of inequality constraints requires constraint_lhs and "
"constraint_rhs."
)
if inequality_constraints and thresholder.lower() not in (
"l1",
"l2",
"weighted_l1",
"weighted_l2",
):
raise ValueError(
"Use of inequality constraints requires a convex regularizer."
)
self.inequality_constraints = inequality_constraints
self.equality_constraints = equality_constraints
self.constraint_separation_index = constraint_separation_index
def _update_full_coef_constraints(self, H, x_transpose_y, coef_sparse):
g = x_transpose_y + coef_sparse / self.nu
inv1 = np.linalg.inv(H)
inv1_mod = np.kron(inv1, np.eye(coef_sparse.shape[1]))
inv2 = np.linalg.inv(
self.constraint_lhs.dot(inv1_mod).dot(self.constraint_lhs.T)
)
rhs = g.flatten() + self.constraint_lhs.T.dot(inv2).dot(
self.constraint_rhs - self.constraint_lhs.dot(inv1_mod).dot(g.flatten())
)
rhs = rhs.reshape(g.shape)
return inv1.dot(rhs)
def _create_var_and_part_cost(
self, var_len: int, x_expanded: np.ndarray, y: np.ndarray
) -> Tuple[cp.Variable, cp.Expression]:
xi = cp.Variable(var_len)
cost = cp.sum_squares(x_expanded @ xi - y.flatten())
if self.thresholder.lower() == "l1":
cost = cost + self.threshold * cp.norm1(xi)
elif self.thresholder.lower() == "weighted_l1":
cost = cost + cp.norm1(np.ravel(self.thresholds) @ xi)
elif self.thresholder.lower() == "l2":
cost = cost + self.threshold * cp.norm2(xi) ** 2
elif self.thresholder.lower() == "weighted_l2":
cost = cost + cp.norm2(np.ravel(self.thresholds) @ xi) ** 2
return xi, cost
def _update_coef_cvxpy(self, xi, cost, var_len, coef_prev, tol):
if self.use_constraints:
if self.inequality_constraints and self.equality_constraints:
# Process inequality constraints then equality constraints
prob = cp.Problem(
cp.Minimize(cost),
[
self.constraint_lhs[: self.constraint_separation_index, :] @ xi
<= self.constraint_rhs[: self.constraint_separation_index],
self.constraint_lhs[self.constraint_separation_index :, :] @ xi
== self.constraint_rhs[self.constraint_separation_index :],
],
)
elif self.inequality_constraints:
prob = cp.Problem(
cp.Minimize(cost),
[self.constraint_lhs @ xi <= self.constraint_rhs],
)
else:
prob = cp.Problem(
cp.Minimize(cost),
[self.constraint_lhs @ xi == self.constraint_rhs],
)
else:
prob = cp.Problem(cp.Minimize(cost))
# default solver is SCS/OSQP here but switches to ECOS for L2
try:
prob.solve(
max_iter=self.max_iter,
eps_abs=tol,
eps_rel=tol,
verbose=self.verbose_cvxpy,
)
# Annoying error coming from L2 norm switching to use the ECOS
# solver, which uses "max_iters" instead of "max_iter", and
# similar semantic changes for the other variables.
except (TypeError, ValueError):
try:
prob.solve(max_iters=self.max_iter, verbose=self.verbose_cvxpy)
except cp.error.SolverError:
print("Solver failed, setting coefs to zeros")
xi.value = np.zeros(var_len)
except cp.error.SolverError:
print("Solver failed, setting coefs to zeros")
xi.value = np.zeros(var_len)
if xi.value is None:
warnings.warn(
"Infeasible solve, probably an issue with the regularizer "
" or the constraint that was used.",
ConvergenceWarning,
)
return None
coef_new = (xi.value).reshape(coef_prev.shape)
return coef_new
def _update_sparse_coef(self, coef_full):
"""Update the regularized weight vector"""
if self.thresholds is None:
return super(ConstrainedSR3, self)._update_sparse_coef(coef_full)
else:
coef_sparse = self.prox(coef_full, self.thresholds.T)
self.history_.append(coef_sparse.T)
return coef_sparse
def _objective(self, x, y, q, coef_full, coef_sparse, trimming_array=None):
"""Objective function"""
if q != 0:
print_ind = q % (self.max_iter // 10.0)
else:
print_ind = q
R2 = (y - np.dot(x, coef_full)) ** 2
D2 = (coef_full - coef_sparse) ** 2
if self.use_trimming:
assert trimming_array is not None
R2 *= trimming_array.reshape(x.shape[0], 1)
if self.thresholds is None:
regularization = self.reg(coef_full, self.threshold**2 / self.nu)
if print_ind == 0 and self.verbose:
row = [
q,
np.sum(R2),
np.sum(D2) / self.nu,
regularization,
np.sum(R2) + np.sum(D2) + regularization,
]
print(
"{0:10d} ... {1:10.4e} ... {2:10.4e} ... {3:10.4e}"
" ... {4:10.4e}".format(*row)
)
return 0.5 * np.sum(R2) + 0.5 * regularization + 0.5 * np.sum(D2) / self.nu
else:
regularization = self.reg(coef_full, self.thresholds**2 / self.nu)
if print_ind == 0 and self.verbose:
row = [
q,
np.sum(R2),
np.sum(D2) / self.nu,
regularization,
np.sum(R2) + np.sum(D2) + regularization,
]
print(
"{0:10d} ... {1:10.4e} ... {2:10.4e} ... {3:10.4e}"
" ... {4:10.4e}".format(*row)
)
return 0.5 * np.sum(R2) + 0.5 * regularization + 0.5 * np.sum(D2) / self.nu
def _reduce(self, x, y):
"""
Perform at most ``self.max_iter`` iterations of the SR3 algorithm
with inequality constraints.
Assumes initial guess for coefficients is stored in ``self.coef_``.
"""
if self.initial_guess is not None:
self.coef_ = self.initial_guess
coef_sparse = self.coef_.T
coef_full = coef_sparse.copy()
n_samples, n_features = x.shape
n_targets = y.shape[1]
if self.use_trimming:
trimming_array = np.repeat(1.0 - self.trimming_fraction, n_samples)
self.history_trimming_ = [trimming_array]
if self.use_constraints and self.constraint_order.lower() == "target":
self.constraint_lhs = reorder_constraints(self.constraint_lhs, n_features)
# Precompute some objects for upcoming least-squares solves.
# Assumes that self.nu is fixed throughout optimization procedure.
H = np.dot(x.T, x) + np.diag(np.full(x.shape[1], 1.0 / self.nu))
x_transpose_y = np.dot(x.T, y)
if not self.use_constraints:
cho = cho_factor(H)
if self.inequality_constraints:
# Precompute some objects for optimization
x_expanded = np.zeros((n_samples, n_targets, n_features, n_targets))
for i in range(n_targets):
x_expanded[:, i, :, i] = x
x_expanded = np.reshape(
x_expanded, (n_samples * n_targets, n_targets * n_features)
)
# Print initial values for each term in the optimization
if self.verbose:
row = [
"Iteration",
"|y - Xw|^2",
"|w-u|^2/v",
"R(u)",
"Total Error: |y - Xw|^2 + |w - u|^2 / v + R(u)",
]
print(
"{: >10} ... {: >10} ... {: >10} ... {: >10} ... {: >10}".format(*row)
)
objective_history = []
if self.inequality_constraints:
var_len = coef_sparse.shape[0] * coef_sparse.shape[1]
xi, cost = self._create_var_and_part_cost(var_len, x_expanded, y)
coef_sparse = self._update_coef_cvxpy(
xi, cost, var_len, coef_sparse, self.tol
)
objective_history.append(self._objective(x, y, 0, coef_full, coef_sparse))
else:
for k in range(self.max_iter):
if self.use_trimming:
x_weighted = x * trimming_array.reshape(n_samples, 1)
H = np.dot(x_weighted.T, x) + np.diag(
np.full(x.shape[1], 1.0 / self.nu)
)
x_transpose_y = np.dot(x_weighted.T, y)
if not self.use_constraints:
cho = cho_factor(H)
trimming_grad = 0.5 * np.sum((y - x.dot(coef_full)) ** 2, axis=1)
if self.use_constraints:
coef_full = self._update_full_coef_constraints(
H, x_transpose_y, coef_sparse
)
else:
coef_full = self._update_full_coef(cho, x_transpose_y, coef_sparse)
coef_sparse = self._update_sparse_coef(coef_full)
self.history_.append(np.copy(coef_sparse).T)
if self.use_trimming:
trimming_array = self._update_trimming_array(
coef_full, trimming_array, trimming_grad
)
objective_history.append(
self._objective(x, y, k, coef_full, coef_sparse, trimming_array)
)
else:
objective_history.append(
self._objective(x, y, k, coef_full, coef_sparse)
)
if self._convergence_criterion() < self.tol:
# TODO: Update this for trimming/constraints
break
else:
warnings.warn(
f"ConstrainedSR3 did not converge after {self.max_iter}"
" iterations.",
ConvergenceWarning,
)
if self.use_constraints and self.constraint_order.lower() == "target":
self.constraint_lhs = reorder_constraints(
self.constraint_lhs, n_features, output_order="target"
)
self.coef_ = coef_sparse.T
self.coef_full_ = coef_full.T
if self.use_trimming:
self.trimming_array = trimming_array
self.objective_history = objective_history