from __future__ import annotations
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 reorder_constraints
from ._constrained_sr3 import ConstrainedSR3
[docs]
class StableLinearSR3(ConstrainedSR3):
"""
Sparse relaxed regularized regression for building a-priori
stable linear models. This requires making a matrix negative definite,
which can be challenging. Here we use a similar method to the
TrappingOptimizer algorithm. Linear equality and linear inequality
constraints are both allowed, as in the ConstrainedSR3 optimizer.
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 } Cu = d, Du = e, w negative definite
over u and w, where :math:`R(u)` is a regularization function, C and D are
constraint matrices, and d and e are vectors of values.
NOTE: This optimizer is intended for building purely linear models that
are guaranteed to be stable.
Parameters
----------
regularizer : string, optional (default 'l1')
Regularization function to use. Currently implemented options
are 'l1' (l1 norm), 'l2' (l2 norm), 'weighted_l1' (weighted l1 norm),
and 'weighted_l2' (weighted l2 norm).
Note that the regularizer must be convex here.
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.
inequality_constraints : bool, optional (default False)
If True, CVXPY methods are used to solve the problem.
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.
See base class for additional arguments
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.
"""
def __init__(
self,
reg_weight_lam=0.1,
regularizer="l1",
relax_coeff_nu=1.0,
tol=1e-5,
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,
equality_constraints=False,
inequality_constraints=False,
constraint_separation_index=0,
verbose=False,
verbose_cvxpy=False,
gamma=-1e-8,
unbias=False,
):
super().__init__(
reg_weight_lam=reg_weight_lam,
regularizer=regularizer,
relax_coeff_nu=relax_coeff_nu,
tol=tol,
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,
verbose_cvxpy=verbose_cvxpy,
constraint_lhs=constraint_lhs,
constraint_rhs=constraint_rhs,
constraint_order=constraint_order,
equality_constraints=equality_constraints,
inequality_constraints=inequality_constraints,
constraint_separation_index=constraint_separation_index,
unbias=unbias,
)
self.gamma = gamma
self.alpha_A = relax_coeff_nu
self.max_iter = max_iter
warnings.warn(
"This optimizer is set up to only be used with a purely linear"
" library in the variables. No constant or nonlinear terms!"
)
if not np.isclose(reg_weight_lam, 0.0):
warnings.warn(
"This optimizer uses CVXPY if the reg_weight_lam is nonzero, "
" meaning the optimization will be much slower for large "
"datasets."
)
def _create_var_and_part_cost(
self,
x: np.ndarray,
y: np.ndarray,
coef_sparse: np.ndarray,
coef_neg_def: np.ndarray,
) -> Tuple["cp.Variable", "cp.Expression"]:
xi = cp.Variable(coef_sparse.shape[0] * coef_sparse.shape[1])
cost = cp.sum_squares(x @ xi - y.flatten())
cost = cost + cp.sum_squares(xi - coef_neg_def.flatten()) / (
2 * self.relax_coeff_nu
)
penalty = self._calculate_penalty(
self.regularizer, np.ravel(self.reg_weight_lam), xi
)
return xi, cost + penalty
def _update_coef_cvxpy(self, x, y, coef_sparse, coef_negative_definite):
"""
Update the coefficients using CVXPY. This function is called if
the sparsity weight is nonzero or constraints are used.
"""
xi, cost = self._create_var_and_part_cost(
x, y, coef_sparse, coef_negative_definite
)
if self.use_constraints:
if self.inequality_constraints and self.equality_constraints:
# Process equality constraints then inequality 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))
try:
prob.solve(
max_iter=self.max_iter**2,
eps_abs=self.tol,
eps_rel=self.tol,
verbose=self.verbose_cvxpy,
)
except cp.error.SolverError:
print("Solver failed, setting coefs to zeros")
xi.value = np.zeros(coef_sparse.shape[0] * coef_sparse.shape[1])
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_sparse.shape)
return coef_new
def _update_A(self, A_old, coef_sparse):
"""
Update the auxiliary variable that approximates the coefficients
(which is a matrix of linear coefficients). Taken and slightly altered
from the TrappingOptimizer code.
"""
r = A_old.shape[1]
if A_old.shape[0] == r:
eigvals, eigvecs = np.linalg.eig(A_old.T)
eigPW, eigvecsPW = np.linalg.eig(coef_sparse.T)
else:
eigvals, eigvecs = np.linalg.eig(A_old[:r, :r].T)
eigPW, eigvecsPW = np.linalg.eig(coef_sparse[:r, :r].T)
A = np.diag(eigvals)
for i in range(r):
if np.real(eigvals[i]) > self.gamma:
A[i, i] = self.gamma + np.imag(eigvals[i]) * 1j
if A_old.shape[0] == r:
return np.real(eigvecsPW @ A @ np.linalg.inv(eigvecsPW))
else:
A_temp = np.zeros(A_old.shape)
A_temp[:r, :r] = np.real(eigvecsPW @ A @ np.linalg.inv(eigvecsPW))
A_temp[r:, :r] = A_old[r:, :r]
return A_temp.T
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_negative_definite = 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 optimization
H = np.dot(x.T, x) + np.diag(np.full(x.shape[1], 1.0 / self.relax_coeff_nu))
x_transpose_y = np.dot(x.T, y)
if not self.use_constraints:
cho = cho_factor(H)
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 = []
eigs_history = []
coef_history = []
for k in range(self.max_iter):
if not np.isclose(self.reg_weight_lam, 0.0) or self.use_constraints:
coef_sparse = self._update_coef_cvxpy(
x_expanded, y, coef_sparse, coef_negative_definite
)
else:
coef_sparse = self._update_full_coef(
cho, x_transpose_y, coef_negative_definite
)
coef_negative_definite = self._update_A(
coef_negative_definite
- self.alpha_A
* (coef_negative_definite - coef_sparse)
/ self.relax_coeff_nu,
coef_sparse,
).T
objective_history.append(
self._objective(x, y, k, coef_negative_definite, coef_sparse)
)
eigs_history.append(np.sort(np.linalg.svd(coef_sparse, compute_uv=False)))
coef_history.append(coef_sparse)
if self._convergence_criterion() < self.tol:
# TODO: Update this for trimming/constraints
break
else:
warnings.warn(
f"StableLinearSR3 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_negative_definite.T
if self.use_trimming:
self.trimming_array = trimming_array
self.objective_history = objective_history
self.eigs_history = np.array(eigs_history)
self.coef_history = np.array(coef_history)