# This code is part of a Qiskit project.
#
# (C) Copyright IBM 2018, 2024.
#
# This code is licensed under the Apache License, Version 2.0. You may
# obtain a copy of this license in the LICENSE.txt file in the root directory
# of this source tree or at http://www.apache.org/licenses/LICENSE-2.0.
#
# Any modifications or derivative works of this code must retain this
# copyright notice, and modified files need to carry a notice indicating
# that they have been altered from the originals.
"""Simultaneous Perturbation Stochastic Approximation (SPSA) optimizer.
This implementation allows both standard first-order and second-order SPSA.
"""
from __future__ import annotations
from collections import deque
from collections.abc import Iterator
from typing import Callable, Any, SupportsFloat
import logging
import warnings
from time import time
import scipy
import numpy as np
from qiskit_algorithms.utils import algorithm_globals
from .optimizer import Optimizer, OptimizerSupportLevel, OptimizerResult, POINT
# number of function evaluations, parameters, loss, stepsize, accepted
CALLBACK = Callable[[int, np.ndarray, float, SupportsFloat, bool], None]
TERMINATIONCHECKER = Callable[[int, np.ndarray, float, SupportsFloat, bool], bool]
logger = logging.getLogger(__name__)
[docs]class SPSA(Optimizer):
"""Simultaneous Perturbation Stochastic Approximation (SPSA) optimizer.
SPSA [1] is an gradient descent method for optimizing systems with multiple unknown parameters.
As an optimization method, it is appropriately suited to large-scale population models,
adaptive modeling, and simulation optimization.
.. seealso::
Many examples are presented at the `SPSA Web site <http://www.jhuapl.edu/SPSA>`__.
The main feature of SPSA is the stochastic gradient approximation, which requires only two
measurements of the objective function, regardless of the dimension of the optimization
problem.
Additionally, to standard first-order SPSA, where only gradient information is used, this
implementation also allows second-order SPSA (2-SPSA) [2]. In 2-SPSA we also estimate the
Hessian of the loss with a stochastic approximation and multiply the gradient with the
inverse Hessian to take local curvature into account and improve convergence.
Notably this Hessian estimate requires only a constant number of function evaluations
unlike an exact evaluation of the Hessian, which scales quadratically in the number of
function evaluations.
.. note::
SPSA can be used in the presence of noise, and it is therefore indicated in situations
involving measurement uncertainty on a quantum computation when finding a minimum.
If you are executing a variational algorithm using a Quantum ASseMbly Language (QASM)
simulator or a real device, SPSA would be the most recommended choice among the optimizers
provided here.
The optimization process can include a calibration phase if neither the ``learning_rate`` nor
``perturbation`` is provided, which requires additional functional evaluations.
(Note that either both or none must be set.) For further details on the automatic calibration,
please refer to the supplementary information section IV. of [3].
.. note::
This component has some function that is normally random. If you want to reproduce behavior
then you should set the random number generator seed in the algorithm_globals
(``qiskit_algorithms.utils.algorithm_globals.random_seed = seed``).
Examples:
This short example runs SPSA for the ground state calculation of the ``Z ^ Z``
observable where the ansatz is a ``PauliTwoDesign`` circuit.
.. code-block:: python
import numpy as np
from qiskit_algorithms.optimizers import SPSA
from qiskit.circuit.library import PauliTwoDesign
from qiskit.primitives import Estimator
from qiskit.quantum_info import SparsePauliOp
ansatz = PauliTwoDesign(2, reps=1, seed=2)
observable = SparsePauliOp("ZZ")
initial_point = np.random.random(ansatz.num_parameters)
estimator = Estimator()
def loss(x):
job = estimator.run([ansatz], [observable], [x])
return job.result().values[0]
spsa = SPSA(maxiter=300)
result = spsa.minimize(loss, x0=initial_point)
To use the Hessian information, i.e. 2-SPSA, you can add `second_order=True` to the
initializer of the `SPSA` class, the rest of the code remains the same.
.. code-block:: python
two_spsa = SPSA(maxiter=300, second_order=True)
result = two_spsa.minimize(loss, x0=initial_point)
The `termination_checker` can be used to implement a custom termination criterion.
.. code-block:: python
import numpy as np
from qiskit_algorithms.optimizers import SPSA
def objective(x):
return np.linalg.norm(x) + .04*np.random.rand(1)
class TerminationChecker:
def __init__(self, N : int):
self.N = N
self.values = []
def __call__(self, nfev, parameters, value, stepsize, accepted) -> bool:
self.values.append(value)
if len(self.values) > self.N:
last_values = self.values[-self.N:]
pp = np.polyfit(range(self.N), last_values, 1)
slope = pp[0] / self.N
if slope > 0:
return True
return False
spsa = SPSA(maxiter=200, termination_checker=TerminationChecker(10))
result = spsa.minimize(objective, x0=[0.5, 0.5])
print(f'SPSA completed after {result.nit} iterations')
References:
[1]: J. C. Spall (1998). An Overview of the Simultaneous Perturbation Method for Efficient
Optimization, Johns Hopkins APL Technical Digest, 19(4), 482–492.
`Online at jhuapl.edu. <https://www.jhuapl.edu/SPSA/PDF-SPSA/Spall_An_Overview.PDF>`_
[2]: J. C. Spall (1997). Accelerated second-order stochastic optimization using only
function measurements, Proceedings of the 36th IEEE Conference on Decision and Control,
1417-1424 vol.2. `Online at IEEE.org. <https://ieeexplore.ieee.org/document/657661>`_
[3]: A. Kandala et al. (2017). Hardware-efficient Variational Quantum Eigensolver for
Small Molecules and Quantum Magnets. Nature 549, pages242–246(2017).
`arXiv:1704.05018v2 <https://arxiv.org/pdf/1704.05018v2.pdf#section*.11>`_
"""
def __init__(
self,
maxiter: int = 100,
blocking: bool = False,
allowed_increase: float | None = None,
trust_region: bool = False,
learning_rate: float | np.ndarray | Callable[[], Iterator] | None = None,
perturbation: float | np.ndarray | Callable[[], Iterator] | None = None,
last_avg: int = 1,
resamplings: int | dict[int, int] = 1,
perturbation_dims: int | None = None,
second_order: bool = False,
regularization: float | None = None,
hessian_delay: int = 0,
lse_solver: Callable[[np.ndarray, np.ndarray], np.ndarray] | None = None,
initial_hessian: np.ndarray | None = None,
callback: CALLBACK | None = None,
termination_checker: TERMINATIONCHECKER | None = None,
) -> None:
r"""
Args:
maxiter: The maximum number of iterations. Note that this is not the maximal number
of function evaluations.
blocking: If True, only accepts updates that improve the loss (up to some allowed
increase, see next argument).
allowed_increase: If ``blocking`` is ``True``, this argument determines by how much
the loss can increase with the proposed parameters and still be accepted.
If ``None``, the allowed increases is calibrated automatically to be twice the
approximated standard deviation of the loss function.
trust_region: If ``True``, restricts the norm of the update step to be :math:`\leq 1`.
learning_rate: The update step is the learning rate is multiplied with the gradient.
If the learning rate is a float, it remains constant over the course of the
optimization. If a NumPy array, the :math:`i`-th element is the learning rate for
the :math:`i`-th iteration. It can also be a callable returning an iterator which
yields the learning rates for each optimization step.
If ``learning_rate`` is set ``perturbation`` must also be provided.
perturbation: Specifies the magnitude of the perturbation for the finite difference
approximation of the gradients. See ``learning_rate`` for the supported types.
If ``perturbation`` is set ``learning_rate`` must also be provided.
last_avg: Return the average of the ``last_avg`` parameters instead of just the
last parameter values.
resamplings: The number of times the gradient (and Hessian) is sampled using a random
direction to construct a gradient estimate. Per default the gradient is estimated
using only one random direction. If an integer, all iterations use the same number
of resamplings. If a dictionary, this is interpreted as
``{iteration: number of resamplings per iteration}``.
perturbation_dims: The number of perturbed dimensions. Per default, all dimensions
are perturbed, but a smaller, fixed number can be perturbed. If set, the perturbed
dimensions are chosen uniformly at random.
second_order: If True, use 2-SPSA instead of SPSA. In 2-SPSA, the Hessian is estimated
additionally to the gradient, and the gradient is preconditioned with the inverse
of the Hessian to improve convergence.
regularization: To ensure the preconditioner is symmetric and positive definite, the
identity times a small coefficient is added to it. This generator yields that
coefficient.
hessian_delay: Start multiplying the gradient with the inverse Hessian only after a
certain number of iterations. The Hessian is still evaluated and therefore this
argument can be useful to first get a stable average over the last iterations before
using it as preconditioner.
lse_solver: The method to solve for the inverse of the Hessian. Per default an
exact LSE solver is used, but can e.g. be overwritten by a minimization routine.
initial_hessian: The initial guess for the Hessian. By default the identity matrix
is used.
callback: A callback function passed information in each iteration step. The
information is, in this order: the number of function evaluations, the parameters,
the function value, the stepsize, whether the step was accepted.
termination_checker: A callback function executed at the end of each iteration step. The
arguments are, in this order: the parameters, the function value, the number
of function evaluations, the stepsize, whether the step was accepted. If the callback
returns True, the optimization is terminated.
To prevent additional evaluations of the objective method, if the objective has not yet
been evaluated, the objective is estimated by taking the mean of the objective
evaluations used in the estimate of the gradient.
Raises:
ValueError: If ``learning_rate`` or ``perturbation`` is an array with less elements
than the number of iterations.
"""
super().__init__()
# general optimizer arguments
self.maxiter = maxiter
self.trust_region = trust_region
self.callback = callback
self.termination_checker = termination_checker
# if learning rate and perturbation are arrays, check they are sufficiently long
for attr, name in zip([learning_rate, perturbation], ["learning_rate", "perturbation"]):
if isinstance(attr, (list, np.ndarray)):
if len(attr) < maxiter:
raise ValueError(f"Length of {name} is smaller than maxiter ({maxiter}).")
self.learning_rate = learning_rate
self.perturbation = perturbation
# SPSA specific arguments
self.blocking = blocking
self.allowed_increase = allowed_increase
self.last_avg = last_avg
self.resamplings = resamplings
self.perturbation_dims = perturbation_dims
# 2-SPSA specific arguments
if regularization is None:
regularization = 0.01
self.second_order = second_order
self.hessian_delay = hessian_delay
self.lse_solver = lse_solver
self.regularization = regularization
self.initial_hessian = initial_hessian
# runtime arguments
self._nfev: int | None = None # the number of function evaluations
self._smoothed_hessian: np.ndarray | None = None # smoothed average of the Hessians
[docs] @staticmethod
def calibrate(
loss: Callable[[np.ndarray], float],
initial_point: np.ndarray,
c: float = 0.2,
stability_constant: float = 0,
target_magnitude: float | None = None, # 2 pi / 10
alpha: float = 0.602,
gamma: float = 0.101,
modelspace: bool = False,
max_evals_grouped: int = 1,
) -> tuple[Callable, Callable]:
r"""Calibrate SPSA parameters with a power series as learning rate and perturbation coeffs.
The power series are:
.. math::
a_k = \frac{a}{(A + k + 1)^\alpha}, c_k = \frac{c}{(k + 1)^\gamma}
Args:
loss: The loss function.
initial_point: The initial guess of the iteration.
c: The initial perturbation magnitude.
stability_constant: The value of `A`.
target_magnitude: The target magnitude for the first update step, defaults to
:math:`2\pi / 10`.
alpha: The exponent of the learning rate power series.
gamma: The exponent of the perturbation power series.
modelspace: Whether the target magnitude is the difference of parameter values
or function values (= model space).
max_evals_grouped: The number of grouped evaluations supported by the loss function.
Defaults to 1, i.e. no grouping.
Returns:
tuple(generator, generator): A tuple of power series generators, the first one for the
learning rate and the second one for the perturbation.
"""
logger.info("SPSA: Starting calibration of learning rate and perturbation.")
if target_magnitude is None:
target_magnitude = 2 * np.pi / 10
dim = len(initial_point)
# compute the average magnitude of the first step
steps = 25
points = []
for _ in range(steps):
# compute the random direction
pert = bernoulli_perturbation(dim)
points += [initial_point + c * pert, initial_point - c * pert]
losses = _batch_evaluate(loss, points, max_evals_grouped)
avg_magnitudes = 0.0
for i in range(steps):
delta = losses[2 * i] - losses[2 * i + 1]
avg_magnitudes += np.abs(delta / (2 * c))
avg_magnitudes /= steps
if modelspace:
a = target_magnitude / (avg_magnitudes**2)
else:
a = target_magnitude / avg_magnitudes
# compute the rescaling factor for correct first learning rate
if a < 1e-10:
warnings.warn(f"Calibration failed, using {target_magnitude} for `a`")
a = target_magnitude
logger.info("Finished calibration:")
logger.info(
" -- Learning rate: a / ((A + n) ^ alpha) with a = %s, A = %s, alpha = %s",
a,
stability_constant,
alpha,
)
logger.info(" -- Perturbation: c / (n ^ gamma) with c = %s, gamma = %s", c, gamma)
# set up the power series
def learning_rate():
return powerseries(a, alpha, stability_constant)
def perturbation():
return powerseries(c, gamma)
return learning_rate, perturbation
[docs] @staticmethod
def estimate_stddev(
loss: Callable[[np.ndarray], float],
initial_point: np.ndarray,
avg: int = 25,
max_evals_grouped: int = 1,
) -> float:
"""Estimate the standard deviation of the loss function."""
losses = _batch_evaluate(loss, avg * [initial_point], max_evals_grouped)
return np.std(losses)
@property
def settings(self) -> dict[str, Any]:
# if learning rate or perturbation are custom iterators expand them
if callable(self.learning_rate):
iterator = self.learning_rate()
learning_rate = np.array([next(iterator) for _ in range(self.maxiter)])
else:
learning_rate = self.learning_rate # type: ignore[assignment]
if callable(self.perturbation):
iterator = self.perturbation()
perturbation = np.array([next(iterator) for _ in range(self.maxiter)])
else:
perturbation = self.perturbation # type: ignore[assignment]
return {
"maxiter": self.maxiter,
"learning_rate": learning_rate,
"perturbation": perturbation,
"trust_region": self.trust_region,
"blocking": self.blocking,
"allowed_increase": self.allowed_increase,
"resamplings": self.resamplings,
"perturbation_dims": self.perturbation_dims,
"second_order": self.second_order,
"hessian_delay": self.hessian_delay,
"regularization": self.regularization,
"lse_solver": self.lse_solver,
"initial_hessian": self.initial_hessian,
"callback": self.callback,
"termination_checker": self.termination_checker,
}
def _point_sample(self, loss, x, eps, delta1, delta2):
"""A single sample of the gradient at position ``x`` in direction ``delta``."""
# points to evaluate
points = [x + eps * delta1, x - eps * delta1]
self._nfev += 2
if self.second_order:
points += [x + eps * (delta1 + delta2), x + eps * (-delta1 + delta2)]
self._nfev += 2
# batch evaluate the points (if possible)
values = _batch_evaluate(loss, points, self._max_evals_grouped)
plus = values[0]
minus = values[1]
gradient_sample = (plus - minus) / (2 * eps) * delta1
hessian_sample = None
if self.second_order:
diff = (values[2] - plus) - (values[3] - minus)
diff /= 2 * eps**2
rank_one = np.outer(delta1, delta2)
hessian_sample = diff * (rank_one + rank_one.T) / 2
return np.mean(values), gradient_sample, hessian_sample
def _point_estimate(self, loss, x, eps, num_samples):
"""The gradient estimate at point x."""
# set up variables to store averages
value_estimate = 0
gradient_estimate = np.zeros(x.size)
hessian_estimate = np.zeros((x.size, x.size))
# iterate over the directions
deltas1 = [
bernoulli_perturbation(x.size, self.perturbation_dims) for _ in range(num_samples)
]
if self.second_order:
deltas2 = [
bernoulli_perturbation(x.size, self.perturbation_dims) for _ in range(num_samples)
]
else:
deltas2 = None
for i in range(num_samples):
delta1 = deltas1[i]
delta2 = deltas2[i] if self.second_order else None
value_sample, gradient_sample, hessian_sample = self._point_sample(
loss, x, eps, delta1, delta2
)
value_estimate += value_sample
gradient_estimate += gradient_sample
if self.second_order:
hessian_estimate += hessian_sample
return (
value_estimate / num_samples,
gradient_estimate / num_samples,
hessian_estimate / num_samples,
)
def _compute_update(self, loss, x, k, eps, lse_solver):
# compute the perturbations
if isinstance(self.resamplings, dict):
num_samples = self.resamplings.get(k, 1)
else:
num_samples = self.resamplings
# accumulate the number of samples
value, gradient, hessian = self._point_estimate(loss, x, eps, num_samples)
# precondition gradient with inverse Hessian, if specified
if self.second_order:
smoothed = k / (k + 1) * self._smoothed_hessian + 1 / (k + 1) * hessian
self._smoothed_hessian = smoothed
if k > self.hessian_delay:
spd_hessian = _make_spd(smoothed, self.regularization)
# solve for the gradient update
gradient = np.real(lse_solver(spd_hessian, gradient))
return value, gradient
[docs] def minimize(
self,
fun: Callable[[POINT], float],
x0: POINT,
jac: Callable[[POINT], POINT] | None = None,
bounds: list[tuple[float, float]] | None = None,
) -> OptimizerResult:
# ensure learning rate and perturbation are correctly set: either none or both
# this happens only here because for the calibration the loss function is required
x0 = np.asarray(x0)
if self.learning_rate is None and self.perturbation is None:
get_eta, get_eps = self.calibrate(fun, x0, max_evals_grouped=self._max_evals_grouped)
else:
get_eta, get_eps = _validate_pert_and_learningrate(
self.perturbation, self.learning_rate
)
eta, eps = get_eta(), get_eps()
lse_solver = self.lse_solver
if self.lse_solver is None:
lse_solver = np.linalg.solve
# prepare some initials
x = np.asarray(x0)
if self.initial_hessian is None:
self._smoothed_hessian = np.identity(x.size)
else:
self._smoothed_hessian = self.initial_hessian
self._nfev = 0
# if blocking is enabled we need to keep track of the function values
if self.blocking:
fx = fun(x) # pylint: disable=invalid-name
self._nfev += 1
if self.allowed_increase is None:
self.allowed_increase = 2 * self.estimate_stddev(
fun, x, max_evals_grouped=self._max_evals_grouped
)
logger.info("SPSA: Starting optimization.")
start = time()
# keep track of the last few steps to return their average
last_steps = deque([x])
# use a local variable and while loop to keep track of the number of iterations
# if the termination checker terminates early
k = 0
while k < self.maxiter:
k += 1
iteration_start = time()
# compute update
fx_estimate, update = self._compute_update(fun, x, k, next(eps), lse_solver)
# trust region
if self.trust_region:
norm = np.linalg.norm(update)
if norm > 1: # stop from dividing by 0
update = update / norm
# compute next parameter value
update = update * next(eta)
x_next = x - update
fx_next = None
# blocking
if self.blocking:
self._nfev += 1
fx_next = fun(x_next)
if fx + self.allowed_increase <= fx_next: # accept only if loss improved
if self.callback is not None:
self.callback(
self._nfev, # number of function evals
x_next, # next parameters
fx_next, # loss at next parameters
np.linalg.norm(update), # size of the update step
False,
) # not accepted
logger.info(
"Iteration %s/%s rejected in %s.",
k,
self.maxiter + 1,
time() - iteration_start,
)
continue
fx = fx_next # pylint: disable=invalid-name
logger.info(
"Iteration %s/%s done in %s.", k, self.maxiter + 1, time() - iteration_start
)
if self.callback is not None:
# if we didn't evaluate the function yet, do it now
if not self.blocking:
self._nfev += 1
fx_next = fun(x_next)
self.callback(
self._nfev, # number of function evals
x_next, # next parameters
fx_next, # loss at next parameters
np.linalg.norm(update), # size of the update step
True,
) # accepted
# update parameters
x = x_next
# update the list of the last ``last_avg`` parameters
if self.last_avg > 1:
last_steps.append(x_next)
if len(last_steps) > self.last_avg:
last_steps.popleft()
if self.termination_checker is not None:
fx_check = fx_estimate if fx_next is None else fx_next
if self.termination_checker(
self._nfev, x_next, fx_check, np.linalg.norm(update), True
):
logger.info("terminated optimization at {k}/{self.maxiter} iterations")
break
logger.info("SPSA: Finished in %s", time() - start)
if self.last_avg > 1:
x = np.mean(np.asarray(last_steps), axis=0)
result = OptimizerResult()
result.x = x
result.fun = fun(x)
result.nfev = self._nfev
result.nit = k
return result
[docs] def get_support_level(self):
"""Get the support level dictionary."""
return {
"gradient": OptimizerSupportLevel.ignored,
"bounds": OptimizerSupportLevel.ignored,
"initial_point": OptimizerSupportLevel.required,
}
def bernoulli_perturbation(dim, perturbation_dims=None):
"""Get a Bernoulli random perturbation."""
if perturbation_dims is None:
return 1 - 2 * algorithm_globals.random.binomial(1, 0.5, size=dim)
pert = 1 - 2 * algorithm_globals.random.binomial(1, 0.5, size=perturbation_dims)
indices = algorithm_globals.random.choice(
list(range(dim)), size=perturbation_dims, replace=False
)
result = np.zeros(dim)
result[indices] = pert
return result
def powerseries(eta=0.01, power=2, offset=0):
"""Yield a series decreasing by a power law."""
n = 1
while True:
yield eta / ((n + offset) ** power)
n += 1
def constant(eta=0.01):
"""Yield a constant series."""
while True:
yield eta
def _batch_evaluate(function, points, max_evals_grouped, unpack_points=False):
"""Evaluate a function on all points with batches of max_evals_grouped.
The points are a list of inputs, as ``[in1, in2, in3, ...]``. If the individual
inputs are tuples (because the function takes multiple inputs), set ``unpack_points`` to ``True``.
"""
# if the function cannot handle lists of points as input, cover this case immediately
if max_evals_grouped is None or max_evals_grouped == 1:
# support functions with multiple arguments where the points are given in a tuple
return [
function(*point) if isinstance(point, tuple) else function(point) for point in points
]
num_points = len(points)
# get the number of batches
num_batches = num_points // max_evals_grouped
if num_points % max_evals_grouped != 0:
num_batches += 1
# split the points
batched_points = np.array_split(np.asarray(points), num_batches)
results = []
for batch in batched_points:
if unpack_points:
batch = _repack_points(batch)
results += _as_list(function(*batch))
else:
results += _as_list(function(batch))
return results
def _as_list(obj):
"""Convert a list or numpy array into a list."""
return obj.tolist() if isinstance(obj, np.ndarray) else obj
def _repack_points(points):
"""Turn a list of tuples of points into a tuple of lists of points.
E.g. turns
[(a1, a2, a3), (b1, b2, b3)]
into
([a1, b1], [a2, b2], [a3, b3])
where all elements are np.ndarray.
"""
num_sets = len(points[0]) # length of (a1, a2, a3)
return ([x[i] for x in points] for i in range(num_sets))
def _make_spd(matrix, bias=0.01):
identity = np.identity(matrix.shape[0])
psd = scipy.linalg.sqrtm(matrix.dot(matrix))
return psd + bias * identity
def _validate_pert_and_learningrate(perturbation, learning_rate):
if learning_rate is None or perturbation is None:
raise ValueError("If one of learning rate or perturbation is set, both must be set.")
if isinstance(perturbation, float):
def get_eps():
return constant(perturbation)
elif isinstance(perturbation, (list, np.ndarray)):
def get_eps():
return iter(perturbation)
else:
get_eps = perturbation
if isinstance(learning_rate, float):
def get_eta():
return constant(learning_rate)
elif isinstance(learning_rate, (list, np.ndarray)):
def get_eta():
return iter(learning_rate)
else:
get_eta = learning_rate
return get_eta, get_eps