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SPSA

qiskit.algorithms.optimizers.SPSA(maxiter=100, blocking=False, allowed_increase=None, trust_region=False, learning_rate=None, perturbation=None, last_avg=1, resamplings=1, perturbation_dims=None, second_order=False, regularization=None, hessian_delay=0, lse_solver=None, initial_hessian=None, callback=None, termination_checker=None) GitHub(opens in a new tab)

Bases: 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.

See also

Many examples are presented at the SPSA Web site(opens in a new tab).

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 an OpenQASM simulator or a real device, SPSA would be the most recommended choice among the optimizers provided here.

The optimization process can includes 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.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.

import numpy as np
from qiskit.algorithms.optimizers import SPSA
from qiskit.circuit.library import PauliTwoDesign
from qiskit.opflow import Z, StateFn
 
ansatz = PauliTwoDesign(2, reps=1, seed=2)
observable = Z ^ Z
initial_point = np.random.random(ansatz.num_parameters)
 
def loss(x):
    bound = ansatz.assign_parameters(x)
    return np.real((StateFn(observable, is_measurement=True) @ StateFn(bound)).eval())
 
spsa = SPSA(maxiter=300)
result = spsa.optimize(ansatz.num_parameters, loss, initial_point=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.

two_spsa = SPSA(maxiter=300, second_order=True)
result = two_spsa.optimize(ansatz.num_parameters, loss, initial_point=initial_point)

The termination_checker can be used to implement a custom termination criterion.

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))
parameters, value, niter = spsa.optimize(2, objective, initial_point=[0.5, 0.5])
print(f'SPSA completed after {niter} 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.(opens in a new tab)

[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.(opens in a new tab)

[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(opens in a new tab)

Parameters

  • maxiter (int(opens in a new tab)) – The maximum number of iterations. Note that this is not the maximal number of function evaluations.
  • blocking (bool(opens in a new tab)) – If True, only accepts updates that improve the loss (up to some allowed increase, see next argument).
  • allowed_increase (float(opens in a new tab) | None) – 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 (bool(opens in a new tab)) – If True, restricts the norm of the update step to be 1\leq 1.
  • learning_rate (float(opens in a new tab) | np.ndarray | Callable[[], Iterator] | None) – 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 ii-th element is the learning rate for the ii-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 (float(opens in a new tab) | np.ndarray | Callable[[], Iterator] | None) – 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 (int(opens in a new tab)) – Return the average of the last_avg parameters instead of just the last parameter values.
  • resamplings (int(opens in a new tab) |dict(opens in a new tab)[int(opens in a new tab), int(opens in a new tab)]) – 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 (int(opens in a new tab) | None) – 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 (bool(opens in a new tab)) – 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 (float(opens in a new tab) | None) – 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 (int(opens in a new tab)) – 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 (Callable[[np.ndarray, np.ndarray], np.ndarray] | None) – 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 (np.ndarray | None) – The initial guess for the Hessian. By default the identity matrix is used.
  • callback (CALLBACK | None) – 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 (TERMINATIONCHECKER | None) – 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(opens in a new tab) – If learning_rate or perturbation is an array with less elements than the number of iterations.


Attributes

bounds_support_level

Returns bounds support level

gradient_support_level

Returns gradient support level

initial_point_support_level

Returns initial point support level

is_bounds_ignored

Returns is bounds ignored

is_bounds_required

Returns is bounds required

is_bounds_supported

Returns is bounds supported

is_gradient_ignored

Returns is gradient ignored

is_gradient_required

Returns is gradient required

is_gradient_supported

Returns is gradient supported

is_initial_point_ignored

Returns is initial point ignored

is_initial_point_required

Returns is initial point required

is_initial_point_supported

Returns is initial point supported

setting

Return setting

settings


Methods

calibrate

static calibrate(loss, initial_point, c=0.2, stability_constant=0, target_magnitude=None, alpha=0.602, gamma=0.101, modelspace=False, max_evals_grouped=1)

Calibrate SPSA parameters with a powerseries as learning rate and perturbation coeffs.

The powerseries are:

ak=a(A+k+1)α,ck=c(k+1)γa_k = \frac{a}{(A + k + 1)^\alpha}, c_k = \frac{c}{(k + 1)^\gamma}

Parameters

Returns

A tuple of powerseries generators, the first one for the

learning rate and the second one for the perturbation.

Return type

tuple(opens in a new tab)(generator, generator)

estimate_stddev

static estimate_stddev(loss, initial_point, avg=25, max_evals_grouped=1)

Estimate the standard deviation of the loss function.

Return type

float(opens in a new tab)

get_support_level

get_support_level()

Get the support level dictionary.

gradient_num_diff

static gradient_num_diff(x_center, f, epsilon, max_evals_grouped=None)

We compute the gradient with the numeric differentiation in the parallel way, around the point x_center.

Parameters

  • x_center (ndarray) – point around which we compute the gradient
  • f (func) – the function of which the gradient is to be computed.
  • epsilon (float(opens in a new tab)) – the epsilon used in the numeric differentiation.
  • max_evals_grouped (int(opens in a new tab)) – max evals grouped, defaults to 1 (i.e. no batching).

Returns

the gradient computed

Return type

grad

minimize

minimize(fun, x0, jac=None, bounds=None)

Minimize the scalar function.

Parameters

Returns

The result of the optimization, containing e.g. the result as attribute x.

Return type

OptimizerResult

optimize

optimize(num_vars, objective_function, gradient_function=None, variable_bounds=None, initial_point=None)

Perform optimization.

Deprecated since version 0.21.0

The method qiskit.algorithms.optimizers.spsa.SPSA.optimize() is deprecated as of qiskit-terra 0.21.0. It will be removed no earlier than 3 months after the release date. Instead, use SPSA.minimize as a replacement, which supports the same arguments but follows the interface of scipy.optimize and returns a complete result object containing additional information.

Parameters

Returns

point, value, nfev

point: is a 1D numpy.ndarray[float] containing the solution value: is a float with the objective function value nfev: number of objective function calls made if available or None

Return type

tuple(opens in a new tab)

print_options()

Print algorithm-specific options.

set_max_evals_grouped

set_max_evals_grouped(limit)

Set max evals grouped

set_options

set_options(**kwargs)

Sets or updates values in the options dictionary.

The options dictionary may be used internally by a given optimizer to pass additional optional values for the underlying optimizer/optimization function used. The options dictionary may be initially populated with a set of key/values when the given optimizer is constructed.

Parameters

kwargs (dict(opens in a new tab)) – options, given as name=value.

wrap_function

static wrap_function(function, args)

Wrap the function to implicitly inject the args at the call of the function.

Parameters

Returns

wrapper

Return type

function_wrapper

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