class GradientDescent(maxiter=100, learning_rate=0.01, tol=1e-07, callback=None, perturbation=None)[source]#

Bases: SteppableOptimizer

The gradient descent minimization routine.

For a function \(f\) and an initial point \(\vec\theta_0\), the standard (or “vanilla”) gradient descent method is an iterative scheme to find the minimum \(\vec\theta^*\) of \(f\) by updating the parameters in the direction of the negative gradient of \(f\)

\[\vec\theta_{n+1} = \vec\theta_{n} - \eta_n \vec\nabla f(\vec\theta_{n}),\]

for a small learning rate \(\eta_n > 0\).

You can either provide the analytic gradient \(\vec\nabla f\) as jac in the minimize() method, or, if you do not provide it, use a finite difference approximation of the gradient. To adapt the size of the perturbation in the finite difference gradients, set the perturbation property in the initializer.

This optimizer supports a callback function. If provided in the initializer, the optimizer will call the callback in each iteration with the following information in this order: current number of function values, current parameters, current function value, norm of current gradient.


A minimum example that will use finite difference gradients with a default perturbation of 0.01 and a default learning rate of 0.01.

from qiskit_algorithms.optimizers import GradientDescent

def f(x):
    return (np.linalg.norm(x) - 1) ** 2

initial_point = np.array([1, 0.5, -0.2])

optimizer = GradientDescent(maxiter=100)

result = optimizer.minimize(fun=fun, x0=initial_point)

print(f"Found minimum {result.x} at a value"
    "of {} using {result.nfev} evaluations.")

An example where the learning rate is an iterator and we supply the analytic gradient. Note how much faster this convergences (i.e. less nfev) compared to the previous example.

from qiskit_algorithms.optimizers import GradientDescent

def learning_rate():
    power = 0.6
    constant_coeff = 0.1
    def power_law():
        n = 0
        while True:
            yield constant_coeff * (n ** power)
            n += 1

    return power_law()

def f(x):
    return (np.linalg.norm(x) - 1) ** 2

def grad_f(x):
    return 2 * (np.linalg.norm(x) - 1) * x / np.linalg.norm(x)

initial_point = np.array([1, 0.5, -0.2])

optimizer = GradientDescent(maxiter=100, learning_rate=learning_rate)
result = optimizer.minimize(fun=fun, jac=grad_f, x0=initial_point)

print(f"Found minimum {result.x} at a value"
"of {} using {result.nfev} evaluations.")

An other example where the evaluation of the function has a chance of failing. The user, with specific knowledge about his function can catch this errors and handle them before passing the result to the optimizer.

import random
import numpy as np
from qiskit_algorithms.optimizers import GradientDescent

def objective(x):
    if random.choice([True, False]):
        return None
        return (np.linalg.norm(x) - 1) ** 2

def grad(x):
    if random.choice([True, False]):
        return None
        return 2 * (np.linalg.norm(x) - 1) * x / np.linalg.norm(x)

initial_point = np.random.normal(0, 1, size=(100,))

optimizer = GradientDescent(maxiter=20)
optimizer.start(x0=initial_point, fun=objective, jac=grad)

while optimizer.continue_condition():
    ask_data = optimizer.ask()
    evaluated_gradient = None

    while evaluated_gradient is None:
        evaluated_gradient = grad(ask_data.x_center)
        optimizer.state.njev += 1

    optimizer.state.nit += 1

    tell_data = TellData(eval_jac=evaluated_gradient)
    optimizer.tell(ask_data=ask_data, tell_data=tell_data)

result = optimizer.create_result()

Users that aren’t dealing with complicated functions and who are more familiar with step by step optimization algorithms can use the step() method which wraps the ask() and tell() methods. In the same spirit the method minimize() will optimize the function and return the result.

To see other libraries that use this interface one can visit:

  • maxiter (int) – The maximum number of iterations.

  • learning_rate (float | list[float] | np.ndarray | Callable[[], Generator[float, None, None]]) – A constant, list, array or factory of generators yielding learning rates for the parameter updates. See the docstring for an example.

  • tol (float) – If the norm of the parameter update is smaller than this threshold, the optimizer has converged.

  • perturbation (float | None) – If no gradient is passed to minimize() the gradient is approximated with a forward finite difference scheme with perturbation perturbation in both directions (defaults to 1e-2 if required). Ignored when we have an explicit function for the gradient.


ValueError – If learning_rate is an array and its length is less than maxiter.



Returns bounds support level


Returns gradient support level


Returns initial point support level


Returns is bounds ignored


Returns is bounds required


Returns is bounds supported


Returns is gradient ignored


Returns is gradient required


Returns is gradient supported


Returns is initial point ignored


Returns is initial point required


Returns is initial point supported


Returns the perturbation.

This is the perturbation used in the finite difference gradient approximation.


Return setting


Return the current state of the optimizer.


Returns the tolerance of the optimizer.

Any step with smaller stepsize than this value will stop the optimization.



Returns an object with the data needed to evaluate the gradient.

If this object contains a gradient function the gradient can be evaluated directly. Otherwise approximate it with a finite difference scheme.

Return type:



Condition that indicates the optimization process should come to an end.

When the stepsize is smaller than the tolerance, the optimization process is considered finished.


True if the optimization process should continue, False otherwise.

Return type:



Creates a result of the optimization process.

This result contains the best point, the best function value, the number of function/gradient evaluations and the number of iterations.


The result of the optimization process.

Return type:



Evaluates the gradient.

It does so either by evaluating an analytic gradient or by approximating it with a finite difference scheme. It will either add 1 to the number of gradient evaluations or add N+1 to the number of function evaluations (Where N is the dimension of the gradient).


ask_data (AskData) – It contains the point where the gradient is to be evaluated and the gradient function or, in its absence, the objective function to perform a finite difference approximation.


The data containing the gradient evaluation.

Return type:



Get the support level dictionary.

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.

  • x_center (ndarray) – point around which we compute the gradient

  • f (func) – the function of which the gradient is to be computed.

  • epsilon (float) – the epsilon used in the numeric differentiation.

  • max_evals_grouped (int) – max evals grouped, defaults to 1 (i.e. no batching).


the gradient computed

Return type:


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

Minimizes the function.

For well behaved functions the user can call this method to minimize a function. If the user wants more control on how to evaluate the function a custom loop can be created using ask() and tell() and evaluating the function manually.

  • fun (Callable[[POINT], float]) – Function to minimize.

  • x0 (POINT) – Initial point.

  • jac (Callable[[POINT], POINT] | None) – Function to compute the gradient.

  • bounds (list[tuple[float, float]] | None) – Bounds of the search space.


Object containing the result of the optimization.

Return type:



Print algorithm-specific options.


Set max evals grouped


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.


kwargs (dict) – options, given as name=value.

start(fun, x0, jac=None, bounds=None)[source]#

Populates the state of the optimizer with the data provided and sets all the counters to 0.

  • fun (Callable[[POINT], float]) – Function to minimize.

  • x0 (POINT) – Initial point.

  • jac (Callable[[POINT], POINT] | None) – Function to compute the gradient.

  • bounds (list[tuple[float, float]] | None) – Bounds of the search space.


Performs one step in the optimization process.

This method composes ask(), evaluate(), and tell() to make a “step” in the optimization process.

tell(ask_data, tell_data)[source]#

Updates x by an amount proportional to the learning rate and value of the gradient at that point.

  • ask_data (AskData) – The data used to evaluate the function.

  • tell_data (TellData) – The data from the function evaluation.


ValueError – If the gradient passed doesn’t have the right dimension.

static wrap_function(function, args)#

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

  • function (func) – the target function

  • args (tuple) – the args to be injected



Return type: