qml.AdagradOptimizer

class AdagradOptimizer(stepsize=0.01, eps=1e-08)

Bases: pennylane.optimize.gradient_descent.GradientDescentOptimizer

Gradient-descent optimizer with past-gradient-dependent learning rate in each dimension.

Adagrad adjusts the learning rate for each parameter \(x_i\) in \(x\) based on past gradients. We therefore have to consider each parameter update individually,

\[x^{(t+1)}_i = x^{(t)}_i - \eta_i^{(t+1)} \partial_{w_i} f(x^{(t)}),\]

where the gradient is replaced by a (scalar) partial derivative.

The learning rate in step \(t\) is given by

\[\eta_i^{(t+1)} = \frac{ \eta_{\mathrm{init}} }{ \sqrt{a_i^{(t+1)} + \epsilon } }, ~~~ a_i^{(t+1)} = \sum_{k=1}^t (\partial_{x_i} f(x^{(k)}))^2.\]

The offset \(\epsilon\) avoids division by zero.

\(\eta\) is the step size, a user defined parameter.

Parameters

• stepsize (float) – the user-defined hyperparameter \(\eta\)

• eps (float) – offset \(\epsilon\) added for numerical stability

Methods

apply_grad(grad, x)	Update the variables x to take a single optimization step.
compute_grad(objective_fn, x[, grad_fn])	Compute gradient of the objective_fn at the point x and return it along with the
reset()	Reset optimizer by erasing memory of past steps.
step(objective_fn, x[, grad_fn])	Update x with one step of the optimizer.
step_and_cost(objective_fn, x[, grad_fn])	Update x with one step of the optimizer and return the corresponding objective function value prior to the step.
update_stepsize(stepsize)	Update the initialized stepsize value \(\eta\).

apply_grad(grad, x)

Update the variables x to take a single optimization step. Flattens and unflattens the inputs to maintain nested iterables as the parameters of the optimization.

Parameters

• grad (array) – The gradient of the objective function at point \(x^{(t)}\): \(\nabla f(x^{(t)})\)

• x (array) – the current value of the variables \(x^{(t)}\)

Returns

the new values \(x^{(t+1)}\)

Return type

array

static compute_grad(objective_fn, x, grad_fn=None)

Compute gradient of the objective_fn at the point x and return it along with the

objective function forward pass (if available).

Parameters

• objective_fn (function) – the objective function for optimization

• x (array) – NumPy array containing the current values of the variables to be updated

• grad_fn (function) – Optional gradient function of the objective function with respect to the variables x. If None, the gradient function is computed automatically.

Returns

The NumPy array containing the gradient \(\nabla f(x^{(t)})\) and the

objective function output. If grad_fn is provided, the objective function will not be evaluted and instead None will be returned.

Return type

tuple

reset()

Reset optimizer by erasing memory of past steps.

step(objective_fn, x, grad_fn=None)

Update x with one step of the optimizer.

Parameters

• objective_fn (function) – the objective function for optimization

• x (array) – NumPy array containing the current values of the variables to be updated

• grad_fn (function) – Optional gradient function of the objective function with respect to the variables x. If None, the gradient function is computed automatically.

Returns

the new variable values \(x^{(t+1)}\)

Return type

array

step_and_cost(objective_fn, x, grad_fn=None)

Update x with one step of the optimizer and return the corresponding objective function value prior to the step.

Parameters

• objective_fn (function) – the objective function for optimization

• x (array) – NumPy array containing the current values of the variables to be updated

• grad_fn (function) – Optional gradient function of the objective function with respect to the variables x. If None, the gradient function is computed automatically.

Returns

the new variable values \(x^{(t+1)}\) and the objective function output

prior to the step

Return type

tuple

update_stepsize(stepsize)

Update the initialized stepsize value \(\eta\).

This allows for techniques such as learning rate scheduling.

Parameters

stepsize (float) – the user-defined hyperparameter \(\eta\)