# qml.NesterovMomentumOptimizer¶

class NesterovMomentumOptimizer(stepsize=0.01, momentum=0.9)[source]

Bases: pennylane.optimize.momentum.MomentumOptimizer

Nesterov Momentum works like the Momentum optimizer, but shifts the current input by the momentum term when computing the gradient of the objective function:

$a^{(t+1)} = m a^{(t)} + \eta \nabla f(x^{(t)} - m a^{(t)}).$

The user defined parameters are:

• $$\eta$$: the step size

• $$m$$: the momentum

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

• momentum (float) – user-defined hyperparameter $$m$$

 apply_grad(grad, args) Update the trainable args to take a single optimization step. compute_grad(objective_fn, args, kwargs[, …]) Compute gradient of the objective function at at the shifted point $$(x - m\times\text{accumulation})$$ and return it along with the objective function forward pass (if available). Reset optimizer by erasing memory of past steps. step(objective_fn, *args[, grad_fn]) Update trainable arguments with one step of the optimizer. step_and_cost(objective_fn, *args[, grad_fn]) Update trainable arguments 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, args)

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

Parameters
• grad (tuple [array]) – the gradient of the objective function at point $$x^{(t)}$$: $$\nabla f(x^{(t)})$$.

• args (tuple) – the current value of the variables $$x^{(t)}$$.

Returns

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

Return type

list [array]

compute_grad(objective_fn, args, kwargs, grad_fn=None)[source]

Compute gradient of the objective function at at the shifted point $$(x - m\times\text{accumulation})$$ and return it along with the objective function forward pass (if available).

Parameters
• objective_fn (function) – the objective function for optimization.

• args (tuple) – tuple of NumPy arrays containing the current values for the objection function.

• kwargs (dict) – keyword arguments for the objective function.

• grad_fn (function) – optional gradient function of the objective function with respect to the variables x. If None, the gradient function is computed automatically. Must return the same shape of tuple [array] as the autograd derivative.

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 [array]

reset()

Reset optimizer by erasing memory of past steps.

step(objective_fn, *args, grad_fn=None, **kwargs)

Update trainable arguments with one step of the optimizer.

Parameters
• objective_fn (function) – the objective function for optimization

• *args – Variable length argument list for objective function

• grad_fn (function) – optional gradient function of the objective function with respect to the variables x. If None, the gradient function is computed automatically. Must return the same shape of tuple [array] as the autograd derivative.

• **kwargs – variable length of keyword arguments for the objective function

Returns

the new variable values $$x^{(t+1)}$$. If single arg is provided, list [array] is replaced by array.

Return type

list [array]

step_and_cost(objective_fn, *args, grad_fn=None, **kwargs)

Update trainable arguments 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

• *args – variable length argument list for objective function

• grad_fn (function) – optional gradient function of the objective function with respect to the variables *args. If None, the gradient function is computed automatically. Must return the same shape of tuple [array] as the autograd derivative.

• **kwargs – variable length of keyword arguments for the objective function

Returns

the new variable values $$x^{(t+1)}$$ and the objective function output prior to the step. If single arg is provided, list [array] is replaced by array.

Return type

tuple[list [array], float]

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 $$\eta$$