qml.AdamOptimizer

class AdamOptimizer(stepsize=0.01, beta1=0.9, beta2=0.99, eps=1e-08)

Bases: pennylane.optimize.gradient_descent.GradientDescentOptimizer

Gradient-descent optimizer with adaptive learning rate, first and second moment.

Adaptive Moment Estimation uses a step-dependent learning rate, a first moment \(a\) and a second moment \(b\), reminiscent of the momentum and velocity of a particle:

\[x^{(t+1)} = x^{(t)} - \eta^{(t+1)} \frac{a^{(t+1)}}{\sqrt{b^{(t+1)}} + \epsilon },\]

where the update rules for the two moments are given by

\[\begin{split}a^{(t+1)} &= \beta_1 a^{(t)} + (1-\beta_1) \nabla f(x^{(t)}),\\ b^{(t+1)} &= \beta_2 b^{(t)} + (1-\beta_2) (\nabla f(x^{(t)}))^{\odot 2},\\ \eta^{(t+1)} &= \eta \frac{\sqrt{(1-\beta_2^{t+1})}}{(1-\beta_1^{t+1})}.\end{split}\]

Above, \(( \nabla f(x^{(t-1)}))^{\odot 2}\) denotes the element-wise square operation, which means that each element in the gradient is multiplied by itself. The hyperparameters \(\beta_1\) and \(\beta_2\) can also be step-dependent. Initially, the first and second moment are zero.

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

For more details, see arXiv:1412.6980.

Parameters

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

• beta1 (float) – hyperparameter governing the update of the first and second moment

• beta2 (float) – hyperparameter governing the update of the first and second moment

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

Note

When using torch, tensorflow or jax interfaces, refer to Gradients and training for suitable optimizers.

Attributes

fm	Returns estimated first moments of gradient
sm	Returns estimated second moments of gradient
t	Returns accumulated timesteps

fm

Returns estimated first moments of gradient

sm

Returns estimated second moments of gradient

t

Returns accumulated timesteps

Methods

apply_grad(grad, args)	Update the variables args to take a single optimization step.
compute_grad(objective_fn, args, kwargs[, ...])	Compute the gradient of the objective function at the given point and return it along with the objective function forward pass (if available).
reset()	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.

apply_grad(grad, args)

Update the variables 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[ndarray]) – 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

static compute_grad(objective_fn, args, kwargs, grad_fn=None)

Compute the gradient of the objective function at the given point 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 parameters 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 args. If None, the gradient function is computed automatically. Must return the same shape of tuple [array] as the autograd derivative.

Returns

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 evaluated 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 a tuple[array] with the same number of elements as *args. Each array of the tuple should have the same shape as the corresponding argument.

• **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 a tuple[array] with the same number of elements as *args. Each array of the tuple should have the same shape as the corresponding argument.

• **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]

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