Root mean squared propagation optimizer

RMSProp (short for Root Mean Squared Propagation) is an adaptive learning rate optimization method that uses a moving average of the squared gradients to normalize the gradient. The idea behind RMSProp is to divide the learning rate by a running average of the magnitudes of recent gradients for a given parameter. This helps prevent the learning rate from being too high or too low, and can speed up convergence. The update rule for RMSProp is given by:

\[ \vec\gamma^{(k+1)} = \vec\gamma^{(k)} - \frac{\alpha}{\sqrt{\vec{v}^{(k)} + \epsilon}} \odot \vec\nabla f(\vec\gamma^{(k)}), \]

where \(\odot\) denotes element-wise multiplication, \(\alpha\) is the step size, \(\epsilon\) is a small hyperparameter to avoid division by zero, and \(\vec{v}^{(k)}\) is a running average of the squared gradients:

\[ \vec{v}^{(k+1)} = \rho \vec{v}^{(k)} + (1-\rho) (\vec\nabla f(\vec\gamma^{(k+1)}))^2, \]

where \(\rho\) is a hyperparameter that controls the weight given to past gradients in the moving average. \(\rho\) is also called decay.

OpenQAOA example

In the code below it is shown how to run QAOA with RMSProp optimizer, using a step size \(\alpha=0.001\), decay \(\rho=0.9\), constant \(\epsilon=10^{-7}\) and approximating the Jacobian with finite difference method.

from openqaoa import QAOA 

# create the QAOA object
q = QAOA()

# set optimizer and properties
q.set_classical_optimizer(
    method='rmsprop', 
    jac="finite_difference", 
    optimizer_options=dict(
        stepsize=0.001,
        decay=0.9,
        eps=1e-07
    )
)

# compile and optimize using the chosen optimizer
q.compile(problem)
q.optimize()