       &\rule{110mm}{0.4pt}                                                                 \\
            &\textbf{for} \: t=1 \: \textbf{to} \: \ldots \: \textbf{do}                         \\
            &\hspace{5mm}g_t           \leftarrow   \nabla_{\theta} f_t (\theta_{t-1})           \\
            &\hspace{5mm}if \: \lambda \neq 0                                                    \\
            &\hspace{10mm} g_t \leftarrow g_t + \lambda  \theta_{t-1}                            \\
            &\hspace{5mm}v_t           \leftarrow   \alpha v_{t-1} + (1 - \alpha) g^2_t
                \hspace{8mm}                                                                     \\
            &\hspace{5mm} \tilde{v_t} \leftarrow v_t                                             \\
            &\hspace{5mm}if \: centered                                                          \\
            &\hspace{10mm} g^{ave}_t \leftarrow g^{ave}_{t-1} \alpha + (1-\alpha) g_t            \\
            &\hspace{10mm} \tilde{v_t} \leftarrow \tilde{v_t} -  \big(g^{ave}_{t} \big)^2        \\
            &\hspace{5mm}if \: \mu > 0                                                           \\
            &\hspace{10mm} \textbf{b}_t\leftarrow \mu \textbf{b}_{t-1} +
                g_t/ \big(\sqrt{\tilde{v_t}} +  \epsilon \big)                                   \\
            &\hspace{10mm} \theta_t \leftarrow \theta_{t-1} - \gamma \textbf{b}_t                \\
            &\hspace{5mm} else                                                                   \\
            &\hspace{10mm}\theta_t      \leftarrow   \theta_{t-1} -
                \gamma  g_t/ \big(\sqrt{\tilde{v_t}} + \epsilon \big)  \hspace{3mm}              \\
            &\rule{110mm}{0.4pt}                                                          \\[-1.ex]
            &\bf{return} \:  \theta_t                                                     \\[-1.ex]
            &\rule{110mm}{0.4pt}                                                          \\[-1.ex]
       \end{aligned}

    For further details regarding the algorithm we refer to
    `lecture notes <https://www.cs.toronto.edu/~tijmen/csc321/slides/lecture_slides_lec6.pdf>`_ by G. Hinton.
    and centered version `Generating Sequences
    With Recurrent Neural Networks <https://arxiv.org/pdf/1308.0850v5.pdf>`_.
    The implementation here takes the square root of the gradient average before
    adding epsilon (note that TensorFlow interchanges these two operations). The effective
    learning rate is thus :math:`\gamma/(\sqrt{v} + \epsilon)` where :math:`\gamma`
    is the scheduled learning rate and :math:`v` is the weighted moving average
    of the squared gradient.
    aŸ