    v_0 \leftarrow 0 \text{ ( second moment)},                                       \\
            &\hspace{18mm} \rho_{\infty} \leftarrow 2/(1-\beta_2) -1                      \\[-1.ex]
            &\rule{110mm}{0.4pt}  \\
            &\textbf{for} \: t=1 \: \textbf{to} \: \ldots \: \textbf{do}                         \\
            &\hspace{6mm}g_t           \leftarrow   \nabla_{\theta} f_t (\theta_{t-1})           \\
            &\hspace{5mm} \textbf{if} \: \lambda \neq 0                                          \\
            &\hspace{10mm} g_t \leftarrow g_t + \lambda \theta_{t-1}                             \\
            &\hspace{6mm}m_t           \leftarrow   \beta_1 m_{t-1} + (1 - \beta_1) g_t          \\
            &\hspace{6mm}v_t           \leftarrow   \beta_2 v_{t-1} + (1-\beta_2) g^2_t          \\
            &\hspace{6mm}\widehat{m_t} \leftarrow   m_t/\big(1-\beta_1^t \big)                   \\
            &\hspace{6mm}\rho_t \leftarrow \rho_{\infty} -
                2 t \beta^t_2 /\big(1-\beta_2^t \big)                                    \\[0.1.ex]
            &\hspace{6mm}\textbf{if} \: \rho_t > 5                                               \\
            &\hspace{12mm} l_t \leftarrow \frac{\sqrt{ (1-\beta^t_2) }}{ \sqrt{v_t} +\epsilon  } \\
            &\hspace{12mm} r_t \leftarrow
      \sqrt{\frac{(\rho_t-4)(\rho_t-2)\rho_{\infty}}{(\rho_{\infty}-4)(\rho_{\infty}-2) \rho_t}} \\
            &\hspace{12mm}\theta_t \leftarrow \theta_{t-1} - \gamma \widehat{m_t} r_t l_t        \\
            &\hspace{6mm}\textbf{else}                                                           \\
            &\hspace{12mm}\theta_t \leftarrow \theta_{t-1} - \gamma \widehat{m_t}                \\
            &\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 `On the variance of the adaptive learning rate and beyond`_.

    This implementation uses the same weight_decay implementation as Adam (were the weight_decay is applied
    to the gradient) and not the one from AdamW (were weight_decay is applied to the update). This
    is different from the `author's implementation`_.
    a