    \begin{equation}\label{eq:learn_a}
        \alpha^{(i)}_{mt}
            \begin{cases}
                \alpha^{(i)}_{m\{t-1\}}+1 \textrm{ if } U_m(\mathbf{d}_{t}) \geq U_m(\mathbf{d}_{t-1}),\\
                \alpha^{(i)}_{m\{t-1\}} \textrm{ otherwise.}
            \end{cases}
    \end{equation}
