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