$$
\begin{array}{l} \theta_ {t + 1} = \arg \max  _ {\theta} \frac {1}{\alpha_ {0} (x , a)} \int_ {\mathcal {X}, \mathcal {A}} \left(\rho_ {M ^ {*}} ^ {\kappa} (x, a) \int_ {\mathcal {X}} M ^ {*} \left(x ^ {\prime} | x, a\right) f \left(\frac {\rho_ {M _ {\theta_ {t}}} ^ {\kappa} \left(x , a , x ^ {\prime}\right)}{\rho_ {M ^ {*}} ^ {\kappa} \left(x , a , x ^ {\prime}\right)}\right) \log M _ {\theta} \left(x ^ {\prime} | x, a\right) d x ^ {\prime} - \right. \\ \rho_ {M ^ {*}} ^ {\kappa} (x, a) f \left(\frac {\rho_ {M _ {\theta_ {t}}} ^ {\kappa} (x , a)}{\rho_ {M ^ {*}} ^ {\kappa} (x , a)}\right) \int_ {\mathcal {X}} M ^ {*} \left(x ^ {\prime} | x, a\right) \log M _ {\theta} \left(x ^ {\prime} | x, a\right) d x ^ {\prime} + \\ \rho_ {M ^ {*}} ^ {\kappa} (x, a) H _ {M ^ {*}} (x, a) \int_ {\mathcal {X}} M ^ {*} \left(x ^ {\prime} | x, a\right) \log M _ {\theta} \left(x ^ {\prime} | x, a\right) d x ^ {\prime}) d a d x \\ \approx \arg \max _ {\theta} \int_ {\mathcal {X}, \mathcal {A}} \left(\rho_ {M _ {\theta_ {t}}} ^ {\kappa} (x, a) \int_ {\mathcal {X}} M _ {\theta_ {t}} (x ^ {\prime} | x, a) f ^ {\prime} \left(\frac {\rho_ {M _ {\theta_ {t}}} ^ {\kappa} (x , a , x ^ {\prime})}{\rho_ {M ^ {*}} ^ {\kappa} (x , a , x ^ {\prime})}\right) \log M _ {\theta} (x ^ {\prime} | x, a) d x ^ {\prime} - \right. \\ \rho_ {M ^ {*}} ^ {\kappa} (x, a) \int_ {\mathcal {X}} M ^ {*} \left(x ^ {\prime} | x, a\right) \left(f ^ {\prime} \left(\frac {\rho_ {M _ {\theta_ {t}}} ^ {\kappa} \left(x , a , x ^ {\prime}\right)}{\rho_ {M ^ {*}} ^ {\kappa} \left(x , a , x ^ {\prime}\right)}\right) - f (1)\right) \log M _ {\theta} \left(x ^ {\prime} | x, a\right) d x ^ {\prime} - \\ \rho_ {M _ {\theta_ {t}}} ^ {\kappa} (x, a) f ^ {\prime} \left(\frac {\rho_ {M _ {\theta_ {t}}} ^ {\kappa} (x , a)}{\rho_ {M ^ {*}} ^ {\kappa} (x , a)}\right) \int_ {\mathcal {X}} M ^ {*} \left(x ^ {\prime} | x, a\right) \log M _ {\theta} \left(x ^ {\prime} | x, a\right) d x ^ {\prime} + \\ \rho_ {M ^ {*}} ^ {\kappa} (x, a) \left(f ^ {\prime} \left(\frac {\rho_ {M _ {\theta_ {t}}} ^ {\kappa} (x , a)}{\rho_ {M ^ {*}} ^ {\kappa} (x , a)}\right) - f (1)\right) \int_ {\mathcal {X}} M ^ {*} \left(x ^ {\prime} | x, a\right) \log M _ {\theta} \left(x ^ {\prime} | x, a\right) d x ^ {\prime} + \\ \rho_ {M ^ {*}} ^ {\kappa} (x, a) H _ {M ^ {*}} (x, a) \int_ {\mathcal {X}} M ^ {*} \left(x ^ {\prime} | x, a\right) \log M _ {\theta} \left(x ^ {\prime} | x, a\right) d x ^ {\prime}) d a d x \\ = \arg \max  _ {\theta} \int_ {\mathcal {X}, \mathcal {A}, \mathcal {X}} \frac {1}{\alpha_ {0} (x , a)} \rho_ {M _ {\theta_ {t}}} ^ {\kappa} (x, a, x) \left(f ^ {\prime} \left(\frac {\rho_ {M _ {\theta_ {t}}} ^ {\kappa} (x , a , x ^ {\prime})}{\rho_ {M ^ {*}} ^ {\kappa} (x , a , x ^ {\prime})}\right) - f ^ {\prime} \left(\frac {\rho_ {M _ {\theta_ {t}}} ^ {\kappa} (x , a)}{\rho_ {M ^ {*}} ^ {\kappa} (x , a)}\right)\right) \log M _ {\theta} (x ^ {\prime} | x, a) d x ^ {\prime} d a d x + \\ \int_ {\mathcal {X}, \mathcal {A}, \mathcal {X}} \frac {1}{\alpha_ {0} (x , a)} \rho_ {M ^ {*}} ^ {\kappa} (x, a, x ^ {\prime}) \left(f ^ {\prime} \left(\frac {\rho_ {M _ {\theta_ {t}}} ^ {\kappa} (x , a)}{\rho_ {M ^ {*}} ^ {\kappa} (x , a)}\right) - f ^ {\prime} \left(\frac {\rho_ {M _ {\theta_ {t}}} ^ {\kappa} (x , a , x ^ {\prime})}{\rho_ {M ^ {*}} ^ {\kappa} (x , a , x ^ {\prime})}\right) + H _ {M ^ {*}} (x, a)\right) \log M _ {\theta} (x ^ {\prime} | x, a) d x ^ {\prime} d a d x. \\ \end{array}
$$