# Proof that the addition of a baseline to the REINFORCE algorithm reduces the variance

A widely used variation of REINFORCE is to subtract a baseline value $$b$$ from the return $$G_t$$ to reduce the variance of gradient estimation, such that

\begin{align} \nabla_\theta J(\theta) & \propto \sum_s d(s|\pi_\theta) \sum_a (q_\pi(s,a)-b(s)) \nabla_\theta \pi(a|s,\theta) \\ \end{align}

I haven't found any proof that the baseline reduces the variance of the gradient estimation, is there one?