A Implementation of PS CD Algorithm

In this section, we provide two different ways to prove Theorem 2. The first one is more straightforward and directly differentiates through the term To solve this issue, we introduce the following variational representation: Lemma 1. With Jensen's inequality, we have: log null null As introduced in Equation (9) in Section 2.3, the divergence corresponding to the This is a direct consequence of Lemma 2. It can also be verified by checking the PS-CD Lemma 3. When 1 γ < 0, we have: S We first make the following assumption, which is similar to the one used in [4, 47]: Assumption 1. The assumption is typically easy to enforce in practice. In this section, we analyze the convergence property of the PS-CD algorithm presented in Algorithm 1. We have the following theorem that characterizes the convergence property of Algorithm 2: Theorem 5. Monte Carlo estimation will incur additional approximation error.