# Where does entropy enter in Soft Actor-Critic?

I am currently trying to understand SAC (Soft Actor-Critic), and I am thinking of it as a basic actor-critic with the entropy included. However, I expected the entropy to appear in the Q-function. From SpinningUp-SAC, it looks like the entropy is entering through the value-function, so I'm thinking it enters by the $$\log \pi_{\phi}(a_t \mid s_t)$$ in the value function?

I'm a little stuck on understanding SAC, can anyone confirm/explain this to me?

Also, side-note question: is being a soft agent equivalent to including entropy in one of the object functions?

In the answer I'll be using notation similar to the one from the SAC paper. If we look at the standard objective function for policy gradient methods we have \begin{align} J_\pi &= V_\pi(s_t)\\ &= \mathbb E_{a_t \sim \pi(a|s_t)}[Q(s_t, a_t)]\\ &= \mathbb E_{a_t \sim \pi(a|s_t)}[ \mathbb E_{s_{t+1} \sim p(s|s_t, a_t)} [r(s_t, a_t) + V(s_{t+1})]]\\ &= \mathbb E_{a_t \sim \pi(a|s_t)}[ \mathbb E_{s_{t+1} \sim p(s|s_t, a_t)} [r(s_t, a_t) + \mathbb E_{a_{t+1} \sim \pi(a|s_{t+1})}[ \mathbb E_{s_{t+2} \sim p(s|s_{t+1}, a_{t+1})} [r(s_{t+1}, a_{t+1}) + V(s_{t+2})]]]]\\ &\cdots\\ &= \sum_t \mathbb E_{(a_t, s_t) \sim \rho_\pi} [r( s_t, a_t)] \end{align} If you keep unwinding this $$V(s_{t+i})$$ you will get expected sum of rewards. We can define soft state value as \begin{align} V(s_t) &= \mathbb E_{a_t \sim \pi(a|s_t)}[Q(s_t, a_t) + \mathcal H(\cdot|s_t)]\\ &= \mathbb E_{a_t \sim \pi(a|s_t)}[Q(s_t, a_t) + \mathbb E_{a \sim \pi(a|s_t)}[-\log(\pi(a|s_t))]]\\ &= \mathbb E_{a_t \sim \pi(a|s_t)}[Q(s_t, a_t) - \log(\pi(a_t|s_t))] \end{align} third equality comes from the fact that $$\mathbb E_{a \sim \pi(a|s_t)}[-\log(\pi(a|s_t))]$$ is nonrandom so its the same thing as if we are sampling over $$\pi$$ only once.
In maximum entropy framework objective function would then be \begin{align} J_\pi &= V_\pi(s_t)\\ &= \mathbb E_{a_t \sim \pi(a|s_t)}[Q(s_t, a_t) - \log(\pi(a_t|s_t))]\\ &= \mathbb E_{a_t \sim \pi(a|s_t)}[ \mathbb E_{s_{t+1} \sim p(s|s_t, a_t)} [r(s_t, a_t) - \log(\pi(a_t|s_t)) + V(s_{t+1})]]\\ & \cdots\\ &= \sum_t \mathbb E_{(a_t, s_t) \sim \rho_\pi} [r(s_t, a_t) -\log(\pi(a_t|s_t))]\\ &= \sum_t \mathbb E_{(a_t, s_t) \sim \rho_\pi} [r(s_t, a_t) + \mathcal H(\cdot|s_t)] \end{align}