I'm employing the Actor-Critic algorithm. The critic network approximates the action-value function, i.e. $Q(s, a)$, which determines how good a particular state is, when provided with an action.
$Q(s, a)$ is approximated using the backpropagation of the temporal difference error (TD error). We can understand that $Q(s, a)$ has been approximated properly when TD error is minimized, i.e. when it is saturated at lower values.
My question is, when exactly can you say that $Q(s, a)$ is approximated properly, if you don't have TD error, i.e. if you have to plot the graph between $Q(s, a)$ vs episodes, then what would be the optimal behaviour?
Will it be increasing exponential with saturation value around reward values, or increasing exponential with saturation (around any value)?
Follow up: What can be the possible mistake, if the output of Q-value function is around 5x the rewards, and not saturating?