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I'm struggling to understand the difference between actor-critic and advantage actor-critic.

At least, I know they are different from asynchronous advantage actor-critic (A3C), as A3C adds an asynchronous mechanism that uses multiple worker agents interacting with their own copy of the environment and reports the gradient to the global agent.

But what is the difference between the actor-critic and advantage actor-critic (A2C)? Is it simply with or without advantage function? But, then, does the actor-critic have any other implementation except for the use of advantage function?

Or maybe are they synonyms and actor-critic is just a shorthand for A2C?

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Actor-Critic is not just a single algorithm, it should be viewed as a "family" of related techniques. They're all techniques based on the policy gradient theorem, which train some form of critic that computes some form of value estimate to plug into the update rule as a lower-variance replacement for the returns at the end of an episode. They all perform "bootstrapping" by using some sort of prediction of value.

Advantage Actor-Critic specifically uses estimates of the advantage function $A(s, a) = V(s) - Q(s, a)$ for its bootstrapping, whereas "actor-critic" without the "advantage" qualifier is not specific; it could be a trained $V(s)$ function, it could be some sort of estimate of $Q(s, a)$, it could be a variety of things.

In practice, the critic of Advantage Actor-Critic methods actually can just be trained to predict $V(s)$. Combined with an empirically observed reward $r$, they can then compute the advantage estimate $A(s, a) = r + \gamma V(s') - V(s)$.

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    $\begingroup$ Nice clarification. Do you know an example of an actor-critic method that is not advantage actor-critic? Also, would it be fair to say that PPO is a type of advantage actor-critic method? $\endgroup$ – Mei Zhang Feb 18 '19 at 10:06
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    $\begingroup$ @MeiZhang Some options are listed here, but I don't think they have clear names really. Advantages are certainly the most commonly-used. Sure, PPO could be viewed as an advantage actor-critic method. $\endgroup$ – Dennis Soemers Feb 18 '19 at 10:35
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According to Sutton and Barto, they are the same thing. Note 13.5-6 (page 338) of their Reinforcement Learning: An Introduction, 2nd Edition book:

Actor-critic methods are sometimes referred to as advantage actor-critic methods in the literature

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Though the word "Advantage" in the actor-critic realm has been used to refer to the difference between the state value and the state action value, A2C brings in the ideas of A3C. In A3C, several worker networks interact with different copies of the environment (asynchronous learning) and update a master network after a set if steps. This was meant to solve instability issues associated with both temporal difference update method and correlations within neural network generated prediction and target values. However it was noticed by OpenAI that there was no need of the asynchrony, i.e. there was no practical benefit of having different worker networks. Instead, they had the same copy of the network that interacted with different copies of the environment (one works from the beginning, another working backwards from the end) and they update at once without the master lagging behind like in A3C. The removal of asynchrony gave rise to A2C.

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