# What is Advantage in Reinforcement Learning?

Created at 2018-12-21 Updated at 2018-12-21 Category Reinforcement Learning

In reinforcement learning, an agent behaves according to a stochastic policy $\pi$, the values of the state-action pair $\left (s, a \right )$ and the state $s$ are defined as follows

$$Q^{\pi}(s, a) = \mathbb{E} [R_t | s_t = s, a_t = a, \pi ],$$
and

$$V^{\pi}(s) = \mathbb{E}_{a \sim \pi(s)} [Q^{\pi}(s, a)]$$

The former measures the value of choosing a particular action given a state, however the later shows how good it is to be in a particular state $s$, one can easily find that it is the expectation of all actions $a \sim \pi( \cdot )$ under a given state $s$ and a policy $\pi( \cdot )$

The Advantage function can be defined as

$$A^{\pi} (s, a) = Q^{\pi} (s, a) - V^{\pi} (s)$$

Note that $\mathbb{E}_{a \sim \pi(s)} [A^{\pi}(s, a)] = 0$. The advantage function substracts the value of the state from $Q$ function to gain a meatures dipicting the relative significance of curren action $a$. More intuitively, $V$ serves as a baseline and advantage defines the relative importance of actions, that is to say, a good action can gain a large advantage.

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