# What is the Q function and what is the V function in reinforcement learning?

It seems to me that the $V$ function can be easily expressed by the $Q$ function and thus the $V$ function seems to be superfluous to me. However, I'm new to reinforcement learning so I guess I got something wrong.

## Definitions

Q- and V-learning are in the context of Markov Decision Processes. A MDP is a 5-tuple $(S, A, P, R, \gamma)$ with

• $S$ is a set of states (typically finite)
• $A$ is a set of actions (typically finite)
• $P(s, s', a) = P(s_{t+1} = s' | s_t = s, a_t = a)$ is the probability to get from state $s$ to state $s'$ with action $a$.
• $R(s, s', a) \in \mathbb{R}$ is the immediate reward after going from state $s$ to state $s'$ with action $a$. (It seems to me that usually only $s'$ matters).
• $\gamma \in [0, 1]$ is called discount factor and determines if one focuses on immediate rewards ($\gamma = 0$), the total reward ($\gamma = 1$) or some trade-off.

A policy $\pi$, according to Reinforcement Learning: An Introduction by Sutton and Barto is a function $\pi: S \rightarrow A$ (this could be probabilistic).

According to Mario Martins slides, the $V$ function is $$V^\pi(s) = E_\pi \{R_t | s_t = s\} = E_\pi \{\sum_{k=0}^\infty \gamma^k r_{t+k+1} | s_t = s\}$$ and the Q function is $$Q^\pi(s, a) = E_\pi \{R_t | s_t = s, a_t = a\} = E_\pi \{\sum_{k=0}^\infty \gamma^k r_{t+k+1} | s_t = s, a_t=a\}$$

## My thoughts

The $V$ function states what the expected overall value (not reward!) of a state $s$ under the policy $\pi$ is.

The $Q$ function states what the value of a state $s$ and an action $a$ under the policy $\pi$ is.

This means, $$Q^\pi(s, \pi(s)) = V^\pi(s)$$

Right? So why do we have the value function at all? (I guess I mixed up something)

Q-values are a great way to the make actions explicit so you can deal with problems where the transition function is not available (model-free). However, when your action-space is large, things are not so nice and Q-values are not so convenient. Think of a huge number of action or even continuous action-spaces.

From a sampling perspective, the dimensionality of Q(s,a) is higher than V(s) so it might get harder to get enough (s,a) samples in comparison with (s). If you have access to the transition function sometimes V is good.

There are also other uses where both are combined. For instance, the advantage function where A(s,a)=Q(s,a)-V(s). If you are interested, you can find a recent example using advantage functions here:

Wang Z, de Freitas N, Lanctot M. Dueling Network Architectures for Deep Reinforcement Learning. arXiv:151106581

Available from: http://arxiv.org/abs/1511.06581

$V^\pi(s)$ is the state-value function of MDP (Markov Decision Process). It's the expected return starting from state S following policy $\pi$.

If you look at the expression: $$V^\pi(s) = E_\pi \{G_t | s_t = s\}$$ $G_t$ is the total DISCOUNTED reward from time step t, as opposed to $R_t$ which is an immediate return. Here you are taking the expectation of ALL actions according to the policy $\pi$.

$Q^\pi(s, a)$ is the action-value function. It is the expected return starting from state S, following policy $\pi$, taking action a. It's focusing on the particular action at the particular state.

$$Q^\pi(s, a) = E_\pi \{G_t | s_t = s, a_t = a\}$$

If you want to draw the relationship between $Q^\pi$ and $V^\pi$ (the value of being in that state), this is the relationship:

$$V^\pi(s) = \sum_{a ∈ A} \pi (a|s) * Q^\pi(a,s)$$

You sum every action-value multiplied by the probability to commit that action (the policy). If you think of the Grid world example, you multiply the probability of (up/down/right/left) with the one step ahead state value of (up/down/right/left)

• This is the most concise answer. – IzRey Aug 7 '18 at 23:03

I found the clearest explanation of Q-learning and how it works in Tom Mitchell's book "Machine Learning" (1997), ch. 13, which is downloadable. The computation of the Q function

$$Q(s,a ) = r(s,a ) + \gamma V^{*}(\delta(s,a))$$

is done by rewriting it in recursive form as $$Q(s, a) = r(s, a) + \gamma \max_{a'} Q(\delta(s, a), a')$$

using the defn of the V* function: $V^{*}(s)= \max_{a'} Q(s,a')$

This may seem an odd recursion at first because its expressing the Q value of an action in the current state in terms of the best Q value of a successor state, but it makes sense when you look at how the backup process uses it: The exploration process stops when it reaches a goal state and collects the reward, which becomes that final transition's Q value. Now in a subsequent training episode, when the exploration process reaches that predecessor state, the backup process uses the above equality to update the current Q value of the predecessor state. Next time its predecessor is visited that state's Q value gets updated, and so on back down the line (Mitchell's book describes a more efficient way of doing this by storing all the computations and replaying them later). Provided every state is visited infinitely often this process eventually computes the optimal Q

Sometimes you will see a learning rate $\alpha$ applied to control how much Q actually gets updated: $$Q(s, a) = (1-\alpha)Q(s, a) + \alpha(r(s, a) + \gamma \max_{a'} Q(s',a'))$$ $$= Q(s, a) + \alpha(r(s, a) + \gamma \max_{a'} Q(s',a') - Q(s,a))$$ Notice now that the update to the Q value does depend on the current Q value. Mitchell's book also explains why that is and why you need $\alpha$: its for stochastic MDPs. Without $\alpha$, every time a state,action pair was attempted there would be a different reward so the Q^ function would bounce all over the place and not converge. $\alpha$ is there so that as the new knowledge is only accepted in part. Initially $\alpha$ is set high so that the current (mostly random values) of Q are less influential. $\alpha$ is decreased as training progresses, so that new updates have less and less influence, and now Q learning converges

Here is a more detailed explanation of the relationship between state value and action value in Aaron's answer. Let's first take a look at the definitions of value function and action value function under policy $$\pi$$: \begin{align} &v_{\pi}(s)=E{\left[G_t|S_t=s\right]} \\ &q_{\pi}(s,a)=E{\left[G_t|S_t=s, A_t=a\right]} \end{align} where $$G_t=\sum_{k=0}^{\infty}\gamma^kR_{t+k+1}$$ is the return at time $$t$$. The relationship between these two value functions can be derived as \begin{align} v_{\pi}(s)&=E{\left[G_t|S_t=s\right]} \nonumber \\ &=\sum_{g_t} p(g_t|S_t=s)g_t \nonumber \\ &= \sum_{g_t}\sum_{a}p(g_t, a|S_t=s)g_t \nonumber \\ &= \sum_{a}p(a|S_t=s)\sum_{g_t}p(g_t|S_t=s, A_t=a)g_t \nonumber \\ &= \sum_{a}p(a|S_t=s)E{\left[G_t|S_t=s, A_t=a\right]} \nonumber \\ &= \sum_{a}p(a|S_t=s)q_{\pi}(s,a) \end{align} The above equation is important. It describes the relationship between two fundamental value functions in reinforcement learning. It is valid for any policy. Moreover, if we have a deterministic policy, then $$v_{\pi}(s)=q_{\pi}(s,\pi(s))$$. Hope this is helpful for you. (to see more about Bellman optimality equation https://stats.stackexchange.com/questions/347268/proof-of-bellman-optimality-equation/370198#370198)

The value function is an abstract formulation of utility. And the Q-function is used for the Q-learning algorithm.

• For the context of this question, the $V$ and $Q$ are different. – Siong Thye Goh Oct 24 '18 at 7:01