What is the difference between V(s) and Q(s,a) in reinforcement learning?
answer
- one scores situations, one scores decisions
- both are expected future totals
- the second one pins the first action
- average over the policy to convert
- choosing an action without a model
basics
~20 sV(s) is the expected return from state s when the agent follows its policy from there on. Q(s,a) is the expected return from taking action a in state s first, then following the same policy.
solid answer
~50 sBoth are expected returns — the discounted sum of all future reward — but they condition on different things. `V(s)` conditions on being in state s and letting the policy choose every action, including the next one. `Q(s,a)` pins the first action to a and only then hands control back to the policy, so it scores a decision rather than a situation. They are tied together by `V(s) = sum over a of pi(a|s) * Q(s,a)`; for a deterministic policy that collapses to V(s) = Q(s, pi(s)). The practical consequence is that Q supports action selection on its own — compare the Q values in a state and pick the largest — whereas ranking actions from V requires knowing where each action leads and what it pays. In an elevator dispatcher, V scores the whole building state; Q scores "send car 3 to floor 8" from that state.
go deeper
Be ready to state both definitions cleanly as expected future totals, and to say which one conditions on the first action. Getting the words 'expected return' rather than 'reward' right is most of the mark here.
Explain the conversion V(s) = sum over a of pi(a|s) * Q(s,a) and why the reverse direction needs the transition dynamics. Interviewers at this level expect you to know both are indexed by a policy.
Show you can pick the right representation for a real system: Q where the agent must choose without a simulator, V where you only need to score situations or where the action set is far too large to tabulate.
Own the tradeoff between representation cost and decision usefulness — |S| versus |S|*|A| numbers, data needed per action, and whether the team should invest in a model that makes cheap state values actionable instead.
## The objects being defined A **policy** `pi` is the agent's rule for choosing actions: `pi(a|s)` is the probability of taking action a in state s (a deterministic policy just puts all the probability on one action). The **return** from time t is the discounted total of everything that follows: `G_t = R_{t+1} + g*R_{t+2} + g^2*R_{t+3} + ...`, where g is the discount factor between 0 and 1. Both value functions are expectations of that same quantity — they differ only in what is held fixed. **State value.** `V_pi(s) = E_pi[G_t | S_t = s]`. Read it as: standing in state s, following policy pi from now on, how much total discounted reward do I expect? It is one number per state. **Action value.** `Q_pi(s,a) = E_pi[G_t | S_t = s, A_t = a]`. Read it as: standing in state s, taking action a *right now* even if the policy would not have chosen it, and following pi afterwards. It is one number per state-action pair. ## How they relate Averaging Q over what the policy would actually do recovers V: `V_pi(s) = sum over a of pi(a|s) * Q_pi(s,a)` and going the other way requires one step of the environment's dynamics: `Q_pi(s,a) = E[ R + g * V_pi(S') | s, a ]` where S' is the next state. That asymmetry is the whole point. Turning Q into V needs only the policy, which the agent already has. Turning V into a comparison between actions needs the transition probabilities and the reward for each action — a **model** of the environment. An agent holding only V and no model can say "this situation is worth 12" but cannot say which of its three actions is best. ## Why control usually wants Q Given Q, greedy action selection is a table lookup: `pick argmax over a of Q(s,a)`. Given V, the same choice needs a one-step lookahead, `argmax over a of E[R + g*V(S') | s,a]`, which is only available if you can predict R and S'. This is why action values are the natural currency for an agent that must act without a simulator of its world, while state values are natural for evaluating or comparing situations. ## What each costs A tabular V has one entry per state; a tabular Q has one per state-action pair, so |S| versus |S|*|A| numbers. Q is correspondingly more expensive to store and needs more experience to estimate, because every action in every state needs its own evidence rather than sharing one number per state. That is a real tradeoff, not a formality: in a dispatcher with a large action set, most (state, action) pairs may never be visited. ## An example An elevator controller in an office tower: the state bundles the current car positions, their directions, and the waiting calls. `V(state)` is a single score for that whole configuration — useful for saying "the morning rush leaves the building in a worse position than the afternoon lull". `Q(state, send car 3 to floor 8)` scores one dispatch decision from that configuration, and the set of Q values across candidate dispatches is exactly what a controller needs to choose. Note that Q for a *bad* first action is still well defined: it assumes the mistake is made once and then the usual policy resumes, which is precisely what makes Q useful for comparing alternatives. ## Common confusions Neither function is the immediate reward. `Q(s,a)` is not "the reward for doing a"; it is the reward for doing a plus everything the resulting state is worth thereafter. A state that pays nothing can have a very high value if it leads somewhere lucrative. Neither is a probability either — values are on the reward's scale and can be negative or larger than one, unless the problem happens to use a terminal 0/1 reward with no discounting, in which case V does coincide with a success probability. Finally, both are indexed by a policy. `V_pi` and `Q_pi` describe *that* policy's behaviour, not the best possible behaviour; the starred versions `V*` and `Q*` describe the optimum and are different objects.
- How do you recover V from Q for a given policy?Average Q over the policy's action probabilities: `V_pi(s) = sum over a of pi(a|s) * Q_pi(s,a)`. For a deterministic policy this is just `V_pi(s) = Q_pi(s, pi(s))` — the action value of the one action the policy takes. Going the other direction is not free: recovering Q from V needs the transition probabilities and rewards for one step.
- Why is Q the more useful object for an agent with no model of the environment?Because acting greedily on Q is a lookup — compare Q(s,a) across the actions available and take the largest. Acting greedily on V needs `argmax over a of E[R + g*V(S')|s,a]`, which requires knowing where each action leads and what it pays. Without those dynamics, V ranks states but cannot rank actions.
- What does Q(s,a) mean when a is an action the policy would never choose?It is still well defined: take a once, then follow the policy from whatever state results. That counterfactual is exactly what makes Q useful — it prices alternatives the current policy ignores, which is how you can tell that the current choice is or is not the best one.
- How much more data does estimating Q take than estimating V?Roughly a factor of the action-set size. V has one number per state; Q has one per state-action pair, so each action in each state needs its own evidence instead of sharing a single per-state estimate. With a large action set, many pairs may be visited rarely or never.
V rates a chess position with you to move; Q rates each individual move you could make from that position.
saying these in an interview costs you the question
- Says Q(s,a) is the immediate reward for taking a
- Thinks V(s) is the reward received in state s
- Claims a state paying no reward must have value zero
- Says you can rank actions from V without a model
- Treats V and Q as probabilities of success