Reinforcement Learning

Q-Learning and SARSA

Q-learning and SARSA both update the action just taken; SARSA bootstraps from the next action the behavior policy sampled, while Q-learning bootstraps from the largest next action value in the current table.

status: reviewimportance: importantdifficulty 4/5math: graduateread: 22mlive demo

Back to Reinforcement Learning Next: Function Approximation in RL (review)

Concept Structure

Q-Learning and SARSA

01Intuition

Start with the picture, metaphor, or geometric mechanism.

02Math

Make the objects explicit and connect them with notation.

03Code

Mirror the equations with runnable implementation details.

04Interactive Demo

Manipulate the mechanism and watch the idea respond.

1prerequisites

2next concepts

4related links

Learner Contract

What this page should let you do.

You are here becauseQ-learning and SARSA both update the action just taken; SARSA bootstraps from the next action the behavior policy sampled, while Q-learning bootstraps from the largest next action value in the current table.

This Reinforcement Learning concept is the current idea: keep the same invariant visible across Intuition, Math, Code, Interactive Demo.

Before thisTemporal-Difference Learning (review)

1 prerequisite listed; refresh them before leaning on the math or code.

Temporal-Difference Learning (review)

By the end4/4 sections ready | runnable code expected | live demo

Explain the mechanism, trace the main notation, and test one prediction in the live demo.

Do this firstIntuition

Read the intuition before the notation; the math should name a mechanism you already felt.

Then go nextFunction Approximation in RL (review)

Follow this edge after making one prediction here; the next page should reuse the result, not restart the route.

Test the linkManipulate one control and predict the visible change.Then continue to Function Approximation in RL (review)

Claim/source review status

Substantive review recorded

1/1 claims have bounded review metadata; still check caveats and source scope.Metadata-derived; review may be AI-assisted. Not a human certification.

Claims1/1 reviewed

Sources3 cited

Codeattached

Demolive

Reviewed2026-07-03

Updatedpage 2026-07-03

Learning item flow

4/4 sections readyAsk about this Research room

ConceptQ-Learning and SARSAReinforcement Learning EquationQ-Learning and SARSA equation 1Exact equation CodeQ-Learning and SARSA runnable code 1Exact runnable code DemoQ-Learning and SARSA interactive demoInteractive demo ClaimIn finite tabular one-step TD control, SARSA updates Q(S_t,A_t) towar...Exact claim check SourceReinforcement Learning: An Introduction, Second EditionExact source

ConceptQ-Learning and SARSAReinforcement Learning

3 sources attachedLocal snapshot ready

concept:reinforcement-learning/q-learning-sarsa

Codenearby Predictbefore reveal Roomitem handoff

01IntuitionStart with the picture, metaphor, or geometric mechanism.02MathMake the objects explicit and connect them with notation.03CodeMirror the equations with runnable implementation details.04Interactive DemoManipulate the mechanism and watch the idea respond.

Intuition

Build the mental picture first so the rest of the page has something to attach to.

Section prompt

Temporal-difference learning updated a state value. Control needs one more question:

If I was in this state and took this action, how good was that action?

That is why Q-learning and SARSA learn $Q(s,a)$ , an action value, instead of only $V(s)$ . Both methods observe one transition,

S_t, A_t, R_{t+1}, S_{t+1},

then update the value of the action that was just taken.

The split happens at the next state. SARSA asks, "What action did my behavior policy actually sample next?" Q-learning asks, "What is the best next action according to my current table?"

When exploration chooses the same action that the table already thinks is best, the two updates match. When exploration chooses a different action, SARSA learns about the exploratory path it is actually following, while Q-learning learns toward the greedy target path.

Math

Translate the story into symbols, assumptions, and a derivation you can inspect.

Section prompt

Equation 1 $Q(S_t,A_t) \leftarrow Q(S_t,A_t) + \alpha \left[ R_{t+1} + \gamma Q(S_{t+1},A_{t+1}) - Q(S_t,A_t) \right].$ Equation 2 $(S_t,A_t,R_{t+1},S_{t+1},A_{t+1}).$

Assume a finite discounted MDP and a tabular action-value estimate $Q(s,a)$ . At time $t$ , the agent observes state $S_t$ , takes action $A_t$ , receives reward $R_{t+1}$ , and lands in $S_{t+1}$ .

SARSA is on-policy one-step TD control. It samples the next action $A_{t+1}$ from the current behavior policy, then updates with

Q(S_t,A_t) \leftarrow Q(S_t,A_t) + \alpha \left[ R_{t+1} + \gamma Q(S_{t+1},A_{t+1}) - Q(S_t,A_t) \right].

The name SARSA comes from the quintuple

(S_t,A_t,R_{t+1},S_{t+1},A_{t+1}).

Q-learning is off-policy one-step TD control. It can act with an exploratory behavior policy, but its target uses the greedy next action under the current table:

Q(S_t,A_t) \leftarrow Q(S_t,A_t) + \alpha \left[ R_{t+1} + \gamma \max_a Q(S_{t+1},a) - Q(S_t,A_t) \right].

If $S_{t+1}$ is terminal, the bootstrap term is zero for both algorithms.

The important comparison is not "which algorithm is always better." It is what each update is estimating from the same experience:

SARSA target: reward plus the sampled next action value.
Q-learning target: reward plus the largest next action value.
Behavior policy: the policy that generated the data.
Target policy: the policy the update is evaluating or improving toward.

Code

Keep the implementation aligned with the notation so the algorithm is legible.

Section prompt

Runnable code 1gamma = 0.90 alpha = 0.40 current_q = 1.10 reward = 0.20 next_q = { "repeat": 2.40, # greedy...python

gamma = 0.90
alpha = 0.40
current_q = 1.10
reward = 0.20

next_q = {
    "repeat": 2.40,  # greedy under the table
    "ask": 1.00,     # sampled by exploration
    "skim": 0.60,
}

sampled_next_action = "ask"

sarsa_target = reward + gamma * next_q[sampled_next_action]
q_learning_target = reward + gamma * max(next_q.values())

for name, target in [
    ("SARSA", sarsa_target),
    ("Q-learning", q_learning_target),
]:
    error = target - current_q
    updated = current_q + alpha * error
    print(name, "target", round(target, 3), "error", round(error, 3), "new Q", round(updated, 3))

The same observed transition can produce different targets because the next action used by the bootstrap is different.

Interactive Demo

Use direct manipulation to connect the explanation to a moving system.

Section prompt

Use the Q-learning vs SARSA Target Lab to predict which update changes the current action value more.

Choose a transition case, inspect the current $Q(S_t,A_t)$ and the next-state action values, then commit to whether SARSA, Q-learning, or neither should update more. After reveal, the lab shows the sampled next action, the max next action, both targets, both TD errors, and the new value each algorithm would write.

The terminal case and sampled-greedy case are there on purpose: they show when the algorithms collapse to the same one-step target.

Live Concept Demo

Explore Q-Learning and SARSA

The stage is code-native and interactive. Use it to test the explanation against the mechanism.

difficulty 4/5graduatecode-aligned

Demo Prediction Checkpoint

Manipulate one control and predict the visible change.

Commit to what Q-Learning and SARSA should make visible before reading the result.

After The First Pass

Turn the concept into an inspected object.

Once the invariant is visible in the intuition, math, code, and demo, use these panels to inspect the mechanism visually, check source support, practice the idea, and attach a grounded research question.

Mechanism Storyboard

See the idea move before the page explains it

Prediction open01 / Intuition

Prediction lens

Start with the picture, metaphor, or geometric mechanism.

Commit first

Before reading further, choose the kind of change Q-Learning and SARSA should make visible.

Visual Inquiry

Make the image answer a mathematical question

4/4 stages readyLive demo connected

Visual cueWhich visible object should carry the first intuition?

Inspection depth2/4

Prediction

Which visible object should carry the first intuition?

Commit first

Pick the cue that should make Q-Learning and SARSA easier to reason about before the page gives the answer.

Source Grounding

Canonical references for the mechanism on this page.

book · 2018Reinforcement Learning: An Introduction, Second EditionRichard S. Sutton and Andrew G. Barto

Canonical source for SARSA as on-policy TD control, Q-learning as off-policy TD control, the sampled next action versus max-over-actions distinction, and cliff-walking behavior contrast.

Open source

course-notes · 2026Stanford CS234: Lecture 4, Model-Free Control and Function ApproximationStanford CS234

Graduate RL source for on-policy/off-policy definitions, tabular Q-learning update, SARSA update, epsilon-greedy behavior policy, and function-approximation caveats.

Open source

course-notes · 2026UC Berkeley CS188: Model-Free LearningUC Berkeley CS188

Course-note source for model-free learning, active Q-learning, feedback-driven policy improvement, and the exploration requirement.

Open source

Claim Review

Q-learning and SARSA both update the action just taken; SARSA bootstraps from the next action the behavior policy sampled, while Q-learning bootstraps from the largest next action value in the current table.

Status1 substantive review recorded

Claims without a substantive review badge still need exact source-support review.

Sources3 references

sutton-barto-2018-rl, stanford-cs234-model-free-control, berkeley-cs188-model-free-learning

Local checks4 local checks

Use equations, runnable code, and demos to check whether the source support is operational.

Substantively reviewedIn finite tabular one-step TD control, SARSA updates Q(S_t,A_t) toward R_{t+1} + gamma Q(S_{t+1},A_{t+1}) using the next action actually sampled by the behavior policy, while Q-learning updates Q(S_t,A_t) toward R_{t+1} + gamma max_a Q(S_{t+1},a), making the target greedy even when the behavior policy explores.Claim metadata: source checked

The sources support finite tabular one-step action-value TD control where both algorithms update the visited state-action pair, but SARSA bootstraps from the sampled next action and Q-learning bootstraps from the greedy next action.

Sources: Reinforcement Learning: An Introduction, Second Edition, Stanford CS234: Lecture 4, Model-Free Control and Function Approximation, UC Berkeley CS188: Model-Free LearningFinite tabular teaching MDP only; excludes Expected SARSA, n-step methods, eligibility traces, Double Q-learning, DQN, replay buffers, target networks, continuous spaces, and function-approximation convergence failures beyond a caveat note.A bounded review summary is present; still check caveats and exact reference scope.

Checked Sutton/Barto Chapter 6 for SARSA's quintuple update and Q-learning's off-policy max-over-next-actions update; checked Stanford CS234 Lecture 4 for on/off-policy definitions, Q-learning and SARSA equations, epsilon-greedy behavior, and tabular convergence caveats; checked Berkeley CS188 for model-free Q-learning as active reinforcement learning with exploration. GPT Pro publication critique remains pending because 127.0.0.1:51672 is unavailable.

Reviewer: codex-local-source-review; reviewed 2026-07-03

source-span-sutton-barto-2018-rl source-span-stanford-cs234-model-free-control source-span-berkeley-cs188-model-free-learning Equation 1 Equation 2 Runnable code 1 Interactive demo

Source support candidates

book 2018Reinforcement Learning: An Introduction, Second Edition

Canonical source for SARSA as on-policy TD control, Q-learning as off-policy TD control, the sampled next action versus max-over-actions distinction, and cliff-walking behavior contrast.

course-notes 2026Stanford CS234: Lecture 4, Model-Free Control and Function Approximation

Graduate RL source for on-policy/off-policy definitions, tabular Q-learning update, SARSA update, epsilon-greedy behavior policy, and function-approximation caveats.

course-notes 2026UC Berkeley CS188: Model-Free Learning

Course-note source for model-free learning, active Q-learning, feedback-driven policy improvement, and the exploration requirement.

Mechanism checks

Equation 1

Q(S_t,A_t) \leftarrow Q(S_t,A_t) + \alpha \left[ R_{t+1} + \gamma Q(S_{t+1},A_{t+1}) - Q(S_t,A_t) \right].

Equation 2

(S_t,A_t,R_{t+1},S_{t+1},A_{t+1}).

Runnable code 1gamma = 0.90 alpha = 0.40 current_q = 1.10 reward = 0.20 next_q = { "repeat": 2.40, # greedy...Demo stateLive mechanism probe

Practice Loop

Try the idea before it explains itself

Readiness0/3 checks ready

Predict

Before touching the demo, predict one visible change that should happen in Q-Learning and SARSA.

Hint 1

Reveal when your model needs a nudge.

Hint 2

Reveal when your model needs a nudge.

Hint 3

Reveal when your model needs a nudge.

Your answer canvas

Local checks

Claim

A concrete answer is on the canvas.

Mechanism

The answer names why the claim should hold.

Bridge

It touches the page context or a neighboring idea.

Misconception check

Grounded research drawerClose

ConceptQ-Learning and SARSAReinforcement Learning

Runnable code comparisonQ-Learning and SARSA runnable code 1gamma = 0.90 alpha = 0.40 current_q = 1.10 reward = 0.20 next_q = { "repeat": 2.40, # greedy...Prediction before revealQ-Learning and SARSA interactive demoManipulate one control and predict the visible change.

Grounded room questionWhat is the smallest example that makes Q-Learning and SARSA click without losing the math?Local snapshot ready

Research Room

Attach the question to a claim, equation, code, or demo

Pick the concept, equation, source, runnable code, claim, misconception, or demo state before asking for help. The handoff keeps that page item in context.

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conceptReinforcement Learning

Q-Learning and SARSA

Attached question

What is the smallest example that makes Q-Learning and SARSA click without losing the math?

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Evidence to inspect

References to inspect: attached references on this page.
Definition, prerequisite, and contrast concept links
The equation or runnable code that makes the concept operational
One demo state that shows the invariant instead of a slogan

What would resolve this

The learner can state the mechanism in their own words
The learner can name the prerequisite that would repair confusion
The learner can predict how the mechanism changes under one perturbation

Grounded AI handoff

I am working in Continuous Function's research reading room. Object: concept - Q-Learning and SARSA Selected item key: recorded for copy. Context: Reinforcement Learning Page anchor: recorded for copy. Open question: What is the smallest example that makes Q-Learning and SARSA click without losing the math? Evidence to inspect: - References to inspect: attached references on this page. - Definition, prerequisite, and contrast concept links - The equation or runnable code that makes the concept operational - One demo state that shows the invariant instead of a slogan What would resolve this: - The learner can state the mechanism in their own words - The learner can name the prerequisite that would repair confusion - The learner can predict how the mechanism changes under one perturbation Answer as a careful research tutor: stay source-grounded, separate verified evidence from assumptions, name the relevant math objects, and end with one next action.

View it in context

concept/concept-notebook/reinforcement-learning/q-learning-sarsa
concept:reinforcement-learning/q-learning-sarsa

Learning Map

Before / Now / Try / Next

BeforeTemporal-Difference Learning (review)

NowIntuition -> Math -> Code -> Demo

TryManipulate one control and predict the visible change.

NextFunction Approximation in RL (review)

Intuitionready
Mathready
Codeready
Interactive Demoready

AI Companion

Ask beside the selected object

Your question

GoalComfortStyleStuck on

Context prompt

You are my AI learning companion for Continuous Function. Current context: Reinforcement Learning concept. Learning surface: Q-Learning and SARSA. What this page says: Q-learning and SARSA both update the action just taken; SARSA bootstraps from the next action the behavior policy sampled, while Q-learning bootstraps from the largest next action value in the current table. Current section: Intuition, math, code, and interactive demo. Suggested next step: Manipulate one control and predict the visible change.. Learner goal: Understand the idea. Learner comfort level: New to this. Preferred explanation style: Visual first. Task: Explain the central idea in plain language, then restate it with the exact math objects from the page. Answer in a way that helps me learn: ask one clarifying question only if needed, use intuition before notation, and end with one thing I should try on the page.

Domain

Reinforcement Learning

reinforcement-learningq-learningsarsatd-controlaction-valuesoff-policy-learning

Prerequisites

Temporal-Difference Learning (review)

Leads To

Function Approximation in RL (review)Deep Q-Learning (review)

Bellman Equations (review)Dynamic Programming and Value Iteration (review)Policy Iteration (review)Monte Carlo Reinforcement Learning (review)

Within this domain