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	<meta charset="utf-8" /><meta name="hf:doc:metadata" content="{"title":"A Q-Learning example","local":"q-learning-example","sections":[{"title":"Step 1: Initialize the Q-table","local":"step1","sections":[],"depth":2},{"title":"Step 2: Choose an action using the Epsilon Greedy Strategy","local":"step2","sections":[],"depth":2},{"title":"Step 3: Perform action At, get Rt+1 and St+1","local":"step3","sections":[],"depth":2},{"title":"Step 4: Update Q(St, At)","local":"step4","sections":[],"depth":2},{"title":"Step 2: Choose an action using the Epsilon Greedy Strategy","local":"step2-2","sections":[],"depth":2},{"title":"Step 3: Perform action At, get Rt+1 and St+1","local":"step3-3","sections":[],"depth":2},{"title":"Step 4: Update Q(St, At)","local":"step4-4","sections":[],"depth":2}],"depth":1}">
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	<link rel="modulepreload" href="/docs/deep-rl-course/pr_592/en/_app/immutable/chunks/EditOnGithub.858acfec.js"><!-- HEAD_svelte-u9bgzb_START --><meta name="hf:doc:metadata" content="{"title":"A Q-Learning example","local":"q-learning-example","sections":[{"title":"Step 1: Initialize the Q-table","local":"step1","sections":[],"depth":2},{"title":"Step 2: Choose an action using the Epsilon Greedy Strategy","local":"step2","sections":[],"depth":2},{"title":"Step 3: Perform action At, get Rt+1 and St+1","local":"step3","sections":[],"depth":2},{"title":"Step 4: Update Q(St, At)","local":"step4","sections":[],"depth":2},{"title":"Step 2: Choose an action using the Epsilon Greedy Strategy","local":"step2-2","sections":[],"depth":2},{"title":"Step 3: Perform action At, get Rt+1 and St+1","local":"step3-3","sections":[],"depth":2},{"title":"Step 4: Update Q(St, At)","local":"step4-4","sections":[],"depth":2}],"depth":1}"><!-- HEAD_svelte-u9bgzb_END --> <p></p> <h1 class="relative group"><a id="q-learning-example" class="header-link block pr-1.5 text-lg no-hover:hidden with-hover:absolute with-hover:p-1.5 with-hover:opacity-0 with-hover:group-hover:opacity-100 with-hover:right-full" href="#q-learning-example"><span><svg class="" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" role="img" width="1em" height="1em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 256 256"><path d="M167.594 88.393a8.001 8.001 0 0 1 0 11.314l-67.882 67.882a8 8 0 1 1-11.314-11.315l67.882-67.881a8.003 8.003 0 0 1 11.314 0zm-28.287 84.86l-28.284 28.284a40 40 0 0 1-56.567-56.567l28.284-28.284a8 8 0 0 0-11.315-11.315l-28.284 28.284a56 56 0 0 0 79.196 79.197l28.285-28.285a8 8 0 1 0-11.315-11.314zM212.852 43.14a56.002 56.002 0 0 0-79.196 0l-28.284 28.284a8 8 0 1 0 11.314 11.314l28.284-28.284a40 40 0 0 1 56.568 56.567l-28.285 28.285a8 8 0 0 0 11.315 11.314l28.284-28.284a56.065 56.065 0 0 0 0-79.196z" fill="currentColor"></path></svg></span></a> <span>A Q-Learning example</span></h1> <p data-svelte-h="svelte-9fmuny">To better understand Q-Learning, let’s take a simple example:</p> <img src="https://huggingface.co/datasets/huggingface-deep-rl-course/course-images/resolve/main/en/unit3/Maze-Example-2.jpg" alt="Maze-Example"> <ul data-svelte-h="svelte-n6lz09"><li>You’re a mouse in this tiny maze. You always <strong>start at the same starting point.</strong></li> <li>The goal is <strong>to eat the big pile of cheese at the bottom right-hand corner</strong> and avoid the poison. After all, who doesn’t like cheese?</li> <li>The episode ends if we eat the poison, <strong>eat the big pile of cheese</strong>, or if we take more than five steps.</li> <li>The learning rate is 0.1</li> <li>The discount rate (gamma) is 0.99</li></ul> <img src="https://huggingface.co/datasets/huggingface-deep-rl-course/course-images/resolve/main/en/unit3/q-ex-1.jpg" alt="Maze-Example"> <p data-svelte-h="svelte-dok1jh">The reward function goes like this:</p> <ul data-svelte-h="svelte-93cdg2"><li><strong>+0:</strong> Going to a state with no cheese in it.</li> <li><strong>+1:</strong> Going to a state with a small cheese in it.</li> <li><strong>+10:</strong> Going to the state with the big pile of cheese.</li> <li><strong>-10:</strong> Going to the state with the poison and thus dying.</li> <li><strong>+0</strong> If we take more than five steps.</li></ul> <img src="https://huggingface.co/datasets/huggingface-deep-rl-course/course-images/resolve/main/en/unit3/q-ex-2.jpg" alt="Maze-Example"> <p data-svelte-h="svelte-pbs6nn">To train our agent to have an optimal policy (so a policy that goes right, right, down), <strong>we will use the Q-Learning algorithm</strong>.</p> <h2 class="relative group"><a id="step1" class="header-link block pr-1.5 text-lg no-hover:hidden with-hover:absolute with-hover:p-1.5 with-hover:opacity-0 with-hover:group-hover:opacity-100 with-hover:right-full" href="#step1"><span><svg class="" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" role="img" width="1em" height="1em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 256 256"><path d="M167.594 88.393a8.001 8.001 0 0 1 0 11.314l-67.882 67.882a8 8 0 1 1-11.314-11.315l67.882-67.881a8.003 8.003 0 0 1 11.314 0zm-28.287 84.86l-28.284 28.284a40 40 0 0 1-56.567-56.567l28.284-28.284a8 8 0 0 0-11.315-11.315l-28.284 28.284a56 56 0 0 0 79.196 79.197l28.285-28.285a8 8 0 1 0-11.315-11.314zM212.852 43.14a56.002 56.002 0 0 0-79.196 0l-28.284 28.284a8 8 0 1 0 11.314 11.314l28.284-28.284a40 40 0 0 1 56.568 56.567l-28.285 28.285a8 8 0 0 0 11.315 11.314l28.284-28.284a56.065 56.065 0 0 0 0-79.196z" fill="currentColor"></path></svg></span></a> <span>Step 1: Initialize the Q-table</span></h2> <img src="https://huggingface.co/datasets/huggingface-deep-rl-course/course-images/resolve/main/en/unit3/Example-1.jpg" alt="Maze-Example"> <p data-svelte-h="svelte-9m32nr">So, for now, <strong>our Q-table is useless</strong>; we need <strong>to train our Q-function using the Q-Learning algorithm.</strong></p> <p data-svelte-h="svelte-29jh6q">Let’s do it for 2 training timesteps:</p> <p data-svelte-h="svelte-51ll5c">Training timestep 1:</p> <h2 class="relative group"><a id="step2" class="header-link block pr-1.5 text-lg no-hover:hidden with-hover:absolute with-hover:p-1.5 with-hover:opacity-0 with-hover:group-hover:opacity-100 with-hover:right-full" href="#step2"><span><svg class="" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" role="img" width="1em" height="1em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 256 256"><path d="M167.594 88.393a8.001 8.001 0 0 1 0 11.314l-67.882 67.882a8 8 0 1 1-11.314-11.315l67.882-67.881a8.003 8.003 0 0 1 11.314 0zm-28.287 84.86l-28.284 28.284a40 40 0 0 1-56.567-56.567l28.284-28.284a8 8 0 0 0-11.315-11.315l-28.284 28.284a56 56 0 0 0 79.196 79.197l28.285-28.285a8 8 0 1 0-11.315-11.314zM212.852 43.14a56.002 56.002 0 0 0-79.196 0l-28.284 28.284a8 8 0 1 0 11.314 11.314l28.284-28.284a40 40 0 0 1 56.568 56.567l-28.285 28.285a8 8 0 0 0 11.315 11.314l28.284-28.284a56.065 56.065 0 0 0 0-79.196z" fill="currentColor"></path></svg></span></a> <span>Step 2: Choose an action using the Epsilon Greedy Strategy</span></h2> <p data-svelte-h="svelte-1hg390z">Because epsilon is big (= 1.0), I take a random action. In this case, I go right.</p> <img src="https://huggingface.co/datasets/huggingface-deep-rl-course/course-images/resolve/main/en/unit3/q-ex-3.jpg" alt="Maze-Example"> <h2 class="relative group"><a id="step3" class="header-link block pr-1.5 text-lg no-hover:hidden with-hover:absolute with-hover:p-1.5 with-hover:opacity-0 with-hover:group-hover:opacity-100 with-hover:right-full" href="#step3"><span><svg class="" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" role="img" width="1em" height="1em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 256 256"><path d="M167.594 88.393a8.001 8.001 0 0 1 0 11.314l-67.882 67.882a8 8 0 1 1-11.314-11.315l67.882-67.881a8.003 8.003 0 0 1 11.314 0zm-28.287 84.86l-28.284 28.284a40 40 0 0 1-56.567-56.567l28.284-28.284a8 8 0 0 0-11.315-11.315l-28.284 28.284a56 56 0 0 0 79.196 79.197l28.285-28.285a8 8 0 1 0-11.315-11.314zM212.852 43.14a56.002 56.002 0 0 0-79.196 0l-28.284 28.284a8 8 0 1 0 11.314 11.314l28.284-28.284a40 40 0 0 1 56.568 56.567l-28.285 28.285a8 8 0 0 0 11.315 11.314l28.284-28.284a56.065 56.065 0 0 0 0-79.196z" fill="currentColor"></path></svg></span></a> <span>Step 3: Perform action At, get Rt+1 and St+1</span></h2> <p>By going right, I get a small cheese, so<!-- HTML_TAG_START --><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>R</mi><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">R_{t+1} = 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8917em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0077em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span><!-- HTML_TAG_END --> and I’m in a new state.</p> <img src="https://huggingface.co/datasets/huggingface-deep-rl-course/course-images/resolve/main/en/unit3/q-ex-4.jpg" alt="Maze-Example"> <h2 class="relative group"><a id="step4" class="header-link block pr-1.5 text-lg no-hover:hidden with-hover:absolute with-hover:p-1.5 with-hover:opacity-0 with-hover:group-hover:opacity-100 with-hover:right-full" href="#step4"><span><svg class="" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" role="img" width="1em" height="1em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 256 256"><path d="M167.594 88.393a8.001 8.001 0 0 1 0 11.314l-67.882 67.882a8 8 0 1 1-11.314-11.315l67.882-67.881a8.003 8.003 0 0 1 11.314 0zm-28.287 84.86l-28.284 28.284a40 40 0 0 1-56.567-56.567l28.284-28.284a8 8 0 0 0-11.315-11.315l-28.284 28.284a56 56 0 0 0 79.196 79.197l28.285-28.285a8 8 0 1 0-11.315-11.314zM212.852 43.14a56.002 56.002 0 0 0-79.196 0l-28.284 28.284a8 8 0 1 0 11.314 11.314l28.284-28.284a40 40 0 0 1 56.568 56.567l-28.285 28.285a8 8 0 0 0 11.315 11.314l28.284-28.284a56.065 56.065 0 0 0 0-79.196z" fill="currentColor"></path></svg></span></a> <span>Step 4: Update Q(St, At)</span></h2> <p>We can now update<!-- HTML_TAG_START --><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi><mo stretchy="false">(</mo><msub><mi>S</mi><mi>t</mi></msub><mo separator="true">,</mo><msub><mi>A</mi><mi>t</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Q(S_t, A_t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">Q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><!-- HTML_TAG_END --> using our formula.</p> <img src="https://huggingface.co/datasets/huggingface-deep-rl-course/course-images/resolve/main/en/unit3/q-ex-5.jpg" alt="Maze-Example"> <img src="https://huggingface.co/datasets/huggingface-deep-rl-course/course-images/resolve/main/en/unit3/Example-4.jpg" alt="Maze-Example"> <p data-svelte-h="svelte-120ic1h">Training timestep 2:</p> <h2 class="relative group"><a id="step2-2" class="header-link block pr-1.5 text-lg no-hover:hidden with-hover:absolute with-hover:p-1.5 with-hover:opacity-0 with-hover:group-hover:opacity-100 with-hover:right-full" href="#step2-2"><span><svg class="" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" role="img" width="1em" height="1em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 256 256"><path d="M167.594 88.393a8.001 8.001 0 0 1 0 11.314l-67.882 67.882a8 8 0 1 1-11.314-11.315l67.882-67.881a8.003 8.003 0 0 1 11.314 0zm-28.287 84.86l-28.284 28.284a40 40 0 0 1-56.567-56.567l28.284-28.284a8 8 0 0 0-11.315-11.315l-28.284 28.284a56 56 0 0 0 79.196 79.197l28.285-28.285a8 8 0 1 0-11.315-11.314zM212.852 43.14a56.002 56.002 0 0 0-79.196 0l-28.284 28.284a8 8 0 1 0 11.314 11.314l28.284-28.284a40 40 0 0 1 56.568 56.567l-28.285 28.285a8 8 0 0 0 11.315 11.314l28.284-28.284a56.065 56.065 0 0 0 0-79.196z" fill="currentColor"></path></svg></span></a> <span>Step 2: Choose an action using the Epsilon Greedy Strategy</span></h2> <p data-svelte-h="svelte-14kapt5"><strong>I take a random action again, since epsilon=0.99 is big</strong>. (Notice we decay epsilon a little bit because, as the training progress, we want less and less exploration).</p> <p data-svelte-h="svelte-7qp2oa">I took the action ‘down’. <strong>This is not a good action since it leads me to the poison.</strong></p> <img src="https://huggingface.co/datasets/huggingface-deep-rl-course/course-images/resolve/main/en/unit3/q-ex-6.jpg" alt="Maze-Example"> <h2 class="relative group"><a id="step3-3" class="header-link block pr-1.5 text-lg no-hover:hidden with-hover:absolute with-hover:p-1.5 with-hover:opacity-0 with-hover:group-hover:opacity-100 with-hover:right-full" href="#step3-3"><span><svg class="" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" role="img" width="1em" height="1em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 256 256"><path d="M167.594 88.393a8.001 8.001 0 0 1 0 11.314l-67.882 67.882a8 8 0 1 1-11.314-11.315l67.882-67.881a8.003 8.003 0 0 1 11.314 0zm-28.287 84.86l-28.284 28.284a40 40 0 0 1-56.567-56.567l28.284-28.284a8 8 0 0 0-11.315-11.315l-28.284 28.284a56 56 0 0 0 79.196 79.197l28.285-28.285a8 8 0 1 0-11.315-11.314zM212.852 43.14a56.002 56.002 0 0 0-79.196 0l-28.284 28.284a8 8 0 1 0 11.314 11.314l28.284-28.284a40 40 0 0 1 56.568 56.567l-28.285 28.285a8 8 0 0 0 11.315 11.314l28.284-28.284a56.065 56.065 0 0 0 0-79.196z" fill="currentColor"></path></svg></span></a> <span>Step 3: Perform action At, get Rt+1 and St+1</span></h2> <p>Because I ate poison, <strong>I get<!-- HTML_TAG_START --><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>R</mi><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mo>−</mo><mn>10</mn></mrow><annotation encoding="application/x-tex">R_{t+1} = -10</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8917em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0077em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">10</span></span></span></span><!-- HTML_TAG_END -->, and I die.</strong></p> <img src="https://huggingface.co/datasets/huggingface-deep-rl-course/course-images/resolve/main/en/unit3/q-ex-7.jpg" alt="Maze-Example"> <h2 class="relative group"><a id="step4-4" class="header-link block pr-1.5 text-lg no-hover:hidden with-hover:absolute with-hover:p-1.5 with-hover:opacity-0 with-hover:group-hover:opacity-100 with-hover:right-full" href="#step4-4"><span><svg class="" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" role="img" width="1em" height="1em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 256 256"><path d="M167.594 88.393a8.001 8.001 0 0 1 0 11.314l-67.882 67.882a8 8 0 1 1-11.314-11.315l67.882-67.881a8.003 8.003 0 0 1 11.314 0zm-28.287 84.86l-28.284 28.284a40 40 0 0 1-56.567-56.567l28.284-28.284a8 8 0 0 0-11.315-11.315l-28.284 28.284a56 56 0 0 0 79.196 79.197l28.285-28.285a8 8 0 1 0-11.315-11.314zM212.852 43.14a56.002 56.002 0 0 0-79.196 0l-28.284 28.284a8 8 0 1 0 11.314 11.314l28.284-28.284a40 40 0 0 1 56.568 56.567l-28.285 28.285a8 8 0 0 0 11.315 11.314l28.284-28.284a56.065 56.065 0 0 0 0-79.196z" fill="currentColor"></path></svg></span></a> <span>Step 4: Update Q(St, At)</span></h2> <img src="https://huggingface.co/datasets/huggingface-deep-rl-course/course-images/resolve/main/en/unit3/q-ex-8.jpg" alt="Maze-Example"> <p data-svelte-h="svelte-1heq24d">Because we’re dead, we start a new episode. But what we see here is that, <strong>with two explorations steps, my agent became smarter.</strong></p> <p data-svelte-h="svelte-1eluo1b">As we continue exploring and exploiting the environment and updating Q-values using the TD target, the <strong>Q-table will give us a better and better approximation. At the end of the training, we’ll get an estimate of the optimal Q-function.</strong></p> <a class="!text-gray-400 !no-underline text-sm flex items-center not-prose mt-4" href="https://github.com/huggingface/deep-rl-class/blob/main/units/en/unit2/q-learning-example.mdx" target="_blank"><span data-svelte-h="svelte-1kd6by1"><</span> <span data-svelte-h="svelte-x0xyl0">></span> <span data-svelte-h="svelte-1dajgef"><span class="underline ml-1.5">Update</span> on GitHub</span></a> <p></p>

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