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		<link rel="modulepreload" href="/docs/deep-rl-course/pr_676/en/_app/immutable/chunks/MermaidChart.svelte_svelte_type_style_lang.3fce6c88.js"><!-- HEAD_svelte-u9bgzb_START --><meta name="hf:doc:metadata" content="{&quot;title&quot;:&quot;Visualize the Clipped Surrogate Objective Function&quot;,&quot;local&quot;:&quot;visualize-the-clipped-surrogate-objective-function&quot;,&quot;sections&quot;:[{&quot;title&quot;:&quot;Case 1 and 2: the ratio is between the range&quot;,&quot;local&quot;:&quot;case-1-and-2-the-ratio-is-between-the-range&quot;,&quot;sections&quot;:[],&quot;depth&quot;:2},{&quot;title&quot;:&quot;Case 3 and 4: the ratio is below the range&quot;,&quot;local&quot;:&quot;case-3-and-4-the-ratio-is-below-the-range&quot;,&quot;sections&quot;:[],&quot;depth&quot;:2},{&quot;title&quot;:&quot;Case 5 and 6: the ratio is above the range&quot;,&quot;local&quot;:&quot;case-5-and-6-the-ratio-is-above-the-range&quot;,&quot;sections&quot;:[],&quot;depth&quot;:2}],&quot;depth&quot;:1}"><!-- HEAD_svelte-u9bgzb_END -->      <p></p>  <div class="items-center shrink-0 min-w-[100px] max-sm:min-w-[50px] justify-end ml-auto flex" style="float: right; margin-left: 10px; display: inline-flex; position: relative; z-index: 10;"><div class="inline-flex rounded-md max-sm:rounded-sm"><button class="inline-flex items-center gap-1 max-sm:gap-0.5 h-6 max-sm:h-5 px-2 max-sm:px-1.5 text-[11px] max-sm:text-[9px] font-medium text-gray-800 border border-r-0 rounded-l-md max-sm:rounded-l-sm border-gray-200 bg-white hover:shadow-inner dark:border-gray-850 dark:bg-gray-950 dark:text-gray-200 dark:hover:bg-gray-800" aria-live="polite"><span class="inline-flex items-center justify-center rounded-md p-0.5 max-sm:p-0"><svg class="w-3 h-3 max-sm:w-2.5 max-sm:h-2.5" xmlns="http://www.w3.org/2000/svg" aria-hidden="true" fill="currentColor" focusable="false" role="img" width="1em" height="1em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 32 32"><path d="M28,10V28H10V10H28m0-2H10a2,2,0,0,0-2,2V28a2,2,0,0,0,2,2H28a2,2,0,0,0,2-2V10a2,2,0,0,0-2-2Z" transform="translate(0)"></path><path d="M4,18H2V4A2,2,0,0,1,4,2H18V4H4Z" transform="translate(0)"></path><rect fill="none" width="32" height="32"></rect></svg></span> <span>Copy page</span></button> <button class="inline-flex items-center justify-center w-6 max-sm:w-5 h-6 max-sm:h-5 disabled:pointer-events-none text-sm text-gray-500 hover:text-gray-700 dark:hover:text-white rounded-r-md max-sm:rounded-r-sm border border-l transition border-gray-200 bg-white hover:shadow-inner dark:border-gray-850 dark:bg-gray-950 dark:text-gray-200 dark:hover:bg-gray-800" aria-haspopup="menu" aria-expanded="false" aria-label="Open copy menu"><svg class="transition-transform text-gray-400 overflow-visible w-3 h-3 max-sm:w-2.5 max-sm:h-2.5 rotate-0" width="1em" height="1em" viewBox="0 0 12 7" fill="none" xmlns="http://www.w3.org/2000/svg"><path d="M1 1L6 6L11 1" stroke="currentColor"></path></svg></button></div> </div>  <h1 class="relative group"><a id="visualize-the-clipped-surrogate-objective-function" 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="#visualize-the-clipped-surrogate-objective-function"><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>Visualize the Clipped Surrogate Objective Function</span></h1> <p data-svelte-h="svelte-8fcaqu">Don’t worry. <strong>It’s normal if this seems complex to handle right now</strong>. But we’re going to see what this Clipped Surrogate Objective Function looks like, and this will help you to visualize better what’s going on.</p> <figure class="image table text-center m-0 w-full" data-svelte-h="svelte-1kvrq9v"><img src="https://huggingface.co/datasets/huggingface-deep-rl-course/course-images/resolve/main/en/unit9/recap.jpg" alt="PPO"> <figcaption><a href="https://fse.studenttheses.ub.rug.nl/25709/1/mAI_2021_BickD.pdf">Table from &quot;Towards Delivering a Coherent Self-Contained
    Explanation of Proximal Policy Optimization&quot; by Daniel Bick</a></figcaption></figure> <p data-svelte-h="svelte-kt354t">We have six different situations. Remember first that we take the minimum between the clipped and unclipped objectives.</p>  <h2 class="relative group"><a id="case-1-and-2-the-ratio-is-between-the-range" 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="#case-1-and-2-the-ratio-is-between-the-range"><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>Case 1 and 2: the ratio is between the range</span></h2> <p>In situations 1 and 2, <strong data-svelte-h="svelte-w52w77">the clipping does not apply since the ratio is between the range</strong><!-- HTML_TAG_START --><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mn>1</mn><mo>−</mo><mi>ϵ</mi><mo separator="true">,</mo><mn>1</mn><mo>+</mo><mi>ϵ</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> [1 - \epsilon, 1 + \epsilon] </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="mopen">[</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ϵ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">ϵ</span><span class="mclose">]</span></span></span></span><!-- HTML_TAG_END --></p> <p data-svelte-h="svelte-1cqqr4g">In situation 1, we have a positive advantage: the <strong>action is better than the average</strong> of all the actions in that state. Therefore, we should encourage our current policy to increase the probability of taking that action in that state.</p> <p data-svelte-h="svelte-jfeco7">Since the ratio is between intervals, <strong>we can increase our policy’s probability of taking that action at that state.</strong></p> <p data-svelte-h="svelte-uloelw">In situation 2, we have a negative advantage: the action is worse than the average of all actions at that state. Therefore, we should discourage our current policy from taking that action in that state.</p> <p data-svelte-h="svelte-1ffs0w0">Since the ratio is between intervals, <strong>we can decrease the probability that our policy takes that action at that state.</strong></p>  <h2 class="relative group"><a id="case-3-and-4-the-ratio-is-below-the-range" 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="#case-3-and-4-the-ratio-is-below-the-range"><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>Case 3 and 4: the ratio is below the range</span></h2> <figure class="image table text-center m-0 w-full" data-svelte-h="svelte-1kvrq9v"><img src="https://huggingface.co/datasets/huggingface-deep-rl-course/course-images/resolve/main/en/unit9/recap.jpg" alt="PPO"> <figcaption><a href="https://fse.studenttheses.ub.rug.nl/25709/1/mAI_2021_BickD.pdf">Table from &quot;Towards Delivering a Coherent Self-Contained
    Explanation of Proximal Policy Optimization&quot; by Daniel Bick</a></figcaption></figure> <p>If the probability ratio is lower than<!-- HTML_TAG_START --><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mn>1</mn><mo>−</mo><mi>ϵ</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> [1 - \epsilon] </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="mopen">[</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">ϵ</span><span class="mclose">]</span></span></span></span><!-- HTML_TAG_END -->, the probability of taking that action at that state is much lower than with the old policy.</p> <p data-svelte-h="svelte-1x6bxjm">If, like in situation 3, the advantage estimate is positive (A&gt;0), then <strong>you want to increase the probability of taking that action at that state.</strong></p> <p data-svelte-h="svelte-1l49ylb">But if, like situation 4, the advantage estimate is negative, <strong>we don’t want to decrease further</strong> the probability of taking that action at that state. Therefore, the gradient is = 0 (since we’re on a flat line), so we don’t update our weights.</p>  <h2 class="relative group"><a id="case-5-and-6-the-ratio-is-above-the-range" 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="#case-5-and-6-the-ratio-is-above-the-range"><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>Case 5 and 6: the ratio is above the range</span></h2> <figure class="image table text-center m-0 w-full" data-svelte-h="svelte-1kvrq9v"><img src="https://huggingface.co/datasets/huggingface-deep-rl-course/course-images/resolve/main/en/unit9/recap.jpg" alt="PPO"> <figcaption><a href="https://fse.studenttheses.ub.rug.nl/25709/1/mAI_2021_BickD.pdf">Table from &quot;Towards Delivering a Coherent Self-Contained
    Explanation of Proximal Policy Optimization&quot; by Daniel Bick</a></figcaption></figure> <p>If the probability ratio is higher than<!-- HTML_TAG_START --><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mn>1</mn><mo>+</mo><mi>ϵ</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> [1 + \epsilon] </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="mopen">[</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">ϵ</span><span class="mclose">]</span></span></span></span><!-- HTML_TAG_END -->, the probability of taking that action at that state in the current policy is <strong data-svelte-h="svelte-qu8nx6">much higher than in the former policy.</strong></p> <p data-svelte-h="svelte-1ptdkag">If, like in situation 5, the advantage is positive, <strong>we don’t want to get too greedy</strong>. We already have a higher probability of taking that action at that state than the former policy. Therefore, the gradient is = 0 (since we’re on a flat line), so we don’t update our weights.</p> <p data-svelte-h="svelte-yugfug">If, like in situation 6, the advantage is negative, we want to decrease the probability of taking that action at that state.</p> <p data-svelte-h="svelte-1eb9ayw">So if we recap, <strong>we only update the policy with the unclipped objective part</strong>. When the minimum is the clipped objective part, we don’t update our policy weights since the gradient will equal 0.</p> <p data-svelte-h="svelte-pg5tgk">So we update our policy  only if:</p> <ul><li>Our ratio is in the range<!-- HTML_TAG_START --><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mn>1</mn><mo>−</mo><mi>ϵ</mi><mo separator="true">,</mo><mn>1</mn><mo>+</mo><mi>ϵ</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> [1 - \epsilon, 1 + \epsilon] </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="mopen">[</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ϵ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">ϵ</span><span class="mclose">]</span></span></span></span><!-- HTML_TAG_END --></li> <li data-svelte-h="svelte-59wfuc">Our ratio is outside the range, but <strong>the advantage leads to getting closer to the range</strong> <ul><li>Being below the ratio but the advantage is &gt; 0</li> <li>Being above the ratio but the advantage is &lt; 0</li></ul></li></ul> <p><strong data-svelte-h="svelte-xl5clc">You might wonder why, when the minimum is the clipped ratio, the gradient is 0.</strong> When the ratio is clipped, the derivative in this case will not be the derivative of the<!-- 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><mi>t</mi></msub><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>∗</mo><msub><mi>A</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex"> r_t(\theta) * 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"><span class="mord mathnormal" style="margin-right:0.02778em;">r</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:-0.0278em;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="mopen">(</span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></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></span></span><!-- HTML_TAG_END -->   but the derivative of either<!-- HTML_TAG_START --><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>ϵ</mi><mo stretchy="false">)</mo><mo>∗</mo><msub><mi>A</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex"> (1 - \epsilon)* 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="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">ϵ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></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></span></span><!-- HTML_TAG_END --> or the derivative of<!-- HTML_TAG_START --><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>ϵ</mi><mo stretchy="false">)</mo><mo>∗</mo><msub><mi>A</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex"> (1 + \epsilon)* 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="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">ϵ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></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></span></span><!-- HTML_TAG_END --> which both = 0.</p> <p>To summarize, thanks to this clipped surrogate objective, <strong data-svelte-h="svelte-vpub9x">we restrict the range that the current policy can vary from the old one.</strong> Because we remove the incentive for the probability ratio to move outside of the interval since the clip forces the gradient to be zero. If the ratio is &gt;<!-- HTML_TAG_START --><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>+</mo><mi>ϵ</mi></mrow><annotation encoding="application/x-tex"> 1 + \epsilon </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">ϵ</span></span></span></span><!-- HTML_TAG_END --> or &lt;<!-- HTML_TAG_START --><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>−</mo><mi>ϵ</mi></mrow><annotation encoding="application/x-tex"> 1 - \epsilon </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">ϵ</span></span></span></span><!-- HTML_TAG_END --> the gradient will be equal to 0.</p> <p data-svelte-h="svelte-d7ieux">The final Clipped Surrogate Objective Loss for PPO Actor-Critic style looks like this, it’s a combination of Clipped Surrogate Objective function, Value Loss Function and Entropy bonus:</p> <img src="https://huggingface.co/datasets/huggingface-deep-rl-course/course-images/resolve/main/en/unit9/ppo-objective.jpg" alt="PPO objective"> <p data-svelte-h="svelte-enoain">That was quite complex. Take time to understand these situations by looking at the table and the graph. <strong>You must understand why this makes sense.</strong> If you want to go deeper, the best resource is the article <a href="https://fse.studenttheses.ub.rug.nl/25709/1/mAI_2021_BickD.pdf" rel="nofollow">“Towards Delivering a Coherent Self-Contained Explanation of Proximal Policy Optimization” by Daniel Bick, especially part 3.4</a>.</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/unit8/visualize.mdx" target="_blank"><svg class="mr-1" xmlns="http://www.w3.org/2000/svg" aria-hidden="true" fill="currentColor" focusable="false" role="img" width="1em" height="1em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 32 32"><path d="M31,16l-7,7l-1.41-1.41L28.17,16l-5.58-5.59L24,9l7,7z"></path><path d="M1,16l7-7l1.41,1.41L3.83,16l5.58,5.59L8,23l-7-7z"></path><path d="M12.419,25.484L17.639,6.552l1.932,0.518L14.351,26.002z"></path></svg> <span data-svelte-h="svelte-zjs2n5"><span class="underline">Update</span> on GitHub</span></a>  <p></p> 
			
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