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| <title>Scaling Laws for Neural Language Models</title> |
| <meta name="description" content="An empirical study of scaling behavior across model size, data, and compute budgets."> |
| <meta name="author" content="Alice Martin, Bob Chen, Carol Wu"> |
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| <meta property="og:title" content="Scaling Laws for Neural Language Models"> |
| <meta property="og:description" content="An empirical study of scaling behavior across model size, data, and compute budgets."> |
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| <meta property="article:published_time" content="Apr. 13, 2026"> |
| <meta property="article:author" content="Alice Martin"> |
| <meta property="article:author" content="Bob Chen"> |
| <meta property="article:author" content="Carol Wu"> |
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| <header class="article-header"> |
| <h1>Scaling Laws for Neural Language Models</h1> |
| <div class="authors"><span class="author"><a href="https://example.com/alice" class="author-link" target="_blank" rel="noopener">Alice Martin</a><span class="author-affiliation">Hugging Face</span></span><span class="author-sep">, </span><span class="author"><span class="author-name">Bob Chen</span><span class="author-affiliation">Hugging Face, MIT</span></span><span class="author-sep">, </span><span class="author"><span class="author-name">Carol Wu</span><span class="author-affiliation">MIT</span></span></div> |
| <p class="date">April 13, 2026</p> |
| </header> |
|
|
| <div class="tiptap"> |
| <h1>Welcome to the Collaborative Editor</h1><p>This demo article showcases <strong>every content type</strong> available in the editor. Feel free to edit, delete, or rewrite anything — the AI assistant in the left panel can help you.</p><h2>Text formatting</h2><p>You can make text <strong>bold</strong>, <em>italic</em>, or <s>strikethrough</s>. Combine them for <strong><em>bold italic</em></strong>. Use <code>inline code</code> for technical terms, and add <a target="_blank" rel="noopener noreferrer nofollow" class="editor-link" href="https://huggingface.co">links</a> to external resources.</p><h2>Lists</h2><h3>Unordered list</h3><ul><li><p>First item with some context</p></li><li><p>Second item — supports <strong>rich formatting</strong> inside</p></li><li><p>Third item with <code>inline code</code></p></li></ul><h3>Ordered list</h3><ol><li><p>Prepare the dataset</p></li><li><p>Fine-tune the model</p></li><li><p>Evaluate on the test split</p></li><li><p>Deploy to production</p></li></ol><h2>Blockquote</h2><blockquote><p>"The best way to predict the future is to invent it." — Alan Kay</p></blockquote><h2>Code block</h2><pre><code class="language-python">import torch |
| from transformers import AutoModelForCausalLM, AutoTokenizer |
|
|
| model_id = "meta-llama/Llama-3-8B" |
| tokenizer = AutoTokenizer.from_pretrained(model_id) |
| model = AutoModelForCausalLM.from_pretrained( |
| model_id, |
| torch_dtype=torch.bfloat16, |
| device_map="auto", |
| ) |
|
|
| prompt = "The future of AI is" |
| inputs = tokenizer(prompt, return_tensors="pt").to(model.device) |
| outputs = model.generate(**inputs, max_new_tokens=100) |
| print(tokenizer.decode(outputs[0], skip_special_tokens=True))</code></pre><h2>Table</h2><table style="min-width: 100px;"><colgroup><col style="min-width: 25px;"><col style="min-width: 25px;"><col style="min-width: 25px;"><col style="min-width: 25px;"></colgroup><tbody><tr><th colspan="1" rowspan="1"><p>Model</p></th><th colspan="1" rowspan="1"><p>Parameters</p></th><th colspan="1" rowspan="1"><p>MMLU</p></th><th colspan="1" rowspan="1"><p>HellaSwag</p></th></tr><tr><td colspan="1" rowspan="1"><p>LLaMA 3 8B</p></td><td colspan="1" rowspan="1"><p>8B</p></td><td colspan="1" rowspan="1"><p>66.6</p></td><td colspan="1" rowspan="1"><p>82.0</p></td></tr><tr><td colspan="1" rowspan="1"><p>LLaMA 3 70B</p></td><td colspan="1" rowspan="1"><p>70B</p></td><td colspan="1" rowspan="1"><p>79.5</p></td><td colspan="1" rowspan="1"><p>87.3</p></td></tr><tr><td colspan="1" rowspan="1"><p>GPT-4</p></td><td colspan="1" rowspan="1"><p>~1.8T</p></td><td colspan="1" rowspan="1"><p>86.4</p></td><td colspan="1" rowspan="1"><p>95.3</p></td></tr><tr><td colspan="1" rowspan="1"><p>Mixtral 8x7B</p></td><td colspan="1" rowspan="1"><p>46.7B</p></td><td colspan="1" rowspan="1"><p>70.6</p></td><td colspan="1" rowspan="1"><p>84.4</p></td></tr></tbody></table><h2>Mathematics</h2><p>Inline math works with double dollars: the quadratic formula is <span data-latex="x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}" data-type="inline-math"></span> and Euler's identity is <span data-latex="e^{i\pi} + 1 = 0" data-type="inline-math"></span>.</p><p>Block equations use triple dollars. Here is a standard autoregressive loss:</p><div data-latex="\mathcal{L}(\theta) = -\sum_{t=1}^{T} \log P(x_t \mid x_{<t}; \theta)" data-type="block-math"></div><p>And the full variational trajectory decomposition:</p><div data-latex="\begin{aligned} \log p_\theta(\mathcal{D}) &= \log \sum_{i=0}^N p_\theta((o,a)_i) \\ &= \log \sum_{i=0}^N \int_{\text{supp}(Z)} p_\theta((o,a)_i \mid z)\, p(z) \\ &= \log \sum_{i=0}^N \int_{\text{supp}(Z)} \frac{q_\theta(z \mid (o,a)_i)}{q_\theta(z \mid (o,a)_i)} \cdot p_\theta((o,a)_i \mid z)\, p(z) \\ &= \log \sum_{i=0}^N \mathbb{E}_{z \sim p_\theta(\bullet \mid (o,a)_i)} \left[ \frac{p(z)}{q_\theta(z \mid (o,a)_i)} \cdot p_\theta((o,a)_i \mid z) \right] \end{aligned}" data-type="block-math"></div><h2>Scientific references</h2><p>The editor supports academic citations. The Transformer architecture <span key="vaswani2017" label="[vaswani2017]" data-type="citation" class="citation-node">[vaswani2017]</span> revolutionized natural language processing by introducing the self-attention mechanism. Large-scale pretraining <span key="devlin2019" label="[devlin2019]" data-type="citation" class="citation-node">[devlin2019]</span> demonstrated that unsupervised objectives on massive corpora produce powerful general-purpose representations.</p><p>More recently, scaling laws <span key="kaplan2020" label="[kaplan2020]" data-type="citation" class="citation-node">[kaplan2020]</span> have shown predictable relationships between model size, dataset size, and performance.</p><hr><h2>Horizontal rule</h2><p>Use a horizontal rule to visually separate sections:</p><hr><h2>Nested content</h2><p>Lists can contain multiple levels of formatting. Blockquotes can hold structured content:</p><blockquote><p><strong>Note:</strong> This editor supports real-time collaboration. Open this page in another tab to see cursors sync live.</p></blockquote><h3>A deeper heading level</h3><p>Heading levels go from H1 down to H3, giving you a clear document hierarchy. The floating menu on the left lets you insert any block type, and the bubble toolbar appears when you select text.</p><h2>Custom components</h2><p>The editor supports rich custom components from the research article template. Use the <code>/</code> slash menu to insert them.</p><details data-component="accordion"> |
| <summary>Details</summary> |
| <div class="accordion-content"><p>The model was fine-tuned using LoRA adapters with rank 16 on 4× A100 GPUs. Training took approximately 12 hours on the full dataset. We used a cosine learning rate schedule with warm-up over the first 10% of steps.</p></div> |
| </details><div title="Important" emoji="💡" variant="info" data-component="note"><p>All experiments were conducted with mixed-precision (bf16) training. Results may vary slightly with different random seeds.</p></div><div author="Ilya Sutskever" source="NeurIPS 2024 keynote" data-component="quoteBlock"><p>If you have a very large neural network and you train it on a very large dataset, you get very good results. It really is that simple.</p></div><div title="Warning" emoji="⚠️" variant="danger" data-component="note"><p>Do not use these scaling estimates for production capacity planning without accounting for inference overhead and memory constraints.</p></div><div data-component="wide"><p>This content stretches beyond the normal column width, useful for wide tables, figures or visualizations that need extra horizontal space.</p></div><div data-component="fullWidth"><p>This content spans the entire page width. It is ideal for large figures, panoramic images, or full-bleed data visualizations.</p></div><div data-component="sidenote"><p>Scaling laws were first systematically studied in the context of neural machine translation before being generalized to language models.</p></div><div id="fig-scaling" caption="Log-linear relationship between compute budget and model performance across three orders of magnitude." data-component="reference"><p>[ Figure placeholder — insert an image or chart here ]</p></div><div src="d3-scaling-chart.html" title="Interactive scaling law visualization" desc="Explore how model size, data, and compute interact." wide="false" downloadable="true" data-component="htmlEmbed"></div><p>The concept of <span term="Scaling law" definition="A power-law relationship between model size, dataset size, compute budget, and performance." data-type="glossary" class="glossary-node" title="A power-law relationship between model size, dataset size, compute budget, and performance." tabindex="0">Scaling law</span> has become central to modern AI research. Early work<span content="Hestness et al. (2017) observed similar scaling behavior in vision and translation tasks before the scaling laws paper formalized the framework." data-type="footnote" class="footnote-node" title="Hestness et al. (2017) observed similar scaling behavior in vision and translation tasks before the scaling laws paper formalized the framework." tabindex="0"><sup class="footnote-marker">*</sup></span> suggested these relationships hold across modalities.</p><div layout="2-column" gap="medium" data-component="stack"><div data-type="stack-column"><p><strong>Column A</strong></p><p>Multi-column layouts let you place content side by side. Each column is fully editable.</p></div><div data-type="stack-column"><p><strong>Column B</strong></p><p>Use the layout selector in the header to switch between 2, 3, or 4 columns.</p></div></div><div username="tfrere" name="Thibaud Frere" url="https://huggingface.co/tfrere" data-component="hfUser"></div><h2>What's next?</h2><p>Try asking the AI assistant to:</p><ul><li><p>Rewrite a paragraph in a different tone</p></li><li><p>Expand a section with more detail</p></li><li><p>Fix grammar or spelling errors</p></li><li><p>Translate content to another language</p></li><li><p>Add a new section on a specific topic</p></li></ul><p>Select some text and use the quick actions, or type a message in the chat panel. All AI edits can be undone with <code>Cmd+Z</code>.</p><div data-type="bibliography" class="bibliography-block"><h2 class="bibliography-title">References</h2><div class="bibliography-content"><div class="csl-bib-body"> |
| <div data-csl-entry-id="devlin2019" class="csl-entry">Devlin, J., Chang, M.-W., Lee, K., & Toutanova, K. (2019). BERT: Pre-training of deep bidirectional transformers for language understanding. <i>Proceedings of NAACL-HLT</i>, 4171–4186. https://doi.org/10.18653/v1/N19-1423</div> |
| <div data-csl-entry-id="kaplan2020" class="csl-entry">Kaplan, J., McCandlish, S., Henighan, T., Brown, T. B., Chess, B., Child, R., Gray, S., Radford, A., Wu, J., & Amodei, D. (2020). Scaling laws for neural language models. <i>arXiv Preprint arXiv:2001.08361</i>. https://doi.org/10.48550/arXiv.2001.08361</div> |
| <div data-csl-entry-id="vaswani2017" class="csl-entry">Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A. N., Kaiser, L., & Polosukhin, I. (2017). Attention is all you need. <i>Advances in Neural Information Processing Systems</i>, <i>30</i>. https://doi.org/10.48550/arXiv.1706.03762</div> |
| </div></div></div><p></p> |
| </div> |
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| var ulStack = [document.createElement('ul')]; |
| nav.appendChild(ulStack[0]); |
| var levelOf = function(tag) { return tag==='H2'?2:tag==='H3'?3:4; }; |
| var prev = 2, hIdx = 0; |
| headingsArr.forEach(function(h) { |
| var lvl = levelOf(h.tagName); |
| while (lvl > prev) { var ul = document.createElement('ul'); var last = ulStack[ulStack.length-1].lastElementChild; if (last) last.appendChild(ul); ulStack.push(ul); prev++; } |
| while (lvl < prev) { ulStack.pop(); prev--; } |
| var li = document.createElement('li'); |
| var a = document.createElement('a'); |
| a.href = '#'+h.id; a.textContent = h.textContent; |
| li.appendChild(a); |
| li.setAttribute('data-heading-idx', String(hIdx++)); |
| ulStack[ulStack.length-1].appendChild(li); |
| }); |
| |
| if (holder) holder.appendChild(nav); |
| var navClone = nav.cloneNode(true); |
| if (holderMobile) holderMobile.appendChild(navClone); |
| |
| |
| nav.classList.add('toc-collapsible'); |
| navClone.classList.add('toc-collapsible'); |
| |
| var allLinks = [].concat( |
| holder ? Array.from(holder.querySelectorAll('a')) : [], |
| holderMobile ? Array.from(holderMobile.querySelectorAll('a')) : [] |
| ); |
| |
| |
| var isMobile = window.matchMedia('(max-width: 1100px)'); |
| |
| function getItemsWithChildren(navEl) { |
| if (!navEl) return []; |
| return Array.from(navEl.querySelectorAll('li[data-heading-idx]')).filter(function(li) { |
| return li.querySelector(':scope > ul'); |
| }); |
| } |
| |
| function getAncestorIndices(items, targetIdx) { |
| var toExpand = {}; |
| var activeLi = null; |
| function find(li) { |
| if (Number(li.getAttribute('data-heading-idx')) === targetIdx) return li; |
| var ul = li.querySelector(':scope > ul'); |
| if (!ul) return null; |
| var children = ul.querySelectorAll(':scope > li[data-heading-idx]'); |
| for (var i = 0; i < children.length; i++) { var f = find(children[i]); if (f) return f; } |
| return null; |
| } |
| for (var i = 0; i < items.length; i++) { activeLi = find(items[i]); if (activeLi) break; } |
| if (!activeLi) return toExpand; |
| toExpand[targetIdx] = true; |
| var cur = activeLi; |
| while (cur) { |
| var parent = cur.parentElement ? cur.parentElement.closest('li[data-heading-idx]') : null; |
| if (parent) { toExpand[Number(parent.getAttribute('data-heading-idx'))] = true; cur = parent; } |
| else break; |
| } |
| return toExpand; |
| } |
| |
| function applyCollapseState(navEl, activeIdx) { |
| if (!navEl) return; |
| var items = getItemsWithChildren(navEl); |
| var ancestors = getAncestorIndices(items, activeIdx); |
| |
| items.forEach(function(li) { |
| var sub = li.querySelector(':scope > ul'); |
| if (!sub) return; |
| var idx = Number(li.getAttribute('data-heading-idx')); |
| var allDesc = li.querySelectorAll('li[data-heading-idx]'); |
| var related = [idx]; |
| allDesc.forEach(function(d) { related.push(Number(d.getAttribute('data-heading-idx'))); }); |
| var shouldExpand = related.some(function(i) { return ancestors[i]; }); |
| |
| if (shouldExpand) { |
| li.classList.remove('collapsed'); |
| sub.style.height = 'auto'; |
| } else { |
| li.classList.add('collapsed'); |
| sub.style.height = '0px'; |
| } |
| }); |
| } |
| |
| function expandAll(navEl) { |
| if (!navEl) return; |
| getItemsWithChildren(navEl).forEach(function(li) { |
| li.classList.remove('collapsed'); |
| var sub = li.querySelector(':scope > ul'); |
| if (sub) sub.style.height = 'auto'; |
| }); |
| } |
| |
| |
| function initCollapse() { |
| if (isMobile.matches) { |
| expandAll(holder ? holder.querySelector('nav') : null); |
| expandAll(holderMobile ? holderMobile.querySelector('nav') : null); |
| } else { |
| applyCollapseState(holder ? holder.querySelector('nav') : null, 0); |
| applyCollapseState(holderMobile ? holderMobile.querySelector('nav') : null, 0); |
| } |
| } |
| initCollapse(); |
| |
| isMobile.addEventListener('change', function() { |
| if (isMobile.matches) { |
| expandAll(holder ? holder.querySelector('nav') : null); |
| expandAll(holderMobile ? holderMobile.querySelector('nav') : null); |
| } else { |
| applyCollapseState(holder ? holder.querySelector('nav') : null, prevActiveIdx); |
| applyCollapseState(holderMobile ? holderMobile.querySelector('nav') : null, prevActiveIdx); |
| } |
| }); |
| |
| |
| var OFFSET = 60, lastScrollTime = 0, prevActiveIdx = 0; |
| |
| function onScroll() { |
| var now = performance.now(); |
| if (now - lastScrollTime < 50) return; |
| lastScrollTime = now; |
| requestAnimationFrame(function() { |
| var activeIdx = -1, activeId = null; |
| for (var i = headingsArr.length - 1; i >= 0; i--) { |
| if (headingsArr[i].getBoundingClientRect().top - OFFSET <= 0) { |
| activeIdx = i; activeId = headingsArr[i].id; break; |
| } |
| } |
| allLinks.forEach(function(l) { |
| if (activeId && l.getAttribute('href') === '#'+activeId) l.classList.add('active'); |
| else l.classList.remove('active'); |
| }); |
| if (activeIdx !== prevActiveIdx && activeIdx >= 0 && !isMobile.matches) { |
| prevActiveIdx = activeIdx; |
| applyCollapseState(holder ? holder.querySelector('nav') : null, activeIdx); |
| } |
| if (activeIdx >= 0) prevActiveIdx = activeIdx; |
| }); |
| } |
| window.addEventListener('scroll', onScroll, { passive: true }); |
| onScroll(); |
| |
| |
| var sidebar = document.querySelector('.toc-mobile-sidebar'); |
| var backdrop = document.querySelector('.toc-mobile-backdrop'); |
| var toggleBtn = document.querySelector('.toc-mobile-toggle'); |
| var closeBtn = document.querySelector('.toc-mobile-sidebar__close'); |
| |
| function openSidebar() { |
| sidebar.classList.add('open'); backdrop.classList.add('open'); |
| toggleBtn.setAttribute('aria-expanded','true'); |
| document.body.style.overflow = 'hidden'; |
| requestAnimationFrame(function() { |
| var active = sidebar.querySelector('a.active'); |
| if (active) { |
| var body = sidebar.querySelector('.toc-mobile-sidebar__body'); |
| if (body) body.scrollTop = Math.max(0, active.offsetTop - body.offsetTop - body.clientHeight/3); |
| } |
| }); |
| } |
| function closeSidebar() { |
| sidebar.classList.remove('open'); backdrop.classList.remove('open'); |
| toggleBtn.setAttribute('aria-expanded','false'); |
| document.body.style.overflow = ''; |
| } |
| if (toggleBtn) toggleBtn.addEventListener('click', openSidebar); |
| if (closeBtn) closeBtn.addEventListener('click', closeSidebar); |
| if (backdrop) backdrop.addEventListener('click', closeSidebar); |
| if (holderMobile) holderMobile.addEventListener('click', function(e) { if (e.target.closest && e.target.closest('a')) closeSidebar(); }); |
| document.addEventListener('keydown', function(e) { if (e.key==='Escape' && sidebar.classList.contains('open')) closeSidebar(); }); |
| |
| |
| if (window.location.hash) { |
| var target = document.querySelector(window.location.hash); |
| if (target) setTimeout(function() { target.scrollIntoView({block:'start'}); }, 100); |
| } |
| window.addEventListener('popstate', function() { |
| var h = window.location.hash; |
| if (h) { var t = document.querySelector(h); if (t) t.scrollIntoView({block:'start'}); } |
| else window.scrollTo({top:0}); |
| }); |
| })(); |
| </script> |
| </body> |
| </html> |
