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<h1 class="title is-2 publication-title" style="display: inline;">Kinetics: Rethinking Test-Time Scaling Laws</h1>
<br><br>
<div class="is-size-5 publication-authors">
<span class="author-block"><a href="https://ranonrkm.github.io/" target="_blank">Ranajoy Sadhukhan</a><sup>*</sup>,</span>
<span class="author-block"><a href="https://dreaming-panda.github.io/" target="_blank">Zhuoming Chen</a><sup>*</sup>,</span>
<span class="author-block"><a href="http://zhenghaizhong.com/" target="_blank">Haizhong Zheng</a>,</span>
<span class="author-block"><a href="" target="_blank">Yang Zhou</a>,</span>
<span class="author-block"><a href="https://strubell.github.io/" target="_blank">Emma Strubell</a>,</span>
<span class="author-block"><a href="https://www.andrew.cmu.edu/user/beidic/" target="_blank">Beidi Chen</a></span>
</div>
<div class="is-size-5 publication-authors">
<span class="affliation">
<small>
Carnegie Mellon University
</small>
</span>
<span class="eql-cntrb">
<small><br><sup>*</sup>Indicates Equal Contribution</small>
</span>
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<a href="https://arxiv.org/abs/2408.11049" target="_blank" class="external-link button is-normal is-rounded is-dark">
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<span>arXiv</span>
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<h4 class="title is-5" style="text-align: center; color: #4b6cb7;">
Kinetics Scaling Law
</h4>
<a href="#figure1">
<img src="static/images/F1A.png" alt="Kinetics Scaling" style="width: 100%; border-radius: 6px;">
</a>
<figcaption style="text-align: center; margin-top: 8px; font-size: 14px; color: #555;">
<em>Figure 1: On AIME24, compute-optimal model choices can be up to 3× costlier than those guided by the Kinetics scaling law.</em>
</figcaption>
</div>
<!-- Figure 2 -->
<div id="figure2" style="flex: 1 1 48%; background: #fff; padding: 16px; border-radius: 8px; box-shadow: 0 4px 12px rgba(0,0,0,0.05);">
<h4 class="title is-5" style="text-align: center; color: #4b6cb7;">
Sparse Scaling Advantage
</h4>
<a href="#figure2">
<img src="static/images/F1B.png" alt="Sparse Scaling" style="width: 100%; border-radius: 6px;">
</a>
<figcaption style="text-align: center; margin-top: 8px; font-size: 14px; color: #555;">
<em>Figure 2: Sparse attention unlocks stronger scaling by enabling longer sequences and more parallel samples.</em>
</figcaption>
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<p>
<strong style="font-weight: 900;color: #0f598a">TL;DR:</strong> We introduce <strong>Kinetics</strong>, which challenges the traditional test-time scaling (TTS) laws by adopting a <strong>practical efficiency</strong> perspective. It reveals that prior compute-optimal approaches overlook major <em>key-value memory access bottlenecks</em> in various TTS strategies. By jointly considering memory and compute, the <strong>Kinetics scaling law</strong> shows that <em>it is more efficient to scale model size up to a threshold before investing more compute in test-time scaling</em>. Additionally, Kinetics promotes <strong>sparse attention</strong> to establish an even better scalability.
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<h2 class="title is-3" style="text-align: center;">
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Introduction
</h2>
<div class="content has-text-justified">
<div style="background: #ffffff; border-left: 4px solid #4b6cb7; border-radius: 6px; padding: 16px 24px; font-size: 16px; line-height: 1.6; color: #333;">
<p>
The existing compute-optimal test-time scaling law often favors smaller models with more test-time compute. We revisit this paradigm from a <em style="color: #4b6cb7;">practical efficiency</em> perspective, uncovering that the effectiveness of small models is often <strong style="color: #4b6cb7;">overestimated</strong> due to overlooked <strong style="color: #222;">key-value (KV) memory bottlenecks</strong> introduced by inference strategies such as Best-of-N or long CoT chains.
</p>
<p>
To address this, we propose the <strong><em style="color: #4b6cb7;">Kinetics Scaling Law</em></strong>, which jointly considers both compute and memory costs. Unlike prior work, it shows that test-time compute is best allocated to scaling model size—up to a threshold (e.g., <strong>14B</strong> parameters for Qwen3)—before increasing generation length. <a href="#figure1" style="color: #2c6eab; text-decoration: underline;">(See Figure 1)</a>
</p>
<p>
Furthermore, we demonstrate that <strong style="color: #4b6cb7;">sparse attention</strong> unlocks new scaling opportunities by mitigating KV memory overhead, enabling <em style="color: #27ae60;">longer generations</em> and <em style="color: #27ae60;">more parallel reasoning trials</em> within the same budget. This leads to substantial gains in test-time accuracy and efficiency. Empirically, we show that sparse attention models consistently outperform dense counterparts, achieving over <strong style="color: #27ae60;">60 points</strong> gains in low-cost regimes and over <strong style="color: #27ae60;">5 points</strong> gains in high-cost regimes for problem-solving accuracy on <em style="color: #27ae60;">AIME</em>, encompassing evaluations on state-of-the-art MoEs. <a href="#figure2" style="color: #2c6eab; text-decoration: underline;">(See Figure 2)</a>. <em>These results suggest that sparse attention is essential for realizing the full potential of test-time scaling because, unlike training—where parameter scaling saturates, test-time accuracy continues to improve through increased generation.</em>
</p>
<p>
These insights form the foundation of <strong>Kinetics</strong>, a new perspective on scaling that aligns resource allocation more closely with real-world inference constraints.
</p>
</div>
</div>
</div>
</div>
</div>
</section>
<!-- Section: Motivation -->
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<h2 class="title is-3" style="text-align: center;">
<img src="static/images/icons/Idea.png" style="height: 50px; display: inline; vertical-align: middle;"/>
Rethinking Test-Time Scaling Law
</h2>
<div class="content has-text-justified">
<h4 class="title is-5">
<img src="static/images/icons/MathematicsCompass.png" style="height: 36px; display: inline; vertical-align: middle;"/>
A Holistic Cost Model for TTS
</h4>
<!-- <p>Given a speculation length γ for a sequence of length S and batch size B, let T<sub>T</sub>(B, S, 1) and T<sub>D</sub>(B, S, 1) denote the target and draft decoding latencies, respectively. The verification time, T<sub>V</sub>(B, S, γ), is the time it takes the target model to verify the γ speculated tokens in a single forward pass. Given an acceptance rate α ∈ [0, 1] and speculation length γ, Ω(γ, α) represents the expected number of tokens generated in one verification step, as described by <a style="color: #209CEE" href="https://arxiv.org/abs/2211.17192">Leviathan et al.</a>:</p> -->
<p>
As a first step, we revisit the cost model to understand the relative importance of compute and memory in TTS and how it can affect the choice of optimal model size and TTS configurations like chain-of-thought (CoT) length or number of trials.
</p>
<details style="margin-bottom: 24px; border: 1px solid #ddd; border-radius: 6px; padding: 12px 16px; background-color: #f9f9fb;">
<summary style="cursor: pointer; font-weight: 600; font-size: 16px; color: #2a63b2;">
View Full Cost Model Derivation
</summary>
<div style="background: #ffffff; border-radius: 6px; padding: 24px; font-size: 16px; line-height: 1.6; color: #333;">
<p>
We analyze test-time inference cost using a model that combines both compute and memory access, and define an equivalent cost function using <strong>eFLOPs</strong>, that unifies both kinds of costs in a single metric.
</p>
<p><strong>Definitions:</strong></p>
<ul style="list-style-type: none; padding-left: 0; line-height: 1.6;">
<li><strong>P</strong>: model parameter size</li>
<li><strong>N</strong>: number of reasoning trials</li>
<li><strong>L<sub>in</sub></strong>: input (prefix) length</li>
<li><strong>L<sub>out</sub></strong>: outout (generation) length</li>
<li><strong>D</strong>: attention head dimension</li>
<li><strong>r</strong>: GQA group size</li>
<li><strong>I</strong>: arithmetic intensity of hardware (FLOPs per GB/s bandwidth)</li>
</ul>
<hr style="margin: 24px 0; border: none; border-top: 1px solid #ccc;">
<div style="background: #ffffff; padding: 24px; font-size: 16px; line-height: 1.6; color: #333;">
<p><strong>Computation Cost:</strong></p>
<p style="text-align: center; font-size: 18px;">
<span style="color: #2a63b2;">C<sub>comp</sub> = C<sub>param-compute</sub></span> +
<span style="color: #2a63b2;">C<sub>attn-compute</sub></span> =
<span style="color: #2a63b2;">2PNL<sub>out</sub></span> +
<span style="color: #2a63b2;">(2rNL<sub>in</sub>DL<sub>out</sub> + rNDL<sub>out</sub><sup>2</sup>)</span>
</p>
<p><strong>Memory Access Cost:</strong></p>
<p style="text-align: center; font-size: 18px;">
<span style="color: #20704c;">C<sub>mem</sub> = C<sub>param-mem</sub></span> +
<span style="color: #20704c;">C<sub>attn-mem</sub></span> =
<span style="color: #20704c;">2PL<sub>out</sub></span> +
<span style="color: #20704c;">(2L<sub>in</sub>DL<sub>out</sub> + NDL<sub>out</sub><sup>2</sup>)</span>
</p>
<p>
Also, we consider the prompt KV cache is reused across <em>N</em> reasoning trials.
</p>
<h5 class="title is-5">eFLOPs (Equivalent FLOPs)</h5>
<p>eFLOPs scales memory cost by the arithmetic intensity of the hardware to unify compute and memory costs in a single metric.</p>
<div style="text-align: center; font-size: 18px; padding: 12px 0;">
eFLOPs = <span style="color: #2a63b2;">C<sub>comp</sub></span> +
<span style="color: #20704c;">I × C<sub>mem</sub></span>
</div>
<p>
where <strong>I</strong> is the arithmetic intensity of the hardware (e.g., 562.5 for NVIDIA B200).
</p>
<h5 class="title is-5">Final Cost Model:</h5>
<p>
In real-world deployment, model parameters are amortized across large batches, rendering
<span style="color: #20704c;">C<sub>param-mem</sub></span> insignificant.
</p>
<div style="text-align: center; font-size: 18px; padding: 12px 0;">
C<sub>TTS</sub> =
<span style="color: #2a63b2;">2NPL<sub>out</sub> + 2rNL<sub>in</sub>DL<sub>out</sub> + rNDL<sub>out</sub><sup>2</sup></span> +
<span style="color: #20704c;">2IL<sub>in</sub>DL<sub>out</sub> + INDL<sub>out</sub><sup>2</sup></span>
</div>
<p><strong>Key Insight:</strong> In long chains-of-thought (CoTs), attention-related costs dominate. We define the ratio:</p>
<div style="text-align: center; font-size: 18px; padding: 12px 0;">
Φ = [2rL<sub>in</sub>D + (rD + ID)L<sub>out</sub>] / 2P
</div>
<p>
When L<sub>out</sub> ≥ 4096, Φ can exceed 10–1000×, indicating that memory-bound attention dominates over parameter-bound compute.
</p>
</div>
<span style="display: inline-block; margin-top: 6px; padding: 6px 10px; background-color: #e8f0fe; border-left: 4px solid #4b6cb7; border-radius: 4px; font-weight: 500;">
<strong>Implication:</strong> As generation length increases, inference bottlenecks shift from linear <em>L<sub>out</sub>P</em> to quadratic <em>L<sub>out</sub><sup>2</sup>D</em> attention terms — eliciting the need for attention efficiency.
</span>
</p>
</div>
</details>
<br>
<h4 class="title is-5">
<img src="static/images/icons/Stonks_emoji.png" style="height: 36px; display: inline; vertical-align: middle;"/>
Kinetics Scaling Law
</h4>
<p>
We compare the TTS scaling obtained from existing FLOPs-based analysis and our eFLOPs-based analysis. For each cost model, we identify the most optimal configuration for each individual task and average their accuracy scores under the same cost budget (FLOPs or eFLOPs).
<div style="background: #ffffff; padding: 24px; font-size: 16px; line-height: 1.6; color: #333;">
<figure>
<img src="static/images/presenterationn-1.jpg" alt="Pareto Frontier Comparison" style="width: 100%; height: auto; margin: 0;" />
<figcaption style="margin-top: 10px; text-align: center;">
<strong>Figure 3:</strong> Comparison of AIME24 pareto frontiers for long CoT-based Qwen3 TTS under FLOPs <strong>(ab)</strong> and eFLOPs <strong>(cd)</strong> cost models. Optimal model choices are highlighted in <strong>(ac)</strong>, while optimal CoT lengths are shown in <strong>(bd)</strong>. Similar observations hold for <em>Best-of-N</em> scaling.
</figcaption>
</figure>
<span style="display: block; width: 100%; margin-top: 6px; padding: 6px 10px; background-color: #e8f0fe; border-left: 4px solid #4b6cb7; border-radius: 4px; font-weight: 500;">
<strong>Takeaway:</strong>
<ol>
<li>
A FLOPs-based cost model tends to favor smaller models with more test-time compute (e.g., more trials or longer generations), until their accuracy gains begin to plateau.
</li>
<li>
In contrast, the eFLOPs-based model suggests that scaling up model size is more effective than increasing test-time compute—even in low-accuracy regimes. We find that only beyond an emergent size (14B for Qwen3 series), longer CoT begin to outperform further parameter scaling. Therefore, even under limited budgets, it is often better to choose a larger model upfront.
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Expand to understand why FLOPs-based and eFLOPs-based scaling laws diverge
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Why do they diverge?
</h5>
<p style="margin-bottom: 16px;">
The <strong>“smallness” of smaller models</strong> is deceptive. As shown in <strong>Figure 4a</strong>, their KV cache footprint can be significantly large—even larger in proportion than that of much bigger models. This issue is exacerbated by the <strong>quadratic cost dependency on generation length</strong>, which the traditional FLOPs-based cost model fails to capture.
</p>
<p style="margin-bottom: 16px;">
The plots below highlight this discrepancy through an <strong>iso-cost analysis</strong> showing how cost budgets influence the optimal combination of model size and CoT length.
In <strong>Figure 4b</strong>, the <span style="color: #4b6cb7;"><strong>Iso-FLOPs contours</strong></span> are nearly vertical—indicating that model size plays a dominant role. In contrast, the <span style="color: #2f8f5b;"><strong>Iso-eFLOPs contours</strong></span> in <strong>Figure 4c</strong> appear more horizontal—revealing that optimal generation length is more responsive to cost budgets under the eFLOPs model.
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<img src="static/images/kv_trend-1.jpg" alt="KV Trend" style="width: 100%; height: auto; border-radius: 6px;" />
<p style="margin-top: 8px; font-size: 14px; color: #666;">Figure 4a: KV memory trend across model sizes</p>
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<img src="static/images/dotline1-1.jpg" alt="Iso-FLOPs" style="width: 100%; height: auto; border-radius: 6px;" />
<p style="margin-top: 8px; font-size: 14px; color: #666;">Figure 4b: Iso-FLOPs cost contours</p>
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<img src="static/images/dotline2-1.jpg" alt="Iso-eFLOPs" style="width: 100%; height: auto; border-radius: 6px;" />
<p style="margin-top: 8px; font-size: 14px; color: #666;">Figure 4c: Iso-eFLOPs cost contours</p>
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<strong>KV memory scales sublinearly with model size:</strong>
Smaller models can have substantial memory footprints due to disproportionately large KV caches. For instance, Qwen3-0.6B needs 3.5GB of KV cache for 32K tokens, while Qwen3-32B uses only 8GB. Empirically, doubling parameters increases KV memory by only ~1.2×.
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<strong>Quadratic cost penalizes long generations:</strong>
Under the <em>L<sup>2</sup>D</em> model, generation cost grows faster than model size. As a result, small models can no longer compensate for limited capacity by simply generating longer outputs—especially for complex reasoning tasks.
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</section>
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Kinetics Sparse Test-Time Scaling
</h2>
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<p>
Our cost analysis hints at the importance of attention efficiency in TTS. Our sparse scaling law studies the effectiveness of attention sparsity and how it can further reshape the dense TTS pareto frontier. <em>To see the full potential of sparse attention, we first study <strong>oracle top-k attention</strong>.</em>
<h4 style="margin-bottom: 16px;">
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Sparse attention significantly enhances problem-solving performance
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<p style="margin-top: 8px; font-size: 14px; color: #666;">Figure 5a: Best-of-N scaling comparison between oracle top-k and dense attention</p>
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<img src="static/images/aime24_qwen3-8b_scaling_logerr_extended_tradeoff-1.jpg" alt="topk-benefits" style="width: 100%; height: auto; border-radius: 6px;" />
<p style="margin-top: 8px; font-size: 14px; color: #666;">Figure 5b: Best-of-N using Qwen3-8B </p>
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<div style="text-align: center;">
<img src="static/images/aime24_qwen3-32b_scaling_logerr_extended_tradeoff-1.jpg" alt="topk-benefits" style="width: 100%; height: auto; border-radius: 6px;" />
<p style="margin-top: 8px; font-size: 14px; color: #666;">Figure 5c: Best-of-N using Qwen3-32B </p>
</div>
<div style="text-align: center;">
<img src="static/images/AIME24-topk-genlen-1.jpg" alt="topk-benefits" style="width: 100%; height: auto; border-radius: 6px;" />
<p style="margin-top: 8px; font-size: 14px; color: #666;">Figure 5d: Long-CoT scaling comparison between oracle top-k and dense attention</p>
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<div style="text-align: center;">
<img src="static/images/aime24_qwen3-8b_acc_vs_genlen-1.jpg" alt="topk-benefits" style="width: 100%; height: auto; border-radius: 6px;" />
<p style="margin-top: 8px; font-size: 14px; color: #666;">Figure 5e: Long-CoT using Qwen3-8B </p>
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<div style="text-align: center;">
<img src="static/images/aime24_qwen3-32b_acc_vs_genlen-1.jpg" alt="topk-benefits" style="width: 100%; height: auto; border-radius: 6px;" />
<p style="margin-top: 8px; font-size: 14px; color: #666;">Figure 5f: Long-CoT using Qwen3-32B </p>
</div>
</div>
<br>
<p>
<strong>Figure 5: Figures 5a and 5d</strong> show that sparse attention significantly improves performance and scalability, enabling higher accuracy at lower compute budgets.
<strong>Figures 5b, 5c, 5e, and 5f</strong> highlight that both 8B and 32B models benefit from sparsity, with <em>50–60 percentage point</em> gains in low-cost settings and consistent <em>~5 point</em> improvements even in high-cost regimes.
Notably, sparse models reach these performance levels at much lower costs —for context, 10<sup>5</sup> Tera-eFLOPs is just 22 seconds of B200 usage.
</p>
</div>
<br>
<h4 style="margin-bottom: 16px;">
<img src="static/images/icons/Idea.png" style="height: 36px; display: inline; vertical-align: middle;"/>
Can sparse attention reshape the Kinetics scaling law?
</h4>
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<p>
To understand the importance of attention sparsity, we include KV sparsification into the list of TTS variables. Our holistic analysis reveals that,
<p>
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With increasing cost budget, it is more beneficial to scale generation tokens than KV cache budget.
</span>
</p>
In other words, optimal KV cache budget changes very slowly with increasing cost budget. Consequently, <strong style="color: #2f8f5b;">it allows us to generate substantiallymore tokens at the same cost budget, resulting in higher performance</strong>. Importantly, the original <em>quadratic</em> cost scaling with generation length is replaced by a <em>linear dependency</em>, significantly reducing cost for longer generations.
</p>
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<img src="static/images/presenterationn7-1.jpg" alt="topk-benefits" style="width: 100%; height: auto; border-radius: 6px" />
<p style="margin-top: 8px; font-size: 14px; color: #666;">Figure 6: Compared to the dense scaling, small models (0.6B, 1.7B, 4B) are more effective with sparse attention. In other words, they occupy more space in the Pareto Frontier</p>
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<h4 style="margin-bottom: 16px;">
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Exploring tractable sparse attention algorithms
</h4>
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<p>
We explore two tractable alternatives to oracle top-k attention: <strong>block top-k attention</strong> and <strong>sliding window attention</strong>. Although sliding window attention is easier to implement and has zero search overhead, its performance is extremely poor. Block top-k attention demonstrates comparable scaling to the oracle top-k attention, improving accuracy by 45 points in the low-cost regime and achieving equivalent accuracy while using 8.58X fewer resources compared to dense attention..
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<img src="static/images/aime24_qwen3-8b_sparse_method_comparison-1.jpg" alt="sparse comparison trials" style="width: 100%; height: auto; border-radius: 6px;" />
<p style="margin-top: 8px; font-size: 14px; color: #666; text-align: center;">Figure 7a: Sparse algorithm comparison</p>
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<div>
<img src="static/images/AIME24-blocktopk-trial-1.jpg" alt="block topk trial" style="width: 100%; height: auto; border-radius: 6px;" />
<p style="margin-top: 8px; font-size: 14px; color: #666; text-align: center;">Figure 7b: Block top-k Best-of-N scaling</p>
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<img src="static/images/AIME24-blocktopk-genlen-1.jpg" alt="block topk genlen" style="width: 100%; height: auto; border-radius: 6px;" />
<p style="margin-top: 8px; font-size: 14px; color: #666; text-align: center;">Figure 7c: Block top-k Long-CoT scaling</p>
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<p style="margin-top: 20px;">
<strong>Figure 7:</strong> <strong>(a)</strong> illustrates how block top-k attention scaling closely follows the oracle top-k scaling for the Qwen3-8B model on AIME24 tasks. <strong>(b)</strong> and <strong>(c)</strong> illustrate the Pareto frontiers of block top-k attention.
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</section>
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Conclusion and Future Work
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<p>
In this post, we introduced the Kinetics Scaling Law, emphasizing that attention cost, not parameter count, is the dominant factor at test time, fundamentally reshaping the traditional scaling landscape. We further demonstrated that sparse attention is crucial for achieving more effective and scalable test-time scaling. While our discussion focused on a simple sparse attention algorithm like block top-k attention, we anticipate that more advanced algorithms will approach or even outperform oracle top-k scaling. Moreover, sparse attention drastically reduces inference cost, enabling more reasoning trials and longer generations. This unlocks greater flexibility in configuring TTS strategies within a fixed resource. Overall, we believe the Kinetics Scaling Law serves as a guiding principle for end-to-end design in agent deployment, model architectures, LLM serving systems, and hardware.
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<img src="static/images/icons/Kinetics.png" alt="<i>TriForce</i>" width="300" height="300" />
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</section>
<!-- Section: References -->
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<h2 class="title">BibTeX</h2>
<pre><code>@misc{chen2024magicdecbreakinglatencythroughputtradeoff,
title={Kinetics: Rethinking Test-Time Scaling Laws},
author={Ranajoy Sadhukhan and Zhuoming Chen and Haizhong Zheng and Yang Zhou and Emma Strubell and Beidi Chen},
year={2024},
eprint={2408.11049},
archivePrefix={arXiv},
primaryClass={cs.CL},
url={https://arxiv.org/abs/2408.11049},
}</code></pre>
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