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test_time_compute_scaling [2026/03/24 17:05] – Create page: Test-Time Compute Scaling with researched content agenttest_time_compute_scaling [2026/03/24 17:44] (current) – Add LaTeX math formatting for best-of-N, beam search, compute-optimal allocation agent
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 Traditional scaling laws focus on pretraining: more parameters, more data, more FLOPs. Test-time compute scaling introduces a complementary axis -- scaling compute at inference. The key insight from Snell et al. (arXiv:2408.03314) and subsequent work (arXiv:2501.02497) is that there exist **compute-optimal strategies** for how to spend inference FLOPs, analogous to Chinchilla-optimal training. Traditional scaling laws focus on pretraining: more parameters, more data, more FLOPs. Test-time compute scaling introduces a complementary axis -- scaling compute at inference. The key insight from Snell et al. (arXiv:2408.03314) and subsequent work (arXiv:2501.02497) is that there exist **compute-optimal strategies** for how to spend inference FLOPs, analogous to Chinchilla-optimal training.
  
-The inference-to-pretraining token ratio **R** determines which strategy dominates: +The inference-to-pretraining token ratio $R = \frac{\text{inference tokens}}{\text{pretraining tokens}}$ determines which strategy dominates: 
-  * **R << 1** (few queries): Test-time compute excels; smaller models with heavy TTS outperform 14x larger models +  * $R \ll 1$ (few queries): Test-time compute excels; smaller models with heavy TTS outperform 14x larger models 
-  * **R >> 1** (high-volume production): Pretraining larger models is more cost-effective+  * $R \gg 1$ (high-volume production): Pretraining larger models is more cost-effective
  
 ===== Core Techniques ===== ===== Core Techniques =====
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 ==== Best-of-N Sampling ==== ==== Best-of-N Sampling ====
  
-Generate N candidate responses in parallel, then select the highest-scoring one using a verifier (typically a Process Reward Model). This provides broad coverage but is compute-intensive for large N and less effective on difficult prompts compared to adaptive methods.+Generate $N$ candidate responses in parallel, then select the highest-scoring one using a verifier (typically a Process Reward Model). The expected quality scales as: 
 + 
 +$$\mathbb{E}\!\left[\max_{i=1}^{N} r(y_i)\right] \geq \mathbb{E}[r(y)]$$ 
 + 
 +with diminishing marginal returns as $N$ increases. This provides broad coverage but is compute-intensive for large $N$ and less effective on difficult prompts compared to adaptive methods.
  
 <code python> <code python>
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 ==== Beam Search over Thoughts ==== ==== Beam Search over Thoughts ====
  
-Maintain a beam of top-k candidate reasoning paths (chains-of-thought), iteratively expanding and pruning based on Process Reward Model scores. This sequential refinement outperforms best-of-N by focusing compute where it matters most:+Maintain a beam of top-$k$ candidate reasoning paths (chains-of-thought), iteratively expanding and pruning based on Process Reward Model scores. This sequential refinement outperforms best-of-N by focusing compute where it matters most:
  
   - Generate initial candidates   - Generate initial candidates
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   - Expand each, rescore, prune again   - Expand each, rescore, prune again
   - Repeat until completion   - Repeat until completion
 +
 +At each step $t$, the beam retains the top-$k$ partial trajectories by cumulative PRM score:
 +
 +$$\mathcal{B}_t = \text{top-}k\!\left\{\tau_{1:t} : \sum_{i=1}^{t} r(s_i, a_i)\right\}$$
  
 Beam search achieves **4x better efficiency** than best-of-N baselines in FLOPs-matched comparisons. Beam search achieves **4x better efficiency** than best-of-N baselines in FLOPs-matched comparisons.
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 ===== Compute-Optimal Strategies ===== ===== Compute-Optimal Strategies =====
  
-The compute-optimal approach estimates prompt difficulty (e.g., via pass@1 rate) and allocates compute adaptively:+The compute-optimal approach estimates prompt difficulty (e.g., via pass@1 rate $p$) and allocates compute adaptively. The optimal number of samples $N^*$ for a given compute budget $C$ satisfies: 
 + 
 +$$N^*(p, C) = \arg\max_N \; P(\text{at least one correct} \mid N) = \arg\max_N \; \left[1 - (1-p)^N\right] \quad \text{s.t.} \; N \cdot c_{\text{gen}} \leq C$$ 
 + 
 +where $c_{\text{gen}}$ is the cost per generation. This yields:
  
-  * **Easy prompts**: Favor iterative self-revision with minimal overhead+  * **Easy prompts** ($p$ high): Favor iterative self-revision with minimal overhead ($N^*$ small)
   * **Medium prompts**: Use moderate beam search with PRM guidance   * **Medium prompts**: Use moderate beam search with PRM guidance
-  * **Hard prompts**: Deploy full beam search or parallel sampling with maximum budget+  * **Hard prompts** ($p$ low): Deploy full beam search or parallel sampling with maximum budget ($N^*$ large)
  
 This adaptive allocation yields dramatically better efficiency than uniform compute budgets across all prompts. This adaptive allocation yields dramatically better efficiency than uniform compute budgets across all prompts.
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