TrustSafeAI/Test-Time-Calibration
0
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class="container is-max-desktop">52 <div class="columns is-centered">53 <div class="column has-text-centered">54 <h1 class="title is-1 publication-title">CarBoN: Calibrated Best-of-N Sampling<br>Improves Test-time Reasoning</h1>55 <div class="is-size-5 publication-authors">56 <span class="author-block">57 <a href="https://sites.google.com/view/yungchentang" target="_blank" rel="noopener noreferrer">Yung-Chen Tang</a><sup>1,2</sup>,58 </span>59 <span class="author-block">60 <a href="https://sites.google.com/site/pinyuchenpage/home" target="_blank" rel="noopener noreferrer">Pin-Yu Chen</a><sup>3</sup>,61 </span>62 <span class="author-block">63 <a href="https://people.epfl.ch/andrea.cavallaro?lang=en" target="_blank" rel="noopener noreferrer">Andrea Cavallaro</a><sup>1,2</sup>64 </span>65 </div>66 <div class="is-size-5 publication-authors">67 <span class="author-block"><sup>1</sup>EPFL,</span>68 <span class="author-block"><sup>2</sup>Idiap Research Institute,</span>69 <span class="author-block"><sup>3</sup>IBM Research</span>70 </div>71 72 <div class="column has-text-centered">73 <div class="publication-links">74 <!-- arXiv Link. -->75 <span class="link-block">76 <a href="https://arxiv.org/abs/2510.15674" target="_blank" rel="noopener noreferrer"77 class="external-link button is-normal is-rounded is-dark">78 <span class="icon">79 <i class="ai ai-arxiv"></i>80 </span>81 <span>arXiv</span>82 </a>83 </span>84 <!-- Code Link. -->85 <span class="link-block">86 <a href="#"87 class="external-link button is-normal is-rounded is-dark">88 <span class="icon">89 <i class="fab fa-github"></i>90 </span>91 <span>Code</span>92 </a>93 </span>94 </div>95 96 </div>97 </div>98 </div>99 </div>100 </div>101</section>102 103 104<section class="section">105 <div class="container is-max-desktop">106 <!-- Abstract. -->107 <div class="columns is-centered has-text-centered">108 <div class="column is-four-fifths">109 <h2 class="title is-3">Abstract</h2>110 <div class="content has-text-justified">111 <p>112 Allocating more computation during inference time (test-time scaling) improves language model performance, especially for reasoning tasks.113 However, popular methods like Best-of-N sampling often show diminishing returns as N increases.114 To address this inefficiency, we introduce a general <strong>test-time calibration framework</strong> that adaptively modifies the model toward high-reward reasoning paths, with theoretical guarantees of improving the lower bound of expected reward under finite sampling, all without large language model (LLM) retraining.115 Within this framework, we propose <strong>CarBoN</strong> (Calibrated Best-of-N), a two-phase method that first explores the solution space and then learns a calibration of the logits via an input-specific temperature T and additive shift vector δ, guiding generation toward more reliable reasoning.116 </p>117 <p>118 Experiments on MATH-500 and AIME-2024 show that CarBoN improves efficiency, with up to 4× fewer rollouts to reach the same accuracy, while often achieving higher accuracy under fixed budgets.119 We also analyze the complementary roles of T and δ in balancing output diversity and correctness, and demonstrate that the framework also generalizes to step-level sampling strategies such as beam search.120 </p>121 </div>122 </div>123 </div>124 <!--/ Abstract. -->125 126 </div>127</section>128 129 130<section class="section">131 <div class="container is-max-desktop">132 133 <!-- Motivation. -->134 <div class="columns is-centered">135 <div class="column is-full-width">136 <h2 class="title is-3">Why calibration for test-time scaling (TTS)? A motivating example of reward-based binary search.</h2>137 138 <div class="content has-text-justified">139 <p>140 Let the task be finding a target in [0,10<sup>4</sup>]. 141 Calibration means that before each search step the model can query <i>n</i> candidate points for reward, where <i>n</i> denotes the number of reward queries per step.142 The reward is defined as the inverse distance to the target plus noise.143 The baseline binary search, corresponding to naive TTS (<i>n</i>=0), requires 13.3 steps on average. Increasing <i>n</i> significantly accelerates convergence: for example, with <i>n</i>=16 the search depth is reduced by up to 74% (see Figure 1, left).144 Figure 1 (right) shows an example run where calibration quickly converges to the target, while vanilla binary search continues oscillating.145 This example highlights that reward feedback for calibration reshapes the sampling distribution and motivates its use for TTS.146 </p>147 </div>148 149 <!-- Figure 1 -->150 <div class="content has-text-centered">151 <img src="./static/images/figure_1.png" alt="Reward-guided calibration accelerates binary search" style="width: 100%; height: auto; margin: 20px 0; display: block;">152 <p class="has-text-justified" style="margin: 0;">153 <strong>Figure 1. Reward-guided calibration accelerates binary search.</strong>154 Left: Increasing per-step noisy reward (inverse-distance signal + noise) lowers average search steps versus vanilla.155 Right: Example showing reward guidance converges early; vanilla keeps oscillating.156 </p>157 </div>158 <!--/ Figure 1 -->159 160 </div>161 </div>162 <!--/ Motivation. -->163 164 165 </div>166</section>167 168 169<section class="section">170 <div class="container is-max-desktop">171 172 <!-- Framework Overview. -->173 <div class="columns is-centered">174 <div class="column is-full-width">175 <h2 class="title is-3">Overview of Test-time Calibration Framework</h2>176 177 <div class="content has-text-justified">178 <p>179 Building on this principle, our framework reuses sampled completions that are normally discarded in parallel sampling methods to extract reward signals and perform calibration.180 Within this framework, we introduce <strong>CarBoN</strong> (Calibrated Best-of-N). Without modifying the original LLM, our framework allocates part of the budget to exploration and calibration, then focuses the remaining budget on high-scoring regions using logit calibration.181 Reusing high-scoring answer selected by reward model enhances answer quality and query efficiency, under the same inference budget.182 </p>183 </div>184 185 <!-- Figure 2 -->186 <div class="content has-text-centered">187 <img src="./static/images/figure_2.png" alt="Test-time calibration framework and MATH-500 results" style="width: 100%; height: auto; margin: 20px 0; display: block;">188 <p class="has-text-justified" style="margin: 0;">189 <strong>Figure 2. (a) Test-time calibration framework.</strong> With a rollout budget N = N<sub>1</sub> + N<sub>2</sub>, the model first explores by generating and scoring N<sub>1</sub> candidate responses. The model then learns calibration parameters (δ, T) from high-scoring responses, using them to adjust the logits for the remaining N<sub>2</sub> generations. The final answer is selected from all N candidates.190 <strong>(b) MATH-500 Results.</strong> CarBoN improves weighted Best-of-N accuracy across four models. For all models, calibrated accuracy at N=64 (orange dash line) matches or exceeds uncalibrated accuracy at N=256, corresponding to up to a 4× reduction in rollout budgets. Notably, with Qwen2.5-Math-1.5B-Instruct at N=64, CarBoN surpasses GPT-4o (red dashed line), while uncalibrated Best-of-N with N=256 does not.191 </p>192 </div>193 <!--/ Figure 2 -->194 195 </div>196 </div>197 <!--/ Framework Overview. -->198 199 200 </div>201</section>202 203 204<section class="section">205 <div class="container is-max-desktop">206 207 <!-- Theoretical Guarantees. -->208 <div class="columns is-centered">209 <div class="column is-full-width">210 <h2 class="title is-3">Theoretical Analysis</h2>211 212 <div class="content has-text-justified">213 <p>214 We provide theoretical foundations for our calibration framework through three formal results. 215 <strong>Lemma 1</strong> establishes the existence of joint calibration parameters \((\delta^*, T^*)\) that provably improve the model's predictive distribution when the base model is not perfectly calibrated. 216 <strong>Theorem 1</strong> proves that applying such calibration strictly increases the expected reward under finite Best-of-\(N\) sampling, by showing that calibration achieves first-order stochastic dominance—making high-reward outputs more likely. 217 <strong>Corollary 1</strong> demonstrates that discarding exploration samples and using only calibrated samples is strictly suboptimal. 218 Together, these results rigorously justify why test-time calibration works and why both phases are essential.219 </p>220 </div>221 222 <!-- Lemma 1 -->223 <details style="margin-bottom: 1rem; border: 1px solid #dbdbdb; border-radius: 6px; padding: 1rem;">224 <summary style="cursor: pointer; font-weight: bold; font-size: 1.1rem; color: #3273dc;">225 Lemma 1: Existence of an Improving Joint Solution \((\delta, T)\)226 </summary>227 <div style="margin-top: 1rem; padding-left: 1rem; border-left: 3px solid #3273dc;">228 <p>229 Let the joint loss function be \(\mathcal{L}(\delta, T) = \mathbb{E}_{y \sim \mathcal{D}_\text{calib}(x)} \left[ -\log p_\theta(y \mid x; \delta, T) \right]\).230 Let \(\bar{p}_{\theta}\) be the model's average predictive distribution and \(\bar{p}_{\text{target}}\) be the empirical average one-hot distribution, both averaged over all generation steps in the calibration set \(\mathcal{D}_\text{calib}(x)\).231 Suppose the base model is not perfectly calibrated in the sense that at least one of the following conditions holds: (1) \(\bar{p}_{\theta} \neq \bar{p}_{\text{target}}\), or (2) the average logit of ground-truth tokens does not equal the average expected logit.232 </p>233 <p>234 Then there exists a joint solution \((\delta, T) \in \mathbb{R}^D \times (0, \infty)\), where \((\delta, T) \neq (\mathbf{0}, 1)\), such that the loss is strictly reduced: 235 \[\mathcal{L}(\delta, T) < \mathcal{L}(\mathbf{0}, 1)\]236 </p>237 </div>238 </details>239 240 <!-- Theorem 1 -->241 <details style="margin-bottom: 1rem; border: 1px solid #dbdbdb; border-radius: 6px; padding: 1rem;">242 <summary style="cursor: pointer; font-weight: bold; font-size: 1.1rem; color: #3273dc;">243 Theorem 1: Joint Calibration \((\delta, T)\) Improves Expected Reward from Best-of-\(N\) Sampling244 </summary>245 <div style="margin-top: 1rem; padding-left: 1rem; border-left: 3px solid #3273dc;">246 <p>247 Let \(p_{\theta}(y \mid x; \delta, T)\) be the model's probability distribution over outputs \(y \in \mathcal{Y}\), parameterized by a calibration vector \(\delta\) and a temperature \(T\). Let the base model be configured with parameters \((\mathbf{0}, T_{base})\) for some \(T_{base} > 0\).248 Let \(R(x,y)\) be a reward function, and assume there exists a unique output \(y^* \in \mathcal{Y}\) with a strictly maximum reward, i.e., \(r^* = R(x,y^*) > \max_{y \neq y^*} R(x,y) = r_{\text{other\_max}}\).249 </p>250 <p>251 We consider cases where joint calibration with parameters \((\delta^*, T^*)\) improves upon the base model by increasing the probability of the unique optimal output:252 \[p_{\theta}(y^* \mid x; \delta^*, T^*) > p_{\theta}(y^* \mid x; \mathbf{0}, T_{base})\]253 </p>254 <p>255 Then, for any \(n \geq 1\) within the remaining inference budget after calibration, the lower bound on the expected best-of-\(N\) reward under the jointly calibrated model is strictly greater than that of the base model.256 Specifically, let \(R_{LB}(p) = r^* - (1-p)^n (r^* - r_{\text{other\_max}})\) be a valid lower bound for the expected best-of-\(N\) reward, where \(p\) is the probability of sampling \(y^*\). The improvement in this lower bound is strictly positive:257 \[\Delta_{R_{LB}}(x,n) = R_{LB}(p_{\theta}(y^* \mid x; \delta^*, T^*)) - R_{LB}(p_{\theta}(y^* \mid x; \mathbf{0}, T_{base})) > 0\]258 </p>259 </div>260 </details>261 262 <!-- Corollary 1 -->263 <details style="margin-bottom: 1rem; border: 1px solid #dbdbdb; border-radius: 6px; padding: 1rem;">264 <summary style="cursor: pointer; font-weight: bold; font-size: 1.1rem; color: #3273dc;">265 Corollary 1: Sub-optimality of Exploitation Alone266 </summary>267 <div style="margin-top: 1rem; padding-left: 1rem; border-left: 3px solid #3273dc;">268 <p>269 The final candidate is selected by maximizing \(R(x, y)\) over a set of candidates \(\mathcal{Y}\). Since \(\mathcal{Y}_{\text{exploit}}\) is a subset of the union \(\mathcal{Y} = \mathcal{Y}_{\text{explore}} \cup \mathcal{Y}_{\text{exploit}}\), the strategy of only selecting from \(\mathcal{Y}_{\text{exploit}}\) is sub-optimal compared to selecting from the union.270 </p>271 <p>272 This is because the maximum reward achievable from the union is greater than or equal to the maximum reward achievable from the exploitation set alone:273 \[\max_{y \in \mathcal{Y}_{\text{explore}} \cup \mathcal{Y}_{\text{exploit}}} R(x, y) \geq \max_{y \in \mathcal{Y}_{\text{exploit}}} R(x, y)\]274 </p>275 <p>276 Therefore, including exploration samples in the final candidate pool is essential for maximizing expected reward.277 </p>278 </div>279 </details>280 281 </div>282 </div>283 <!--/ Theoretical Guarantees. -->284 285 286 </div>287</section>288 289 290<section class="section">291 <div class="container is-max-desktop">292 293 <!-- Empirical Results. -->294 <div class="columns is-centered">295 <div class="column is-full-width">296 <h2 class="title is-3">Empirical Results</h2>297 298 <div class="content has-text-justified">299 <p>300 We evaluate CarBoN on MATH-500 and AIME-2024 benchmarks across multiple language models. The results demonstrate that CarBoN consistently improves upon standard Best-of-\(N\) sampling, with calibrated accuracy at \(N=64\) matching or exceeding uncalibrated results at \(N=256\) (\(4\times\) reduction in rollout budget).301 </p>302 </div>303 304 <!-- Table 1: MATH-500 Results -->305 <h3 class="title is-4" style="margin-top: 2rem;">MATH-500 Results</h3>306 <div style="max-width: 800px; margin: 0 auto 2rem auto;">307 <p style="font-size: 0.95rem; margin-bottom: 0.5rem;">308 <strong>Table 1.</strong> Accuracy (%) of four models on MATH-500, comparing Weighted Best-of-<span style='font-style:italic;'>N</span> methods before and after calibration. CarBoN enables further improvements beyond the plateau of standard Best-of-<span style='font-style:italic;'>N</span>, with calibrated accuracy at <span style='font-style:italic;'>N</span>=64 exceeding the uncalibrated results at <span style='font-style:italic;'>N</span>=256. Bold indicates better accuracy for each <span style='font-style:italic;'>N</span>.309 </p>310 <div style="overflow-x: auto;">311 <table class="table is-bordered is-striped is-narrow is-hoverable" style="margin: 0 auto; width: 100%; min-width: 600px;">312 <thead>313 <tr>314 <th style="text-align: center; width: 180px;">Model</th>315 <th style="text-align: center; width: 110px;">Method</th>316 <th style="text-align: center; width: 70px;">N=8</th>317 <th style="text-align: center; width: 70px;">N=16</th>318 <th style="text-align: center; width: 70px;">N=32</th>319 <th style="text-align: center; width: 70px;">N=64</th>320 <th style="text-align: center; width: 70px;">N=128</th>321 <th style="text-align: center; width: 70px;">N=256</th>322 </tr>323 </thead>324 <tbody>325 <tr>326 <td rowspan="2" style="vertical-align: middle; text-align: left; background-color: white;">Llama-3.2-1B-Instruct</td>327 <td style="text-align: left; background-color: white;">Best-of-<span style='font-style:italic;'>N</span></td>328 <td style="text-align: center; background-color: white;">42.0</td>329 <td style="text-align: center; background-color: white;">44.6</td>330 <td style="text-align: center; background-color: white;">47.8</td>331 <td style="text-align: center; background-color: white;">48.6</td>332 <td style="text-align: center; background-color: white;">50.6</td>333 <td style="text-align: center; background-color: white;">50.8</td>334 </tr>335 <tr>336 <td style="text-align: left; background-color: white;">CarBoN</td>337 <td style="text-align: center; background-color: white;"><strong>43.0</strong></td>338 <td style="text-align: center; background-color: white;"><strong>45.6</strong></td>339 <td style="text-align: center; background-color: white;"><strong>48.4</strong></td>340 <td style="text-align: center; background-color: white;"><strong>51.0</strong></td>341 <td style="text-align: center; background-color: white;"><strong>51.8</strong></td>342 <td style="text-align: center; background-color: white;"><strong>51.8</strong></td>343 </tr>344 <tr>345 <td rowspan="2" style="vertical-align: middle; text-align: left; background-color: #f9f9f9;">Llama-3.2-3B-Instruct</td>346 <td style="text-align: left; background-color: #f9f9f9;">Best-of-<span style='font-style:italic;'>N</span></td>347 <td style="text-align: center; background-color: #f9f9f9;">56.8</td>348 <td style="text-align: center; background-color: #f9f9f9;">58.2</td>349 <td style="text-align: center; background-color: #f9f9f9;">59.6</td>350 <td style="text-align: center; background-color: #f9f9f9;">61.6</td>351 <td style="text-align: center; background-color: #f9f9f9;">61.8</td>352 <td style="text-align: center; background-color: #f9f9f9;">62.2</td>353 </tr>354 <tr>355 <td style="text-align: left; background-color: #f9f9f9;">CarBoN</td>356 <td style="text-align: center; background-color: #f9f9f9;"><strong>57.6</strong></td>357 <td style="text-align: center; background-color: #f9f9f9;"><strong>59.0</strong></td>358 <td style="text-align: center; background-color: #f9f9f9;"><strong>60.8</strong></td>359 <td style="text-align: center; background-color: #f9f9f9;"><strong>62.2</strong></td>360 <td style="text-align: center; background-color: #f9f9f9;"><strong>63.2</strong></td>361 <td style="text-align: center; background-color: #f9f9f9;"><strong>63.4</strong></td>362 </tr>363 <tr>364 <td rowspan="2" style="vertical-align: middle; text-align: left; background-color: white;">Qwen2.5-1.5B-Instruct</td>365 <td style="text-align: left; background-color: white;">Best-of-<span style='font-style:italic;'>N</span></td>366 <td style="text-align: center; background-color: white;"><strong>56.4</strong></td>367 <td style="text-align: center; background-color: white;">57.6</td>368 <td style="text-align: center; background-color: white;">61.4</td>369 <td style="text-align: center; background-color: white;">62.0</td>370 <td style="text-align: center; background-color: white;">62.6</td>371 <td style="text-align: center; background-color: white;">62.2</td>372 </tr>373 <tr>374 <td style="text-align: left; background-color: white;">CarBoN</td>375 <td style="text-align: center; background-color: white;">55.0</td>376 <td style="text-align: center; background-color: white;"><strong>60.0</strong></td>377 <td style="text-align: center; background-color: white;"><strong>61.8</strong></td>378 <td style="text-align: center; background-color: white;"><strong>62.4</strong></td>379 <td style="text-align: center; background-color: white;"><strong>64.0</strong></td>380 <td style="text-align: center; background-color: white;"><strong>64.4</strong></td>381 </tr>382 <tr>383 <td rowspan="2" style="vertical-align: middle; text-align: left; background-color: #f9f9f9;">Qwen2.5-Math-1.5B-Instruct</td>384 <td style="text-align: left; background-color: #f9f9f9;">Best-of-<span style='font-style:italic;'>N</span></td>385 <td style="text-align: center; background-color: #f9f9f9;">73.6</td>386 <td style="text-align: center; background-color: #f9f9f9;">75.4</td>387 <td style="text-align: center; background-color: #f9f9f9;"><strong>76.4</strong></td>388 <td style="text-align: center; background-color: #f9f9f9;">75.6</td>389 <td style="text-align: center; background-color: #f9f9f9;">76.4</td>390 <td style="text-align: center; background-color: #f9f9f9;">76.8</td>391 </tr>392 <tr>393 <td style="text-align: left; background-color: #f9f9f9;">CarBoN</td>394 <td style="text-align: center; background-color: #f9f9f9;"><strong>74.2</strong></td>395 <td style="text-align: center; background-color: #f9f9f9;"><strong>76.0</strong></td>396 <td style="text-align: center; background-color: #f9f9f9;"><strong>76.4</strong></td>397 <td style="text-align: center; background-color: #f9f9f9;"><strong>77.2</strong></td>398 <td style="text-align: center; background-color: #f9f9f9;"><strong>77.2</strong></td>399 <td style="text-align: center; background-color: #f9f9f9;"><strong>77.8</strong></td>400 </tr>401 </tbody>402 </table>403 </div>404 </div>405 406 <!-- Table 2: AIME-2024 Results -->407 <h3 class="title is-4" style="margin-top: 3rem;">AIME-2024 Results</h3>408 <div style="max-width: 800px; margin: 0 auto 2rem auto;">409 <p style="font-size: 0.95rem; margin-bottom: 0.5rem;">410 <strong>Table 2.</strong> Correct answers (out of 30) on the AIME-2024 benchmark for two math-specialized models, comparing Best-of-<span style='font-style:italic;'>N</span> and CarBoN across different rollout budgets. CarBoN enables further improvements beyond the plateau of standard Best-of-<span style='font-style:italic;'>N</span>. Bold numbers indicate the higher number of correct answers for each <span style='font-style:italic;'>N</span>.411 </p>412 <div style="overflow-x: auto;">413 <table class="table is-bordered is-striped is-narrow is-hoverable" style="margin: 0 auto; width: 100%; min-width: 600px;">414 <thead>415 <tr>416 <th style="text-align: center; width: 180px;">Model</th>417 <th style="text-align: center; width: 110px;">Method</th>418 <th style="text-align: center; width: 70px;">N=16</th>419 <th style="text-align: center; width: 70px;">N=32</th>420 <th style="text-align: center; width: 70px;">N=64</th>421 <th style="text-align: center; width: 70px;">N=128</th>422 <th style="text-align: center; width: 70px;">N=256</th>423 </tr>424 </thead>425 <tbody>426 <tr>427 <td rowspan="2" style="vertical-align: middle; text-align: left; background-color: white;">Qwen2.5-Math-1.5B-Instruct</td>428 <td style="text-align: left; background-color: white;">Best-of-<span style='font-style:italic;'>N</span></td>429 <td style="text-align: center; background-color: white;">4/30</td>430 <td style="text-align: center; background-color: white;">5/30</td>431 <td style="text-align: center; background-color: white;">6/30</td>432 <td style="text-align: center; background-color: white;">6/30</td>433 <td style="text-align: center; background-color: white;">6/30</td>434 </tr>435 <tr>436 <td style="text-align: left; background-color: white;">CarBoN</td>437 <td style="text-align: center; background-color: white;">4/30</td>438 <td style="text-align: center; background-color: white;">5/30</td>439 <td style="text-align: center; background-color: white;">6/30</td>440 <td style="text-align: center; background-color: white;"><strong>7/30</strong></td>441 <td style="text-align: center; background-color: white;"><strong>7/30</strong></td>442 </tr>443 <tr>444 <td rowspan="2" style="vertical-align: middle; text-align: left; background-color: #f9f9f9;">Qwen2.5-Math-7B-Instruct</td>445 <td style="text-align: left; background-color: #f9f9f9;">Best-of-<span style='font-style:italic;'>N</span></td>446 <td style="text-align: center; background-color: #f9f9f9;">5/30</td>447 <td style="text-align: center; background-color: #f9f9f9;">5/30</td>448 <td style="text-align: center; background-color: #f9f9f9;">6/30</td>449 <td style="text-align: center; background-color: #f9f9f9;">6/30</td>450 <td style="text-align: center; background-color: #f9f9f9;">6/30</td>451 </tr>452 <tr>453 <td style="text-align: left; background-color: #f9f9f9;">CarBoN</td>454 <td style="text-align: center; background-color: #f9f9f9;">5/30</td>455 <td style="text-align: center; background-color: #f9f9f9;"><strong>6/30</strong></td>456 <td style="text-align: center; background-color: #f9f9f9">6/30</td>457 <td style="text-align: center; background-color: #f9f9f9">6/30</td>458 <td style="text-align: center; background-color: #f9f9f9;"><strong>7/30</strong></td>459 </tr>460 </tbody>461 </table>462 </div>463 </div>464 465 <!-- Generalizing Test-time Calibration Beyond Best-of-N -->466 <h3 class="title is-4" style="margin-top: 3rem;">Generalizing Test-time Calibration Beyond Best-of-<span style='font-style:italic;'>N</span></h3>467 <div class="content has-text-justified" style="margin-bottom: 1rem;">468 <p>469 Test-time calibration is not limited to Best-of-<span style='font-style:italic;'>N</span> sampling. We show that calibrating step-level strategies such as beam search can further improve sample efficiency and accuracy. As shown below, calibrated beam search achieves similar or better performance with fewer candidates, demonstrating the generality of our approach.470 </p>471 </div>472 <div style="max-width: 800px; margin: 0 auto 2rem auto;">473 <p style="font-size: 0.95rem; margin-bottom: 0.5rem;">474 <strong>Table 3.</strong> Accuracy (%) of standard and calibrated beam search on the MATH-500 benchmark. Calibrated beam search generally improves test-time reasoning performance, especially for larger <span style='font-style:italic;'>N</span>.475 </p>476 <div style="overflow-x: auto;">477 <table class="table is-bordered is-striped is-narrow is-hoverable" style="margin: 0 auto; width: 100%; min-width: 600px;">478 <thead>479 <tr>480 <th style="text-align: center; width: 180px;">Model</th>481 <th style="text-align: center; width: 170px;">Method</th>482 <th style="text-align: center; width: 70px;">N=8</th>483 <th style="text-align: center; width: 70px;">N=16</th>484 <th style="text-align: center; width: 70px;">N=32</th>485 <th style="text-align: center; width: 70px;">N=64</th>486 </tr>487 </thead>488 <tbody>489 <tr>490 <td rowspan="2" style="vertical-align: middle; text-align: left; background-color: white;">Llama-3.2-1B-Instruct</td>491 <td style="text-align: left; background-color: white;">Beam Search</td>492 <td style="text-align: center; background-color: white;">56.0</td>493 <td style="text-align: center; background-color: white;">58.4</td>494 <td style="text-align: center; background-color: white;">60.4</td>495 <td style="text-align: center; background-color: white;">62.2</td>496 </tr>497 <tr>498 <td style="text-align: left; background-color: white;">Calibrated Beam Search</td>499 <td style="text-align: center; background-color: white;"><strong>57.2</strong></td>500 <td style="text-align: center; background-color: white;"><strong>60.0</strong></td>501 <td style="text-align: center; background-color: white;"><strong>62.2</strong></td>502 <td style="text-align: center; background-color: white;"><strong>64.2</strong></td>503 </tr>504 <tr>505 <td rowspan="2" style="vertical-align: middle; text-align: left; background-color: #f9f9f9;">Qwen2.5-Math-1.5B-Instruct</td>506 <td style="text-align: left; background-color: #f9f9f9;">Beam Search</td>507 <td style="text-align: center; background-color: #f9f9f9;"><strong>79.0</strong></td>508 <td style="text-align: center; background-color: #f9f9f9;">79.2</td>509 <td style="text-align: center; background-color: #f9f9f9;">80.2</td>510 <td style="text-align: center; background-color: #f9f9f9;">81.4</td>511 </tr>512 <tr>513 <td style="text-align: left; background-color: #f9f9f9;">Calibrated Beam Search</td>514 <td style="text-align: center; background-color: #f9f9f9;">78.6</td>515 <td style="text-align: center; background-color: #f9f9f9;"><strong>79.6</strong></td>516 <td style="text-align: center; background-color: #f9f9f9;"><strong>81.2</strong></td>517 <td style="text-align: center; background-color: #f9f9f9;"><strong>82.8</strong></td>518 </tr>519 </tbody>520 </table>521 </div>522 </div>523 <div class="content has-text-justified" style="margin-top: 2rem;">524 <p>525 For more detailed ablation studies and analysis, please refer to our paper.526 </p>527 </div>528 529 </div>530 </div>531 <!--/ Empirical Results. -->532 533 </div>534</section>535 536 537<section class="section" id="BibTeX">538 <div class="container is-max-desktop content">539 <h2 class="title">BibTeX</h2>540 <pre><code>@misc{tang2025carboncalibratedbestofnsampling,541 title={CarBoN: Calibrated Best-of-N Sampling Improves Test-time Reasoning}, 542 author={Yung-Chen Tang and Pin-Yu Chen and Andrea Cavallaro},543 year={2025},544 eprint={2510.15674},545 archivePrefix={arXiv},546 primaryClass={cs.LG},547 url={https://arxiv.org/abs/2510.15674}, 548}549</code></pre>550 </div>551</section>552 553 554<footer class="footer">555 <div class="container">556 <div class="columns is-centered">557 <div class="column is-8">558 <div class="content">559 <p>560 This website is licensed under a <a rel="license"561 href="http://creativecommons.org/licenses/by-sa/4.0/">Creative562 Commons Attribution-ShareAlike 4.0 International License</a>.563 </p>564 <p>565 This means you are free to borrow the <a566 href="https://github.com/nerfies/nerfies.github.io">source code</a> of this website,567 we just ask that you link back to this page in the footer.568 Please remember to remove the analytics code included in the header of the website which569 you do not want on your website.570 </p>571 </div>572 </div>573 </div>574 </div>575</footer>576 577</body>578</html>579