Update index.html
Browse files- index.html +368 -274
index.html
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<head>
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<meta charset="utf-8">
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<meta name="description"
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content="
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<meta name="keywords" content="Nerfies, D-NeRF, NeRF">
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<meta name="viewport" content="width=device-width, initial-scale=1">
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<title>
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<link href="https://fonts.googleapis.com/css?family=Google+Sans|Noto+Sans|Castoro"
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rel="stylesheet">
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<link rel="stylesheet" href="
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<link rel="stylesheet" href="
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<link rel="stylesheet" href="./static/css/bulma-slider.min.css">
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<link rel="stylesheet" href="./static/css/fontawesome.all.min.css">
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<link rel="stylesheet"
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href="https://cdn.jsdelivr.net/gh/jpswalsh/academicons@1/css/academicons.min.css">
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<script src="https://ajax.googleapis.com/ajax/libs/jquery/3.5.1/jquery.min.js"></script>
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<script
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<script src="./static/js/bulma-slider.min.js"></script>
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<script src="./static/js/index.js"></script>
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</head>
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<body>
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<div class="container is-max-desktop">
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<div class="columns is-centered">
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<div class="column has-text-centered">
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<h1 class="title is-1 publication-title">
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<div class="is-size-5 publication-authors">
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<span class="author-block">
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<a href="https://
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<span class="author-block">
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<a href="https://utkarshsinha.com" target="_blank">Utkarsh Sinha</a><sup>2</sup>,</span>
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<span class="author-block">
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<a href="https://jonbarron.info" target="_blank">Jonathan T. Barron</a><sup>2</sup>,
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</span>
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<span class="author-block">
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<a href="http://sofienbouaziz.com" target="_blank">Sofien Bouaziz</a><sup>2</sup>,
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</span>
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<span class="author-block">
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<a href="https://www.
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</span>
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<span class="author-block">
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<a href="https://homes.cs.washington.edu/~seitz/" target="_blank">Steven M. Seitz</a><sup>1,2</sup>,
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</span>
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<span class="author-block">
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<a href="http://www.ricardomartinbrualla.com" target="_blank">Ricardo Martin-Brualla</a><sup>2</sup>
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</span>
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</div>
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<div class="is-size-5 publication-authors">
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<span class="author-block"><sup>1</sup>
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<span class="author-block"><sup>2</sup>Google Research</span>
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</div>
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<div class="column has-text-centered">
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<div class="publication-links">
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<!-- PDF Link. -->
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<span class="link-block">
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<a href="https://arxiv.org/pdf/
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class="external-link button is-normal is-rounded is-dark">
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<span class="icon">
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<i class="fas fa-file-pdf"></i>
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</a>
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</span>
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<span class="link-block">
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<a href="https://arxiv.org/abs/
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class="external-link button is-normal is-rounded is-dark">
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<span class="icon">
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<i class="ai ai-arxiv"></i>
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<span>arXiv</span>
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</a>
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</span>
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<!-- Video Link. -->
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<span class="link-block">
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<a href="https://www.youtube.com/watch?v=MrKrnHhk8IA" target="_blank"
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class="external-link button is-normal is-rounded is-dark">
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<span class="icon">
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<i class="fab fa-youtube"></i>
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</span>
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<span>Video</span>
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</a>
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</span>
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<!-- Code Link. -->
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<span class="link-block">
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<a href="https://github.com/
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class="external-link button is-normal is-rounded is-dark">
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<span class="icon">
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<i class="fab fa-github"></i>
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<span>Code</span>
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</a>
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</span>
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<!-- Dataset Link. -->
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<span class="link-block">
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<a href="https://github.com/google/nerfies/releases/tag/0.1" target="_blank"
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class="external-link button is-normal is-rounded is-dark">
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<span class="icon">
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<i class="far fa-images"></i>
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</span>
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<span>Data</span>
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</a>
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</div>
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</div>
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</div>
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</section>
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</div>
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</section>
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<
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<div class="hero-body">
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<div id="results-carousel" class="carousel results-carousel">
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<div class="item item-steve">
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<video poster="" id="steve" autoplay controls muted loop playsinline height="100%">
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<source src="./static/videos/steve.mp4"
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type="video/mp4">
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</video>
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type="video/mp4">
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</video>
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<video poster="" id="shiba" autoplay controls muted loop playsinline height="100%">
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type="video/mp4">
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</video>
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<div class="item item-fullbody">
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<video poster="" id="fullbody" autoplay controls muted loop playsinline height="100%">
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<source src="./static/videos/fullbody.mp4"
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type="video/mp4">
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<video poster="" id="blueshirt" autoplay controls muted loop playsinline height="100%">
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<source src="./static/videos/blueshirt.mp4"
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type="video/mp4">
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<video poster="" id="mask" autoplay controls muted loop playsinline height="100%">
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type="video/mp4">
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</video>
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<video poster="" id="coffee" autoplay controls muted loop playsinline height="100%">
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<source src="./static/videos/coffee.mp4"
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type="video/mp4">
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</video>
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<video poster="" id="toby" autoplay controls muted loop playsinline height="100%">
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<source src="./static/videos/toby2.mp4"
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type="video/mp4">
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</video>
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</div>
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</div>
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</div>
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</section>
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<section class="section">
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<div class="container is-max-desktop">
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<!-- Abstract. -->
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<h2 class="title is-3">Abstract</h2>
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<div class="content has-text-justified">
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<p>
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</p>
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<p>
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propose a coarse-to-fine optimization method for coordinate-based models that allows for
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more robust optimization.
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By adapting principles from geometry processing and physical simulation to NeRF-like
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models, we propose an elastic regularization of the deformation field that further
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improves robustness.
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</p>
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<p>
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images of the same pose at different viewpoints. We show that our method faithfully
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reconstructs non-rigidly deforming scenes and reproduces unseen views with high
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fidelity.
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</p>
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</div>
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</div>
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</div>
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<!--/ Abstract. -->
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<!-- Paper video. -->
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<div class="columns is-centered has-text-centered">
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<div class="column is-four-fifths">
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<h2 class="title is-3">Video</h2>
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<div class="publication-video">
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<iframe src="https://www.youtube.com/embed/MrKrnHhk8IA?rel=0&showinfo=0"
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frameborder="0" allow="autoplay; encrypted-media" allowfullscreen></iframe>
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</div>
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</div>
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</div>
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<!--/ Paper video. -->
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</div>
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</section>
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<section class="section">
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<div class="container is-max-desktop">
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<div class="columns is-centered">
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</div>
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</div>
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<!--/ Visual Effects. -->
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<!-- Matting. -->
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<div class="column">
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<h2 class="title is-3">Matting</h2>
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<div class="columns is-centered">
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<div class="column content">
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<p>
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As a byproduct of our method, we can also solve the matting problem by ignoring
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samples that fall outside of a bounding box during rendering.
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</p>
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<video id="matting-video" controls playsinline height="100%">
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<source src="./static/videos/matting.mp4"
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type="video/mp4">
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</video>
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</div>
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</div>
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<div class="column is-full-width">
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<
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<!-- Interpolating. -->
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<h3 class="title is-4">Interpolating states</h3>
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<div class="content has-text-justified">
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<p>
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</div>
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<img src="./static/images/interpolate_end.jpg"
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class="interpolation-image"
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alt="Interpolation end reference image."/>
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<p class="is-bold">End Frame</p>
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</div>
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</div>
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<br/>
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<!--/ Interpolating. -->
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<h3 class="title is-4">Re-rendering the input video</h3>
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<h1 class="title is-1 publication-title">Certified Self-Consistency: Statistical Guarantees and Test-Time Training for Reliable Reasoning in LLMs</h1>
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<a href="https://paulaoak.github.io/">Paula Cordero-Encinar</a><sup>1</sup>,</span>
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<span class="author-block"><sup>1</sup>Imperial College London</span>
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<span class="has-text-weight-bold">TLDR:</span> We provide a unified statistical framework of when and why self-consistency yields certifiable reliability in reasoning models, and how test-time adaptation can further reduce the computational cost of this certification.
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<img src="condorcet_framework.png" alt="Certified self-consistency workflow" style="width: 100%;">
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<figcaption style="color:#6b7280; font-size: 0.9rem; margin-top: 8px;">
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Given a prompt, the model generates multiple reasoning rollouts from the
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reference distribution \(\pi_{\mathrm{ref}}(\cdot|{pr})\).
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The resulting terminal answers are aggregated via majority voting, viewed
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as mode estimation under sampling uncertainty.
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The Martingale Majority Certificate (MMC) monitors the empirical margin and
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provides an <em>anytime-valid</em> stopping rule for certification.
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Test-time training with SNR or entropy-based adaptation sharpens the
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terminal distribution, thereby increasing the
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signal-to-noise ratio (SNR) and reducing the number of samples required for
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certification.
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<img src="mmc_point_shared.gif" alt="MMC stopping rule in action" style="width: 80%;">
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<figcaption style="color:#6b7280; font-size: 0.9rem; margin-top: 8px;">
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MMC stopping rule in action.
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<h2 class="title is-3">Abstract</h2>
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Recent advances such as self-consistency and test-time reinforcement learning (TTRL) improve the
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reliability of large language models (LLMs) without additional supervision, yet their underlying
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mechanisms and statistical guarantees remain poorly understood.
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We present a unified framework for certifiable inference in LLMs, showing that majority voting provides a
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statistical certificate of self-consistency: under mild assumptions, the aggregated answer coincides with
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the mode of the model’s terminal distribution with high probability. We derive finite-sample and anytime-valid
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concentration bounds that quantify this confidence, and introduce the Martingale Majority Certificate (MMC), a
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sequential stopping rule that adaptively determines when sufficient samples have been drawn.
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We further prove that label-free post-training methods such as TTRL implicitly sharpen the answer distribution
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by exponentially tilting it toward its mode, thereby reducing the number of samples required for certification.
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Building on this insight, we propose new post-training objectives that explicitly optimise this trade-off between
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sharpness and bias. Together, these results explain and connect two central test-time scaling strategies,
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self-consistency and TTRL, within a single statistical framework for label-free, certifiable reliability in
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reasoning LLMs.
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<h3 class="title is-4">Setting</h3>
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<p>
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LLM rollouts can be formalised as a stochastic decoding process
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\[
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(Y_t)_{t \ge 0}, \quad Y_t \in \mathcal{V},
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\]
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where \( \mathcal{V} \) is the vocabulary and the process is initialised by a prompt \( pr \).
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At each step the model samples
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\[
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Y_{t+1} \sim \pi_\phi(\cdot \mid Y_{\le t}, pr),
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\]
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from a conditional policy parametrised by weights \( \phi \).
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The <em>thinking phase</em> consists of the random evolution of this sequence until a termination token is produced,
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at which point the model emits the response, starting from a random stopping time \( \tau \).
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We denote by
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\[
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X := g(Y_{\tau:}) \in \mathcal{A}
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\]
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the canonicalised terminal answer, obtained by applying a deterministic extraction map \( g \).
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The induced terminal distribution \( \mathbf{p} = \mathrm{Law}(X) \) over the answer set \( \mathcal{A} \) captures the model’s epistemic uncertainty about its own final output.
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In an ideal reasoning model, we would like rollouts to exhibit rich variability in \( Y_{1:\tau-1} \) (the reasoning trajectories), yet concentrate mass in the final answer \( X \) (the outcome).
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That is, we seek <em>diversity over reasoning paths, but consistency over terminal responses</em>.
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</p>
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<p>
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In supervised or verifier-equipped settings, correctness can be externally validated.
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In open-ended reasoning tasks, such supervision is unavailable.
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In the absence of external rewards, a model must act relative to its own uncertainty.
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Letting \( a \in \mathcal{A} \) denote the chosen output and \( X \sim \mathbf{p} \) the stochastic model response, the expected 0–1 loss is \( \mathbb{E}[1\{a \neq X\}] \).
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The Bayes-optimal decision minimising this loss is the mode
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</p>
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<p>
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\[
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c^\star = \arg\max_j p_j,
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\]
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</p>
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<p>
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which corresponds to the model’s most probable self-consistent answer.
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Hence, under symmetric loss, recovering the mode is the optimal <em>model-relative</em> prediction.
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When a verifier is absent, certifying that a model’s reported answer coincides with this mode provides a natural measure of reliability.
|
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</p>
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+
</div>
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+
<h3 class="title is-4">Statistical Certificates of Self-Consistency</h3>
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<div class="content has-text-justified">
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<p>
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In practice, the terminal probabilities \( \mathbf{p} \) are unknown and can be estimated only through multiple
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independent rollouts \( X_1,\ldots,X_n \).
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The simplest estimator of the mode is the <em>majority vote</em>
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</p>
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<p>
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\[
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+
\widehat{c}_n := \arg\max_j \hat{p}_{n,j},
|
| 241 |
+
\qquad
|
| 242 |
+
\hat{p}_{n,j} = \frac{1}{n}\sum_{i=1}^{n}\mathbf{1}\{X_i=j\}.
|
| 243 |
+
\]
|
| 244 |
+
</p>
|
| 245 |
+
|
| 246 |
+
<p>
|
| 247 |
+
This estimator forms the basis of <em>self-consistency</em> test-time scaling.
|
| 248 |
+
From a statistical standpoint, majority voting is the Bayes-optimal estimator of \( c^\star \) under 0--1 loss,
|
| 249 |
+
and an associated upper bound on \( \mathbb{P}[\widehat{c}_n \neq c^\star] \) provides a
|
| 250 |
+
<em>statistical certificate of self-consistency</em>: a quantitative guarantee that the aggregated answer
|
| 251 |
+
coincides with the mode of the terminal law \( \mathbf{p} \) with high probability.
|
| 252 |
+
</p>
|
| 253 |
+
|
| 254 |
+
<p>
|
| 255 |
+
Under standard regularity conditions the majority-vote estimator is consistent, \( \Pr[\widehat{c}_n = c^\star] \to 1 \) as \( n \to \infty \).
|
| 256 |
+
<strong>A more practical question concerns the finite-sample regime: how large must \( n \) be to guarantee, with
|
| 257 |
+
confidence \( 1-\varepsilon \), that \( \widehat{c}_n \) already equals \( c^\star \)?</strong>
|
| 258 |
+
</p>
|
| 259 |
+
|
| 260 |
+
<p>
|
| 261 |
+
To address this, we derive finite-sample and asymptotic certificates, leveraging Hoeffding, Bernstein,
|
| 262 |
+
Chernoff–Markov, and Sanov concentration bounds for the error probability \( \mathbb{P}[\widehat{c}_n \neq c^\star] \).
|
| 263 |
+
These bounds clarify how reliability scales with the ensemble size and with the <em>mode margin</em>
|
| 264 |
+
\( \delta = p_{c^\star} - p_{j^\star} \), i.e., the gap between the top two answer probabilities.
|
| 265 |
+
</p>
|
| 266 |
+
|
| 267 |
+
<p>
|
| 268 |
+
If the probabilities \( p_j \) were known, one could invert these bounds to determine the number of samples required
|
| 269 |
+
to achieve a desired confidence \( 1-\varepsilon \).
|
| 270 |
+
In reality, both \( p_j \) and \( \delta \) must be estimated on the fly.
|
| 271 |
+
This motivates a <em>sequential</em> formulation: <strong>as rollouts arrive, can we determine adaptively when the current majority
|
| 272 |
+
is statistically reliable?</strong>
|
| 273 |
+
|
| 274 |
+
We introduce the <em>Martingale Majority Certificate (MMC)</em>, a sequential procedure that adaptively tests whether the empirical leader remains significantly ahead of its nearest rival and
|
| 275 |
+
of all others combined. This guarantees that at the (random) stopping time \( \tau \), majority vote coincides with the true mode with high probability:
|
| 276 |
+
</p>
|
| 277 |
+
|
| 278 |
+
<p>
|
| 279 |
+
\[
|
| 280 |
+
\Pr[\widehat{c}_{n_\tau} \neq c^\star] \le \varepsilon,
|
| 281 |
+
\]
|
| 282 |
+
</p>
|
| 283 |
+
|
| 284 |
+
<p>
|
| 285 |
+
thus providing an <em>anytime-valid certificate</em> of model self-consistency.
|
| 286 |
+
</p>
|
| 287 |
</div>
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|
| 288 |
|
| 289 |
+
<h3 class="title is-4">Martingale Majority Certificate Stopping Rule</h3>
|
| 290 |
+
<div class="content has-text-justified">
|
| 291 |
+
<p>
|
| 292 |
+
Our proposed stopping rule adaptively decides when to stop sampling rollouts while controlling the error of returning the empirical majority.
|
| 293 |
+
</p>
|
| 294 |
+
<p>
|
| 295 |
+
The central challenge in the LLM setting is the potentially large number of possible outcomes.
|
| 296 |
+
A naive stopping rule would require pairwise comparisons of the empirical probabilities across all classes
|
| 297 |
+
\( i \neq j \), \( i,j \in \{1, \dots, k\} \), which becomes computationally prohibitive as \( k \) grows.
|
| 298 |
+
</p>
|
| 299 |
+
|
| 300 |
+
<p>
|
| 301 |
+
To address this, we exploit the observation that the mass of the terminal law is typically concentrated on a few classes \( m \ll k \).
|
| 302 |
+
Thus, instead of considering all classes individually, we aggregate votes into three categories:
|
| 303 |
+
<ul>
|
| 304 |
+
<li>the current leader \( \widehat{c}_n \),</li>
|
| 305 |
+
<li>the runner-up</li>
|
| 306 |
+
<li>all the <em>others</em>.</li>
|
| 307 |
+
</ul>
|
| 308 |
+
</p>
|
| 309 |
+
<p>
|
| 310 |
+
Accordingly, we perform two tests: leader vs runner-up and leader vs <em>others</em>.
|
| 311 |
+
</p>
|
| 312 |
+
<div style="text-align:center; margin: 24px 0;">
|
| 313 |
+
<img src="mmc_algorithm.png" alt="MMC algorithm" width="70%">
|
| 314 |
+
</div>
|
| 315 |
</div>
|
| 316 |
</div>
|
| 317 |
</div>
|
| 318 |
+
</div>
|
| 319 |
+
</section>
|
| 320 |
|
| 321 |
+
<section class="section">
|
| 322 |
+
<div class="container is-max-desktop">
|
| 323 |
<div class="columns is-centered">
|
| 324 |
<div class="column is-full-width">
|
| 325 |
+
<h3 class="title is-4">Optimising Sample Efficiency with Test-Time Training</h3>
|
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|
| 326 |
<div class="content has-text-justified">
|
| 327 |
<p>
|
| 328 |
+
Our ultimate goal is to minimise the number of samples required from the LLM for the majority vote
|
| 329 |
+
to return the correct answer with high confidence \(1-\varepsilon\). The expected stopping time of the MMC scales approximately as
|
| 330 |
+
<span id="eq-expected_number_samples">
|
| 331 |
+
\[
|
| 332 |
+
N \;\approx\;
|
| 333 |
+
\frac{2(p_{\hat c}+p_{j^\star})}{(p_{\hat c}-p_{j^\star})^{2}} \,\log \frac{1}{\varepsilon},
|
| 334 |
+
\]
|
| 335 |
+
</span>
|
| 336 |
+
so that small mode margins
|
| 337 |
+
<span>\( \delta = p_{\hat c}-p_{j^\star} \)</span>
|
| 338 |
+
lead to rapidly increasing sample requirements.
|
| 339 |
+
</p>
|
| 340 |
+
<p>
|
| 341 |
+
<strong>The key question is whether test-time adaptation can reshape the terminal distribution to enlarge this margin, thereby improving sample efficiency.</strong>
|
| 342 |
+
</p>
|
| 343 |
+
<p>
|
| 344 |
+
We show that the optimal policy corresponding to the KL-regularised objective proposed in <a href="https://arxiv.org/pdf/2504.16084">TTRL</a> is an exponentially tilted version of the base model.
|
| 345 |
+
Decreasing the regularisation parameter consistently increases the margin and reduces the number of samples required for certification.
|
| 346 |
</p>
|
| 347 |
+
<p><strong style="font-size: 1.3em;">Two new test-time RL objectives</strong></p>
|
| 348 |
+
|
| 349 |
+
<p>
|
| 350 |
+
We introduce two label-free group-level rewards designed to optimise the trade-off between sharpness
|
| 351 |
+
and bias. Let \( \mathbf{X} = (X_1, \dots, X_n) \) be a set of answers arising from rollouts
|
| 352 |
+
\( \mathbf{Y} =(Y_1, \ldots, Y_n) \) for a given prompt, with \( \widehat{c}_n \) denoting the majority vote
|
| 353 |
+
and \( j_n^\star \) the runner-up. Define \( N_j = \sum_i \mathbf{1}\{X_i=j\} \).
|
| 354 |
+
</p>
|
| 355 |
+
|
| 356 |
+
<ol class="objective-list">
|
| 357 |
+
<li>
|
| 358 |
+
<span class="objective-title">SNR-based reward.</span>
|
| 359 |
+
<p>
|
| 360 |
+
Directly leveraging the SNR as a driving factor in the efficiency of the MMC scheme we introduce the first reward
|
| 361 |
+
</p>
|
| 362 |
+
<p>
|
| 363 |
+
\[
|
| 364 |
+
r^{(1)}_n(\mathbf{Y})
|
| 365 |
+
= \widehat{\mathrm{SNR}}(\Delta_{j^\star_n})(\mathbf{X})
|
| 366 |
+
= \frac{(N_{\widehat c_n}-N_{j^\star_n})^{2}}
|
| 367 |
+
{n \left(N_{\widehat c_n}+N_{j^\star_n}\right)
|
| 368 |
+
-(N_{\widehat c_n}-N_{j^\star_n})^{2}}
|
| 369 |
+
\;\xrightarrow[n\to\infty]{}\;
|
| 370 |
+
\mathrm{SNR}(\Delta_{j^\star_n}).
|
| 371 |
+
\]
|
| 372 |
+
</p>
|
| 373 |
+
<p>
|
| 374 |
+
This objective aims to directly maximise \( \text{SNR}(\Delta_{j_n^\star}) \), which is equivalent to minimising the expected
|
| 375 |
+
number of samples required to obtain statistical certificates for the majority vote.
|
| 376 |
+
</p>
|
| 377 |
+
</li>
|
| 378 |
+
|
| 379 |
+
<li>
|
| 380 |
+
<span class="objective-title">Entropy-based reward.</span>
|
| 381 |
+
<p>
|
| 382 |
+
As we want to encourage a more peaked terminal distribution, another natural option is negative entropy, i.e.
|
| 383 |
+
</p>
|
| 384 |
+
<p>
|
| 385 |
+
\[
|
| 386 |
+
r^{(2)}_n(\mathbf{Y})
|
| 387 |
+
= \widehat H_n(\mathbf{X})
|
| 388 |
+
= \sum_{j:N_j>0}\frac{N_j}{n} \log \frac{N_j}{n}
|
| 389 |
+
\;\xrightarrow[n\to\infty]{}\;
|
| 390 |
+
\sum_j p_j \log p_j = -H(p).
|
| 391 |
+
\]
|
| 392 |
+
</p>
|
| 393 |
+
<p>
|
| 394 |
+
Maximising \( \widehat H_n \) <em>minimises</em> the Shannon entropy of the answer
|
| 395 |
+
distribution, encouraging a sharper, lower-entropy terminal distribution.
|
| 396 |
+
🚨<strong>Important:</strong> The tempering sharpens only the distribution of final answers, not the full sequence distribution.
|
| 397 |
+
This gives us the best of both worlds: promoting certainty when providing a final answer, but permitting exploration of diverse
|
| 398 |
+
pathways during the chain-of-thought reasoning process.
|
| 399 |
+
</p>
|
| 400 |
+
</li>
|
| 401 |
+
</ol>
|
| 402 |
+
<div style="text-align:center; margin: 24px 0;">
|
| 403 |
+
<img src="ttt_performance_math500.png" alt="Performance TTT" width="100%">
|
| 404 |
+
<figcaption style="color:#6b7280; font-size: 0.9rem; margin-top: 8px;">
|
| 405 |
+
Pass@1 performance after test-time training with SNR and entropy-based rewards relative to the base models.
|
| 406 |
+
</figcaption>
|
| 407 |
</div>
|
|
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|
| 408 |
|
|
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|
|
| 409 |
<p>
|
| 410 |
+
We observe in the table below that the number of samples required under the MMC stopping rule decreases after applying test-time training, relative to the pre-trained model.
|
| 411 |
+
That is, test-time training sharpens the terminal answer distribution, increasing the mode margin and thus reducing the number of samples required for certification.
|
| 412 |
</p>
|
| 413 |
+
<div style="text-align:center; margin: 24px 0;">
|
| 414 |
+
<img src="table_mmc.png" alt="Performance TTT" width="75%">
|
| 415 |
+
<figcaption style="color:#6b7280; font-size: 0.9rem; margin-top: 8px;">
|
| 416 |
+
Majority vote accuracy and required number of samples under the MMC stopping rule (✅) at confidence levels 0.1 and 0.4 for the pre-trained model and after test-time training with SNR-based rewards. Performance is compared to that obtained using the full sample budget (❌).
|
| 417 |
+
</figcaption>
|
| 418 |
+
</div>
|
| 419 |
+
</ul>
|
| 420 |
</div>
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|
| 421 |
</div>
|
| 422 |
</div>
|
| 423 |
+
</div>
|
| 424 |
+
</section>
|
| 425 |
|
| 426 |
+
<section class="section">
|
| 427 |
+
<div class="container is-max-desktop">
|
| 428 |
<div class="columns is-centered">
|
| 429 |
<div class="column is-full-width">
|
| 430 |
+
<h3 class="title is-4">SNR as a label-free estimator of task difficulty</h3>
|
|
|
|
| 431 |
<div class="content has-text-justified">
|
| 432 |
<p>
|
| 433 |
+
Our experiments reveal a notable empirical regularity: the
|
| 434 |
+
<em>signal-to-noise ratio</em> (SNR) of the margin variable
|
| 435 |
+
\(\Delta_{j^\star} = \mathbf 1\{X = c^\star\} - \mathbf 1\{X = j^\star\}\),
|
| 436 |
+
which quantifies the sharpness of the model’s terminal answer distribution,
|
| 437 |
+
correlates strongly with external measures of problem difficulty.
|
| 438 |
+
Across the MATH-500 benchmark, harder problems exhibit systematically lower and more variable SNR values,
|
| 439 |
+
while easier problems yield sharply peaked distributions concentrated around a single answer.
|
| 440 |
</p>
|
| 441 |
<p>
|
| 442 |
+
This behaviour is non-trivial: the model has no access to ground-truth difficulty labels, yet its own epistemic
|
| 443 |
+
uncertainty, reflected in the variability of its rollouts, aligns closely with these labels.
|
| 444 |
+
<strong>This suggests an emergent form of calibration in reasoning LLMs</strong>:
|
| 445 |
+
without explicit supervision or external verification, models appear to ''know when they do not know.''
|
| 446 |
+
In statistical terms, the SNR acts as a label-free proxy for epistemic uncertainty and, consequently, for task difficulty.
|
| 447 |
</p>
|
| 448 |
+
<div style="text-align:center; margin: 24px 0;">
|
| 449 |
+
<img src="QWEN-MATH-1.5B_violin_maj100_SNR.png" alt="SNR distribution qwen-math-1.5B." style="width: 48%;margin-right: 1%;">
|
| 450 |
+
<img src="QWEN-MATH-7B_violin_maj100_SNR.png" alt="SNR distribution qwen-math-7B." style="width: 48%;margin-left: 1%;">
|
| 451 |
+
<figcaption style="color:#6b7280; font-size: 0.9rem; margin-top: 8px;">
|
| 452 |
+
Distribution of the estimated SNR when using MMC stopping rule with \(\varepsilon = 0.1\) and \(N_{\text{budget}}=100\). Results are obtained after applying test-time training with SNR-based rewards.</figcaption>
|
| 453 |
+
</div>
|
| 454 |
+
</div>
|
| 455 |
+
</div>
|
| 456 |
+
</div>
|
| 457 |
+
</div>
|
| 458 |
+
</section>
|
| 459 |
+
|
| 460 |
+
<section class="section">
|
| 461 |
+
<div class="container is-max-desktop">
|
| 462 |
+
<div class="columns is-centered">
|
| 463 |
+
<div class="column is-full-width">
|
| 464 |
+
<h3 class="title is-4">Conclusion</h3>
|
| 465 |
+
|
| 466 |
+
<div class="content has-text-justified">
|
| 467 |
<p>
|
| 468 |
+
<strong>Our results unify several strands of recent work on reliable inference in LLMs, self-consistency,
|
| 469 |
+
adaptive compute allocation, and test-time reinforcement learning (TTRL), under a common
|
| 470 |
+
statistical perspective.</strong> Through this lens, majority voting emerges naturally as a means of estimating the mode of the terminal distribution.
|
| 471 |
+
The validity of the majority vote as an estimate of the mode can be certified by finite-sample and asymptotic bounds. The Martingale Majority Certificate (MMC)
|
| 472 |
+
extends this view by providing an operational test-time algorithm that determines, from model
|
| 473 |
+
rollouts alone, when a response is statistically self-consistent.
|
| 474 |
</p>
|
| 475 |
<p>
|
| 476 |
+
Furthermore, <strong>we shed light on the underlying mechanism by which TTRL and related post-training
|
| 477 |
+
approaches improve reasoning reliability: KL-regularised optimisation corresponds to an
|
| 478 |
+
exponential tilting of the terminal law, sharpening it around its mode and increasing the
|
| 479 |
+
signal-to-noise ratio (SNR) of the margin variable.</strong> This insight explains empirical observations of
|
| 480 |
+
enhanced consistency after test-time adaptation, and motivates new label-free objectives such as
|
| 481 |
+
our SNR- and entropy-based rewards, which explicitly target this trade-off between sharpness and
|
| 482 |
+
bias. Unlike prior work that tunes temperature or per-token distributions, our formulation operates
|
| 483 |
+
on the terminal marginal, preserving exploration during reasoning while promoting confidence in the
|
| 484 |
+
final answer.
|
| 485 |
</p>
|
| 486 |
</div>
|
| 487 |
</div>
|
| 488 |
</div>
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|
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|
| 489 |
</div>
|
| 490 |
</section>
|
| 491 |
|
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|
|
| 492 |
<section class="section" id="BibTeX">
|
| 493 |
<div class="container is-max-desktop content">
|
| 494 |
<h2 class="title">BibTeX</h2>
|
| 495 |
+
<pre><code>@article{corderoencinar2025certified,
|
| 496 |
+
author = {Paula Cordero-Encinar and Andrew B. Duncan},
|
| 497 |
+
title = {Certified Self-Consistency: Statistical Guarantees and Test-Time Training for Reliable Reasoning in LLMs},
|
| 498 |
+
journal = {arXiv:2510.17472},
|
| 499 |
+
year = {2025},
|
| 500 |
}</code></pre>
|
| 501 |
</div>
|
| 502 |
</section>
|
| 503 |
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|
| 504 |
<footer class="footer">
|
| 505 |
<div class="container">
|
| 506 |
<div class="content has-text-centered">
|
| 507 |
+
<a class="icon-link" href="https://arxiv.org/pdf/2510.17472" class="external-link">
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|
|
| 508 |
<i class="fas fa-file-pdf"></i>
|
| 509 |
</a>
|
| 510 |
+
<a class="icon-link" href="https://github.com/paulaoak/certified_self_consistency" class="external-link">
|
| 511 |
<i class="fab fa-github"></i>
|
| 512 |
</a>
|
| 513 |
</div>
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| 515 |
<div class="column is-8">
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| 516 |
<div class="content">
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<p>
|
| 518 |
+
This website template is borrowed from <a href="https://nerfies.github.io/">Nerfies</a>,
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