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arXiv 2605.20105cs.LG
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最优表示尺寸:预训练和线性探测的高维分析

Optimal Representation Size: High-Dimensional Analysis of Pretraining and Linear Probing

Gatsby Computational Neuroscience Unit(Gatsby计算神经科学单元);University College London(伦敦大学学院);Institute of Science and Technology Austria(奥地利科学与技术研究所);Sainsbury Wellcome Centre(萨金斯-韦尔科姆中心)

Valentina Njaradi, Clémentine Dominé, Rachel Swanson, Marco Mondelli, Andrew Saxe

AI总结:

本文研究了预训练和线性探测过程中的最优表示尺寸问题,通过高维分析揭示了表示维度、未标记和标记样本数量以及任务对齐性对训练和泛化误差的影响,提出了在不同预训练和下游数据条件下优化表示尺寸的条件。

摘要:

学习从有限数据中泛化是人工和生物系统面临的基本挑战。一种常见策略是从大量未标记数据中提取可重用的结构,从而高效适应新任务。这种两阶段范式现在已成为现代训练流水线的标准,即预训练后进行微调或线性探测。我们为这一过程提供了一个分析模型:结构提取被形式化为主成分分析,而下游学习则被建模为对单独标记数据集的线性回归。在高维情况下,我们推导出训练和泛化误差的精确表达式,展示了其对表示维度、未标记和标记样本数量以及任务对齐性的依赖性。我们的结果表明,预训练表示强烈影响下游泛化,我们将其最优表示尺寸作为任务参数的函数进行表征:在大量预训练数据但稀缺下游数据时,最大压缩表示最优;而在预训练数据有限时,高维表示泛化更好。此外,我们建立了预训练和监督之间的精确权衡,量化了需要多少未标记数据来替代一个标记样本。除了我们理想化的模型外,我们在自编码器和预训练大语言模型中也观察到相似的现象。总体而言,我们强调优化表示尺寸至关重要,给出了压缩预训练时提高泛化的条件。

英文摘要:

Learning to generalise from limited data is a fundamental challenge for both artificial and biological systems. A common strategy is to extract reusable structure from abundant unlabelled data, enabling efficient adaptation to new tasks from limited labelled data. This two-stage paradigm is now standard in modern training pipelines, where pretraining is followed by fine-tuning or linear probing. We provide an analytical model of this process: structure extraction is formalized as principal component analysis on unlabelled data, and downstream learning as linear regression on a separate labelled dataset. In the high-dimensional regime, we derive exact expressions for training and generalisation error showcasing their dependence on representation dimensionality, unlabelled and labelled sample sizes, and task alignment. Our results show that pretrained representations strongly influence downstream generalisation, and we characterize the optimal representation size as a function of task parameters: with abundant pretraining data but scarce downstream data, maximally compressed representations are optimal, whereas with limited pretraining data, higher-dimensional representations generalise better. Furthermore, we establish an exact trade-off between pretraining and supervision, quantifying how much unlabelled data is required to replace a single labelled sample. Beyond our idealised model, we observe similar phenomenology in autoencoders and pretrained LLMs. Altogether, we highlight that optimising representation size is critical, giving conditions for when compression during pretraining improves generalisation.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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