Understanding whether fine-tuning elicits latent capabilities or teaches new ones is a fundamental question for language model evaluation and safety. We develop a formal information-theoretic framework for quantifying how much predictive structure fine-tuning extracts from the train dataset and writes into a model's parameters. Our central quantity, Excess Description Length (EDL), is defined via prequential coding and measures the gap between the bits required to encode training labels sequentially using an evolving model (trained online) and the residual encoding cost under the final trained model. We establish that EDL is non-negative in expectation, converges to surplus description length in the infinite-data limit, and provides bounds on expected generalization gain. Through a series of toy models, we clarify common confusions about information in learning: why random labels yield EDL near zero, how a single example can eliminate many bits of uncertainty about the underlying rule(s) that describe the data distribution, why structure learned on rare inputs contributes proportionally little to expected generalization, and how format learning creates early transients distinct from capability acquisition. This framework provides rigorous foundations for the empirical observation that capability elicitation and teaching exhibit qualitatively distinct scaling signatures.
核心贡献 · Key contributions
通过预序编码形式化定义了超额描述长度（EDL），用于量化微调过程中吸收的预测信息。 Formalizes Excess Description Length (EDL) via prequential coding to quantify predictive information absorbed during fine-tuning.
证明了 EDL 在期望下非负、渐近收敛于剩余描述长度，并约束了期望泛化增益。 Proves EDL is non-negative in expectation, converges to surplus description length asymptotically, and bounds expected generalization gain.
澄清了常见混淆：随机标签产生接近零的 EDL、单个示例可消除大量假设不确定性、稀有输入对期望泛化贡献小、格式学习产生早期瞬态。 Clarifies common confusions: random labels yield near-zero EDL, single examples can eliminate many bits of hypothesis uncertainty, rare inputs contribute little to expected generalization, and format learning creates early transients.
为通过不同的 EDL 缩放特征区分能力激发与教学提供了严格基础。 Provides rigorous foundations for distinguishing capability elicitation from teaching via distinct EDL scaling signatures.
将 EDL 确立为剩余描述长度的有限数据类比，无需预言机即可从标准训练日志计算。 Establishes EDL as a finite-data analog to surplus description length, computable from standard training logs without oracle access.
引入玩具模型说明激发（每示例 EDL 单调递减）与教学（每示例 EDL 递增）动态。 Introduces toy models illustrating elicitation (monotonically decreasing per-example EDL) vs. teaching (increasing per-example EDL) dynamics.
局限 · Limitations
EDL 依赖于训练算法，使得跨不同算法的比较复杂化。 EDL depends on the training algorithm, complicating comparisons across different algorithms.
EDL 不是语义度量；它量化预测改进，但不量化内部表示或理解。 EDL is not a semantic measure; it quantifies predictive improvement but not internal representations or understanding.
该框架聚焦于监督微调；扩展到强化学习、偏好优化及其他后训练方法需要进一步发展。 The framework focuses on supervised fine-tuning; extensions to RL, preference optimization, and other post-training methods require further development.
玩具模型高度简化，可能无法捕捉真实神经网络学习的全部复杂性。 Toy models are highly simplified and may not capture the full complexity of real neural network learning.
对于小数据集，EDL 在不同数据集或顺序间的方差可能很大，需要置信区间以可靠解释。 EDL variance across datasets or orderings can be substantial for small datasets, requiring confidence intervals for reliable interpretation.
论文章节 · Sections（共 38）
摘要Abstract
1 引言1 Introduction
2 信息论背景与定义2 Information-Theoretic Background and Definitions
2.1 设置与符号2.1 Setup and Notation
2.2 交叉熵损失作为压缩2.2 Cross-Entropy Loss as Compression