The Cost of Compression: A Rate-Distortion Limit on Factual Hallucination

压缩的代价：事实幻觉的速率-失真极限

Abstract: Factual hallucination in closed-book question answering is often treated as a coverage problem: a model fails because the relevant fact is absent from its internal memory. This view misses a second source of error. Even when a fact has been observed, finite memory may force it to be stored only approximately.

摘要： 在闭卷问答中，事实幻觉通常被视为一个覆盖率问题：模型失败是因为其内部记忆中缺失了相关事实。这种观点忽略了第二种错误来源。即使事实已被观测到，有限的内存也可能迫使其只能被近似存储。

We study this effect through a simple coverage—compression model of factual recall. We consider an unstructured question-answering task with $N$ possible queries and $K$ possible answers. A learner observes $M$ training facts, compresses them into at most $B$ bits, and answers uniformly drawn test queries without retrieval.

我们通过一个简单的事实回忆“覆盖-压缩”模型来研究这一效应。我们考虑一个非结构化的问答任务，其中包含 $N$ 个可能的查询和 $K$ 个可能的答案。学习者观测 $M$ 个训练事实，将其压缩为最多 $B$ 比特，并在不进行检索的情况下回答均匀抽取的测试查询。

For a uniformly random ground-truth mapping, we prove $\mathcal{E} \geq \frac{M}{N}\delta^\star!\left(\frac{B}{M}\right) + \left(1-\frac{M}{N}\right)\left(1-\frac{1}{K}\right)$, where $\delta^\star(r)$ is the inverse rate-distortion function of a uniform $K$-ary source under zero-one loss. The two terms separate compression distortion on observed facts from missing coverage on unobserved facts.

对于均匀随机的基准真值映射，我们证明了 $\mathcal{E} \geq \frac{M}{N}\delta^\star!\left(\frac{B}{M}\right) + \left(1-\frac{M}{N}\right)\left(1-\frac{1}{K}\right)$，其中 $\delta^\star(r)$ 是在零一损失下均匀 $K$ 元信源的逆速率-失真函数。这两项分别区分了观测事实上的压缩失真与未观测事实上的覆盖缺失。

The bound gives a compact way to reason about selective memory, forced compression, structure, retrieval, abstention, and long-context organization. We study the predicted signatures with theory-implied simulations and controlled fact-injection probes in modern language models that vary fact load and effective trainable memory.

该界限提供了一种简洁的方法来推理选择性记忆、强制压缩、结构、检索、弃权和长上下文组织。我们通过理论推导的模拟以及在现代语言模型中进行的受控事实注入探测，研究了这些预测特征，并改变了事实负载和有效的可训练内存。

The result is not a complete theory of hallucination, but an information-theoretic account of a separable failure mode: lossy recall of observed facts under finite memory.

该结果并非关于幻觉的完整理论，而是对一种可分离失效模式的信息论解释：即在有限内存下对已观测事实的有损回忆。