The Value of a Prompt: An LLM-Relative Kolmogorov-Complexity Approach

In a world where valuable artifacts are increasingly created, completed, or processed by LLMs, the central economic question is not only what the LLM can produce, but what \emph{value} remains in the inputs (i.e., the prompts) we provide to it. Given a prompt, hint, critique, problem statement, or partial solution that helps an LLM produce an artifact $z$---a proof, program, design, or scientific hypothesis---how should we measure the value of that input? Intuitively, an input is valuable when it makes the target artifact easier for the model to generate: either by increasing its sampling probability, or by reducing the thinking time needed to find it. We propose a computational Levin--Kolmogorov complexity approach to this problem, by appropriately replacing the universal Turing machine in the classical definitions by the LLM itself. Concretely, we introduce an LLM-relative notion of \emph{probabilistic Levin--Kolmogorov complexity} $pKt$---treating the model's thinking as the random tape of the program, and charging logarithmically for it in Levin's manner---and define prompt value as algorithmic mutual information with respect to $pKt$. This captures the intuition above: a prompt having $b$ bits of value for an artifact $z$ makes $z$ $2^b$ times ``easier to obtain'', by multiplying the success probability by $2^b$, by dividing the required computation by $2^b$, or by any corresponding tradeoff between probability and computation. In contrast to the classical notion of algorithmic mutual information, ours is efficiently estimable. We additionally show that, under a natural reproduction experiment, a prompt value of \(b\) bits means that reproducing \(z\) without the prompt has median token cost \(2^b\) times that of reproducing it with the prompt.

A Complexity Measure for Active Learning in Multi-group Mean Estimation

We study a \emph{max-risk} objective for active learning in a multi-group mean estimation $d$-armed bandits: a learner adaptively allocates a budget of $T$ samples across $d$ groups to minimize the worst-case uncertainty index $\max_{k\in[d]}σ_k^2/n_k$, where $σ_k$ is the standard deviation of the distribution of arm $d$, and $n_k$ is the number of times arm $d$ is sampled. We develop a local minimax framework and prove the first general lower bound for this objective, valid for any finite-variance hypothesis class. The bound separates difficulty into three orthogonal factors: a \emph{budget} term, a \emph{heteroscedasticity} index measuring how unevenly the uncertainty is spread across arms, and a model-dependent complexity measure, the \emph{Variance Local Curvature} ($\mathrm{VLC}$), which captures how much information a local change of variance creates inside the hypothesis class. For smooth classes, the $\mathrm{VLC}$ is a reparametrization of a variance--Fisher information, with closed-form values for common families. Benchmarking against the strongest available upper bound shows near-optimality up to logarithmic factors in broad regimes, and pinpoints a systematic gap in highly heterogeneous instances. Our proof introduces two key ingredients: a loss-induced $\ell_1$ geometry on the decision space, and a representation-based instance generator that reduces hard-instance construction to an explicit random matrix calculation.