Model Stealing for Any Low-Rank Language Model

Model stealing, where a learner tries to recover an unknown model via carefully chosen queries, is a critical problem in machine learning, as it threatens the security of proprietary models and the privacy of data they are trained on. In recent years, there has been particular interest in stealing large language models (LLMs). In this paper, we aim to build a theoretical understanding of stealing language models by studying a simple and mathematically tractable setting. We study model stealing for Hidden Markov Models (HMMs), and more generally low-rank language models. We assume that the learner works in the conditional query model, introduced by Kakade, Krishnamurthy, Mahajan and Zhang [KKMZ24]. Our main result is an efficient algorithm in the conditional query model, for learning any low-rank distribution. In other words, our algorithm succeeds at stealing any language model whose output distribution is low-rank. This improves upon the result in [KKMZ24] which also requires the unknown distribution to have high "fidelity" -a property that holds only in restricted cases. There are two key insights behind our algorithm: First, we represent the conditional distributions at each timestep by constructing barycentric spanners among a collection of vectors of exponentially large dimension. Second, for sampling from our representation, we iteratively solve a sequence of convex optimization problems that involve projection in relative entropy to prevent compounding of errors over the length of the sequence. This is an interesting example where, at least theoretically, allowing a machine learning model to solve more complex problems at inference time can lead to drastic improvements in its performance.