Understanding Behavior Cloning with Action Quantization

Behavior cloning is a fundamental paradigm in machine learning, enabling policy learning from expert demonstrations across robotics, autonomous driving, and generative models. Autoregressive models like transformer have proven remarkably effective, from large language models (LLMs) to vision-language-action systems (VLAs). However, applying autoregressive models to continuous control requires discretizing actions through quantization, a practice widely adopted yet poorly understood theoretically. This paper provides theoretical foundations for this practice. We analyze how quantization error propagates along the horizon and interacts with statistical sample complexity. We show that behavior cloning with quantized actions and log-loss achieves optimal sample complexity—matching existing lower bounds—and incurs only polynomial horizon dependence on quantization error, provided the dynamics are stable and the policy satisfies a probabilistic smoothness condition. We further characterize when different quantization schemes satisfy or violate these requirements, and propose a model-based augmentation that provably improves the error bound without requiring policy smoothness. Finally, we establish fundamental limits that jointly capture the effects of quantization error and statistical complexity.

Robots and other decision-making systems often learn by copying expert demonstrations. To make this easier for modern generative AI architectures, continuous actions are sometimes converted into discrete symbols, similar to words in a sentence. This compact representation can make learning easier, but it also introduces approximation errors. A fundamental theoretical question is therefore: how much performance can be lost because of this discretization? In this paper, we provide an answer to this question. We show that when the underlying environment and expert demonstrations are sufficiently stable, the loss caused by action discretization is acceptable, and our algorithm is guaranteed to behave well. In contrast, when these stability conditions are violated, we construct hard examples where this approach is bound to fail. We also provide a remedy that is connected to action chunking, a recently popular paradigm in robot learning. Overall, we study a problem that has been widely explored empirically but has received little theoretical understanding, and we take a first step toward filling this gap.