Understanding Behavior Cloning with Action Quantization

TL;DR

This paper provides a theoretical analysis of action quantization in behavior cloning, showing how quantization error affects learning and proposing methods to optimize quantization schemes for stable, efficient policy learning.

Contribution

It offers the first theoretical foundations for action quantization in behavior cloning, analyzing error propagation, sample complexity, and proposing a model-based augmentation to improve performance.

Findings

Quantization with log-loss achieves optimal sample complexity.

Polynomial horizon dependence on quantization error under certain conditions.

Model-based augmentation improves error bounds without policy smoothness.

Abstract

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.