We propose a theoretical framework for studying the imitation of stochastic,
non-Markovian, potentially multi-modal (i.e. "complex" ) expert demonstrations
in nonlinear dynamical systems. Our framework invokes low-level controllers -
either learned or implicit in position-command control - to stabilize imitation
policies around expert demonstrations. We show that with (a) a suitable
low-level stability guarantee and (b) a stochastic continuity property of the
learned policy we call "total variation continuity" (TVC), an imitator that
accurately estimates actions on the demonstrator's state distribution closely
matches the demonstrator's distribution over entire trajectories. We then show
that TVC can be ensured with minimal degradation of accuracy by combining a
popular data-augmentation regimen with a novel algorithmic trick: adding
augmentation noise at execution time. We instantiate our guarantees for
policies parameterized by diffusion models and prove that if the learner
accurately estimates the score of the (noise-augmented) expert policy, then the
distribution of imitator trajectories is close to the demonstrator distribution
in a natural optimal transport distance. Our analysis constructs intricate
couplings between noise-augmented trajectories, a technique that may be of
independent interest. We conclude by empirically validating our algorithmic
recommendations.

我们提出了一个理论框架，用于研究非马尔可夫、潜在多模态（即 “复杂的”）专家示范在非线性动力系统中的模仿。