# Behavior Cloning Under PD Control: A Finite-Horizon Theory of Gain-Dependent Error Amplification

arXiv: https://arxiv.org/abs/2604.14484 · HTML: https://mlsanity.com/paper/2604.14484/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 31 references that are in the map.
- First posted on arXiv 15 Apr 2026.

## Builds on (most cited first, 17 of 17)

- [Diffusion Policy: Visuomotor Policy Learning via Action Diffusion](https://mlsanity.com/paper/2303.04137/) (2023)
- [Learning Fine-Grained Bimanual Manipulation with Low-Cost Hardware](https://mlsanity.com/paper/2304.13705/) (2023)
- [What Matters in Learning from Offline Human Demonstrations for Robot Manipulation](https://mlsanity.com/paper/2108.03298/) (2021)
- [robosuite: A Modular Simulation Framework and Benchmark for Robot Learning](https://mlsanity.com/paper/2009.12293/) (2020)
- [Implicit Behavioral Cloning](https://mlsanity.com/paper/2109.00137/) (2021)
- [Universal Manipulation Interface: In-The-Wild Robot Teaching Without In-The-Wild Robots](https://mlsanity.com/paper/2402.10329/) (2024)
- [A Tour of Reinforcement Learning: The View from Continuous Control](https://mlsanity.com/paper/1806.09460/) (2018)
- [Is Behavior Cloning All You Need? Understanding Horizon in Imitation Learning](https://mlsanity.com/paper/2407.15007/) (2024)
- [Feedback in Imitation Learning: The Three Regimes of Covariate Shift](https://mlsanity.com/paper/2102.02872/) (2021)
- [The Gap Between Model-Based and Model-Free Methods on the Linear Quadratic Regulator: An Asymptotic Viewpoint](https://mlsanity.com/paper/1812.03565/) (2018)
- [Provable Guarantees for Generative Behavior Cloning: Bridging Low-Level Stability and High-Level Behavior](https://mlsanity.com/paper/2307.14619/) (2023)
- [On the Role of the Action Space in Robot Manipulation Learning and Sim-to-Real Transfer](https://mlsanity.com/paper/2312.03673/) (2023)
- [The Pitfalls of Imitation Learning when Actions are Continuous](https://mlsanity.com/paper/2503.09722/) (2025)
- [Torque-based Deep Reinforcement Learning for Task-and-Robot Agnostic Learning on Bipedal Robots Using Sim-to-Real Transfer](https://mlsanity.com/paper/2304.09434/) (2023)
- [Learning Variable Impedance Control for Contact Sensitive Tasks](https://mlsanity.com/paper/1907.07500/) (2019)
- [Butterfly Effects of SGD Noise: Error Amplification in Behavior Cloning and Autoregression](https://mlsanity.com/paper/2310.11428/) (2023)
- [Tune to Learn: How Controller Gains Shape Robot Policy Learning](https://mlsanity.com/paper/2604.02523/) (2026)

## Abstract (arXiv)

Behavior cloning (BC) on position-controlled robots is shaped by the PD loop that executes policy actions. We give a finite-horizon, nonasymptotic analysis of how controller gains affect BC failure. Independent sub-Gaussian action errors propagate through gain-dependent closed-loop dynamics into sub-Gaussian position errors. The resulting failure tail is controlled by controller amplification multiplied by validation loss and generalization slack, so validation loss alone can mis-rank gains. Under shape-preserving upper-bound assumptions, the analysis separates label difficulty, injection strength, and contraction, ranking compliant-overdamped gains as tightest and stiff-underdamped gains as loosest, with the mixed regimes system-dependent. In the canonical scalar second-order PD system, stationary position-error variance increases with stiffness and decreases with damping over the stable range, and exact zero-order-hold discretization inherits the ordering to leading order. This extends the error-attenuation explanation of bronars et al. (2026) to finite-horizon failure bounds.