# Learning the optimal state-feedback via supervised imitation learning

arXiv: https://arxiv.org/abs/1901.02369 · HTML: https://mlsanity.com/paper/1901.02369/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Placed in Imitation learning methods, one of 13 directions in Offline RL and imitation learning from data.
- Builds on 6 of its 33 references that are in the map.
- First posted on arXiv 7 Jan 2019.

## Placed in

- Offline RL and imitation learning from data / [Imitation learning methods](https://mlsanity.com/offlinerl/imitation-learning-methods/)

## Builds on (most cited first, 6 of 6)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Cyclical Learning Rates for Training Neural Networks](https://mlsanity.com/paper/1506.01186/) (2015)
- [Real-time optimal control via Deep Neural Networks: study on landing problems](https://mlsanity.com/paper/1610.08668/) (2016)
- [Machine learning and evolutionary techniques in interplanetary trajectory design](https://mlsanity.com/paper/1802.00180/) (2018)

## Built on by (most cited first, 8 of 8)

- [Neural representation of a time optimal, constant acceleration rendezvous](https://mlsanity.com/paper/2203.15490/) (2022)
- [Selected Trends in Artificial Intelligence for Space Applications](https://mlsanity.com/paper/2212.06662/) (2022)
- [Optimality Principles in Spacecraft Neural Guidance and Control](https://mlsanity.com/paper/2305.13078/) (2023)
- [Neural Network Optimal Feedback Control with Guaranteed Local Stability](https://mlsanity.com/paper/2205.00394/) (2022)
- [Neural network optimal feedback control with enhanced closed loop stability](https://mlsanity.com/paper/2109.07466/) (2021)
- [Guidance and Control Neural Network Acceleration using Memristors](https://mlsanity.com/paper/2509.02369/) (2025)
- [GCImOpt: Learning efficient goal-conditioned policies by imitating optimal trajectories](https://mlsanity.com/paper/2604.22724/) (2026)
- [Pretrained Approximators for Low-Thrust Trajectory Cost and Reachability](https://mlsanity.com/paper/2605.26790/) (2026)

## Abstract (arXiv)

Imitation learning is a control design paradigm that seeks to learn a control policy reproducing demonstrations from expert agents. By substituting expert demonstrations for optimal behaviours, the same paradigm leads to the design of control policies closely approximating the optimal state-feedback. This approach requires training a machine learning algorithm (in our case deep neural networks) directly on state-control pairs originating from optimal trajectories. We have shown in previous work that, when restricted to low-dimensional state and control spaces, this approach is very successful in several deterministic, non-linear problems in continuous-time. In this work, we refine our previous studies using as a test case a simple quadcopter model with quadratic and time-optimal objective functions. We describe in detail the best learning pipeline we have developed, that is able to approximate via deep neural networks the state-feedback map to a very high accuracy. We introduce the use of the softplus activation function in the hidden units of neural networks showing that it results in a smoother control profile whilst retaining the benefits of rectifiers. We show how to evaluate the optimality of the trained state-feedback, and find that already with two layers the objective function reached and its optimal value differ by less than one percent. We later consider also an additional metric linked to the system asymptotic behaviour - time taken to converge to the policy's fixed point. With respect to these metrics, we show that improvements in the mean absolute error do not necessarily correspond to better policies.