# Advancing Trajectory Optimization with Approximate Inference: Exploration, Covariance Control and Adaptive Risk

arXiv: https://arxiv.org/abs/2103.06319 · HTML: https://mlsanity.com/paper/2103.06319/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 36 references that are in the map.
- First posted on arXiv 10 Mar 2021.

## Builds on (most cited first, 7 of 7)

- [Probabilistic Numerics and Uncertainty in Computations](https://mlsanity.com/paper/1506.01326/) (2015)
- [Optimal steering of a linear stochastic system to a final probability distribution](https://mlsanity.com/paper/1408.2222/) (2014)
- [Stochastic Optimal Control as Approximate Input Inference](https://mlsanity.com/paper/1910.03003/) (2019)
- [Nonlinear Covariance Control via Differential Dynamic Programming](https://mlsanity.com/paper/1911.09283/) (2019)
- [Finite-Horizon Covariance Control of Linear Time-Varying Systems](https://mlsanity.com/paper/1707.04729/) (2017)
- [Greedy Finite-Horizon Covariance Steering for Discrete-Time Stochastic Nonlinear Systems Based on the Unscented Transform](https://mlsanity.com/paper/2003.03679/) (2020)
- [T-PFC: A Trajectory-Optimized Perturbation Feedback Control Approach](https://mlsanity.com/paper/1902.01389/) (2019)

## Built on by (most cited first, 3 of 3)

- [Inferring Smooth Control: Monte Carlo Posterior Policy Iteration with Gaussian Processes](https://mlsanity.com/paper/2210.03512/) (2022)
- [Efficient Stochastic Optimal Control through Approximate Bayesian Input Inference](https://mlsanity.com/paper/2105.07693/) (2021)
- [Deterministic Trajectory Optimization through Probabilistic Optimal Control](https://mlsanity.com/paper/2407.13316/) (2024)

## Abstract (arXiv)

Discrete-time stochastic optimal control remains a challenging problem for general, nonlinear systems under significant uncertainty, with practical solvers typically relying on the certainty equivalence assumption, replanning and/or extensive regularization. Control as inference is an approach that frames stochastic control as an equivalent inference problem, and has demonstrated desirable qualities over existing methods, namely in exploration and regularization. We look specifically at the input inference for control (i2c) algorithm, and derive three key characteristics that enable advanced trajectory optimization: An `expert' linear Gaussian controller that combines the benefits of open-loop optima and closed-loop variance reduction when optimizing for nonlinear systems, inherent adaptive risk sensitivity from the inference formulation, and covariance control functionality with only a minor algorithmic adjustment.