# A Derivation of Nesterov's Accelerated Gradient Algorithm from Optimal Control Theory

arXiv: https://arxiv.org/abs/2203.17226 · HTML: https://mlsanity.com/paper/2203.17226/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 18 references that are in the map.
- First posted on arXiv 29 Mar 2022.

## Builds on (most cited first, 4 of 4)

- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [A Differential Equation for Modeling Nesterov's Accelerated Gradient Method: Theory and Insights](https://mlsanity.com/paper/1503.01243/) (2015)
- [Understanding the Acceleration Phenomenon via High-Resolution Differential Equations](https://mlsanity.com/paper/1810.08907/) (2018)
- [An Optimal Control Theory for Accelerated Optimization](https://mlsanity.com/paper/1902.09004/) (2019)

## Abstract (arXiv)

Nesterov's accelerated gradient algorithm is derived from first principles. The first principles are founded on the recently-developed optimal control theory for optimization. This theory frames an optimization problem as an optimal control problem whose trajectories generate various continuous-time algorithms. The algorithmic trajectories satisfy the necessary conditions for optimal control. The necessary conditions produce a controllable dynamical system for accelerated optimization. Stabilizing this system via a quadratic control Lyapunov function generates an ordinary differential equation. An Euler discretization of the resulting differential equation produces Nesterov's algorithm. In this context, this result solves the purported mystery surrounding the algorithm.