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

• URL: http://arxiv.org/abs/2203.17226v1
• Date: Tue, 29 Mar 2022 19:26:20 GMT
• Title: A Derivation of Nesterov's Accelerated Gradient Algorithm from Optimal Control Theory
• Authors: I. M. Ross
• Abstract summary: Nesterov's accelerated gradient algorithm is derived from first principles. An Euler discretization of the resulting differential equation produces Nesterov's algorithm.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: 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.

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