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

TL;DR

This paper derives Nesterov's accelerated gradient algorithm from optimal control theory, providing a first-principles explanation and connecting it to continuous-time dynamical systems.

Contribution

It introduces a novel derivation of Nesterov's algorithm using optimal control principles, linking discrete algorithms to continuous dynamical systems.

Findings

Derives Nesterov's algorithm from optimal control theory

Connects accelerated optimization to controllable dynamical systems

Provides a new perspective on the algorithm's underlying principles

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.