论文标题
Nesterov的加速梯度算法从最佳控制理论推导
A Derivation of Nesterov's Accelerated Gradient Algorithm from Optimal Control Theory
论文作者
论文摘要
Nesterov的加速梯度算法源自第一原理。第一原理建立在最近开发的最佳控制理论以进行优化的基础上。该理论将优化问题置于一个最佳控制问题，其轨迹会产生各种连续的时间算法。算法轨迹满足了最佳控制的必要条件。必要的条件产生可控的动力系统，以加速优化。通过二次控制Lyapunov函数稳定该系统会产生一个普通的微分方程。由此产生的微分方程的Euler离散化会产生Nesterov的算法。在这种情况下，该结果解决了算法围绕的谜团。
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.
