论文标题
通过扩张坐标系中的保护定律对加速梯度方法进行连续分析
Continuous-Time Analysis of Accelerated Gradient Methods via Conservation Laws in Dilated Coordinate Systems
论文作者
论文摘要
我们通过在扩张的坐标系中得出保护法，分析加速梯度方法的连续时间模型。即，我们分析了一些$ w（t）= t^α（x（t）-x_c）$的动力学，而不是分析$ w（t）= t^α（x（t）-x_c）$的动力学，并在此扩张的坐标系统中得出了类似于物理能量的保守数量，类似于物理能量。通过这种方法，我们以简化的方式恢复了许多已知的连续时间分析，并获得了OGM-G的新型连续时间分析，OGM-G是一种加速机制，可有效降低与Nesterov不同的梯度幅度。最后，我们表明，在扩张的坐标系中，半秒符号欧拉离散化导致$ \ MATHCAL {O}（1/k^2）$速率在平滑凸的标准设置中，而没有任何进一步的假设，例如无限可不同的能力。
We analyze continuous-time models of accelerated gradient methods through deriving conservation laws in dilated coordinate systems. Namely, instead of analyzing the dynamics of $X(t)$, we analyze the dynamics of $W(t)=t^α(X(t)-X_c)$ for some $α$ and $X_c$ and derive a conserved quantity, analogous to physical energy, in this dilated coordinate system. Through this methodology, we recover many known continuous-time analyses in a streamlined manner and obtain novel continuous-time analyses for OGM-G, an acceleration mechanism for efficiently reducing gradient magnitude that is distinct from that of Nesterov. Finally, we show that a semi-second-order symplectic Euler discretization in the dilated coordinate system leads to an $\mathcal{O}(1/k^2)$ rate on the standard setup of smooth convex minimization, without any further assumptions such as infinite differentiability.
