$\beta$-High Resolution ODE and Phase Transition between NAG-SC and Heavy Ball Method

In this paper, we study the convergence properties of an algorithm that can be viewed as an interpolation between two gradient based optimization methods, Nesterov's acceleration method for strongly convex functions $(NAG$-$SC)$ and Polyak's heavy ball method. Recent Progress has been made on using High-Resolution ordinary differential equations (ODEs) to distinguish these two fundamentally different methods. The key difference between them can be attributed to the gradient correction term, which is reflected by the Hessian term in the High-Resolution ODE. Our goal is to understand how this term can affect the convergence rate and the choice of our step size. To achieve this goal, we introduce the notion of $\beta$-High Resolution ODE, $0\leq \beta\leq 1$ and prove that within certain range of step size, there is a phase transition happening at $\beta_c$. When $\beta_c\leq\beta\leq 1$, the algorithm associated with $\beta$-High Resolution ODE have the same convergence rate as NAG-SC. When $0\leq \beta\leq \beta_c$, this algorithm will have the slower convergence rate than NAG-SC.