A Unified Model for High-Resolution ODEs: New Insights on Accelerated Methods

TL;DR

This work develops a unified high-resolution ODE framework for accelerated gradient methods by deriving a general HR-ODE from a forced Euler–Lagrange equation and analyzing it with integral quadratic constraints and Lyapunov methods. The framework recovers and tightens the continuous-time behavior of HB, NAG, TM, and QHM, and, via Semi-Implicit Euler discretization, yields a single discrete-time scheme that encompasses these methods while providing improved convergence guarantees, including an exact recovery of NAG and enhanced gradient-norm rates. It also connects rate-matching discretization to NAG, showing how NAG can arise as an accurate discretization of a rate-matching ODE, and offers sharper QHM and TM results within the same unified theory. The results advance theoretical understanding of acceleration in smooth convex and strongly convex settings and provide a versatile blueprint for designing accelerated methods with provable rates, while suggesting future work on non-Euclidean extensions and achieving TM’s best possible rates for its exact HR-ODE.

Abstract

Recent work on high-resolution ordinary differential equations (HR-ODEs) captures fine nuances among different momentum-based optimization methods, leading to accurate theoretical insights. However, these HR-ODEs often appear disconnected, each targeting a specific algorithm and derived with different assumptions and techniques. We present a unifying framework by showing that these diverse HR-ODEs emerge as special cases of a general HR-ODE derived using the Forced Euler-Lagrange equation. Discretizing this model recovers a wide range of optimization algorithms through different parameter choices. Using integral quadratic constraints, we also introduce a general Lyapunov function to analyze the convergence of the proposed HR-ODE and its discretizations, achieving significant improvements across various cases, including new guarantees for the triple momentum method$'$s HR-ODE and the quasi-hyperbolic momentum method, as well as faster gradient norm minimization rates for Nesterov$'$s accelerated gradient algorithm, among other advances.

Figures (5)

• Figure 1: Continuous-time simulation using $f(x)=4(L-\mu)\log(1+e^{-x})+\frac{\mu}{2}x^2$, (left) comparison of trajectories $(X_t,V_t)$ from \ref{['proposed_ODE_conti']} and $(Q_t,J_t)$ from (\ref{['proposed_ODE_conti2']}). The simulation is done for ${L=1,\mu=10^{-2}}$ and $X(0)=Q(0)=10, V(0)=J(0)=0$, (right) trajectory of (\ref{['HR_TM']}) ODE for $L=10, \mu=10^{-3}, \text{ random }Y(0),\dot Y(0)=0$ and its corresponding upper bounds.
• Figure 2: Trajectory of TM method and various proposed ODEs for $f(x_1,x_2)= 5\times 10^{-3}x_1^2+x_2^2$ with starting point $(x_1(0),x_2(0))=(1,1)$. In this figure, (TM LR-ODE) corresponds to the low-resolution ODE of the TM method ((\ref{['TM_new_high_res2']}) with $\sqrt{s}=0$). The step-size was $s=0.16$; (\ref{['HR_TM']}) and (\ref{['TM_new_high_res2']}) were according to Corollaries \ref{['corl_TM']} and \ref{['corollary_hrtm2']} respectively.
• Figure 3: Discrete-time simulation; comparison of QHM method performance under various settings with existing upper bounds for (a) $f(x)=4(L-\mu)\log(1+e^{-x})+\frac{\mu}{2}x^2$, $L=10,\mu=10^{-3}$ and (b) 10-dimensional regularized binary classification problem with logistic loss for random data and labels of length 1000. The regularization parameter is chosen $\mu = 10^{-3}$. QHM parameters are chosen so that the best possible theoretical convergence rate is achieved.
• Figure 4: Positivity of $G(\xi,\kappa)$ for various values of $\kappa$
• Figure 5: Comparison between the NAG, (\ref{['NAG_SIE_shi']}) as Shi-SIE, SIE discretization of (\ref{['GM-ODE']}) as GM-SIE with superscript 1 and 2 when $n'=1,m'=\sqrt{s},q'=2\sqrt{\mu}$ and $n'=1-2\sqrt{\mu s},m'=\sqrt{s},q'=2\sqrt{\mu}$ respectively. The simulation function was $f(x)=4(L-\mu)\log(1+e^{-x})+\frac{\mu}{2}x^2$ with $L=1,\mu=0.01$. (a) The effect of the approximations ($1/(1-\sqrt{\mu s})\approx 1$ in Shi-SIE and coefficient deviation in GM-SIE$^1$) in the ODE trajectories. (b) Different coefficients used for discretizing (\ref{['GM-ODE']}). GM-SIE$^1$ is the SIE discretization of GM-ODE$^1$ (the recovered high-resolution NAG ODE from (\ref{['GM-ODE']})) and GM-SIE$^2$ is the SIE discretization of GM-ODE$^2$ (the ODE used to recover the NAG algorithm).

Theorems & Definitions (44)

• Theorem 1
• proof : Proof of \ref{['Theorem3_1']}
• Remark 1
• Theorem 2
• proof : Proof of \ref{['Theorem_ODE_laborde']}
• Remark 2
• Theorem 3
• proof : Proof of \ref{['Theorem_ODE_Shi']}
• Remark 3
• Remark 4