What does this research mean for the field?

First-order optimization algorithms derived via Runge-Kutta discretization of high-resolution ordinary differential equations achieve an accelerated convergence rate of O(N^(-2s/(s+1))) for convex functions and outperform symplectic Euler discretization for strongly convex functions. Novelty: ClaimNovelty.METHODOLOGICAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

This research aims to enhance the efficiency of first-order optimization algorithms by using Runge-Kutta discretization.

June 26, 2026

Acceleration in convex optimization via Runge-Kutta discretization of high-resolution ordinary differential equations

Key Points

This research aims to enhance the efficiency of first-order optimization algorithms by using Runge-Kutta discretization.
Discretized high-resolution ordinary differential equations to derive first-order optimization algorithms.
Analyzed Lipschitz continuous gradients and differentiability for convex functions.
Conducted numerical experiments contrasting new algorithms with classical methods.
Achieved convergence rate of O(N−2ss+1) for smooth convex functions.
Proved faster convergence rates for strongly convex functions compared to symplectic Euler discretization.
Numerical experiments confirmed superiority over Nesterov's accelerated gradient method.

Abstract

This paper investigates first-order optimization algorithms derived by discretizing two high-resolution ordinary differential equations. For sufficiently smooth objective functions, we demonstrate that acceleration can be achieved via Runge-Kutta integrators. Specifically, for convex objective functions f satisfying the assumptions of Lipschitz continuous gradients and (s+2)-th order differentiability, we prove that the objective function converges to its minimum value with a rate of O(N−2ss+1), where s denotes the order of the Runge-Kutta integrator. For strongly convex objective functions f, under the same assumptions, we further prove that the proposed algorithms achieve a faster convergence rate than the algorithms generated by symplectic Euler discretization. Numerical experiments validate the effectiveness and superiority of the proposed algorithms through comparisons with classical benchmark methods, including Nesterov's accelerated gradient method and the symplectic Euler scheme.

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