Iterate convergence of critical Nesterov acceleration
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Do trajectories of the critical r=3 Nesterov ODE converge to a minimizer of a differentiable convex function with a minimizer? The discrete counterpart asks whether both standard accelerated-gradient iterates xₖ and yₖ converge for finite-dimensional L-smooth convex objectives under the stated growing tₖ schedule.
Why it matters and what counts as progress
Why it matters: Point convergence is stronger than decay of objective values and resolves which minimizer a critical accelerated method approaches.
What counts as progress: Prove convergence of the points, beyond objective-value decay, or extend it to broader schedules and settings.
How the problem developed
- 1983 · Method [1]
Yurii Nesterov Introduced the accelerated gradient method whose critical-schedule point convergence later became the focus of this problem.
- 27 Oct 2025 · Proof claim [1]
Uijeong Jang and Ernest K. Ryu Posted a ChatGPT-assisted proof of point convergence for the classical accelerated gradient method; the paper also develops an ODE convergence result for a convex objective with a minimizer.
Uijeong Jang and Ernest K. Ryu Revised the point-convergence manuscript; a separate concurrent discrete-time convergence preprint appeared in the same period.
AI contribution
Point Convergence of Nesterov's Accelerated Gradient Method: An AI-Assisted Proof
For a differentiable convex function with a minimizer, trajectories of the r=3 Nesterov ODE converge to a minimizer. Jang and Ryu adapt this argument to show that both iterates xₖ and yₖ of standard finite-dimensional Nesterov accelerated gradient converge to one minimizer for an L-smooth convex function under the stated growing tₖ schedule.
Sources
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Publication dates and editorial changes.
GPT-origin argument proves critical Nesterov trajectory convergence