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Point Convergence of Nesterov's Accelerated Gradient Method: An AI-Assisted Proof

Uijeong Jang, Ernest K. Ryu

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 for this record

What was already known

The O(1/k²) objective-value rate was known, but point convergence at the critical r=3 schedule remained open. Results for r>3 and other regimes did not settle this case.

AI involvement AI-centered

GPT-5 Pro supplied the continuous-time proof after extensive interaction and failed attempts. Jang and Ryu filtered and verified it, then developed the discrete adaptation.

The model supplied the central continuous-time proof argument after extensive human direction and filtering; humans translated it to discrete NAG.

Importance Major

Answers the critical point-convergence question for a foundational accelerated method.

Limitations

Jang and Ryu explicitly checked the AI-origin continuous-time proof. The discrete adaptation has no separately documented completed check. A withdrawn r∈[1,3) extension is excluded; a concurrent human discrete-time result exists.

Technical verification Formal (human)

The authors explicitly report verifying a complete GPT-5 Pro proof of Theorem 2 on a later prompted trial. This completed human check covers the AI-origin continuous argument, not a separately documented full check of the discrete adaptation; the r<3 error is a different claim.

Formal checking

Human verification of a complete generated proof obtained on the third trial in October 2025

Checker
Uijeong Jang and Ernest K. Ryu
Scope
Theorem 2 point convergence of the r=3 Nesterov ODE
Basis
Reported by the linked checker
Evidence source

Checking a formal statement does not establish that it matches the original problem.

Human understanding Full

The authors explain the energy-difference argument that forces all cluster points to coincide and adapt it to the discrete algorithm.

Correction and publication history