GPT-5 Pro proves an improved convex-optimisation bound, which humans had already surpassed
OpenAI's Sébastien Bubeck reported that GPT-5 Pro, in about 17 minutes, proved that gradient descent on L-smooth convex functions yields a convex sequence of function values for step sizes up to 1.5/L. The paper's v1 had proved it for 1/L. However, the authors' own v2 had already proved the tight 1.75/L bound.
Key facts
- Problem: for which step sizes η is the optimisation curve of gradient descent convex? v1 proved η ≤ 1/L and gave a counterexample above 1.75/L
- GPT-5 Pro proved η ≤ 1.5/L by a different argument; Bubeck checked it
- The human authors' updated version had already closed the gap at 1.75/L
- Bubeck: 'Claim: gpt-5-pro can prove new interesting mathematics.'
Science result
- Field
- mathematics / convex optimisation
- Problem
- Convexity of the gradient-descent optimisation curve for L-smooth convex functions
- Result
- A correct, novel proof of the bound η ≤ 1.5/L, improving v1's 1/L but weaker than the humans' tight 1.75/L result already posted.
- AI system
- GPT-5 Pro
- Human role
- Human posed the problem and verified the proof
- Verification
- Expert-checked (Bubeck); informal
- Status
- confirmed
- Why surprising
- A general chatbot produced a correct, non-trivial research-level proof in minutes, although the result was already superseded.
What happened
Bubeck gave GPT-5 Pro the open question from v1 of a paper; the model produced a valid proof of an intermediate bound.
Why it matters
It was one of the first widely discussed cases of an LLM producing correct new research-level mathematics. The fact that humans had already done better also foreshadowed later disputes over novelty.
Changelog
- 2026-09-29: created
Related posts (1)
- Sebastien Bubeck Sebastien Bubeck @SebastienBubeck · x · 2025-08-20
Cited as a source by: 2025-08-20-gpt-5-pro-convex-optimization-proof
Related events
- OpenAI launches GPT-5 ★★★★★
- OpenAI researchers claim GPT-5 'solved' 10 Erdős problems; the solutions were already in the literature ★★★
Sources (3)
- officialSébastien Bubeck on X
- discussionwhataifound.org: GPT-5 convex bound
- pressWhat does GPT-5's new math claim actually mean?
id: 2025-08-20-gpt-5-pro-convex-optimization-proof · updated 2026-09-29 · open in the interactive timeline