I used ChatGPT to solve an open problem in convex optimization. *Part I* (1/N)

Problem statement: (2/N)

The proof, cleaned up and typed up by me: (3/N)

Key steps of the proof produced by ChatGPT: https://chatgpt.com/share/68f805f2-b8fc-8010-8df6-20a46bc1df44… (4/N)

My reaction: ChatGPT was really effective at accelerating my progress. This work took about 12 hours, spread over 3 days. In hindsight, the proof is really simple. (5/N)

But I iterated through so many other strategies that didn't pan out, and ChatGPT crucially helped to quickly explore and eliminate those dead-end approaches. Also, the key successful steps were suggested by ChatGPT. (6/N)

ChatGPT did not produce the proof in a single prompt. The process was highly interactive. It generated many arguments, roughly 80% of which were incorrect. (7/N)

Yet some were genuinely novel to me. Whenever I recognized a novel idea, whether correct or only partially so, I distilled the key insight and prompted ChatGPT to develop it further. (8/N)

My contribution: - Filtering out incorrect arguments and accumulating a set of correct facts. - Identifying promising new lines of reasoning and guiding ChatGPT to explore them further - Recognizing when a strategy had been fully explored and deciding when to move on. (9/N)

ChatGPT's contribution: - Producing the final proof argument. - Significantly accelerating my (or our) exploration of the many dead-end arguments, rapidly ruling out approaches that did not work. (10/N)

Plans forward: In my view, this result is already publishable in a respectable optimization theory journal. However, I would like to flesh it out further. Here are the next steps I plan to take. (11/N)

*Part II (forthcoming)* I'll attempt to generalize to the ODE with r>0. The convergence and divergence behavior is already known for r≤1 (divergence) and r>3 (convergence). I'll aim to extend our analysis to the intermediate regime, r∈(1,3). (12/N)

*Part III (forthcoming)* We'll see if I can translate the argument to prove convergence of the discrete-time counterpart, namely of Nesterov's accelerated gradient method. (This is actually the main open problem.) (13/N)

Roadblock: However, I've run out of ChatGPT Pro queries. I'm on the expensive pro-tier plan, but I've already used up my allowance, and it doesn't refresh until next month. Could someone from OpenAI help me out with this? (14/N) @Sebastien Bubeck @Kevin Weil 🇺🇸

Conclusion: I will continue to develop this project and publish the results in a respected optimization theory journal. I'll share updates and future parts (II and III) here on Twitter as the work progresses. (15/N)

ChatGPT is now at the level of solving some math research questions, but you do need an expert guiding it. This exercise was a lot of fun and was highly productive. I also feel I'm getting better at prompting ChatGPT. I'll also try other open and unsolved problems. (16/N, N=16)

The final proof corresponds to a geometric intuition saying that the X(t) must oscillate between z1 and z2 indefinitely, and the difference of energy functions yields an ODE mapping this oscillation onto the line through z1 and z2. This ODE shows the oscillation isn't possible.