Totally agree with @ErnestRyu that AI helpers will become very useful for research. But in the near future the biggest help will be with *informal* math, the kind we work out with our collaborators/grad students on a whiteboard. I already use frontier models to help write/debug lemmas for my papers and lectures. AI is fast, but can also misunderstand. So have to still carefully check the lemma statements and proofs. But already a big productivity boost. (Lean provers will automate the proof checking, but the human will still need to check that the lean formalization accurately captures their intent, which humans will be doing for a while.)

Imagine that projected gradient descent (PGD) was a new method, discovered today. How would that feel? This is a textbook algorithm... What further research, extensions, improvements and variants would this enable? In fact, together with Kaja Gruntkowska and Hanmin Li, we have just discovered a sister method to projected gradient descent -- one of equal conceptual importance. Our method admits the same or very similar guarantees as PGD. However, instead of relying on projections onto the constraint, it relies on linear minimization! You may say: Did you rediscover Frank-Wolfe? No. In contrast to Frank-Wolfe, which uses a global linear minimization oracle (global LMO), our method relies on a local minimization oracle (local LMO). For this reason, we simply call the method "Local LMO" (admittedly, conflating the oracle name with the method name). Frank-Wolfe theory is much more limited to the theory of Local LMO. Here are some key differences: 1) Frank-Wolfe only works if the constraint is bounded, and its convergence theory depends in the diameter of the constraint set. Local LMO works even for unbounded constraints, and its theory does not depend on the diameter of the constraint set. 2) In fact, Local LMO reduces to gradient descent (GD) in the unconstrained case. If the constraint is affine, Local LMO reduces to (preconditioned) GD in the affine space. 3) While Frank-Wolfe does not converge linearly for smooth strongly convex functions, Local LMO does. 4) While Frank-Wolfe does not converge for non-smooth convex problems (its theory depends on a curvature assumption), Local LMO does. arxiv.org/abs/2605.08850

I'm thrilled to welcome @ErnestRyu to our team in @OpenAI !! If you're excited about the progress we've made in making ChatGPT a useful tool for scientists, just wait for what we'll cook for you next year with @ErnestRyu and the rest of the team!

The proof, cleaned up and typed up by me, resolves the 42-year-old open problem: 5/N