Invited talk: Last-Iterate Convergence of Optimistic Multiplicative Weight Update -- Francesco Orabona

Optimistic Gradient Descent Ascent (OGDA) and Optimistic Multiplicative-Weights Update (OMWU) are two very popular algorithms to solve convex/concave saddle-point problems, where OMWU is the non-Euclidean, entropic version of OGDA. It is known since the '80s that the last iterate of OGDA asymptotically converges to a saddle point in smooth problems. On the other hand, it is unknown if OMWU has the same property, and the literature presents only wrong proofs on this problem. In this talk, I show that OMWU converges asymptotically for smooth convex-concave saddle-point problems, with a small enough constant learning rate. This result does not require uniqueness, strict complementarity, an error bound, or initialization near a solution. The main new ingredient of the proof was discovered with assistance from ChatGPT, and I will describe the LLM-assisted proof process as well.