aiqbench analysis
This paper resolves a long-standing theoretical gap by proving that the Optimistic Multiplicative-Weights Update (OMWU) algorithm asymptotically converges to saddle points in smooth convex-concave problems, matching the behavior of its Euclidean counterpart OGDA. The proof establishes convergence without requiring strict complementarity, error bounds, or specific initialization conditions, relying instead on a novel boundary argument for inactive coordinates. Notably, the core of this boundary argument was discovered with assistance from ChatGPT, highlighting an emerging intersection between AI research and algorithmic discovery.
Last-Iterate Convergence of Optimistic Multiplicative Weight Update
arXiv:2606.11773v1 Announce Type: cross Abstract: 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. In this paper, I show that OMWU converges asymptotically for smooth convex-concave saddle-point problems, with a small enough constant learning rate. The result does not require uniqueness, strict complementarity, an error bound, or initialization near a solution. The main new ingredient is a boundary argument showing that every cluster point satisfies the inactive-coordinate KKT inequalities. The boundary argument was discovered with assistance from ChatGPT and is documented in the appendix.
Abstract: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. In this paper, I show that OMWU converges asymptotically for smooth convex-concave saddle-point problems, with a small enough constant learning rate. The result does not require uniqueness, strict complementarity, an error bound, or initialization near a solution. The main new ingredient is a boundary argument showing that every cluster point satisfies the inactive-coordinate KKT inequalities. The boundary argument was discovered with assistance from ChatGPT and is documented in the appendix.
Optimization and Control (math.OC); Machine Learning (cs.LG)
arXiv:2606.11773 [math.OC]
(or arXiv:2606.11773v1 [math.OC] for this version)
https://doi.org/10.48550/arXiv.2606.11773 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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