# Last-Iterate Convergence of Optimistic Multiplicative Weight Update

arXiv: https://arxiv.org/abs/2606.11773 · HTML: https://mlsanity.com/paper/2606.11773/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 10 references that are in the map.
- First posted on arXiv 10 Jun 2026.

## Builds on (most cited first, 8 of 8)

- [Online Learning: A Modern Introduction Using Convex Optimization](https://mlsanity.com/paper/1912.13213/) (2019)
- [Training GANs with Optimism](https://mlsanity.com/paper/1711.00141/) (2017)
- [Optimistic mirror descent in saddle-point problems: Going the extra (gradient) mile](https://mlsanity.com/paper/1807.02629/) (2018)
- [Last-Iterate Convergence: Zero-Sum Games and Constrained Min-Max Optimization](https://mlsanity.com/paper/1807.04252/) (2018)
- [Last-iterate Convergence in Extensive-Form Games](https://mlsanity.com/paper/2106.14326/) (2021)
- [Projected Reflected Gradient Methods for Monotone Variational Inequalities](https://mlsanity.com/paper/1502.04968/) (2015)
- [Last iterate convergence in no-regret learning: constrained min-max optimization for convex-concave landscapes](https://mlsanity.com/paper/2002.06768/) (2020)
- [Fast Last-Iterate Convergence of Learning in Games Requires Forgetful Algorithms](https://mlsanity.com/paper/2406.10631/) (2024)

## Abstract (arXiv)

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.