of GDA is not surprising: It is the analogue of *“Rock! Paper! Scissors! Rock! Paper! Scissors! Rock! Paper!...”* in the eponymous hand game. While gradient descent is a reliable basic *workhorse* for single-agent optimization, GDA can not play the same role for competitive optimization. At the moment, the lack of such a *workhorse* greatly hinders the broader adoption of methods based on competition.

**Existing works:** Most existing approaches to stabilizing GDA follow one of three lines of attack.\
In the special case $f = -g$, the problem can be written as a minimization problem $\min_{x} F(x)$, where $F(x) {\coloneqq}\max_{y} f(x,y)$. For certain structured problems, [@gilpin2007gradient] use techniques from convex optimization [@nesterov2005excessive] to minimize the implicitly defined $F$. For general problems, the two-scale update rules proposed in [@goodfellow2014generative; @heusel2017gans; @metz2016unrolled] can be seen as an attempt to approximate $F$ and its gradients.\
In GDA, players pick their next strategy based on the last strategy picked by the other players. Methods based on *follow the regularized leader* [@shalev2007convex; @grnarova2017online], *fictitious play* [@brown1951iterative], *predictive updates* [@yadav2017stabilizing], *opponent learning awareness* [@foerster2018learning], and *optimism* [@rakhlin2013online; @daskalakis2017training; @mertikopoulos2019optimistic] propose more sophisticated heuristics that the players could use to predict each other’s next move. Algorithmically, many of 