the linear approximation in the single agent setting to the competitive setting: *How to linearize a game?*.

**Linear or Multilinear:** GDA answers the above question by choosing a linear approximation of $f,g: {\mathbb{R}}^m \times {\mathbb{R}}^n \longrightarrow {\mathbb{R}}$. This seemingly natural choice has the flaw that linear functions can not express any interaction between the two players and are thus unable to capture the competitive nature of the underlying problem. From this point of view it is not surprising that the convergent modifications of GDA are, implicitly or explicitly, based on higher order approximations (see also [@li2017limitations]). An equally valid generalization of the linear approximation in the single player setting is to use a *bilinear* approximation in the two-player setting. Since the bilinear approximation is the lowest order approximation that can capture some interaction between the two players, we argue that the natural generalization of gradient descent to competitive optimization is not GDA, but rather the update rule $(x_{k+1},y_{k+1}) = (x_k,y_k) + (x,y)$, where $(x,y)$ is a Nash equilibrium of the game [^2] $$\begin{aligned}
    \begin{split}
    \label{eqn:localgame}
    \min_{x \in {\mathbb{R}}^m} x^{\top} \nabla_x f &+ x^{\top} D_{xy}^2 f y + y^{\top} \nabla_y f + \frac{1}{2\eta} x^{\top} x \\
    \min_{y \in {\mathbb{R}}^n}  y^{\top} \nabla_y 