\end{pmatrix}
    =  
    -
    \begin{pmatrix}
        {\operatorname{Id}}+ \eta D_{xx}^2 f & \eta D_{xy}^2 f \\
        \eta D_{yx}^2 g & {\operatorname{Id}}+ \eta D_{yy}^2 g
    \end{pmatrix}^{-1}
    \begin{pmatrix}
         \nabla_{x} f\\
         \nabla_{y} g
    \end{pmatrix}.$$ For the following reasons we believe that the bilinear approximation is preferable both from a practical and conceptual point of view.

-   *Conditioning of matrix inverse:* One advantage of competitive gradient descent is that in many cases, including all zero-sum games, the condition number of the matrix inverse in Algorithm \[alg:CGD\] is bounded above by $\eta^2 \|D_{xy}\|^2$. If we include the diagonal blocks of the Hessian in a non-convex-concave problem, the matrix can even be singular as soon as $\eta \|D_{xx}^2 f\| \geq 1$ or $\eta \|D_{yy}^2 g\| \geq 1$.

-   *Irrational updates:* We can only expect the update rule  to correspond to a local Nash equilibrium if the problem is convex-concave or $\eta \|D_{xx}^2 f\|, \eta \|D_{yy}^2 g\| < 1$. If these conditions are violated it can instead correspond to the players playing their *worst* as opposed to best strategy based on the quadratic approximation, leading to behavior that contradicts the game-interpretation of the problem.

-   *Lack of regularity:* For the inclusion of the diagonal blocks of the Hessian to be 