spectral function $h_{\pm}(\lambda) {\coloneqq}\min(3\lambda, \lambda)/2$.

\[thm:convergence\] If $f$ is two times continiously differentiable with $L$-Lipschitz continuous mixed Hessian, $f$ is convex-concave or $D_{xx}^2 f, D_{yy}^2f$ are $L$-Lipschitz continuous, and the diagonal blocks of its Hessian are bounded as $\eta \|D_{xx}^2 f\|, \eta \|D_{yy}^2 f\| \leq  1$, we have $$\begin{aligned}
       &\left\|\nabla_x f\left(x_{k+1}, y_{k+1}\right) \right\|^2 + \left\|\nabla_y f\left(x_{k+1}, y_{k+1}\right) \right\|^2 
       - \left\|\nabla_{x} f \right\|^2 - \left\|\nabla_{y} f \right\|^2 \leq \\
       &- \nabla_{x} f^{\top} \left( \eta h_{\pm} \left(D_{xx}^2 f\right) + \bar{D} -32 L \eta^2\|\nabla_{x} f\| \right) \nabla_x f  
    - \nabla_{y} f^{\top} \left( \eta h_{\pm}\left(-D_{yy}^2 f\right) + \tilde{D} -32 L \eta^2 \|\nabla_y f\| \right) \nabla_{y}f
    \end{aligned}$$

Under suitable assumptions on the curvature of $f$, Theorem \[thm:convergence\] implies results on the convergence of CGD.

Under the assumptions of Theorem \[thm:convergence\], if for $\alpha > 0$ $$\Bigl( \eta h_{\pm} \left(D_{xx}^2 f\right) + \bar{D} -32 L \eta^2\|\nabla_{x} f(x_0, y_0)\| \Bigr),
        \Bigl( \eta h_{\pm}\left(-D_{yy}^2 f\right) + \tilde{D} -32 L \eta^2 \|\nabla_y f(x_0, y_0)\| \Bigr) \succeq \alpha {\operatorname{Id}},$$ for all $(x,y) \in {\mathbb{R}}^{m + n}$, then CGD started in $(x_0, y_0)$ converges at exponential rate with exponent $\alpha$ to a critical point.

Furthermore, we can deduce the following local stability result.

Let $(x^*,y^*)$ be a critical point ($(\nabla_{x} f, 