Closed-form worst-case conjecture for gradient descent on $\mathcal{D}_{\mu_g}^{\mu_M}$

Let $N$ be the iteration number, let $h$ be the step size, and let $\mathcal{D}_{\mu_g}^{\mu_M}$ be the operator-composed class defined in the source. For $0\leq\mu_g\leq1$ and $0\leq\mu_M\leq1$, define

$$M^*=\operatorname{proj}_{[\mu_M,1]}\left(\sqrt{\frac{h_0}{h}}\right),$$

where $h_0$ is a solution of

$$(1-\mu_g)(1-\mu_g h_0)^{2N+1}=1-(2N+1)\mu_g h_0,\qquad 0\leq h_0\leq\frac{1}{\mu_g}.$$

Closed-form worst-case conjecture. The worst-case performance satisfies

$$w(\mathcal{D}_{\mu_g}^{\mu_M};h)=\frac12\max\left\{\frac{\mu_g M^{*2}}{\mu_g-1+\left(1-\mu_g M^{*2}h\right)^{-2N}},(1-h)^{2N}\right\}.$$

The formula is obtained by solving the maximization suggested by the one-dimensional conjecture and the cited worst-case formula for smooth strongly convex functions; the supplied text does not establish it.