Strengthened minimax optimal constant-stepsize conjecture for gradient descent

Let $\mathcal{F}_{L,D}$ be the class of pairs $(f,x_0)$ where $f$ is $L$-smooth and convex, has a minimizer $x_\star$ with $\|x_0-x_\star\|\leq D$, and $x_N$ is obtained by $N$ steps of gradient descent with constant stepsize. For any $N,L,D$, let $\alpha(N)\geq1$ be the unique solution of

$$\frac{1}{2(2N\alpha+1)}=\frac{1}{2}(1-\alpha)^{2N},$$

and let $r(N)$ be their common value. Strengthened minimax conjecture. The stepsize $\alpha(N)$ is the unique minimizer of

$$\min_{\tilde\alpha\in\mathbb{R}}\max_{(f,x_0)\in\mathcal{F}_{L,D}} f(x_N)-\inf f,$$

and the optimal value is

$$\min_{\tilde\alpha\in\mathbb{R}}\max_{(f,x_0)\in\mathcal{F}_{L,D}} f(x_N)-\inf f=r(N)LD^2.$$

This strengthens the balancing conjecture by asserting both uniqueness and the exact minimax value for every $N,L,D$; the paper presents it as an open problem.