Drori–Teboulle conjecture on the minimax-optimal constant stepsize for gradient descent
Prove that for any N, L, D, the unique α(N) ≥ 1 solving 1/(2(2Nα + 1)) = (1/2)(1 − α)^{2N} is the unique minimizer of the worst-case final objective gap over all L-smooth convex functions with initial points within distance D of a minimizer when running N steps of gradient descent with constant stepsize α (i.e., x_{k+1} = x_k − (α/L) ∇f(x_k)), and that the minimax value equals r(N) L D^2, where r(N) is the common value of the two sides of the defining equality.
References
\begin{conjecture} [{Conjecture 3.1]\label{conj:main}
For any N,L,D, let α(N)≥1 denote the unique solution of \frac{1}{2(2Nα+1)} = \frac{1}{2}(1-α){2N} and let r(N) denote their common value. Then, α(N) is the unique minimizer of~eq:minimax-design and achieves the value
\begin{equation*}
\min_{\tilde \alpha\in\mathbb{R} \max_{(f,x_0)\in\mathcal{F}_{L,D} f(x_N) - \inf f = r(N) LD2.
\end{equation*}
\end{conjecture}