Establish a matching lower bound for a known initial optimality gap

Establish a lower bound matching the dependence on the known initial optimality-gap bound $\Delta$ for the class $\mathcal P_\Delta$ when $4\varepsilon<\Delta<\frac{L_0}{L_1^2}(e^{L_1R}-1-L_1R)$, and determine the exact complexity dependence on $\Delta/\varepsilon$ when $\varepsilon<\Delta<4\varepsilon$.

Background

The paper studies the class $\mathcal P_\Delta$ of convex $(L_0,L_1)$-smooth functions whose initial optimality gap is at most a known value $\Delta$. It proves matching bounds up to logarithmic factors when $\Delta=4\varepsilon$, but for larger gaps only an upper bound is obtained. The paper identifies two unresolved ranges: intermediate gaps between $4\varepsilon$ and the radial growth bound, and gaps between $\varepsilon$ and $4\varepsilon$, for which the exact complexity dependence has not been determined.