Optimal objective-gap rate for gradient descent stepsizes

Determine whether O(N^{-log_2(1+√2)}) is the optimal worst-case convergence rate for the terminal objective gap f(x_N)−f(x_*) achievable by gradient descent on L-smooth convex functions over all possible choices of stepsizes (including arbitrary iteration-dependent selections).

Background

The paper proves that specific long-step schedules for gradient descent on L-smooth convex functions achieve an objective-gap rate of order N{-log_2(1+√2)} and improve constants versus the silver stepsize schedule. Within a broader context, Altschuler and Parrilo’s work suggests that no gradient descent stepsize scheme can surpass this exponent for objective-gap convergence.

The authors explicitly note that this N{-log_2(1+√2)} rate is conjectured to be optimal among all gradient descent stepsize schemes for objective-gap reduction, motivating a definitive characterization of the best possible exponent independent of the particular stepsize schedule.

References

Note although O(1/N{\log_2()}) is conjectured to be the optimal rate for gradient descent among all possible stepsize selections, it is not optimal among all gradient methods.

— Accelerated Objective Gap and Gradient Norm Convergence for Gradient Descent via Long Steps  (2403.14045 - Grimmer et al., 2024) in Remark (On Optimality of Rates)

However, it remains an open question whether an alternative step-size schedule can improve these upper-bound results.

— Stronger Lower Bounds for (Non-)Anytime Acceleration of Gradient Descent  (2609.04032 - Jung et al., 3 Sep 2026) in Section 1, subsection “Step-Size Schedules for Accelerating GD”

Closing the gaps between these lower bounds and their corresponding upper bounds remains a significant open problem.

— Improved Gradient Descent Lower Bounds Beyond Nesterov  (2609.02855 - Ye et al., 2 Sep 2026) in Section 6, Concluding Remarks

A more structural question is whether \cref{lem:hard-product} is already sufficient to prove an \Omega(n{-\log_2(1+\sqrt{2})}) lower bound matching the silver-stepsize rate.

— Improved Gradient Descent Lower Bounds Beyond Nesterov  (2609.02855 - Ye et al., 2 Sep 2026) in Section 6, Concluding Remarks

The remaining boundary is therefore finer than the exponent: exact finite-horizon minimax behavior beyond these certificate-generated classes, as well as the exact extrema and pivot geometry of the OBS-F phase, remains open.

— Optimal Recursive Composition and Dyadic Phase Laws for Gradient Descent with Predetermined Stepsizes  (2609.11788 - Liu et al., 10 Sep 2026) in Conclusion, final paragraph