Adaptive acceleration without GRAAL’s specific extrapolation

Determine whether the convergence and adaptive properties established for the Accelerated GRAAL algorithm—namely, the monotonic potential decrease for convex continuously differentiable objectives and the optimal O(1/k^2) convergence rate for L-smooth objectives with geometrically growing adaptive stepsizes based on local curvature estimates—can be obtained using baseline algorithms that do not employ the specific GRAAL extrapolation step.

Background

The paper develops Accelerated GRAAL, which integrates Nesterov acceleration with an adaptive stepsize rule guided by local curvature estimates. A key component is the extrapolation step borrowed from the GRAAL methodology, which the authors argue is central to the algorithm’s ability to adapt the stepsize effectively.

While the authors prove optimal convergence guarantees for their algorithm under L-smoothness and provide a general potential decrease for convex objectives, they note uncertainty about whether these results rely fundamentally on the GRAAL-style extrapolation. This raises the question of whether alternative baseline algorithms, lacking this specific extrapolation step, could still achieve the same adaptive and accelerated guarantees.

References

In particular, it is unclear whether our results can be obtained with different baseline algorithms. This remains an interesting open question.

— Nesterov Finds GRAAL: Optimal and Adaptive Gradient Method for Convex Optimization  (2507.09823 - Borodich et al., 13 Jul 2025) in Section 2.1 (Algorithm Development), GRAAL extrapolation paragraph

For general Bregman divergence, however, $\kappa_k$ may grow without bound, which can cause the convergence rates to deteriorate. Identifying global conditions that characterize the behavior of $\kappa_k$ and developing methods to avoid this deterioration remain topics for future work.

— Accelerated Bregman Proximal Gradient Methods from Dual Geometric Perspectives  (2609.34773 - Yamashita et al., 28 Sep 2026) in Section 6, Conclusion

Accelerating our local, linesearch-free methods remains a separate question.

— Linesearch-Free Nonlinearly Preconditioned GD with Semiautomatic Geometry Design  (2609.34288 - Iwade et al., 28 Sep 2026) in Appendix, Section 'Extended Related Work', subsection 'Further adaptive methods'