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.