Strengthen the anytime lower bound by coupling horizons or using multi-step estimates

Determine whether coupling several horizons in the finite-to-anytime transfer, or replacing the terminal-step estimate with a multi-step bound, can yield a stronger anytime lower bound for gradient descent with a single predetermined infinite stepsize schedule.

Background

The paper’s anytime exponent $2(1+\alpha)/(2+\alpha)$ arises from a transfer argument that combines finite-prefix estimates with a terminal-step bound at horizons where the final stepsize is the largest observed so far.

The authors identify the transfer mechanism as a possible source of weakness. The unresolved possibilities are to use information from several horizons jointly or to replace the one-step terminal estimate with an estimate involving multiple steps, potentially improving the $\Omega(n^{-1.2408})$ barrier.