A Strengthened Conjecture on the Minimax Optimal Constant Stepsize for Gradient Descent

TL;DR

This paper strengthens a conjecture about the optimal stepsize for gradient descent, proposing a specific low-rank certificate that enables verification of the conjecture for much larger iteration counts.

Contribution

It introduces a low-rank certificate structure that bypasses SDPs, allowing verification of the conjecture up to 20,160 iterations.

Findings

Verification of the conjecture up to N=20160 iterations

Proposal of a low-rank certificate structure

Enhanced understanding of optimal stepsize for gradient descent

Abstract

Drori and Teboulle [4] conjectured that the minimax optimal constant stepsize for N steps of gradient descent is given by the stepsize that balances performance on Huber and quadratic objective functions. This was numerically supported by semidefinite program (SDP) solves of the associated performance estimation problems up to $N\approx 100$. This note presents a strengthened version of the initial conjecture. Specifically, we conjecture the existence of a certificate for the convergence rate with a very specific low-rank structure. This structure allows us to bypass SDPs and to numerically verify both conjectures up to $N = 20160$.