The Grimmer-Shu-Wang Certificate and the Drori-Teboulle Minimax Nonnegative Constant-Stepsize Bound for N >= 3

TL;DR

This paper proves the existence of a performance-estimation certificate for gradient descent with constant stepsize, establishing bounds for all horizons N >= 3, and confirming the minimax value for these settings.

Contribution

It demonstrates the existence of the Grimmer-Shu-Wang certificate for all N >= 3, confirming the Drori-Teboulle minimax bounds for gradient descent.

Findings

GSW certificate exists for all N >= 3

Confirms Drori-Teboulle upper bounds for gradient descent

Establishes minimax nonnegative constant-stepsize value for all N >= 3

Abstract

We prove, for every horizon N >= 3, the existence of the strengthened low-rank performance-estimation certificate proposed by Grimmer, Shu, and Wang for the Drori-Teboulle minimax nonnegative constant-stepsize problem for gradient descent. Let rho_N in (0,1) be the unique solution of rho_N^{2N}(2N rho_N+2N+1)=1. We show that the GSW certificate equations admit positive vectors a, b, c, d satisfying all residual equations. The proof proceeds through a reduced residual system in the variables d, a simplex existence argument for a positive reduced zero, a terminal residual completion identity, and a tail-square convolution argument proving the cumulative margins that force positivity of the certificate coefficients. Consequently, the GSW low-rank PEP certificate exists for every N >= 3 and yields the Drori-Teboulle upper bound. Together with the one-dimensional quadratic and Huber lower-bound examples, this establishes the Drori-Teboulle minimax nonnegative constant-stepsize value for gradient descent for every N >= 3.