The Method of Ellipcenters for Strongly Convex Functions

TL;DR

The paper extends the Method of Ellipcenters to strongly convex functions with Lipschitz continuous gradients, demonstrating convergence rates comparable to gradient descent and showing potential for faster convergence under certain conditions.

Contribution

It generalizes the Method of Ellipcenters from quadratic to broader strongly convex functions and analyzes its convergence properties.

Findings

ME matches gradient descent with exact line search in convergence rate.

Further per-step improvements occur when gradient directions are linearly independent.

Numerical experiments confirm theoretical convergence predictions.

Abstract

The Method of Ellipcenters (ME), introduced in~\cite{ME2025} for strongly convex quadratic minimization, uses two gradient evaluations per iteration: one at the current iterate and one at a companion point on the same level set. We extend ME to the broader class of strongly convex functions with Lipschitz continuous gradient, and prove that ME matches the convergence rate of gradient descent with exact line search on this class. When the two gradient directions are linearly independent, a midpoint argument exploiting the level-set symmetry yields a further per-step improvement, which is global when the angle between the two gradients is uniformly bounded away from zero. ME also converges in at most two steps in dimension two. Numerical experiments on regularized logistic regression confirm the theoretical predictions.