Deterministic Zeroth-Order Mirror Descent via Vector Fields with A Posteriori Certification

TL;DR

This paper introduces a deterministic zeroth-order mirror descent method using vector fields with a posteriori certification, enabling derivative-free optimization with explicit guarantees and broad applicability to information-geometric algorithms.

Contribution

It develops a unified vector-field-driven mirror descent framework with trajectory-wise certification, applicable to derivative-free oracles and broad classes of algorithms.

Findings

Provides explicit last-iterate guarantees under verifiable inequalities.

Instantiates the theory with finite differences requiring 2d+1 evaluations.

Establishes a geometric property linking Bregman identities and conic dominance.

Abstract

We develop a deterministic zeroth-order mirror descent framework by replacing gradients with a general vector field, yielding a vector-field-driven mirror update that preserves Bregman geometry while accommodating derivative-free oracles. Our analysis provides a unified evaluation template for last-iterate function values under a relative-smoothness-type inequality, with an emphasis on trajectory-wise (a posteriori) certification: whenever a verifiable inequality holds along the realized iterates, we obtain explicit last-iterate guarantees. The framework subsumes a broad class of information-geometric algorithms, including generalized Blahut-Arimoto-type updates, by expressing their dynamics through suitable choices of the vector field. We then instantiate the theory with deterministic central finite differences in moderate dimension, where constructing the vector field via deterministic central finite differences requires 2d off-center function values (and one reusable center value), i.e., 2d+1 evaluations in total, where d is the number of input real numbers. In this deterministic finite-difference setting, the key interface property is not classical convexity alone but a punctured-neighborhood generalized star-convexity condition that isolates an explicit resolution-dependent error floor. Establishing this property for the finite-difference vector field reduces to a robust conic dominance design problem; we give an explicit scaling rule ensuring the required uniform dominance on a circular cone. Together, these results expose a hidden geometric structure linking Bregman telescoping identities, deterministic certification, and robust conic geometry in zeroth-order mirror descent.