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

2026-01-31

AI Summary

This work addresses the challenges of zeroth-order optimization in the absence of gradient information and convexity assumptions by proposing a deterministic mirror descent framework. The method constructs surrogate gradients from vector fields while preserving Bregman geometric structure and introduces a trajectory-level post-hoc verification mechanism to provide explicit convergence guarantees. Innovatively integrating information-geometric algorithms—such as generalized Blahut–Arimoto updates—into the zeroth-order setting, it establishes novel conditions including punctured neighborhood generalized star-convexity and cone dominance, revealing intrinsic connections among Bregman identities, deterministic certification, and robust conic geometry. Requiring only $2d+1$ function evaluations, the approach achieves verifiable convergence with error bounds in moderate dimensions, thereby transcending traditional convexity constraints.

Technology Category

• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• zeroth-order optimization
• mirror descent
• vector fields
• a posteriori certification
• Bregman geometry

Innovation

• zeroth-order optimization
• mirror descent
• vector field
• a posteriori certification
• Bregman geometry