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.

翻译：本文提出了一种确定性零阶镜像下降框架，其核心思想是用一般向量场替代梯度，从而形成一种由向量场驱动的镜像更新方法。该方法在保持Bregman几何结构的同时，兼容无导数优化算子。我们的分析在相对光滑型不等式的条件下，为末点函数值提供了一个统一的评估模板，并特别强调沿轨迹（后验）验证：只要实际迭代点满足一个可验证的不等式，即可获得显式的末点保证。该框架通过选择合适的向量场来表达其动态特性，从而涵盖了一大类信息几何算法，包括广义Blahut-Arimoto型更新。随后，我们在中等维度下采用确定性中心有限差分对理论进行实例化：通过确定性中心有限差分构建向量场需要2d个偏离中心的函数值（以及一个可重复使用的中心值），即总共需要2d+1次函数评估，其中d为输入实数的个数。在此确定性有限差分设定下，关键接口性质并非仅依赖于经典凸性，而是一种具有穿孔邻域的广义星形凸性条件，该条件可分离出显式的分辨率相关误差下限。为有限差分向量场建立此性质可转化为一个鲁棒锥支配设计问题；我们给出了一个显式的缩放规则，确保在圆锥上满足所需的均匀支配性。综上，这些结果揭示了一种隐含的几何结构，该结构在零阶镜像下降中连接了Bregman伸缩恒等式、确定性验证与鲁棒锥几何。