Intrinsic perturbation scale for certified oracle objectives with epigraphic information

TL;DR

This paper proposes a new displacement control method for oracle objectives with epigraphic information, achieving optimal displacement estimates without extrinsic assumptions.

Contribution

It introduces a natural displacement control based on epigraphic information, replacing traditional local uniform value control, and achieves classical displacement estimates under set-based quadratic growth.

Findings

Achieves classical square-root displacement estimate with optimal exponent 1/2.

Works under set-based quadratic growth allowing nonunique minimizers.

Does not require extrinsic assumptions for displacement control.

Abstract

We introduce a natural displacement control for minimizer sets of oracle objectives equipped with certified epigraphic information. Formally, we replace the usual local uniform value control of objective perturbations - uncertifiable from finite pointwise information without additional structure - by the strictly weaker requirement of a cylinder-localized vertical epigraphic control, naturally provided by certified envelopes. Under set-based quadratic growth (allowing nonunique minimizers), this yields the classical square-root displacement estimate with optimal exponent 1/2, without any extrinsic assumption.