Intrinsic perturbation scale for certified oracle objectives with epigraphic information
This provides a theoretical foundation for certified optimization methods in machine learning, though it appears incremental as it refines existing perturbation analysis.
The paper tackles the problem of controlling minimizer displacement for oracle objectives with certified epigraphic information by replacing local uniform value control with cylinder-localized vertical epigraphic control, yielding a square-root displacement estimate with optimal exponent 1/2 under set-based quadratic growth.
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.