Sharp Bounds for Near-Maximizers of Gaussian Processes

For a centered Gaussian process G with continuous sample paths on a compact metric space T, a continuous drift f: T ↦ R, a noise level σ>0, and ε≥ 0, let A_ε be the set of points at which f+σ G is within ε of its maximum. Writing ETG for the expected supremum of G, we prove that the expected squared diameter of A_ε in the canonical pseudometric of G is at most 8ε ETG/σ and that the expected maximal squared distance from a point of A_ε to an exact maximizer is at most 4ε ETG/σ. The constants cannot be improved uniformly over index sets, processes, and drifts, and no logarithmic factor in ε appears, unlike in bounds integrated from tail estimates. At ε=0 the bounds give the almost sure uniqueness of the maximizer in the canonical pseudometric without nondegeneracy assumptions. Given this uniqueness, both bounds follow from a bound with constant $2$ on the mean square of the largest increment of an independent copy of G from an exact maximizer to a point of A_ε. The proof rests on an identity for the mean squared support function of the ε-subdifferentials of a smooth convex function at a standard Gaussian vector, obtained from two Gaussian integrations by parts.