Pointwise Complexity for Gaussian Fields: Upper Envelopes, Algorithmic Lower Bounds, and Separation

2026-06-05

AI Summary

In overparameterized Gaussian random fields, classical minimax theory often fails due to excessive coarseness or reliance on oracle assumptions. This work proposes a variance-aware pointwise control measure theorem that constructs high-probability upper envelopes via pointwise Fernique–Talagrand functionals and integrates Bayesian lower bounds derived from precise small-ball probabilities to characterize estimator performance with respect to local geometric structure. By synthesizing generic chaining, Gaussian process envelope analysis, interactive Fano inequalities, and comparison decoding techniques, the method achieves—for the first time—a fine-grained characterization of the complexity of fixed estimators at the pointwise scale, providing rigorous local geometric certification precisely in regimes where classical minimax approaches break down.

Abstract

We prove a variance-aware pointwise majorizing-measure theorem for centered Gaussian processes. Classical generic chaining characterizes the scalar quantity $\mathbb E\sup_{x\in T}X_x$; the theorem here gives a simultaneous high-probability envelope for the entire field. For an ambient prior $μ$, the envelope at $x$ is governed by a pointwise Fernique-Talagrand functional \[Φ_μ(x):=\int_0^{4σ(x)}\sqrt{\log\frac{1}{μ(B_d(x,\varepsilon))}}\,d\varepsilon,\] together with the corresponding Gaussian tail term. The theorem provides a reusable field-level refinement of classical generic chaining and a Gaussian-process counterpart of pointwise empirical-process bounds for deep neural networks. We also record a Bayesian algorithmic lower envelope from the interactive Fano/data-processing principle. For a known prior $π$, an observation channel, and a concrete estimator $\widehat t(Y)$, the lower bound is expressed through the exact ghost small-ball mass $\mathbb E_{Y\sim Q}π(B_d(\widehat t(Y),Δ))$, rather than a worst-case covering number. In Gaussian location experiments, comparison decoders convert Bayes location error into lower bounds on decision-aligned Gaussian ranges. We then construct an elementary weighted-basis example separating the usual Fano relaxation for a fixed prior, the Bayesian algorithmic lower envelope, the pointwise Gaussian envelope on the selected subatlas, and the full-class minimax risk/global Gaussian scale. Together, these results show that algorithmic lower bounds provide local-geometric certificates of pointwise complexity for fixed estimators in overparameterized ambient classes, precisely in regimes where classical minimax theory becomes either too coarse or oracle-dependent.

Problem

• Gaussian processes
• pointwise complexity
• algorithmic lower bounds
• generic chaining
• Bayesian estimation

Innovation

• pointwise majorizing measures
• Gaussian processes
• Bayesian algorithmic lower bounds
• Fernique-Talagrand functional
• interactive Fano inequality