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

arXiv: https://arxiv.org/abs/2606.07931 · HTML: https://mlsanity.com/paper/2606.07931/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 48 references that are in the map.
- First posted on arXiv 6 Jun 2026.

## Builds on (most cited first, 10 of 10)

- [The Statistical Complexity of Interactive Decision Making](https://mlsanity.com/paper/2112.13487/) (2021)
- [Bilinear Classes: A Structural Framework for Provable Generalization in RL](https://mlsanity.com/paper/2103.10897/) (2021)
- [Best-Arm Identification in Linear Bandits](https://mlsanity.com/paper/1409.6110/) (2014)
- [Overfitting or perfect fitting? Risk bounds for classification and regression rules that interpolate](https://mlsanity.com/paper/1806.05161/) (2018)
- [Tight Guarantees for Interactive Decision Making with the Decision-Estimation Coefficient](https://mlsanity.com/paper/2301.08215/) (2023)
- [Bayesian Design Principles for Frequentist Sequential Learning](https://mlsanity.com/paper/2310.00806/) (2023)
- [Majorizing Measures, Sequential Complexities, and Online Learning](https://mlsanity.com/paper/2102.01729/) (2021)
- [Towards Optimal Problem Dependent Generalization Error Bounds in Statistical Learning Theory](https://mlsanity.com/paper/2011.06186/) (2020)
- [Assouad, Fano, and Le Cam with Interaction: A Unifying Lower Bound Framework and Characterization for Bandit Learnability](https://mlsanity.com/paper/2410.05117/) (2024)
- [Pointwise Generalization in Deep Neural Networks](https://mlsanity.com/paper/2605.18598/) (2026)

## Built on by (most cited first, 2 of 2)

- [Bellman-sufficient Information Complexity](https://mlsanity.com/paper/2606.11171/) (2026)
- [Simultaneous Pointwise Majorization for Mixed Tail Processes with Applications in Gaussian Chaos and Ergodic Diffusions](https://mlsanity.com/paper/2609.01576/) (2026)

## Abstract (arXiv)

We prove a variance-aware pointwise majorizing-measure theorem for centered Gaussian processes. Classical generic chaining characterizes the scalar quantity mathbb Esupx∈ TXx; 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 [Phiμ(x):=int04σ(x)√logfrac1μ(Bd(x,varepsilon)),dvarepsilon,] 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 EY~ Qπ(Bd(widehat t(Y),Delta)), 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 example separating the usual Fano relaxation, the Bayesian algorithmic lower envelope, the pointwise Gaussian envelope, and the full-class minimax risk. Together, these results show that algorithmic lower bounds provide local-geometric validations of pointwise complexity for fixed estimators in overparameterized ambient classes, precisely in regimes where classical minimax theory becomes either too coarse or oracle-dependent. This separation can also be recast in minimax language as penalty-range information relaxation, highlighting an important question of algorithmic robustness for classical high-dimensional models and regularized algorithms.