Uniform mean estimation via generic chaining
Daniel Bartl, Shahar Mendelson
TL;DR
The paper tackles uniform estimation of the mean $\mathbb{E}\,u(f(X))$ over a rich class $F$ of potentially heavy-tailed functions. It introduces an optimal uniform mean estimator $\Psi$ that combines an optimal univariate mean estimator with Talagrand’s generic chaining, producing a subgaussian-type error bound $\sup_{f\in F} |\Psi - \mathbb{E}\,u(f)| \lesssim R(F) \frac{\mathbb{E} \sup_{f\in F} G_f}{\sqrt{N}}$ under minimal assumptions. Key contributions include the precise assumptions and definitions (isomorphic distance $\mathbbm{\rho}$, $R(F)$, $d_F$, and $D^*(F)$), a complete proof framework, and two major applications: a geometric setting for isotropic log-concave measures and a robust covariance-estimation scenario with adversarial corruption. The results yield optimal uniform mean estimation beyond light-tailed settings and extend to corrupted data, with explicit bounds and practical implications for high-dimensional probability and statistics.
Abstract
We introduce an empirical functional $Ψ$ that is an optimal uniform mean estimator: Let $F\subset L_2(μ)$ be a class of mean zero functions, $u$ is a real valued function, and $X_1,\dots,X_N$ are independent, distributed according to $μ$. We show that under minimal assumptions, with $μ^{\otimes N}$ exponentially high probability, \[ \sup_{f\in F} |Ψ(X_1,\dots,X_N,f) - \mathbb{E} u(f(X))| \leq c R(F) \frac{ \mathbb{E} \sup_{f\in F } |G_f| }{\sqrt N}, \] where $(G_f)_{f\in F}$ is the gaussian processes indexed by $F$ and $R(F)$ is an appropriate notion of `diameter' of the class $\{u(f(X)) : f\in F\}$. The fact that such a bound is possible is surprising, and it leads to the solution of various key problems in high dimensional probability and high dimensional statistics. The construction is based on combining Talagrand's generic chaining mechanism with optimal mean estimation procedures for a single real-valued random variable.