[Paper Review] Bounds for Rademacher Processes via Chaining

Johannes Lederer|arXiv (Cornell University)|Oct 27, 2010

Bayesian Methods and Mixture Models9 references3 citations

TL;DR

This paper develops sharp upper bounds for Rademacher processes in high-dimensional settings using chaining techniques, specifically classical entropy bounds and Majorizing Measures. It shows that under strong correlation among covariates, bounds become independent of dimension 𝑙, significantly improving performance in high-dimensional asymptotic regimes where 𝑙 ≫ 𝑛.

ABSTRACT

We study Rademacher processes where the coefficients are functions evaluated at fixed, but arbitrary covariables. Specifically, we assume the function class under consideration to be parametrized by the standard cocube in l dimensions and we are mainly interested in the high-dimensional, asymptotic situation, that is, l as well the number of Rademacher variables n go to infinity with l much larger than n. We refine and apply classical entropy bounds and Majorizing Measures, both going back to the well known idea of chaining. That way, we derive general upper bounds for Rademacher processes. In the linear case and under high correlations, we further improve on these bounds. In particular, we give bounds independent of l for highly correlated covariables.

Motivation & Objective

• To derive tight upper bounds for Rademacher processes in high-dimensional, asymptotic regimes where 𝑙 ≫ 𝑛.
• To refine classical entropy bounds and Majorizing Measures using chaining for function classes parametrized by the 𝑙-dimensional standard cocube.
• To investigate how strong correlations among covariates affect the growth of Rademacher complexity and whether dimension-independent bounds are achievable.
• To demonstrate that Majorizing Measures yield substantially better bounds than classical entropy methods under high correlation.

Proposed method

• Chaining is applied by decomposing the supremum of the Rademacher process into a sum of increments over a sequence of approximating sets.
• Classical entropy bounds are refined using integral forms of covering numbers and logarithmic entropy integrals.
• Majorizing Measures are employed via the construction of a Gaussian process on a metric space with a metric derived from the pseudometric d(𝜃,𝜃′) = ‖(𝜙𝜃(𝑥𝑖)−𝜙𝜃′(𝑥𝑖))‖₂.
• A contraction property is assumed: d(𝜃,𝜃′) ≤ √𝑛𝐴(𝑥)‖𝜃−𝜃′‖₂, which allows bounding the expected supremum via √𝑛 log(𝑙+1)𝐴(𝑥)𝑀.
• For the linear case, the log(𝑛+1) factor is removed, improving the bound to K√𝑛 log(𝑙+1)𝐴(𝑥)𝑀.
• Under strong correlation, the process is reparameterized using orthogonal matrices and a symmetric convex hull of column vectors, leading to dimension-independent bounds via ellipsoidal covering and metric entropy integration.

Experimental results

Research questions

• RQ1 Can classical entropy bounds be refined to yield tighter control of Rademacher processes in high-dimensional settings?
• RQ2 How do strong correlations among covariates affect the growth of Rademacher complexity in high-dimensional asymptotics?
• RQ3 Can Majorizing Measures outperform classical entropy bounds in the context of ℓ¹-penalized function classes?
• RQ4 Under what conditions can Rademacher process bounds be made independent of the dimension 𝑙?
• RQ5 Is it possible to achieve dimension-independent bounds in the linear model when the design matrix columns are highly correlated?

Key findings

• For general function classes with a contraction property, the bound is 𝔼[sup|𝑋𝜃|] ≤ 𝔼|𝑋𝜃₀| + K√𝑛 log(𝑙+1) log(𝑛+1)𝐴(𝑥)𝑀, with a log(𝑛+1) factor.
• In the linear case, the log(𝑛+1) factor can be removed, yielding the improved bound 𝔼[sup|𝑋𝜃|] ≤ K√𝑛 log(𝑙+1)𝐴(𝑥)𝑀.
• Under strong correlation, the bound becomes independent of 𝑙, with the key result being that 𝔼[sup|𝑋𝜃|] ≤ K𝛾₁(S,d₂) for a set S with bounded diameter.
• The use of Majorizing Measures leads to dimension-independent bounds when the columns of the design matrix are collectively enveloped in a small ellipsoid.
• The proof relies on constructing a maximal 𝑀-separated set in an ellipsoid and applying Hoeffding’s inequality and metric entropy integration.
• The final bound depends only on the metric entropy of the ellipsoid and the contraction property, not on the ambient dimension 𝑙.