Brownian Heads for Deep ReLU Representations: Activation Mass and the Cost of Same-Sample Selection
TL;DR
This work studies the conditional empirical Rademacher complexity of deep ReLU representations followed by bounded-norm predictors in additive or L\'evy-Brownian RKHSs, termed Brownian heads, and derives an exact dual identity and sharp bounds in terms of activation mass, the average norm of the observed hidden vectors.
Abstract
Deep representation learning often selects hidden features and fits the final predictor on the same sample, so fixed-feature analysis performed after selection can omit selection cost. We study the conditional empirical Rademacher complexity of deep ReLU representations followed by bounded-norm predictors in additive or L\'evy-Brownian RKHSs, termed Brownian heads. For a fixed representation, we derive an exact dual identity and sharp bounds in terms of activation mass, the average norm of the observed hidden vectors. Under same-sample selection, the representation supremum induces a quadratic Rademacher process. Brownian layer-cake and Gaussian-projection identities reduce it to coordinatewise or signed projected threshold traces, separating realized scale from selection complexity. For samples with pairwise-distinct inputs, explicit scalar ReLU families match the finite-trace and VC rates up to universal constants at the realized trace-and-envelope level. Induced-norm contraction also yields architecture-level bounds for rectangular, rank-deficient ReLU networks. Experiments verify the sharp bounds and rates, exhibit a selection gap at fixed activation mass, and assess the predictive feasibility of Brownian heads.
