Tight Stochastic Sequential Covers for General VC Classes with Unknown i.i.d. Features
Changlong Wu, Wojciech Szpankowski
Abstract
We study hybrid online learning with stochastic features and adversarial labels when the feature distribution is unknown. Prior work gave stochastic sequential covers of log-size $O(d\log^2 T)$ for a general VC class and logarithmic bounds only under stronger star-type structure. We show that finite VC dimension alone is enough: every binary class of VC dimension $d$ admits a distribution-blind exact cover with log-size $O(d\log T+\log(1/δ))$, matching the worst-case $Ω(d\log(T/d))$ dependence on the horizon. The key ingredient is a new local cube weight that counts Boolean subcubes incident to each realizable trace. Unlike the equal splits used in earlier constructions, we split mass unevenly according to these cube weights; a coordinate-deletion inequality shows that VC dimension controls the total cost of these splits. This yields, to our knowledge, the first tight $O((d+1)\log T)$ expected worst-case log-loss regret for general VC partitions with unknown i.i.d. features, and an exact polynomial-time implementation for planar affine halfspaces. We also show that the binary phenomenon does not extend from VC dimension to pseudo-dimension: a fixed pseudo-dimension-one sigmoid class requires $Ω(\log^2 T)$ bits for uniform sequential approximation at accuracy $1/T$.