Tight Stochastic Sequential Covers for General VC Classes with Unknown i.i.d. Features

arXiv:2611.00788v1 Announce Type: new 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 $O(d\log T+\log(1/\delta))$, matching the worst-case $\Omega(d\log(T/d))$ 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 $\Omega(\log^2 T)$ bits for uniform sequential approximation at accuracy $1/T$.