Adversarial Online Classification with a Preview

Abstract

Worst-case online classification is governed by sequential complexity, such as Littlestone dimension, and can be impossible even for statistically simple classes, such as thresholds of VC dimension one. We study a preview model in which an oblivious adversary fixes an entire labeled sequence of length $T$, a uniformly random subset of size $pT$ is revealed before prediction begins, and the remaining $(1-p)T$ examples are then presented in their original adversarial order. Against the best full-sequence hypothesis evaluated on the unrevealed examples, we characterize the dependence on the preview rate $p$: for binary classes of VC dimension $d$, the optimal excess loss is $\Theta (d/p+\sqrt{dT})$, up to the trivial cap at $T$; for multiclass classes we obtain the corresponding $\tilde{O}(d_{\mathrm{DS}}/p+\sqrt{d_{\mathrm{Nat}}T})$ bound with no dependence on the number of labels. Thus a random preview can replace worst-case sequential complexity by classical statistical dimensions without randomizing the online order. To achieve the sharp binary bound, our ChainedPrediction algorithm uses an online analogue of chaining, implemented as a multiscale aggregation algorithm rather than only as an analytic argument.