Adversarial Online Classification with a Preview

We consider binary prediction with i.i.d. contexts from an unknown distribution and adaptively chosen losses. We show that a simple Follow-the-Perturbed-Leader algorithm using a Gaussian perturbation for each observed context achieves $\widetilde O(\sqrt{T\log N})$ regret for a class of $N$ experts, while requiring one optimization-oracle call per round and no explicit enumeration of the class. For an infinite hypothesis class $\mathcal H$, the same algorithm achieves $\widetilde O(\sqrt{T\operatorname{VC}(\mathcal H)})$ regret. This resolves an open problem posed by Lazaric and Munos (2012), showing that hybrid classification is computationally as easy as statistical learning. As an application, we reduce the problem of contextual bandits with $K$ actions to classification through uniform exploration, achieving $\widetilde O(K^{2/3}T^{2/3}(\log N)^{1/3})$ regret. This matches the best known dependence on the horizon while removing the context-distribution access required by prior oracle-efficient methods.

We study online multiclass multicalibration over $T$ rounds, and give an algorithm with beyond-worst-case guarantees. With $n$ possible outcomes, forecasts lie in a simplex of dimension $d=n-1$. In the worst case, cumulative $\ell_1$ multicalibration error scales with $\widetildeΘ(T^{(d+1)/(d+2)})$, which becomes untenable once $d$ exceeds a small constant. Our algorithm adapts to the geometry of the realized sequence: if the sequence of adversarially selected outcome means lies in an affine subspace of dimension $s$, it obtains error $\widetilde O(n^{1/(s+2)}T^{(s+1)/(s+2)})$ with high probability. The algorithm adapts to this structure without knowing the subspace or its dimension. For $T\ge n\ge2$, this rate is minimax optimal up to polylogarithmic factors. For a stochastic instance with a fixed conditional mean, the bound specializes to $\widetilde O(\sqrt{nT})$, giving a square-root dependence on the horizon.

A monotone adversary observes an i.i.d. labeled sample and appends a finite number of further examples of its choice, every one of them labeled correctly by the target hypothesis. The learner sees a uniform shuffle of the combined sample and is scored on the original distribution. Every example is correctly labeled, but the insertions depend on the clean sample, so the combined sample is not exchangeable. Larsen, Pabbaraju, and Shetty, who introduced this model, showed that empirical risk minimization attains expected error $O((d/n)\log(n/d))$ for classes of VC dimension $d$, and that every known optimal learner can be pushed away from the $Θ(d/n)$ rate, optimal for PAC learning. They asked whether the extra logarithm is an artifact of those particular algorithms or an inherent consequence of the lack of exchangeability. We show that this additional cost is inherent beyond VC dimension one. In the worst case over classes of VC dimension $d$ and over known finite insertion budgets, the minimax expected error is $Θ(1/n)$ at $d=1$ and $Θ((d/n)\log(n/d))$ for $d\geq 2$. The same rates hold with Littlestone dimension $d_{\mathrm L}$ in place of $d$, so the clean online-to-batch rate $O(d_{\mathrm L}/n)$ is unattainable as well. Thus, somewhat counterintuitively, adding correctly labeled examples can make learning harder by a logarithmic factor, even for classes that admit finite mistake bounds in online learning. The dimension-one upper bound is achieved by a simple improper learner whose analysis adapts the leave-one-out argument underlying the one-inclusion graph. All of our lower bounds are elementary and come from a single construction: an explicit class and prior on which two target hypothesis, which differ a point of nonnegligible mass, produce the same sample.