Optimal Lower Bounds and New Upper Bounds for Sequential Prediction with Abstention

We study the problem of sequential prediction with abstentions, where a learner faces a stream of i.i.d. data interspersed with adversarial examples, a setting introduced by \cite{goel2024adversarialresiliencesequentialprediction}. The learner can abstain, incurring no penalty on adversarial points, but is penalized for mistakes and for abstaining on i.i.d. points. Prior work left open whether a fundamental gap exists between the known and unknown distribution settings, and their positive results for the unknown case were restricted to simple classes whose structure they heavily exploited. We resolve both of these questions. First, we establish an $\Omega(\sqrt{T})$ lower bound on the error for any learner facing a VC-dimension 1 class when the distribution is unknown, proving the existing algorithms are optimal and demonstrating a quantitative separation from the logarithmic error achievable when the distribution is known. Second, we provide the first sublinear error algorithm for a more complex geometric class, achieving an $\tilde{O}(T^{2/3})$ error bound for biased half-spaces in $\mathbb{R}^2$.