Optimistic Rates for Multiclass PAC Learning

Abstract

Worst-case multiclass bounds do not become smaller when the best classifier is already nearly correct: what is missing is an optimistic rate, a guarantee whose fluctuation scales with the oracle risk itself. For a class of Natarajan dimension d_N and Daniely-Shalev-Shwartz dimension d_DS, the optimal excess risk is known at the two endpoints (d_DS/n realizable, √(d_N/n) + d_DS/n agnostic [HMZ24, CEH+26, Pab26]) and open in between. We close the gap: at every fixed oracle risk L*, the optimal excess risk is Θ̃(√(L* d_N/n) + d_DS/n), uniformly in the alphabet size, attained by a learner that knows neither L* nor the confidence level. The upper bound composes the cover-menu-compression architecture of [CEH+26], at the realizable rate of [Pab26], with a new comparator-facing relative compression theorem: a size-k compression rule that empirically dominates a comparator h has population risk at most L(h) + O(√(L(h)Γ) + Γ) with Γ = (k log n + log(1/δ))/n, without stability; this transfers the comparison principle of the sharp binary theory [MQZ26] while discarding its Boolean-cube geometry, which does not lift to multiclass labels. The lower bound forces both terms using one class and one distribution at every fixed L*, by a pair-Assouad scheme calibrated to L* and a fiber argument on the pseudo-cubes underlying the Natarajan-versus-DS separation of [BCD+22]. Both theorems extend to list learning: against the best r-tuple of hypotheses, the same architecture and the same two engines yield an optimistic rate and a lower bound of the same shape, forcing the fluctuation term that [Pab26] expected to be necessary against list comparators, and removing the factor r from the known realizable list lower bound.