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, $\sqrt{d_N/n}+d_{DS}/n$ agnostic [HMZ24, CEH+26, Pab26]) and open in between. We close the gap: at every fixed oracle risk $L^{\star }$, the optimal excess risk is $\tilde{\Theta }(\sqrt{L^{\star }{d}_N/n}+d_{DS}/n)$, uniformly in the alphabet size, attained by a learner that knows neither $L^{\star }$ 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 $L(h)+O(\sqrt{L(h)\Gamma }+\Gamma )$ with $\Gamma =(k\log n+\log (1/\delta ))/n$, 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 $L^{\star }$, by a pair-Assouad scheme calibrated to $L^{\star }$ 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.

The main theorems are machine-checked in Lean 4; Section D records what is verified and in which form, and the development is available at https://github.com/xiaoyulics/multiclass-pac-learning