Characterization of favourable ERM selection rules

Characterize which pre-declared consistency-type ERM selection rules for thresholds on an unknown total order permit deterministic learning with both logarithmic numbers of oracle calls and logarithmic numbers of mistakes.

Background

The paper shows that deterministic complexity depends strongly on the oracle’s selection rule. Extremal rules force linear total cost, the feasible-median rule permits logarithmic calls and mistakes, and the global-median rule has an intermediate linear lower bound. The sufficient one-pin common-value property does not characterize all such behaviours, so a complete criterion or quantitative helpfulness parameter remains unresolved.

References

The general question, for arbitrary classes and in the non-transductive setting, remains open, as do the following. Characterize favourable selection rules. Determine which pre-declared ERM rules permit deterministic logarithmic-query, logarithmic-mistake learning.

— An Exponential Deterministic--Randomized Gap in ERM-Oracle Complexity for Thresholds on an Unknown Order  (2609.10196 - Li, 9 Sep 2026) in Section 5, Discussion and open problems