Randomization Is Necessary for Learning from Symmetric Equivalence Queries

We resolve the dimension-only derandomization question for general symmetric equivalence queries posed in Learning from Equivalence Queries, Revisited at COLT 2026. Prior work gave a randomized proper learner with expected query complexity $O(d)$ for every finite class of Littlestone dimension $d$. We show that deterministic proper learners cannot match this dimension-only guarantee, even with a fixed target. Under target reselection, symmetric two-point generators can force deterministic complexity linear in the number of concepts even at Littlestone dimension one, while randomized learning uses fewer than three expected queries. For independently resampled distributions over total orders, the extremal deterministic complexity on the singleton class is $\Theta (\sqrt{n})$. A uniform mixture of cyclic rotations has an exact value asymptotic to $\sqrt{\pi n/2}$. A row lower-envelope inequality for the linear-order polytope yields the matching universal $O(\sqrt{n})$ upper bound. For one fixed, known total order, a lexicographic projection of the standard optimal algorithm gives a version-consistent proper learner using at most $\mathrm{Ldim}(\mathcal{H})+1$ queries. Complete classes attain this bound. With a fixed target, arbitrary symmetric generators have a tight worst-case $\Theta (\log n)$ hierarchy on singleton classes. Under the same fixed-target protocol, every finite class admits a generator-dependent $2{\log }_2|\mathcal{H}|+O(1)$ deterministic upper bound. Randomized learning remains $O(\mathrm{Ldim})$ throughout. Dimension alone therefore cannot characterize deterministic EQ learning: the target protocol and feedback structure determine the cost of determinism.