Assouad's lemma

Assouad's lemma lower-bounds minimax risk by embedding a Hamming hypercube of separated parameters whose neighboring statistical experiments are difficult to distinguish.

An Assouad hypercube is a family $(P_{\omega },{\theta }_{\omega }{)}_{\omega \in \{0,1{\}}^k}$ in which loss separates proportionally to Hamming distance while distributions at neighboring vertices remain close in total variation.