Asymmetric Likelihood-Free Hypothesis Testing

In likelihood-free hypothesis testing (LFHT), given samples from two unknown distributions $P,Q$ on $[n]$ with ${\mathrm{d}}_{\mathrm{T}\mathrm{V}}(P,Q)\ge \epsilon$, and from a third distribution $D\in \{P,Q\}$, the goal is to identify whether $D=P$ or $D=Q$. We study asymmetric likelihood-free hypothesis testing, where the sample budgets $s_P$ and $s_Q$ for $P$ and $Q$ may differ, and characterize the feasible triples $(s_P,s_Q,s_D)$ up to logarithmic factors. We establish that unequal reference budgets can give a polynomial saving. For instance, for constant separation, $n$ samples from $P$ and $n^{3\mathrm{/}4}$ from $Q$ allow testing with $\tilde{O}(n^{1\mathrm{/}4})$ samples from $D$, whereas both existing baselines—symmetric LFHT and unequal-sample closeness testing—require $\mathrm{\Omega }(\sqrt{n})$ samples from $D$. We further apply our results to non-adaptive distribution 2-clustering, where the sample allocation among $k$ distributions must be fixed before observing any data. We obtain matching upper and lower bounds for this problem up to logarithmic factors in terms of the domain size, number of distributions, smaller cluster size, and separation, both with one cluster distribution known and with both unknown. Comparing with known bounds for adaptive sample allocation gives a worst-case adaptivity gap of $\tilde{\mathrm{\Theta }}(k^{1\mathrm{/}4})$.