Beyond Talagrand Functions: New Lower Bounds for Testing Monotonicity and Unateness

We prove a lower bound of $\tilde{\Omega}(n^{1/3})$ for the query complexity of any two-sided and adaptive algorithm that tests whether an unknown Boolean function $f:{0,1}^n\rightarrow {0,1}$ is monotone or far from monotone. This improves the recent bound of $\tilde{\Omega}(n^{1/4})$ for the same problem by Belovs and Blais [BB15]. Our result builds on a new family of random Boolean functions that can be viewed as a two-level extension of Talagrand's random DNFs. Beyond monotonicity, we also prove a lower bound of $\tilde{\Omega}(n^{2/3})$ for any two-sided and adaptive algorithm, and a lower bound of $\tilde{\Omega}(n)$ for any one-sided and non-adaptive algorithm for testing unateness, a natural generalization of monotonicity. The latter matches the recent linear upper bounds by Khot and Shinkar [KS15] and by Chakrabarty and Seshadhri [CS16].