We study monotonicity testing of functions $f \colon \{0,1\}^d \to \{0,1\}$ using sample-based algorithms, which are only allowed to observe the value of $f$ on points drawn independently from the uniform distribution. A classic result by Bshouty--Tamon (J. ACM 1996) proved that monotone functions can be learned with error at most $\varepsilon \in (0,1)$ using $\exp(\widetilde{O}(\min\{\frac{1}{\varepsilon}\sqrt{d},d\}))$ samples and it is not hard to show that this bound extends to testing. Prior to our work the only lower bound for this problem was $\Omega(\sqrt{\exp(d)/\varepsilon})$ in the small $\varepsilon$ parameter regime, when $\varepsilon = O(d^{-3/2})$, due to Goldreich--Goldwasser--Lehman--Ron--Samorodnitsky (Combinatorica 2000). Thus, the sample complexity of monotonicity testing was wide open when $\varepsilon$ is much greater than $d^{-3/2}$. We resolve this question, obtaining a nearly tight lower bound of $\exp(\Omega(\min\{\frac{1}{\varepsilon}\sqrt{d},d\}))$ for all $\varepsilon$ at most a sufficiently small constant. In fact, we prove a more general result, showing that the sample complexity of $k$-monotonicity testing and learning for functions $f \colon \{0,1\}^d \to [r]$ is $\exp(\Omega(\min\{\frac{rk}{\varepsilon}\sqrt{d},d\}))$. For testing with one-sided error we show that the sample complexity is $\exp(\Omega(d))$.

Beyond the hypercube, we prove nearly tight bounds (up to polylog factors of $d,k,r,1/\varepsilon$ in the exponent) of $\exp(\widetilde{\Theta}(\min\{\frac{rk}{\varepsilon}\sqrt{d},d\}))$ on the sample complexity of testing and learning measurable $k$-monotone functions $f \colon \mathbb{R}^d \to [r]$ under product distributions. Our upper bound improves upon the previous bound of $\exp(\widetilde{O}(\min\{\frac{k}{\varepsilon^2}\sqrt{d},d\}))$ by Harms--Yoshida (ICALP'22) for Boolean functions ($r=2$).

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A conference version of this paper appeared in the Proceedings of the 28th Internat. Conference on Randomization and Computation (RANDOM'24).