Beyond Talagrand functions: new lower bounds for testing monotonicity and unateness

Abstract

We prove a lower bound ofΩ(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 → {0, 1} is monotone or far from monotone. This improves the recent bound ofΩ(n 1/4 ) for the same problem by Belovs and Blais [BB16]. 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 ofΩ(n 2/3 ) for any two-sided and adaptive algorithm, and a lower bound ofΩ(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 [KS16] and by Chakrabarty and Seshadhri [CS16].

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“…Subsequent to the initial publication of this work [4], the problem of testing unateness of Boolean functions received significant attention. Concurrent with our work, Chen et al [29] proved a lower bound of Ω d 2/3 /log 3 d for adaptive unateness testers of Boolean functions over {0, 1} d . Subsequently, Chen et al [30] gave an adaptive unateness tester with query complexity O d 3/4 /ε 2 for the same class of functions.…”

“…Testing various properties of functions, including monotonicity (see, e. g., [38,52,33,34,47,36,53,35,41,1,42,5,16,12,19,22,17,11,23,24,21,27,26,45,8,9,32,49,29,7,13,25,14,50] and recent surveys [54,55,20]), the Lipschitz property [43,22,31,17,2], bounded-derivative properties [21], linearity [18,6,10,44,56], submodularity [51,58,60,…”

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“…As we will see, it carries any lower bound for testing monotonicity over to testing k-monotonicity, while preserving the characteristics (two-sidedness, adaptivity) of the original lower bound. In particular, combining it with the recent of [19,21], we obtain the following corollary.…”

“…All these lower bounds applying to non-adaptive testers, they only imply an Ω(log d) lower bound for adaptive ones. Recently, Belovs and Blais [7] showed an Ω(d 1/4 ) lower bound for two-sided adaptive testers, i. e., an exponential improvement over the previous bounds, further improved by Chen, Waingarten, and Xie [21] to Ω(d 1/3 ). All mentioned lower bounds hold for constant ε > 0, and are summarized in Table 3.…”

Testing unateness nearly optimally

“…The following conditions on the function f are satisfied by the hard functions in[CWX17a] used for proving thẽ Ω(n 2/3 ) lower bound.…”