Properly learning monotone functions via local reconstruction
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We give a 2^Õ(√(n)/ε)-time algorithm for properly learning monotone Boolean functions under the uniform distribution over {0,1}^n. Our algorithm is robust to adversarial label noise and has a running time nearly matching that of the state-of-the-art improper learning algorithm of Bshouty and Tamon [BT96] and an information-theoretic lower bound of [BCO+15]. Prior to this work, no proper learning algorithm with running time smaller than 2^Ω(n) was known to exist. The core of our proper learner is a local computation algorithm for sorting binary labels on a poset. Our algorithm is built on a body of work on distributed greedy graph algorithms; specifically we rely on a recent work of Ghaffari and Uitto [GU19], which gives an efficient algorithm for computing maximal matchings in a graph in the LCA model of [RTVX11, ARVX11]. The applications of our local sorting algorithm extend beyond learning on the Boolean cube: we also give a tolerant tester for Boolean functions over general posets that distinguishes functions that are ε/3-close to monotone from those that are ε-far. Previous tolerant testers for the Boolean cube only distinguished between ε/Ω(√(n))-close and ε-far.
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