We show that there are monotone data sets that cannot be interpolated by a monotone network of depth 2. On the other hand, we prove that for every monotone data set with n points in Rd, there exists an interpolating monotone network of depth 4 and size O(nd). Our interpolation result implies that every monotone function over [0,1]d can be approximated arbitrarily well by a depth4 monotone network, improving the previous best-known construction of depth d+1.

Backdoor attack is a common threat to deep neural networks.

Ensemble learning is a popular technique for enhancing the performance of machine learning algorithms. It is used to capture a complex model space with simple hypotheses which are often significantly easier to learn, or to increase the accuracy of an otherwise unstable procedure [see 14,27,29,andreferencestherein].