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.

Monotone functions and data sets arise in a variety of applications. We study the interpolation problem for monotone data sets: The input is a monotone data set with $n$ points, and the goal is to find a size and depth efficient monotone neural network with \emph{non negative parameters} and threshold units that interpolates the data set. 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 $\mathbb{R}^d$, 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 depth-4 monotone network, improving the previous best-known construction of depth $d+1$. Finally, building on results from Boolean circuit complexity, we show that the inductive bias of having positive parameters can lead to a super-polynomial blow-up in the number of neurons when approximating monotone functions.

Partially monotone regression is a regression analysis in which the target values are monotonically increasing with respect to a subset of input features. The TensorFlow Lattice library is one of the standard machine learning libraries for partially monotone regression. It consists of several neural network layers, and its core component is the lattice layer. One of the problems of the lattice layer is that it requires the projected gradient descent algorithm with many constraints to train it. Another problem is that it cannot receive a high-dimensional input vector due to the memory consumption. We propose a novel neural network layer, the hierarchical lattice layer (HLL), as an extension of the lattice layer so that we can use a standard stochastic gradient descent algorithm to train HLL while satisfying monotonicity constraints and so that it can receive a high-dimensional input vector. Our experiments demonstrate that HLL did not sacrifice its prediction performance on real datasets compared with the lattice layer.

Monotone functions and data sets arise in a variety of applications. We study the interpolation problem for monotone data sets: The input is a monotone data set with n points, and the goal is to find a size and depth efficient monotone neural network with non negative parameters and threshold units that interpolates the data set. 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 R

Monotone functions and data sets arise in a variety of applications. We study the interpolation problem for monotone data sets: The input is a monotone data set with n points, and the goal is to find a size and depth efficient monotone neural network with non negative parameters and threshold units that interpolates the data set. 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 R

Neural networks with piecewise linear activation functions, such as rectified linear units (ReLU) or maxout, are among the most fundamental models in modern machine learning. We make a step towards proving lower bounds on the size of such neural networks by linking their representative capabilities to the notion of the extension complexity $\mathrm{xc}(P)$ of a polytope $P$, a well-studied quantity in combinatorial optimization and polyhedral geometry. To this end, we propose the notion of virtual extension complexity $\mathrm{vxc}(P)=\min\{\mathrm{xc}(Q)+\mathrm{xc}(R)\mid P+Q=R\}$. This generalizes $\mathrm{xc}(P)$ and describes the number of inequalities needed to represent the linear optimization problem over $P$ as a difference of two linear programs. We prove that $\mathrm{vxc}(P)$ is a lower bound on the size of a neural network that optimizes over $P$. While it remains open to derive strong lower bounds on virtual extension complexity, we show that powerful results on the ordinary extension complexity can be converted into lower bounds for monotone neural networks, that is, neural networks with only nonnegative weights. Furthermore, we show that one can efficiently optimize over a polytope $P$ using a small virtual extended formulation. We therefore believe that virtual extension complexity deserves to be studied independently from neural networks, just like the ordinary extension complexity. As a first step in this direction, we derive an example showing that extension complexity can go down under Minkowski sum.

Partially monotone regression is a regression analysis in which the target values are monotonically increasing with respect to a subset of input features. The TensorFlow Lattice library is one of the standard machine learning libraries for partially monotone regression. It consists of several neural network layers, and its core component is the lattice layer. One of the problems of the lattice layer is that it requires the projected gradient descent algorithm with many constraints to train it. Another problem is that it cannot receive a high-dimensional input vector due to the memory consumption. We propose a novel neural network layer, the hierarchical lattice layer (HLL), as an extension of the lattice layer so that we can use a standard stochastic gradient descent algorithm to train HLL while satisfying monotonicity constraints and so that it can receive a high-dimensional input vector.

Monotone functions and data sets arise in a variety of applications. We study the interpolation problem for monotone data sets: The input is a monotone data set with n points, and the goal is to find a size and depth efficient monotone neural network with \emph{non negative parameters} and threshold units that interpolates the data set. 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 \mathbb{R} d, 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 depth-4 monotone network, improving the previous best-known construction of depth d 1 .

This article introduces a novel approach to learning monotone neural networks through a newly defined penalization loss. The proposed method is particularly effective in solving classes of variational problems, specifically monotone inclusion problems, commonly encountered in image processing tasks. The Forward-Backward-Forward (FBF) algorithm is employed to address these problems, offering a solution even when the Lipschitz constant of the neural network is unknown. Notably, the FBF algorithm provides convergence guarantees under the condition that the learned operator is monotone. Building on plug-and-play methodologies, our objective is to apply these newly learned operators to solving non-linear inverse problems. To achieve this, we initially formulate the problem as a variational inclusion problem. Subsequently, we train a monotone neural network to approximate an operator that may not inherently be monotone. Leveraging the FBF algorithm, we then show simulation examples where the non-linear inverse problem is successfully solved.