A New Computationally Simple Approach for Implementing Neural Networks with Output Hard Constraints

A new computationally simple method of imposing hard convex constraints on the neural network output values is proposed. The key idea behind the method is to map a vector of hidden parameters of the network to a point that is guaranteed to be inside the feasible set defined by a set of constraints. The mapping is implemented by the additional neural network layer with constraints for output. The proposed method is simply extended to the case when constraints are imposed not only on the output vectors, but also on joint constraints depending on inputs. The projection approach to imposing constraints on outputs can simply be implemented in the framework of the proposed method. It is shown how to incorporate different types of constraints into the proposed method, including linear and quadratic constraints, equality constraints, and dynamic constraints, constraints in the form of boundaries. An important feature of the method is its computational simplicity. Complexities of the forward pass of the proposed neural network layer by linear and quadratic constraints are O(n*m) and O(n^2*m), respectively, where n is the number of variables, m is the number of constraints. Numerical experiments illustrate the method by solving optimization and classification problems. The code implementing the method is publicly available

Heterogeneous Treatment Effect with Trained Kernels of the Nadaraya-Watson Regression

A new method for estimating the conditional average treatment effect is proposed in the paper. It is called TNW-CATE (the Trainable Nadaraya-Watson regression for CATE) and based on the assumption that the number of controls is rather large whereas the number of treatments is small. TNW-CATE uses the Nadaraya-Watson regression for predicting outcomes of patients from the control and treatment groups. The main idea behind TNW-CATE is to train kernels of the Nadaraya-Watson regression by using a weight sharing neural network of a specific form. The network is trained on controls, and it replaces standard kernels with a set of neural subnetworks with shared parameters such that every subnetwork implements the trainable kernel, but the whole network implements the Nadaraya-Watson estimator. The network memorizes how the feature vectors are located in the feature space. The proposed approach is similar to the transfer learning when domains of source and target data are similar, but tasks are different. Various numerical simulation experiments illustrate TNW-CATE and compare it with the well-known T-learner, S-learner and X-learner for several types of the control and treatment outcome functions. The code of proposed algorithms implementing TNW-CATE is available in https://github.com/Stasychbr/TNW-CATE