Ordinal Neural Networks Without Iterative Tuning

Ordinal regression (OR) is an important branch of supervised learning in between the multiclass classification and regression. In this paper, the traditional classification scheme of neural network is adapted to learn ordinal ranks. The model proposed imposes monotonicity constraints on the weights connecting the hidden layer with the output layer. To do so, the weights are transcribed using padding variables. This reformulation leads to the so-called inequality constrained least squares (ICLS) problem. Its numerical solution can be obtained by several iterative methods, for example, trust region or line search algorithms. In this proposal, the optimum is determined analytically according to the closed-form solution of the ICLS problem estimated from the Karush–Kuhn–Tucker conditions. Furthermore, following the guidelines of the extreme learning machine framework, the weights connecting the input and the hidden layers are randomly generated, so the final model estimates all its parameters without iterative tuning. The model proposed achieves competitive performance compared with the state-of-the-art neural networks methods for OR.