Deep neural nets with fixed bias configuration

For any given neural network architecture a permutation of weights and biases results in the same functional network. This implies that optimization algorithms used to 'train' or 'learn' the network are faced with a very large number (in the millions even for small networks) of equivalent optimal solutions in the parameter space. To the best of our knowledge, this observation is absent in the literature. In order to narrow down the parameter search space, a novel technique is introduced in order to fix the bias vector configurations to be monotonically increasing. This is achieved by augmenting a typical learning problem with inequality constraints on the bias vectors in each layer. A Moreau-Yosida regularization based algorithm is proposed to handle these inequality constraints and a theoretical convergence of this algorithm is established. Applications of the proposed approach to standard trigonometric functions and more challenging stiff ordinary differential equations arising in chemically reacting flows clearly illustrate the benefits of the proposed approach. Further application of the approach on the MNIST dataset within TensorFlow, illustrate that the presented approach can be incorporated in any of the existing machine learning libraries.