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The design of a neural network is usually carried out by defining the numberof layers, the number of neurons per layer, their connections or synapses, andthe activation function that they will execute. The training process tries tooptimize the weights assigned to those connections, together with the biases ofthe neurons, to better fit the training data. However, the definition of theactivation functions is, in general, determined in the design process and notmodified during the training, meaning that their behavior is unrelated to thetraining data set. In this paper we propose the definition and utilization ofan implicit, parametric, non-linear activation function that adapts its shapeduring the training process. This fact increases the space of parameters tooptimize within the network, but it allows a greater flexibility andgeneralizes the concept of neural networks. Furthermore, it simplifies thearchitectural design since the same activation function definition can beemployed in each neuron, letting the training process to optimize theirparameters and, thus, their behavior. Our proposed activation function comesfrom the definition of the consensus variable from the optimization of a linearunderdetermined problem with an $L_p^q$ regularization term, via theAlternating Direction Method of Multipliers (ADMM). We define the neuralnetworks using this type of activation functions as $pq-$networks. Preliminaryresults show that the use of these neural networks with this type of adaptiveactivation functions reduces the error in regression and classificationexamples, compared to equivalent regular feedforward neural networks with fixedactivation functions.

Consensus Function from an $L_p^q-$norm Regularization Term for its Use as Adaptive Activation Functions in Neural Networks