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

The design of a neural network is usually carried out by defining the number of layers, the number of neurons per layer, their connections or synapses, and the activation function that they will execute. The training process tries to optimize the weights assigned to those connections, together with the biases of the neurons, to better fit the training data. However, the definition of the activation functions is, in general, determined in the design process and not modified during the training, meaning that their behavior is unrelated to the training data set. In this paper we propose the definition and utilization of an implicit, parametric, non-linear activation function that adapts its shape during the training process. This fact increases the space of parameters to optimize within the network, but it allows a greater flexibility and generalizes the concept of neural networks. Furthermore, it simplifies the architectural design since the same activation function definition can be employed in each neuron, letting the training process to optimize their parameters and, thus, their behavior. Our proposed activation function comes from the definition of the consensus variable from the optimization of a linear underdetermined problem with an $L_p^q$ regularization term, via the Alternating Direction Method of Multipliers (ADMM). We define the neural networks using this type of activation functions as $pq-$networks. Preliminary results show that the use of these neural networks with this type of adaptive activation functions reduces the error in regression and classification examples, compared to equivalent regular feedforward neural networks with fixed activation functions.