Empirical Bounds on Linear Regions of Deep Rectifier Networks

We can compare the expressiveness of neural networks that use rectified linear units (ReLUs) by the number of linear regions, which reflect the number of pieces of the piecewise linear functions modeled by such networks. However, enumerating these regions is prohibitive and the known analytical bounds are identical for networks with same dimensions. In this work, we approximate the number of linear regions through empirical bounds based on features of the trained network and probabilistic inference. Our first contribution is a method to sample the activation patterns defined by ReLUs using universal hash functions. This method is based on a Mixed-Integer Linear Programming (MILP) formulation of the network and an algorithm for probabilistic lower bounds of MILP solution sets that we call MIPBound, which is considerably faster than exact counting and reaches values in similar orders of magnitude. Our second contribution is a tighter activation-based bound for the maximum number of linear regions, which is particularly stronger in networks with narrow layers. Combined, these bounds yield a fast proxy for the number of linear regions of a deep neural network.Comment: AAAI 202

FReLU: Flexible Rectified Linear Units for Improving Convolutional Neural Networks

Rectified linear unit (ReLU) is a widely used activation function for deep convolutional neural networks. However, because of the zero-hard rectification, ReLU networks miss the benefits from negative values. In this paper, we propose a novel activation function called \emph{flexible rectified linear unit (FReLU)} to further explore the effects of negative values. By redesigning the rectified point of ReLU as a learnable parameter, FReLU expands the states of the activation output. When the network is successfully trained, FReLU tends to converge to a negative value, which improves the expressiveness and thus the performance. Furthermore, FReLU is designed to be simple and effective without exponential functions to maintain low cost computation. For being able to easily used in various network architectures, FReLU does not rely on strict assumptions by self-adaption. We evaluate FReLU on three standard image classification datasets, including CIFAR-10, CIFAR-100, and ImageNet. Experimental results show that the proposed method achieves fast convergence and higher performances on both plain and residual networks

A survey on modern trainable activation functions

In neural networks literature, there is a strong interest in identifying and defining activation functions which can improve neural network performance. In recent years there has been a renovated interest of the scientific community in investigating activation functions which can be trained during the learning process, usually referred to as "trainable", "learnable" or "adaptable" activation functions. They appear to lead to better network performance. Diverse and heterogeneous models of trainable activation function have been proposed in the literature. In this paper, we present a survey of these models. Starting from a discussion on the use of the term "activation function" in literature, we propose a taxonomy of trainable activation functions, highlight common and distinctive proprieties of recent and past models, and discuss main advantages and limitations of this type of approach. We show that many of the proposed approaches are equivalent to adding neuron layers which use fixed (non-trainable) activation functions and some simple local rule that constraints the corresponding weight layers.Comment: Published in "Neural Networks" journal (Elsevier