论文标题
具有可训练激活和受控Lipschitz常数的深神经网络
Deep Neural Networks with Trainable Activations and Controlled Lipschitz Constant
论文作者
论文摘要
我们引入了一个差异框架，以了解深神经网络的激活功能。我们的目的是增加网络的容量，同时控制输入输出关系的实际Lipschitz常数的上限。为此，我们首先为Lipschitz的神经网络常数建立了全球界限。基于获得的界限，我们为学习激活功能提出了一个变分问题。我们的变异问题是无限维度，并且在计算上不可行。但是，我们证明始终存在具有连续和分段线性（线性调节）激活的解决方案。这将原始问题减少到有限维度最小化，其中激活参数的L1惩罚有利于学习稀疏非线性的学习。我们将我们的方案与标准Relu网络及其变化（PRELU和LeakyRelu）进行比较，并从经验上证明了框架的实际方面。
We introduce a variational framework to learn the activation functions of deep neural networks. Our aim is to increase the capacity of the network while controlling an upper-bound of the actual Lipschitz constant of the input-output relation. To that end, we first establish a global bound for the Lipschitz constant of neural networks. Based on the obtained bound, we then formulate a variational problem for learning activation functions. Our variational problem is infinite-dimensional and is not computationally tractable. However, we prove that there always exists a solution that has continuous and piecewise-linear (linear-spline) activations. This reduces the original problem to a finite-dimensional minimization where an l1 penalty on the parameters of the activations favors the learning of sparse nonlinearities. We numerically compare our scheme with standard ReLU network and its variations, PReLU and LeakyReLU and we empirically demonstrate the practical aspects of our framework.
