On ADMM in Deep Learning: Convergence and Saturation-Avoidance

Abstract: In this paper, we develop an alternating direction method of multipliers (ADMM) for deep neural networks training with sigmoid-type activation functions (called \textit{sigmoid-ADMM pair}), mainly motivated by the gradient-free nature of ADMM in avoiding the saturation of sigmoid-type activations and the advantages of deep neural networks with sigmoid-type activations (called deep sigmoid nets) over their rectified linear unit (ReLU) counterparts (called deep ReLU nets) in terms of approximation. In particular, we prove that the approximation capability of deep sigmoid nets is not worse than that of deep ReLU nets by showing that ReLU activation function can be well approximated by deep sigmoid nets with two hidden layers and finitely many free parameters but not vice-verse. We also establish the global convergence of the proposed ADMM for the nonlinearly constrained formulation of the deep sigmoid nets training from arbitrary initial points to a Karush-Kuhn-Tucker (KKT) point at a rate of order ${\cal O}(1/k)$. Besides sigmoid activation, such a convergence theorem holds for a general class of smooth activations. Compared with the widely used stochastic gradient descent (SGD) algorithm for the deep ReLU nets training (called ReLU-SGD pair), the proposed sigmoid-ADMM pair is practically stable with respect to the algorithmic hyperparameters including the learning rate, initial schemes and the pro-processing of the input data. Moreover, we find that to approximate and learn simple but important functions the proposed sigmoid-ADMM pair numerically outperforms the ReLU-SGD pair.

Jinshan Zeng
Shao-Bo Lin
Yuan Yao
Ding-Xuan Zhou


Generalisation dynamics of online learning in over-parameterised neural networks

Abstract: Deep neural networks achieve stellar generalisation on a variety of problems, despite often being large enough to easily fit all their training data. Here we study the generalisation dynamics of two-layer neural networks in a teacher-student setup, where one network, the student, is trained using stochastic gradient descent (SGD) on data generated by another network, called the teacher. We show how for this problem, the dynamics of SGD are captured by a set of differential equations. In particular, we demonstrate analytically that the generalisation error of the student increases linearly with the network size, with other relevant parameters held constant. Our results indicate that achieving good generalisation in neural networks depends on the interplay of at least the algorithm, its learning rate, the model architecture, and the data set.

Sebastian Goldt
Madhu S. Advani
Andrew M. Saxe
Florent Krzakala
Lenka Zdeborová


Unsupervised Learning of Neural Networks to Explain Neural Networks (extended abstract)

Abstract: This paper presents an unsupervised method to learn a neural network, namely an explainer, to interpret a pre-trained convolutional neural network (CNN), i.e., the explainer uses interpretable visual concepts to explain features in middle conv-layers of a CNN. Given feature maps of a conv-layer of the CNN, the explainer performs like an auto-encoder, which decomposes the feature maps into object-part features. The object-part features are learned to reconstruct CNN features without much loss of information. We can consider the disentangled representations of object parts a paraphrase of CNN features, which help people understand the knowledge encoded by the CNN. More crucially, we learn the explainer via knowledge distillation without using any annotations of object parts or textures for supervision. In experiments, our method was widely used to interpret features of different benchmark CNNs, and explainers significantly boosted the feature interpretability without hurting the discrimination power of the CNNs.


Hierarchical Attentional Hybrid Neural Networks for Document Classification

Abstract: Document classification is a challenging task with important applications. The deep learning approaches to the problem have gained much attention recently. Despite the progress, the proposed models do not incorporate the knowledge of the document structure in the architecture efficiently and not take into account the contexting importance of words and sentences. In this paper, we propose a new approach based on a combination of convolutional neural networks, gated recurrent units, and attention mechanisms for document classification tasks. The main contribution of this work is the use of convolution layers to extract more meaningful, generalizable and abstract features by the hierarchical representation. The proposed method in this paper improves the results of the current attention-based approaches for document classification.

Jader Abreu
Luis Fred
David Macêdo
Cleber Zanchettin

International Conference on Artificial Neural Networks