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Register a new userDissecting Hessian: Understanding Common Structure of Hessian in Neural Networks
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Xingyu Zhu
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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Hessian captures important properties of the deep neural network loss landscape. Previous works have observed low rank structure in the Hessians of neural networks. We make several new observations about the top eigenspace of layer-wise Hessian: top eigenspaces for different models have surprisingly high overlap, and top eigenvectors form low rank matrices when they are reshaped into the same shape as the corresponding weight matrix. Towards formally explaining such structures of the Hessian, we show that the new eigenspace structure can be explained by approximating the Hessian using Kronecker factorization; we also prove the low rank structure for random data at random initialization for over-parametrized two-layer neural nets. Our new understanding can explain why some of these structures become weaker when the network is trained with batch normalization. The Kronecker factorization also leads to better explicit generalization bounds.
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Regularizing the input gradient has shown to be effective in promoting the robustness of neural networks. The regularization of the inputs Hessian is therefore a natural next step. A key challenge here is the computational complexity. Computing the Hessian of inputs is computationally infeasible. In this paper we propose an efficient algorithm to train deep neural networks with Hessian operator-norm regularization. We analyze the approach theoretically and prove that the Hessian operator norm relates to the ability of a neural network to withstand an adversarial attack. We give a preliminary experimental evaluation on the MNIST and FMNIST datasets, which demonstrates that the new regularizer can, indeed, be feasible and, furthermore, that it increases the robustness of neural networks over input gradient regularization.
Continuous deep learning architectures have recently re-emerged as Neural Ordinary Differential Equations (Neural ODEs). This infinite-depth approach theoretically bridges the gap between deep learning and dynamical systems, offering a novel perspective. However, deciphering the inner working of these models is still an open challenge, as most applications apply them as generic black-box modules. In this work we open the box, further developing the continuous-depth formulation with the aim of clarifying the influence of several design choices on the underlying dynamics.
The rapid development of computer hardware and Internet technology makes large scale data dependent models computationally tractable, and opens a bright avenue for annotating images through innovative machine learning algorithms. Semi-supervised learning (SSL) has consequently received intensive attention in recent years and has been successfully deployed in image annotation. One representative work in SSL is Laplacian regularization (LR), which smoothes the conditional distribution for classification along the manifold encoded in the graph Laplacian, however, it has been observed that LR biases the classification function towards a constant function which possibly results in poor generalization. In addition, LR is developed to handle uniformly distributed data (or single view data), although instances or objects, such as images and videos, are usually represented by multiview features, such as color, shape and texture. In this paper, we present multiview Hessian regularization (mHR) to address the above two problems in LR-based image annotation. In particular, mHR optimally combines multiple Hessian regularizations, each of which is obtained from a particular view of instances, and steers the classification function which varies linearly along the data manifold. We apply mHR to kernel least squares and support vector machines as two examples for image annotation. Extensive experiments on the PASCAL VOC07 dataset validate the effectiveness of mHR by comparing it with baseline algorithms, including LR and HR.
In a recurrent setting, conventional approaches to neural architecture search find and fix a general model for all data samples and time steps. We propose a novel algorithm that can dynamically search for the structure of cells in a recurrent neural network model. Based on a combination of recurrent and recursive neural networks, our algorithm is able to construct customized cell structures for each data sample and time step, allowing for a more efficient architecture search than existing models. Experiments on three common datasets show that the algorithm discovers high-performance cell architectures and achieves better prediction accuracy compared to the GRU structure for language modelling and sentiment analysis.
Quantization is an effective method for reducing memory footprint and inference time of Neural Networks, e.g., for efficient inference in the cloud, especially at the edge. However, ultra low precision quantization could lead to significant degradation in model generalization. A promising method to address this is to perform mixed-precision quantization, where more sensitive layers are kept at higher precision. However, the search space for a mixed-precision quantization is exponential in the number of layers. Recent work has proposed HAWQ, a novel Hessian based framework, with the aim of reducing this exponential search space by using second-order information. While promising, this prior work has three major limitations: (i) HAWQV1 only uses the top Hessian eigenvalue as a measure of sensitivity and do not consider the rest of the Hessian spectrum; (ii) HAWQV1 approach only provides relative sensitivity of different layers and therefore requires a manual selection of the mixed-precision setting; and (iii) HAWQV1 does not consider mixed-precision activation quantization. Here, we present HAWQV2 which addresses these shortcomings. For (i), we perform a theoretical analysis showing that a better sensitivity metric is to compute the average of all of the Hessian eigenvalues. For (ii), we develop a Pareto frontier based method for selecting the exact bit precision of different layers without any manual selection. For (iii), we extend the Hessian analysis to mixed-precision activation quantization. We have found this to be very beneficial for object detection. We show that HAWQV2 achieves new state-of-the-art results for a wide range of tasks.
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