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Register a new userDeep Neural Networks Are Congestion Games: From Loss Landscape to Wardrop Equilibrium and Beyond
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Ievgen Redko
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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The theoretical analysis of deep neural networks (DNN) is arguably among the most challenging research directions in machine learning (ML) right now, as it requires from scientists to lay novel statistical learning foundations to explain their behaviour in practice. While some success has been achieved recently in this endeavour, the question on whether DNNs can be analyzed using the tools from other scientific fields outside the ML community has not received the attention it may well have deserved. In this paper, we explore the interplay between DNNs and game theory (GT), and show how one can benefit from the classic readily available results from the latter when analyzing the former. In particular, we consider the widely studied class of congestion games, and illustrate their intrinsic relatedness to both linear and non-linear DNNs and to the properties of their loss surface. Beyond retrieving the state-of-the-art results from the literature, we argue that our work provides a very promising novel tool for analyzing the DNNs and support this claim by proposing concrete open problems that can advance significantly our understanding of DNNs when solved.
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Understanding the structure of loss landscape of deep neural networks (DNNs)is obviously important. In this work, we prove an embedding principle that the loss landscape of a DNN contains all the critical points of all the narrower DNNs. More precisely, we propose a critical embedding such that any critical point, e.g., local or global minima, of a narrower DNN can be embedded to a critical point/hyperplane of the target DNN with higher degeneracy and preserving the DNN output function. The embedding structure of critical points is independent of loss function and training data, showing a stark difference from other nonconvex problems such as protein-folding. Empirically, we find that a wide DNN is often attracted by highly-degenerate critical points that are embedded from narrow DNNs. The embedding principle provides an explanation for the general easy optimization of wide DNNs and unravels a potential implicit low-complexity regularization during the training. Overall, our work provides a skeleton for the study of loss landscape of DNNs and its implication, by which a more exact and comprehensive understanding can be anticipated in the near
Miscalibration - a mismatch between a models confidence and its correctness - of Deep Neural Networks (DNNs) makes their predictions hard to rely on. Ideally, we want networks to be accurate, calibrated and confident. We show that, as opposed to the standard cross-entropy loss, focal loss [Lin et. al., 2017] allows us to learn models that are already very well calibrated. When combined with temperature scaling, whilst preserving accuracy, it yields state-of-the-art calibrated models. We provide a thorough analysis of the factors causing miscalibration, and use the insights we glean from this to justify the empirically excellent performance of focal loss. To facilitate the use of focal loss in practice, we also provide a principled approach to automatically select the hyperparameter involved in the loss function. We perform extensive experiments on a variety of computer vision and NLP datasets, and with a wide variety of network architectures, and show that our approach achieves state-of-the-art calibration without compromising on accuracy in almost all cases. Code is available at https://github.com/torrvision/focal_calibration.
Deep neural networks often consist of a great number of trainable parameters for extracting powerful features from given datasets. On one hand, massive trainable parameters significantly enhance the performance of these deep networks. On the other hand, they bring the problem of over-fitting. To this end, dropout based methods disable some elements in the output feature maps during the training phase for reducing the co-adaptation of neurons. Although the generalization ability of the resulting models can be enhanced by these approaches, the conventional binary dropout is not the optimal solution. Therefore, we investigate the empirical Rademacher complexity related to intermediate layers of deep neural networks and propose a feature distortion method (Disout) for addressing the aforementioned problem. In the training period, randomly selected elements in the feature maps will be replaced with specific values by exploiting the generalization error bound. The superiority of the proposed feature map distortion for producing deep neural network with higher testing performance is analyzed and demonstrated on several benchmark image datasets.
We study how permutation symmetries in overparameterized multi-layer neural networks generate `symmetry-induced critical points. Assuming a network with $ L $ layers of minimal widths $ r_1^*, ldots, r_{L-1}^* $ reaches a zero-loss minimum at $ r_1^*! cdots r_{L-1}^*! $ isolated points that are permutations of one another, we show that adding one extra neuron to each layer is sufficient to connect all these previously discrete minima into a single manifold. For a two-layer overparameterized network of width $ r^*+ h =: m $ we explicitly describe the manifold of global minima: it consists of $ T(r^*, m) $ affine subspaces of dimension at least $ h $ that are connected to one another. For a network of width $m$, we identify the number $G(r,m)$ of affine subspaces containing only symmetry-induced critical points that are related to the critical points of a smaller network of width $r<r^*$. Via a combinatorial analysis, we derive closed-form formulas for $ T $ and $ G $ and show that the number of symmetry-induced critical subspaces dominates the number of affine subspaces forming the global minima manifold in the mildly overparameterized regime (small $ h $) and vice versa in the vastly overparameterized regime ($h gg r^*$). Our results provide new insights into the minimization of the non-convex loss function of overparameterized neural networks.
Non-recurring traffic congestion is caused by temporary disruptions, such as accidents, sports games, adverse weather, etc. We use data related to real-time traffic speed, jam factors (a traffic congestion indicator), and events collected over a year from Nashville, TN to train a multi-layered deep neural network. The traffic dataset contains over 900 million data records. The network is thereafter used to classify the real-time data and identify anomalous operations. Compared with traditional approaches of using statistical or machine learning techniques, our model reaches an accuracy of 98.73 percent when identifying traffic congestion caused by football games. Our approach first encodes the traffic across a region as a scaled image. After that the image data from different timestamps is fused with event- and time-related data. Then a crossover operator is used as a data augmentation method to generate training datasets with more balanced classes. Finally, we use the receiver operating characteristic (ROC) analysis to tune the sensitivity of the classifier. We present the analysis of the training time and the inference time separately.
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