Deep neural networks (DNNs) perform well on a variety of tasks despite the fact that most used in practice are vastly overparametrized and even capable of perfectly fitting randomly labeled data. Recent evidence suggests that developing "compressible" representations is key for adjusting the complexity of overparametrized networks to the task at hand and avoiding overfitting (Arora et al., 2018; Zhou et al., 2018). In this paper, we provide new empirical evidence that supports this hypothesis, identifying two independent mechanisms that emerge when the network’s width is increased: robustness (having units that can be removed without affecting accuracy) and redundancy (having units with similar activity). In a series of experiments with AlexNet, ResNet and Inception networks in the CIFAR-10 and ImageNet datasets, and also using shallow networks with synthetic data, we show that DNNs consistently increase either their robustness, their redundancy, or both at greater widths for a comprehensive set of hyperparameters. These results suggest that networks in the deep learning regime adjust their effective capacity by developing either robustness or redundancy. Deep neural networks (DNNs) are capable of successfully learning from examples in a wide variety of tasks. Though these networks are typically trained with large amounts of data, the number of free parameters in their architectures is often several orders of magnitude greater than the number of training examples. This overparametrization reflects the ability of DNNs to memorize entire datasets, even with randomized labels . Additionally, large networks not only tend to match the performance of small ones, but often generalize better (e.g. Neyshabur et al. (2017b) ; Frankle & Carbin (2018) ; Neyshabur et al. (2018) ; Novak et al. (2018) ). Figure 1 demonstrates this for a variety of modern networks trained in ImageNet and CIFAR-10. These observations raise the question of how vastly overparametrized networks can perform well in structured tasks without overfitting. While DNNs appear to adapt their capacity to the complexity of the given task, precisely what causes them to do so remains an open question . Several previous studies have aimed to uncover why, out of the many optima an overparametrized network can reach to achieve 100% training accuracy, they tend toward ones that generalize well (Neyshabur et al., 2017b; Zhang et al., 2017; Neyshabur et al., 2018; Novak et al., 2018) often by proving generalization bounds for simple models related to weight matrix norms or Rademacher complexity (Bartlett et al., 2017; Neyshabur et al., 2017a; Arora et al., 2018; Neyshabur et al., 2018) . Frankle & Carbin (2018) , showed that, in certain networks, the crucial computations were performed by sparse subnetworks within them. In doing so, they suggested that large networks tend to perform as well as or better than smaller ones because they more reliably contained fortuitously-initialized "lottery ticket" subnetworks. Here, we focus on the question of why generalization ability does not decrease as a network's degree of overparametrization increases. We investigate two critical properties of DNNs: robustness (how fragile the network is to removal of units) and redundancy (how similar unit activity is). In doing so, we build off of theoretical work by Arora et al. (2018) and Zhou et al. (2018) , connecting the compressibility of DNNs to their non-overfitting behavior. We find that various DNNs train toward regimes with different degrees of robustness and redundancy, but that at least one of the two properties, if not both, consistently emerges as a model's size is increased. Based on these results, we offer interpretations of the various ways in which DNNs may constrain their effective capacity to protect from overfitting. In this work, we empirically analyze models in terms of their activations (Novak et al., 2018; Morcos et al., 2018a; b) which makes our results contextual to input data. Because of this, we are able to scale our analysis to state of the art networks like ResNet18 and Inception-v3. And by focusing not on the broad question of generalization, but on the subproblem of why networks do not perform worse when their size is increased, we are able to show that redundancy and robustness are central to how networks autoregularize. A related branch of work has focused on the relationship between a network's compressibility and its generalization behavior (Zhou et al., 2018) Our results generally validate both of these approaches, but we show that different networks develop different compressible features and to different extents, so we speculate that both pruning unimportant units and compressing redundant units may be complementary tools for developing new compression algorithms. We also show that redundancy is highly sensitive to a network's initialization while its accuracy is not. This suggests that certain compression techniques could be improved greatly by validating over multiple initializations in order to produce maximally redundant models. We also make progress toward tightening our understanding of how compressible DNNs are which Zhou et al. (2018) shows can lead to improved practical generalization bounds. Arora et al. (2014) suggests that redundancy implies robustness, and Morcos et al. (2018b) connects a network's robustness to the flattening of a layers' activation space along the direction of a single activation vector to improved generalization performance. However, our findings suggest that these trends may not hold for all networks and that redundancy and robustness poorly predict generalization. Our work is also related to Maennel et al. (2018) who takes a theoretical approach to show that model networks in the overparametrized regime tend to develop weight vectors that align to a set of discrete directions that are determined by the input data. Our work suggest that their conclusions may retain a high degree of explanatory power in some but not all state of the art cases. Despite a great deal of recent progress, to our knowledge, ours is the first work to date that has quantitatively studied the connections between overparametrization, robustness, and redundancy together. We analyze these phenomena across a wide range of networks which may aid in understanding how well theoretical findings (which are typically based on simple models) generalize to common networks in machine learning. We find that each network we analyze displays unique trends in robustness, compressibility, and similarity, yet that all deep ones develop more redundancy and/or robustness at large model sizes. We also demonstrate that the two are highly dependent on initializations and that high variance increases redundancy in some networks and decrease it in others. Limitations of our work include that we do not analyze cases with varying network depth and the fact that our single-layer MLPs with large initializations trained with high-dimensional, uncorrelated data do not seem to develop either increased robustness or redundancy at large model sizes. However, a recent strand of research has emerged illluminating similarities between deep networks and kernel machines (Belkin et al., 2018; Jacot et al., 2018; Liang & Rakhlin, 2018) and suggesting that networks with high-variance initializations can operate in a kernel-like regime (Chizat et al., 2019; Woodworth et al., 2019) which we suspect relates to these findings for networks initialized with large variance. In this paper, we jointly analyze the robustness and redundancy of deep neural networks with the aim of understanding why generalization ability does not tend to decrease as a network's degree of overparametrization increases. In doing so, we find that robustness and redundancy do not imply each other but that one or the other or both consistently increase alongside overparametrization. We connect these observations to various capacity-constraining features which DNNs may develop in order to support the connection between compressibility and generalization and to shed light on the features networks may develop to avoid overfitting. In doing so, we paint a more complex picture of robustness and redundancy than much previous work has assumed. By illustrating the relationships between these phenomena, we suggest various new research directions in theory of learning and compression. We believe that together, these findings represent a milestone in understanding the emergent properties of overparametrized neural networks. ResNet18s: These networks were off the shelf from He et al. (2016) for the ImageNet dataset. They consisted of an initial convolution and batch norm followed by 4 building block (v1) layers, each with 2 blocks and a fully connected layer leading to a softmax output. All kernel sizes in the initial layers and block layers were of size 7 × 7 and stride 2. All activations were ReLU. In the 1x sized model, the convolutions in the initial and block layers used 64, 64, 128, and 256 filters respectively. After Xavier/Glorot initialization, we trained them for 90 epochs with a default batch size of 256 an initial default learning rate of 1 which decayed by a factor of 10 at epochs 30, 60, and 80. Training was done on the ILSVRC 2012 dataset with approximately 1 million images, and evaluation was done on 50,000 validation images. Optimization was done with SGD using 0.9 momentum. We used batch normalization, data augmentation with random cropping and flipping, and 0.0001 weight decay.
Probing robustness and redundancy in deep neural networks reveals capacity-constraining features which help to explain non-overfitting.
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