Residual networks (ResNet) and weight normalization play an important role in various deep learning applications. However, parameter initialization strategies have not been studied previously for weight normalized networks and, in practice, initialization methods designed for un-normalized networks are used as a proxy. Similarly, initialization for ResNets have also been studied for un-normalized networks and often under simplified settings ignoring the shortcut connection. To address these issues, we propose a novel parameter initialization strategy that avoids explosion/vanishment of information across layers for weight normalized networks with and without residual connections. The proposed strategy is based on a theoretical analysis using mean field approximation.

A BSTRACT It has been shown that instead of learning actual object features, deep networks tend to exploit non-robust (spurious) discriminative features that are shared between training and test sets. Therefore, while they achieve state of the art performance on such test sets, they achieve poor generalization on out of distribution (OOD) samples where the IID (independent, identical distribution) assumption breaks and the distribution of non-robust features shifts. Through theoretical and empirical analysis, we show that this happens because maximum likelihood training (without appropriate regularization) leads the model to depend on all the correlations (including spurious ones) present between inputs and targets in the dataset. We then show evidence that the information bottleneck (IB) principle can address this problem. To do so, we propose a regularization approach based on IB, called Entropy Penalty, that reduces the model's dependence on spurious features-features corresponding to such spurious correlations. This allows deep networks trained with Entropy Penalty to generalize well even under distribution shift of spurious features. As a controlled test-bed for evaluating our claim, we train deep networks with Entropy Penalty on a colored MNIST (C-MNIST) dataset and show that it is able to generalize well on vanilla MNIST, MNIST -M and SVHN datasets in addition to an OOD version of C-MNIST itself. The baseline regularization methods we compare against fail to generalize on this test-bed. An example of non-robust feature is the presence of desert in camel images, which may correlate well with this object class. More realistically, models can learn to exploit the abundance of input-target correlations present in datasets, not all of which may be invariant under different environments. Interestingly, such classifiers can achieve good performance on test sets which share the same non-robust features. However, due to this exploitation, these classifiers perform poorly under distribution shift (Geirhos et al., 2018a; Hendrycks & Dietterich, 2019) because it violates the IID assumption which is the foundation of existing generalization theory (Bartlett & Mendelson, 2002; McAllester, 1999b;a).

Residual networks (ResNet) and weight normalization play an important role in various deep learning applications. However, parameter initialization strategies have not been studied previously for weight normalized networks and, in practice, initialization methods designed for un-normalized networks are used as a proxy. Similarly, initialization for ResNets have also been studied for un-normalized networks and often under simplified settings ignoring the shortcut connection. To address these issues, we propose a novel parameter initialization strategy that avoids explosion/vanishment of information across layers for weight normalized networks with and without residual connections. The proposed strategy is based on a theoretical analysis using mean field approximation. We run over 2,500 experiments and evaluate our proposal on image datasets showing that the proposed initialization outperforms existing initialization methods in terms of generalization performance, robustness to hyper-parameter values and variance between seeds, especially when networks get deeper in which case existing methods fail to even start training. Finally, we show that using our initialization in conjunction with learning rate warmup is able to reduce the gap between the performance of weight normalized and batch normalized networks.

It has been noted in existing literature that over-parameterization in ReLU networks generally leads to better performance. While there could be several reasons for this, we investigate desirable network properties at initialization which may be enjoyed by ReLU networks. Without making any assumption, we derive a lower bound on the layer width of deep ReLU networks whose weights are initialized from a certain distribution, such that with high probability, i) the norm of hidden activation of all layers are roughly equal to the norm of the input, and, ii) the norm of parameter gradient for all the layers are roughly the same. In this way, sufficiently wide deep ReLU nets with appropriate initialization can inherently preserve the forward flow of information and also avoid the gradient exploding/vanishing problem. We further show that these results hold for an infinite number of data samples, in which case the finite lower bound depends on the input dimensionality and the depth of the network. In the case of deep ReLU networks with weight vectors normalized by their norm, we derive an initialization required to tap the aforementioned benefits from over-parameterization without which network fails to learn for large depth.

Recurrent neural networks are known for their notorious exploding and vanishing gradient problem (EVGP). This problem becomes more evident in tasks where the information needed to correctly solve them exist over long time scales, because EVGP prevents important gradient components from being back-propagated adequately over a large number of steps. We introduce a simple stochastic algorithm (h-detach) that is specific to LSTM optimization and targeted towards addressing this problem. Specifically, we show that when the LSTM weights are large, the gradient components through the linear path (cell state) in the LSTM computational graph get suppressed. Based on the hypothesis that these components carry information about long term dependencies (which we show empirically), their suppression can prevent LSTMs from capturing them. Our algorithm prevents gradients flowing through this path from getting suppressed, thus allowing the LSTM to capture such dependencies better. We show significant convergence and generalization improvements using our algorithm on various benchmark datasets. Recurrent Neural Networks (RNNs) (Rumelhart et al. (1986); Elman (1990)) are a class of neural network architectures used for modeling sequential data.

It is well known that over-parametrized deep neural networks (DNNs) are an overly expressive class of functions that can memorize even random data with $100\%$ training accuracy. This raises the question why they do not easily overfit real data. To answer this question, we study deep networks using Fourier analysis. We show that deep networks with finite weights (or trained for finite number of steps) are inherently biased towards representing smooth functions over the input space. Specifically, the magnitude of a particular frequency component ($k$) of deep ReLU network function decays at least as fast as $\mathcal{O}(k^{-2})$, with width and depth helping polynomially and exponentially (respectively) in modeling higher frequencies. This shows for instance why DNNs cannot perfectly \textit{memorize} peaky delta-like functions. We also show that DNNs can exploit the geometry of low dimensional data manifolds to approximate complex functions that exist along the manifold with simple functions when seen with respect to the input space. As a consequence, we find that all samples (including adversarial samples) classified by a network to belong to a certain class are connected by a path such that the prediction of the network along that path does not change. Finally we find that DNN parameters corresponding to functions with higher frequency components occupy a smaller volume in the parameter.

Exploring why stochastic gradient descent (SGD) based optimization methods train deep neural networks (DNNs) that generalize well has become an active area of research. Towards this end, we empirically study the dynamics of SGD when training over-parametrized DNNs. Specifically we study the DNN loss surface along the trajectory of SGD by interpolating the loss surface between parameters from consecutive \textit{iterations} and tracking various metrics during training. We find that the loss interpolation between parameters before and after a training update is roughly convex with a minimum (\textit{valley floor}) in between for most of the training. Based on this and other metrics, we deduce that during most of the training, SGD explores regions in a valley by bouncing off valley walls at a height above the valley floor. This 'bouncing off walls at a height' mechanism helps SGD traverse larger distance for small batch sizes and large learning rates which we find play qualitatively different roles in the dynamics. While a large learning rate maintains a large height from the valley floor, a small batch size injects noise facilitating exploration. We find this mechanism is crucial for generalization because the valley floor has barriers and this exploration above the valley floor allows SGD to quickly travel far away from the initialization point (without being affected by barriers) and find flatter regions, corresponding to better generalization.

Recurrent neural networks (RNNs) are important class of architectures among neural networks useful for language modeling and sequential prediction. However, optimizing RNNs is known to be harder compared to feed-forward neural networks. A number of techniques have been proposed in literature to address this problem. In this paper we propose a simple technique called fraternal dropout that takes advantage of dropout to achieve this goal. Specifically, we propose to train two identical copies of an RNN (that share parameters) with different dropout masks while minimizing the difference between their (pre-softmax) predictions. In this way our regularization encourages the representations of RNNs to be invariant to dropout mask, thus being robust. We show that our regularization term is upper bounded by the expectation-linear dropout objective which has been shown to address the gap due to the difference between the train and inference phases of dropout. We evaluate our model and achieve state-of-the-art results in sequence modeling tasks on two benchmark datasets - Penn Treebank and Wikitext-2. We also show that our approach leads to performance improvement by a significant margin in image captioning (Microsoft COCO) and semi-supervised (CIFAR-10) tasks.

Recurrent neural networks like long short-term memory (LSTM) are important architectures for sequential prediction tasks. LSTMs (and RNNs in general) model sequences along the forward time direction. Bidirectional LSTMs (Bi-LSTMs) on the other hand model sequences along both forward and backward directions and are generally known to perform better at such tasks because they capture a richer representation of the data. In the training of Bi-LSTMs, the forward and backward paths are learned independently. We propose a variant of the Bi-LSTM architecture, which we call Variational Bi-LSTM, that creates a channel between the two paths (during training, but which may be omitted during inference); thus optimizing the two paths jointly. We arrive at this joint objective for our model by minimizing a variational lower bound of the joint likelihood of the data sequence. Our model acts as a regularizer and encourages the two networks to inform each other in making their respective predictions using distinct information. We perform ablation studies to better understand the different components of our model and evaluate the method on various benchmarks, showing state-of-the-art performance.

We study the properties of the endpoint of stochastic gradient descent (SGD). By approximating SGD as a stochastic differential equation (SDE) we consider the Boltzmann-Gibbs equilibrium distribution of that SDE under the assumption of isotropic variance in loss gradients. Through this analysis, we find that three factors - learning rate, batch size and the variance of the loss gradients - control the trade-off between the depth and width of the minima found by SGD, with wider minima favoured by a higher ratio of learning rate to batch size. We have direct control over the learning rate and batch size, while the variance is determined by the choice of model architecture, model parameterization and dataset. In the equilibrium distribution only the ratio of learning rate to batch size appears, implying that the equilibrium distribution is invariant under a simultaneous rescaling of learning rate and batch size by the same amount. We then explore experimentally how learning rate and batch size affect SGD from two perspectives: the endpoint of SGD and the dynamics that lead up to it. For the endpoint, the experiments suggest the endpoint of SGD is invariant under simultaneous rescaling of batch size and learning rate, and also that a higher ratio leads to flatter minima, both findings are consistent with our theoretical analysis. We note experimentally that the dynamics also seem to be invariant under the same rescaling of learning rate and batch size, which we explore showing that one can exchange batch size and learning rate for cyclical learning rate schedule. Next, we illustrate how noise affects memorization, showing that high noise levels lead to better generalization. Finally, we find experimentally that the invariance under simultaneous rescaling of learning rate and batch size breaks down if the learning rate gets too large or the batch size gets too small.