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Register a new userEffect of Depth and Width on Local Minima in Deep Learning
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Kenji Kawaguchi
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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In this paper, we analyze the effects of depth and width on the quality of local minima, without strong over-parameterization and simplification assumptions in the literature. Without any simplification assumption, for deep nonlinear neural networks with the squared loss, we theoretically show that the quality of local minima tends to improve towards the global minimum value as depth and width increase. Furthermore, with a locally-induced structure on deep nonlinear neural networks, the values of local minima of neural networks are theoretically proven to be no worse than the globally optimal values of corresponding classical machine learning models. We empirically support our theoretical observation with a synthetic dataset as well as MNIST, CIFAR-10 and SVHN datasets. When compared to previous studies with strong over-parameterization assumptions, the results in this paper do not require over-parameterization, and instead show the gradual effects of over-parameterization as consequences of general results.
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In this paper, we theoretically prove that adding one special neuron per output unit eliminates all suboptimal local minima of any deep neural network, for multi-class classification, binary classification, and regression with an arbitrary loss function, under practical assumptions. At every local minimum of any deep neural network with these added neurons, the set of parameters of the original neural network (without added neurons) is guaranteed to be a global minimum of the original neural network. The effects of the added neurons are proven to automatically vanish at every local minimum. Moreover, we provide a novel theoretical characterization of a failure mode of eliminating suboptimal local minima via an additional theorem and several examples. This paper also introduces a novel proof technique based on the perturbable gradient basis (PGB) necessary condition of local minima, which provides new insight into the elimination of local minima and is applicable to analyze various models and transformations of objective functions beyond the elimination of local minima.
We formally study how ensemble of deep learning models can improve test accuracy, and how the superior performance of ensemble can be distilled into a single model using knowledge distillation. We consider the challenging case where the ensemble is simply an average of the outputs of a few independently trained neural networks with the SAME architecture, trained using the SAME algorithm on the SAME data set, and they only differ by the random seeds used in the initialization. We empirically show that ensemble/knowledge distillation in deep learning works very differently from traditional learning theory, especially differently from ensemble of random feature mappings or the neural-tangent-kernel feature mappings, and is potentially out of the scope of existing theorems. Thus, to properly understand ensemble and knowledge distillation in deep learning, we develop a theory showing that when data has a structure we refer to as multi-view, then ensemble of independently trained neural networks can provably improve test accuracy, and such superior test accuracy can also be provably distilled into a single model by training a single model to match the output of the ensemble instead of the true label. Our result sheds light on how ensemble works in deep learning in a way that is completely different from traditional theorems, and how the dark knowledge is hidden in the outputs of the ensemble -- that can be used in knowledge distillation -- comparing to the true data labels. In the end, we prove that self-distillation can also be viewed as implicitly combining ensemble and knowledge distillation to improve test accuracy.
Despite the empirical success of using Adversarial Training to defend deep learning models against adversarial perturbations, so far, it still remains rather unclear what the principles are behind the existence of adversarial perturbations, and what adversarial training does to the neural network to remove them. In this paper, we present a principle that we call Feature Purification, where we show one of the causes of the existence of adversarial examples is the accumulation of certain small dense mixtures in the hidden weights during the training process of a neural network; and more importantly, one of the goals of adversarial training is to remove such mixtures to purify hidden weights. We present both experiments on the CIFAR-10 dataset to illustrate this principle, and a theoretical result proving that for certain natural classification tasks, training a two-layer neural network with ReLU activation using randomly initialized gradient descent indeed satisfies this principle. Technically, we give, to the best of our knowledge, the first result proving that the following two can hold simultaneously for training a neural network with ReLU activation. (1) Training over the original data is indeed non-robust to small adversarial perturbations of some radius. (2) Adversarial training, even with an empirical perturbation algorithm such as FGM, can in fact be provably robust against ANY perturbations of the same radius. Finally, we also prove a complexity lower bound, showing that low complexity models such as linear classifiers, low-degree polynomials, or even the neural tangent kernel for this network, CANNOT defend against perturbations of this same radius, no matter what algorithms are used to train them.
The stochastic gradient descent (SGD) method and its variants are algorithms of choice for many Deep Learning tasks. These methods operate in a small-batch regime wherein a fraction of the training data, say $32$-$512$ data points, is sampled to compute an approximation to the gradient. It has been observed in practice that when using a larger batch there is a degradation in the quality of the model, as measured by its ability to generalize. We investigate the cause for this generalization drop in the large-batch regime and present numerical evidence that supports the view that large-batch methods tend to converge to sharp minimizers of the training and testing functions - and as is well known, sharp minima lead to poorer generalization. In contrast, small-batch methods consistently converge to flat minimizers, and our experiments support a commonly held view that this is due to the inherent noise in the gradient estimation. We discuss several strategies to attempt to help large-batch methods eliminate this generalization gap.
Expressive efficiency refers to the relation between two architectures A and B, whereby any function realized by B could be replicated by A, but there exists functions realized by A, which cannot be replicated by B unless its size grows significantly larger. For example, it is known that deep networks are exponentially efficient with respect to shallow networks, in the sense that a shallow network must grow exponentially large in order to approximate the functions represented by a deep network of polynomial size. In this work, we extend the study of expressive efficiency to the attribute of network connectivity and in particular to the effect of overlaps in the convolutional process, i.e., when the stride of the convolution is smaller than its filter size (receptive field). To theoretically analyze this aspect of networks design, we focus on a well-established surrogate for ConvNets called Convolutional Arithmetic Circuits (ConvACs), and then demonstrate empirically that our results hold for standard ConvNets as well. Specifically, our analysis shows that having overlapping local receptive fields, and more broadly denser connectivity, results in an exponential increase in the expressive capacity of neural networks. Moreover, while denser connectivity can increase the expressive capacity, we show that the most common types of modern architectures already exhibit exponential increase in expressivity, without relying on fully-connected layers.
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