Pruning neural network parameters to reduce model size is an area of much interest, but the original motivation for pruning was the prevention of overfitting rather than the improvement of computational efficiency. This motivation is particularly relevant given the perhaps surprising observation that a wide variety of pruning approaches confer increases in test accuracy, even when parameter counts are drastically reduced. To better understand this phenomenon, we analyze the behavior of pruning over the course of training, finding that pruning's effect on generalization relies more on the instability generated by pruning than the final size of the pruned model. We demonstrate that even pruning of seemingly unimportant parameters can lead to such instability, allowing our finding to account for the generalization benefits of modern pruning techniques. Our results ultimately suggest that, counter-intuitively, pruning regularizes through instability and mechanisms unrelated to parameter counts.
As stated in @cite_0 , one role of generalization theory is to provide theoretical insights to guide the search over model classes.'' VC dimension (a measure of model capacity) motivated pruning as a regularizer in the early pruning techniques Optimal Brain Damage (OBD) @cite_30 and Optimal Brain Surgeon (OBS) @cite_2 . The logic was that overfitting can be bounded above by a function of VC dimension, which increases with parameter counts, so fewer parameters guarantees a better bound @cite_2 @cite_15 . Unfortunately, such bounds are so loose that tightening them by reducing parameter counts need not translate to better generalization in practice @cite_15 . Furthermore, more recent generalization bounds suggest that fewer parameters actually makes typical neural networks generalize worse @cite_24 .
@cite_30 @cite_0 @cite_24 @cite_2 @cite_15
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Neural networks exhibit good generalization behavior in the over-parameterized regime, where the number of network parameters exceeds the number of observations. Nonetheless, current generalization bounds for neural networks fail to explain this phenomenon. In an attempt to bridge this gap, we study the problem of learning a two-layer over-parameterized neural network, when the data is generated by a linearly separable function. In the case where the network has Leaky ReLU activations, we provide both optimization and generalization guarantees for over-parameterized networks. Specifically, we prove convergence rates of SGD to a global minimum and provide generalization guarantees for this global minimum that are independent of the network size. Therefore, our result clearly shows that the use of SGD for optimization both finds a global minimum, and avoids overfitting despite the high capacity of the model. This is the first theoretical demonstration that SGD can avoid overfitting, when learning over-specified neural network classifiers. We perform an average case analysis of the generalization dynamics of large neural networks trained using gradient descent. We study the practically-relevant "high-dimensional" regime where the number of free parameters in the network is on the order of or even larger than the number of examples in the dataset. Using random matrix theory and exact solutions in linear models, we derive the generalization error and training error dynamics of learning and analyze how they depend on the dimensionality of data and signal to noise ratio of the learning problem. We find that the dynamics of gradient descent learning naturally protect against overtraining and overfitting in large networks. Overtraining is worst at intermediate network sizes, when the effective number of free parameters equals the number of samples, and thus can be reduced by making a network smaller or larger. Additionally, in the high-dimensional regime, low generalization error requires starting with small initial weights. We then turn to non-linear neural networks, and show that making networks very large does not harm their generalization performance. On the contrary, it can in fact reduce overtraining, even without early stopping or regularization of any sort. We identify two novel phenomena underlying this behavior in overcomplete models: first, there is a frozen subspace of the weights in which no learning occurs under gradient descent; and second, the statistical properties of the high-dimensional regime yield better-conditioned input correlations which protect against overtraining. We demonstrate that naive application of worst-case theories such as Rademacher complexity are inaccurate in predicting the generalization performance of deep neural networks, and derive an alternative bound which incorporates the frozen subspace and conditioning effects and qualitatively matches the behavior observed in simulation. Regularization is key for deep learning since it allows training more complex models while keeping lower levels of overfitting. However, the most prevalent regularizations do not leverage all the capacity of the models since they rely on reducing the effective number of parameters. Feature decorrelation is an alternative for using the full capacity of the models but the overfitting reduction margins are too narrow given the overhead it introduces. In this paper, we show that regularizing negatively correlated features is an obstacle for effective decorrelation and present OrthoReg, a novel regularization technique that locally enforces feature orthogonality. As a result, imposing locality constraints in feature decorrelation removes interferences between negatively correlated feature weights, allowing the regularizer to reach higher decorrelation bounds, and reducing the overfitting more effectively. In particular, we show that the models regularized with OrthoReg have higher accuracy bounds even when batch normalization and dropout are present. Moreover, since our regularization is directly performed on the weights, it is especially suitable for fully convolutional neural networks, where the weight space is constant compared to the feature map space. As a result, we are able to reduce the overfitting of state-of-the-art CNNs on CIFAR-10, CIFAR-100, and SVHN. Regularization is key for deep learning since it allows training more complex models while keeping lower levels of overfitting. However, the most prevalent regularizations do not leverage all the capacity of the models since they rely on reducing the effective number of parameters. Feature decorrelation is an alternative for using the full capacity of the models but the overfitting reduction margins are too narrow given the overhead it introduces. In this paper, we show that regularizing negatively correlated features is an obstacle for effective decorrelation and present OrthoReg, a novel regularization technique that locally enforces feature orthogonality. As a result, imposing locality constraints in feature decorrelation removes interferences between negatively correlated feature weights, allowing the regularizer to reach higher decorrelation bounds, and reducing the overfitting more effectively. In particular, we show that the models regularized with OrthoReg have higher accuracy bounds even when batch normalization and dropout are present. Moreover, since our regularization is directly performed on the weights, it is especially suitable for fully convolutional neural networks, where the weight space is constant compared to the feature map space. As a result, we are able to reduce the overfitting of state-of-the-art CNNs on CIFAR-10, CIFAR-100, and SVHN.