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Title: Three Mechanisms of Weight Decay Regularization

Paper content:
Weight decay is one of the standard tricks in the neural network toolbox, but the reasons for its regularization effect are poorly understood, and recent results have cast doubt on the traditional interpretation in terms of $L_2$ regularization.
Literal weight decay has been shown to outperform $L_2$ regularization for optimizers for which they differ.
We empirically investigate weight decay for three optimization algorithms (SGD, Adam, and K-FAC) and a variety of network architectures.
We identify three distinct mechanisms by which weight decay exerts a regularization effect, depending on the particular optimization algorithm and architecture: (1) increasing the effective learning rate, (2) approximately regularizing the input-output Jacobian norm, and (3) reducing the effective damping coefficient for second-order optimization.
Our results provide insight into how to improve the regularization of neural networks.
Weight decay has long been a standard trick to improve the generalization performance of neural networks (Krogh & Hertz, 1992; Bos & Chug, 1996) by encouraging the weights to be small in magnitude.
It is widely interpreted as a form of L 2 regularization because it can be derived from the gradient of the L 2 norm of the weights in the gradient descent setting.
However, several findings cast doubt on this interpretation:• Weight decay has sometimes been observed to improve training accuracy, not just generalization performance (e.g. Krizhevsky et al. (2012) ).•
Loshchilov & Hutter (2017) found that when using Adam (Kingma & Ba, 2014) as the optimizer, literally applying weight decay (i.e. scaling the weights by a factor less than 1 in each iteration) enabled far better generalization than adding an L 2 regularizer to the training objective.•
Weight decay is widely used in networks with Batch Normalization (BN) (Ioffe & Szegedy, 2015) . In
principle, weight decay regularization should have no effect in this case, since one can scale the weights by a small factor without changing the network's predictions. Hence
, it does not meaningfully constrain the network's capacity.The effect of weight decay remains poorly understood, and we lack clear guidelines for which tasks and architectures it is likely to help or hurt. A better
understanding of the role of weight decay would help us design more efficient and robust neural network architectures.In order to better understand the effect of weight decay, we experimented with both weight decay and L 2 regularization applied to image classifiers using three different optimization algorithms: SGD, Adam, and Kronecker-Factored Approximate Curvature (K-FAC) BID1 . Consistent
with the observations of Loshchilov & Hutter (2017), we found that weight decay consistently outperformed L 2 regularization in cases where they differ. Weight decay
gave an especially strong performance boost to the K-FAC optimizer, and closed most of the generalization gaps between first-and second-order optimizers, as well as between small and large batches. We then investigated
the reasons for weight decay's performance boost. Surprisingly, we identified
three distinct mechanisms by which weight decay has a regularizing effect, depending on the particular algorithm and architecture: Comparison of test accuracy of the networks trained with different optimizers on both CIFAR10 and CIFAR100. We compare Weight Decay regularization
to L2 regularization and the Baseline (which used neither). Here, BN+Aug denotes the use of BN and
data augmentation. K-FAC-G and K-FAC-F denote K-FAC using
Gauss-Newton and Fisher matrices as the preconditioner, respectively. The results suggest that weight decay
leads to improved performance across different optimizers and settings.1. In our experiments with first-order optimization
methods (SGD and Adam) on networks with BN, we found that it acts by way of the effective learning rate. Specifically, weight decay reduces the scale of
the weights, increasing the effective learning rate, thereby increasing the regularization effect of gradient noise BID2 Keskar et al., 2016) . As evidence, we found that almost all of the regularization
effect of weight decay was due to applying it to layers with BN (for which weight decay is meaningless). Furthermore, when we computed the effective learning rate for
the network with weight decay, and applied the same effective learning rate to a network without weight decay, this captured the full regularization effect. 2. We show that when K-FAC is applied to a linear network using
the Gauss-Newton metric (K-FAC-G), weight decay is equivalent to regularizing the squared Frobenius norm of the input-output Jacobian (which was shown by BID3 to improve generalization). Empirically, we found that even for (nonlinear) classification
networks, the Gauss-Newton norm (which K-FAC with weight decay is implicitly regularizing) is highly correlated with the Jacobian norm, and that K-FAC with weight decay significantly reduces the Jacobian norm. 3. Because the idealized, undamped version of K-FAC is invariant
to affine reparameterizations, the implicit learning rate effect described above should not apply. However, in practice the approximate curvature matrix is damped
by adding a multiple of the identity matrix, and this damping is not scale-invariant. We show that without weight decay, the weights grow large, causing
the effective damping term to increase. If the effective damping term grows large enough to dominate the curvature
term, it effectively turns K-FAC into a first-order optimizer. Weight decay keeps the effective damping term small, enabling K-FAC to retain
its second-order properties, and hence improving generalization.Hence, we have identified three distinct mechanisms by which weight decay improves generalization, depending on the optimization algorithm and network architecture. Our results underscore the subtlety and complexity of neural network training
: the final performance numbers obscure a variety of complex interactions between phenomena. While more analysis and experimentation is needed to understand how broadly
each of our three mechanisms applies (and to find additional mechanisms!), our work provides a starting point for understanding practical regularization effects in neural network training.
Despite its long history, weight decay regularization remains poorly understood.
We've identified three distinct mechanisms by which weight decay improves generalization, depending on the architecture and optimization algorithm: increasing the effective learning rate, reducing the Jacobian norm, and reducing the effective damping parameter.
We would not be surprised if there remain additional mechanisms we have not found.The dynamics of neural net training is incredibly complex, and it can be tempting to simply do what works and not look into why.
But we think it is important to at least sometimes dig deeper to determine exactly why an algorithm has the effect that it does.
Some of our analysis may seem mundane, or even tedious, as the interactions between different hyperparameters are not commonly seen as a topic worthy of detailed scientific study.
But our experiments highlight that the dynamics of the norms of weights and curvature matrices, and their interaction with optimization hyperparameters, can have a substantial impact on generalization.
We believe these effects deserve more attention, and would not be surprised if they can help explain the apparent success or failure of other neural net design choices.
We also believe our results highlight the need for automatic adaptation of optimization hyperparameters, to eliminate potential experimental confounds and to allow researchers and practitioners to focus on higher level design issues.Jimmy Ba, Roger Grosse, and James Martens.
Distributed second-order optimization using kroneckerfactored approximations.
2016.

assistant: text: Discusses the effect of weight decay on the training of deep network models with and without batch normalization and when using first/second order optimization methods and hypothesizes that a larger learning rate has a regularization effect.