Using weight decay to penalize the L2 norms of weights in neural networks has been a standard training practice to regularize the complexity of networks. In this paper, we show that a family of regularizers, including weight decay, is ineffective at penalizing the intrinsic norms of weights for networks with positively homogeneous activation functions, such as linear, ReLU and max-pooling functions. As a result of homogeneity, functions specified by the networks are invariant to the shifting of weight scales between layers. The ineffective regularizers are sensitive to such shifting and thus poorly regularize the model capacity, leading to overfitting. To address this shortcoming, we propose an improved regularizer that is invariant to weight scale shifting and thus effectively constrains the intrinsic norm of a neural network. The derived regularizer is an upper bound for the input gradient of the network so minimizing the improved regularizer also benefits the adversarial robustness. Residual connections are also considered and we show that our regularizer also forms an upper bound to input gradients of such a residual network. We demonstrate the efficacy of our proposed regularizer on various datasets and neural network architectures at improving generalization and adversarial robustness.

Estimating the uncertainty of a Bayesian model has been investigated for decades. The model posterior is almost always intractable, such that approximation is necessary. In many real-world cases, even though a decent estimation of the model posterior is obtained, another approximation is required to compute the predictive distribution over the desired output. A common accurate solution is to use Monte Carlo (MC) integration. However, it needs to maintain a large number of samples, evaluate the model repeatedly and average multiple model outputs. In this paper, we propose a method to approximate the probability distribution over the simplex induced by model posterior, enabling tractable computation of the predictive distribution for classification. The aim is to approximate the induced uncertainty of a specific Bayesian model, meanwhile alleviating the heavy workload of MC integration in testing time. Methodologically, we adapt Wasserstein distance to learn the induced conditional distributions, which is novel for Bayesian learning. The proposed method is universally applicable to Bayesian classification models that allow for posterior sampling. Empirical results validate the strong practical performance of our approach.