This work presents a new algorithm for training recurrent neural networks (although ideas are applicable to feedforward networks as well). The algorithm is derived from a theory in nonconvex optimization related to the diffusion equation. The contributions made in this work are two fold. First, we show how some seemingly disconnected mechanisms used in deep learning such as smart initialization, annealed learning rate, layerwise pretraining, and noise injection (as done in dropout and SGD) arise naturally and automatically from this framework, without manually crafting them into the algorithms. Second, we present some preliminary results on comparing the proposed method against SGD. It turns out that the new algorithm can achieve similar level of generalization accuracy of SGD in much fewer number of epochs.
As discussed in Section , smoothing can be considered as means to inject noise into the training process. The idea of noise injection is already used in methods such as SGD or dropout @cite_19 in order to improve learning. The key advantage of our framework for noise injection, however, is that the noise injection can be achieved in closed form and without need of sampling. In order words, we can compute the effect of infinitely many noisified objective functions in closed form. This is similar to the idea of Marginalized Denoising Autoencoders (mDA) @cite_13 , where the effect of infinitely many nosified inputs is marginalized to obtain a closed form expression. However, mDA limits the form of the injected noise. Specifically, the marginalized effect is only computable in a c linear reconstruction setup (nonlinearity is applied only after computation of the marginalized reconstruction). In addition, mDA performs noise injection layer by layer in a greedy fashion. In contrast, our framework is able to compute closed form expression for the entire deep network and allowing full nonconvexity of the associated optimization, up to reasonable approximation.
@cite_19 @cite_13
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Recent development in the field of Deep Learning have exposed the underlying vulnerability of Deep Neural Network (DNN) against adversarial examples. In image classification, an adversarial example is a carefully modified image that is visually imperceptible to the original image but can cause DNN model to misclassify it. Training the network with Gaussian noise is an effective technique to perform model regularization, thus improving model robustness against input variation. Inspired by this classical method, we explore to utilize the regularization characteristic of noise injection to improve DNN's robustness against adversarial attack. In this work, we propose Parametric-Noise-Injection (PNI) which involves trainable Gaussian noise injection at each layer on either activation or weights through solving the min-max optimization problem, embedded with adversarial training. These parameters are trained explicitly to achieve improved robustness. To the best of our knowledge, this is the first work that uses trainable noise injection to improve network robustness against adversarial attacks, rather than manually configuring the injected noise level through cross-validation. The extensive results show that our proposed PNI technique effectively improves the robustness against a variety of powerful white-box and black-box attacks such as PGD, C & W, FGSM, transferable attack and ZOO attack. Last but not the least, PNI method improves both clean- and perturbed-data accuracy in comparison to the state-of-the-art defense methods, which outperforms current unbroken PGD defense by 1.1 and 6.8 on clean test data and perturbed test data respectively using Resnet-20 architecture. Dropout-based regularization methods can be regarded as injecting random noise with pre-defined magnitude to different parts of the neural network during training. It was recently shown that Bayesian dropout procedure not only improves gener- alization but also leads to extremely sparse neural architectures by automatically setting the individual noise magnitude per weight. However, this sparsity can hardly be used for acceleration since it is unstructured. In the paper, we propose a new Bayesian model that takes into account the computational structure of neural net- works and provides structured sparsity, e.g. removes neurons and or convolutional channels in CNNs. To do this we inject noise to the neurons outputs while keeping the weights unregularized. We establish the probabilistic model with a proper truncated log-uniform prior over the noise and truncated log-normal variational approximation that ensures that the KL-term in the evidence lower bound is com- puted in closed-form. The model leads to structured sparsity by removing elements with a low SNR from the computation graph and provides significant acceleration on a number of deep neural architectures. The model is easy to implement as it can be formulated as a separate dropout-like layer. Recent work has shown how denoising and contractive autoencoders implicitly capture the structure of the data-generating density, in the case where the corruption noise is Gaussian, the reconstruction error is the squared error, and the data is continuous-valued. This has led to various proposals for sampling from this implicitly learned density function, using Langevin and Metropolis-Hastings MCMC. However, it remained unclear how to connect the training procedure of regularized auto-encoders to the implicit estimation of the underlying data-generating distribution when the data are discrete, or using other forms of corruption process and reconstruction errors. Another issue is the mathematical justification which is only valid in the limit of small corruption noise. We propose here a different attack on the problem, which deals with all these issues: arbitrary (but noisy enough) corruption, arbitrary reconstruction loss (seen as a log-likelihood), handling both discrete and continuous-valued variables, and removing the bias due to non-infinitesimal corruption noise (or non-infinitesimal contractive penalty). Recent work has shown how denoising and contractive autoencoders implicitly capture the structure of the data-generating density, in the case where the corruption noise is Gaussian, the reconstruction error is the squared error, and the data is continuous-valued. This has led to various proposals for sampling from this implicitly learned density function, using Langevin and Metropolis-Hastings MCMC. However, it remained unclear how to connect the training procedure of regularized auto-encoders to the implicit estimation of the underlying data-generating distribution when the data are discrete, or using other forms of corruption process and reconstruction errors. Another issue is the mathematical justification which is only valid in the limit of small corruption noise. We propose here a different attack on the problem, which deals with all these issues: arbitrary (but noisy enough) corruption, arbitrary reconstruction loss (seen as a log-likelihood), handling both discrete and continuous-valued variables, and removing the bias due to non-infinitesimal corruption noise (or non-infinitesimal contractive penalty).