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"anisotropic noise" has 9 results

A Walk with SGD: How SGD Explores Regions of Deep Network Loss?    

No tl;dr =[

The non-convex nature of the loss landscape of deep neural networks (DNN) lends them the intuition that over the course of training, stochastic optimization algorithms explore different regions of the loss surface by entering and escaping many local minima due to the noise induced by mini-batches. But is this really the case? This question couples the geometry of the DNN loss landscape with how stochastic optimization algorithms like SGD interact with it during training. Answering this question may help us qualitatively understand the dynamics of deep neural network optimization. We show evidence through qualitative and quantitative experiments that mini-batch SGD rarely crosses barriers during DNN optimization. As we show, the mini-batch induced noise helps SGD explore different regions of the loss surface using a seemingly different mechanism. To complement this finding, we also investigate the qualitative reason behind the slowing down of this exploration when using larger batch-sizes. We show this happens because gradients from larger batch-sizes align more with the top eigenvectors of the Hessian, which makes SGD oscillate in the proximity of the parameter initialization, thus preventing exploration.

Identifying Generalization Properties in Neural Networks    

tl;dr a theory connecting Hessian of the solution and the generalization power of the model

While it has not yet been proven, empirical evidence suggests that model generalization is related to local properties of the optima which can be described via the Hessian. We connect model generalization with the local property of a solution under the PAC-Bayes paradigm. In particular, we prove that model generalization ability is related to the Hessian, the higher-order "smoothness" terms characterized by the Lipschitz constant of the Hessian, and the scales of the parameters. Guided by the proof, we propose a metric to score the generalization capability of the model, as well as an algorithm that optimizes the perturbed model accordingly.

The Anisotropic Noise in Stochastic Gradient Descent: Its Behavior of Escaping from Minima and Regularization Effects    

tl;dr We provide theoretical and empirical analysis on the role of anisotropic noise introduced by stochastic gradient on escaping from minima.

Understanding the behavior of stochastic gradient descent (SGD) in the context of deep neural networks has raised lots of concerns recently. Along this line, we theoretically study a general form of gradient based optimization dynamics with unbiased noise, which unifies SGD and standard Langevin dynamics. Through investigating this general optimization dynamics, we analyze the behavior of SGD on escaping from minima and its regularization effects. A novel indicator is derived to characterize the efficiency of escaping from minima through measuring the alignment of noise covariance and the curvature of loss function. Based on this indicator, two conditions are established to show which type of noise structure is superior to isotropic noise in term of escaping efficiency. We further show that the anisotropic noise in SGD satisfies the two conditions, and thus helps to escape from sharp and poor minima effectively, towards more stable and flat minima that typically generalize well. We verify our understanding through comparing this anisotropic diffusion with full gradient descent plus isotropic diffusion (i.e. Langevin dynamics) and other types of position-dependent noise.


tl;dr Large batch size training using adversarial training and second order information

Stochastic Gradient Descent (SGD) methods using randomly selected batches are widely-used to train neural network (NN) models. Performing design exploration to find the best NN for a particular task often requires extensive training with different models on a large dataset, which is very computationally expensive. The most straightforward method to accelerate this computation is to distribute the batch of SGD over multiple processors. However, large batch training often times leads to degradation in accuracy, poor generalization, and even poor robustness to adversarial attacks. Existing solutions for large batch training either do not work or require massive hyper-parameter tuning. To address this issue, we propose a novel large batch training method which combines recent results in adversarial training (to regularize against ``sharp minima'') and second order optimization (to use curvature information to change batch size adaptively during training). We extensively evaluate our method on Cifar-10/100, SVHN, TinyImageNet, and ImageNet datasets, using multiple NNs, including residual networks as well as compressed networks such as SqueezeNext. Our new approach exceeds the performance of the existing solutions in terms of both accuracy and the number of SGD iterations (up to 1\% and $3\times$, respectively). We emphasize that this is achieved without any additional hyper-parameter tuning to tailor our method to any of these experiments.

Exploring Curvature Noise in Large-Batch Stochastic Optimization    

tl;dr Engineer large-batch training such that we retain fast training while achieving better generalization.

Using stochastic gradient descent (SGD) with large batch-sizes to train deep neural networks is an increasingly popular technique. By doing so, one can improve parallelization by scaling to multiple workers (GPUs) and hence leading to significant reductions in training time. Unfortunately, a major drawback is the so-called generalization gap: large-batch training typically leads to a degradation in generalization performance of the model as compared to small-batch training. In this paper, we propose to correct this generalization gap by adding diagonal Fisher curvature noise to large-batch gradient updates. We provide a theoretical analysis of our method in the convex quadratic setting. Our empirical study with state-of-the-art deep learning models shows that our method not only improves the generalization performance in large-batch training but furthermore, does so in a way where the training convergence remains desirable and the training duration is not elongated. We additionally connect our method to recent works on loss surface landscape in the experimental section.

On the Computational Inefficiency of Large Batch Sizes for Stochastic Gradient Descent    

tl;dr Large batch training results in rapidly diminishing returns in wall-clock time to convergence to find a good model.

Increasing the mini-batch size for stochastic gradient descent offers significant opportunities to reduce wall-clock training time, but there are a variety of theoretical and systems challenges that impede the widespread success of this technique (Daset al., 2016; Keskar et al., 2016). We investigate these issues, with an emphasis on time to convergence and total computational cost, through an extensive empirical analysis of network training across several architectures and problem domains, including image classification, image segmentation, and language modeling. Although it is common practice to increase the batch size in order to fully exploit available computational resources, we find a substantially more nuanced picture. Our main finding is that across a wide range of network architectures and problem domains, increasing the batch size beyond a certain point yields no decrease in wall-clock time to convergence for either train or test loss. This batch size is usually substantially below the capacity of current systems. We show that popular training strategies for large batch size optimization begin to fail before we can populate all available compute resources, and we show that the point at which these methods break down depends more on attributes like model architecture and data complexity than it does directly on the size of the dataset.

Fluctuation-dissipation relations for stochastic gradient descent    

tl;dr We prove fluctuation-dissipation relations for SGD, which can be used to (i) adaptively set learning rates and (ii) probe loss surfaces.

The notion of the stationary equilibrium ensemble has played a central role in statistical mechanics. In machine learning as well, training serves as generalized equilibration that drives the probability distribution of model parameters toward stationarity. Here, we derive stationary fluctuation-dissipation relations that link measurable quantities and hyperparameters in the stochastic gradient descent algorithm. These relations hold exactly for any stationary state and can in particular be used to adaptively set training schedule. We can further use the relations to efficiently extract information pertaining to a loss-function landscape such as the magnitudes of its Hessian and anharmonicity. Our claims are empirically verified.


tl;dr Introduce an information theoretic viewpoint on the behavior of deep networks optimization processes and their generalization abilities

Understanding the groundbreaking performance of Deep Neural Networks is one of the greatest challenges to the scientific community today. In this work, we introduce an information theoretic viewpoint on the behavior of deep networks optimization processes and their generalization abilities. By studying the Information Plane, the plane of the mutual information between the input variable and the desired label, for each hidden layer. Specifically, we show that the training of the network is characterized by a rapid increase in the mutual information (MI) between the layers and the target label, followed by a longer decrease in the MI between the layers and the input variable. Further, we explicitly show that these two fundamental information-theoretic quantities correspond to the generalization error of the network, as a result of introducing a new generalization bound that is exponential in the representation compression. The analysis focuses on typical patterns of large-scale problems. For this purpose, we introduce a novel analytic bound on the mutual information between consecutive layers in the network. An important consequence of our analysis is a super-linear boost in training time with the number of non-degenerate hidden layers, demonstrating the computational benefit of the hidden layers.

On the Relation Between the Sharpest Directions of DNN Loss and the SGD Step Length    

tl;dr SGD is steered early on in training towards a region in which its step is too large compared to curvature, which impacts the rest of training.

Training of deep neural networks with Stochastic Gradient Descent (SGD) typically ends in regions of the weight space, where both the generalization properties and the flatness of the local loss curvature depend on the learning rate and the batch size. We discover that a related phenomena happens in the early phase of training and study its consequences. Initially, SGD visits increasingly sharp regions of the loss surface, reaching a maximum sharpness determined by both the learning rate and the batch-size of SGD. At this early peak value, an SGD step is on average too large to minimize the loss along the directions corresponding to the largest eigenvalues of the Hessian (i.e. the sharpest directions). To query the importance of this phenomena for training, we study a variant of SGD using a reduced learning rate along the sharpest directions and show that it can improve training speed while finding both sharper and better--generalizing solution, compared to vanilla SGD. Overall, our results show that the SGD dynamics along the sharpest directions influence the regions of the weight space visited, the overall training speed, and generalization ability.