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# Demystifying Batch Normalization in ReLU Networks: Equivalent Convex Optimization Models and Implicit Regularization.

02 Mar 2021-arXiv: Learning-
Abstract: Batch Normalization (BN) is a commonly used technique to accelerate and stabilize training of deep neural networks. Despite its empirical success, a full theoretical understanding of BN is yet to be developed. In this work, we analyze BN through the lens of convex optimization. We introduce an analytic framework based on convex duality to obtain exact convex representations of weight-decay regularized ReLU networks with BN, which can be trained in polynomial-time. Our analyses also show that optimal layer weights can be obtained as simple closed-form formulas in the high-dimensional and/or overparameterized regimes. Furthermore, we find that Gradient Descent provides an algorithmic bias effect on the standard non-convex BN network, and we design an approach to explicitly encode this implicit regularization into the convex objective. Experiments with CIFAR image classification highlight the effectiveness of this explicit regularization for mimicking and substantially improving the performance of standard BN networks.

Topics: Convex optimization (60%), Gradient descent (53%)
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Tolga Ergen1, Mert Pilanci1Institutions (1)
22 Feb 2020-arXiv: Learning
Abstract: We study regularized deep neural networks (DNNs) and introduce a convex analytic framework to characterize the structure of the hidden layers. We show that a set of optimal hidden layer weights for a norm regularized DNN training problem can be explicitly found as the extreme points of a convex set. For the special case of deep linear networks, we prove that each optimal weight matrix aligns with the previous layers via duality. More importantly, we apply the same characterization to deep ReLU networks with whitened data and prove the same weight alignment holds. As a corollary, we also prove that norm regularized deep ReLU networks yield spline interpolation for one-dimensional datasets which was previously known only for two-layer networks. Furthermore, we provide closed-form solutions for the optimal layer weights when data is rank-one or whitened. The same analysis also applies to architectures with batch normalization even for arbitrary data. Therefore, we obtain a complete explanation for a recent empirical observation termed Neural Collapse where class means collapse to the vertices of a simplex equiangular tight frame.

15 Citations

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Tolga Ergen1, Mert Pilanci1Institutions (1)
25 Feb 2020-arXiv: Learning
Abstract: We develop a convex analytic approach to analyze finite width two-layer ReLU networks. We first prove that an optimal solution to the regularized training problem can be characterized as extreme points of a convex set, where simple solutions are encouraged via its convex geometrical properties. We then leverage this characterization to show that an optimal set of parameters yield linear spline interpolation for regression problems involving one dimensional or rank-one data. We also characterize the classification decision regions in terms of a kernel matrix and minimum $\ell_1$-norm solutions. This is in contrast to Neural Tangent Kernel which is unable to explain predictions of finite width networks. Our convex geometric characterization also provides intuitive explanations of hidden neurons as auto-encoders. In higher dimensions, we show that the training problem can be cast as a finite dimensional convex problem with infinitely many constraints. Then, we apply certain convex relaxations and introduce a cutting-plane algorithm to globally optimize the network. We further analyze the exactness of the relaxations to provide conditions for the convergence to a global optimum. Our analysis also shows that optimal network parameters can be also characterized as interpretable closed-form formulas in some practically relevant special cases.

Topics: Convex set (69%), Convex optimization (67%), Convex geometry (64%) ... show more

14 Citations

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Tolga Ergen1, Mert Pilanci1Institutions (1)
26 Jun 2020-arXiv: Learning
Abstract: We study training of Convolutional Neural Networks (CNNs) with ReLU activations and introduce exact convex optimization formulations with a polynomial complexity with respect to the number of data samples, the number of neurons, and data dimension. More specifically, we develop a convex analytic framework utilizing semi-infinite duality to obtain equivalent convex optimization problems for several two- and three-layer CNN architectures. We first prove that two-layer CNNs can be globally optimized via an $\ell_2$ norm regularized convex program. We then show that multi-layer circular CNN training problems with a single ReLU layer are equivalent to an $\ell_1$ regularized convex program that encourages sparsity in the spectral domain. We also extend these results to three-layer CNNs with two ReLU layers. Furthermore, we present extensions of our approach to different pooling methods, which elucidates the implicit architectural bias as convex regularizers.

Topics: Convex optimization (65%),

5 Citations

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Pichao Wang1, Xue Wang1, Hao Luo1, Jingkai Zhou1  +4 moreInstitutions (1)
Abstract: Vision transformers (ViTs) have been an alternative design paradigm to convolutional neural networks (CNNs). However, the training of ViTs is much harder than CNNs, as it is sensitive to the training parameters, such as learning rate, optimizer and warmup epoch. The reasons for training difficulty are empirically analysed in ~\cite{xiao2021early}, and the authors conjecture that the issue lies with the \textit{patchify-stem} of ViT models and propose that early convolutions help transformers see better. In this paper, we further investigate this problem and extend the above conclusion: only early convolutions do not help for stable training, but the scaled ReLU operation in the \textit{convolutional stem} (\textit{conv-stem}) matters. We verify, both theoretically and empirically, that scaled ReLU in \textit{conv-stem} not only improves training stabilization, but also increases the diversity of patch tokens, thus boosting peak performance with a large margin via adding few parameters and flops. In addition, extensive experiments are conducted to demonstrate that previous ViTs are far from being well trained, further showing that ViTs have great potential to be a better substitute of CNNs.

3 Citations

Open accessPosted Content
12 Jul 2021-arXiv: Learning
Abstract: Generative Adversarial Networks (GANs) are commonly used for modeling complex distributions of data. Both the generators and discriminators of GANs are often modeled by neural networks, posing a non-transparent optimization problem which is non-convex and non-concave over the generator and discriminator, respectively. Such networks are often heuristically optimized with gradient descent-ascent (GDA), but it is unclear whether the optimization problem contains any saddle points, or whether heuristic methods can find them in practice. In this work, we analyze the training of Wasserstein GANs with two-layer neural network discriminators through the lens of convex duality, and for a variety of generators expose the conditions under which Wasserstein GANs can be solved exactly with convex optimization approaches, or can be represented as convex-concave games. Using this convex duality interpretation, we further demonstrate the impact of different activation functions of the discriminator. Our observations are verified with numerical results demonstrating the power of the convex interpretation, with applications in progressive training of convex architectures corresponding to linear generators and quadratic-activation discriminators for CelebA image generation. The code for our experiments is available at this https URL.

3 Citations

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Open accessProceedings Article
Kaiming He1, Xiangyu Zhang1, Shaoqing Ren1, Jian Sun1Institutions (1)
27 Jun 2016-
Abstract: Deeper neural networks are more difficult to train. We present a residual learning framework to ease the training of networks that are substantially deeper than those used previously. We explicitly reformulate the layers as learning residual functions with reference to the layer inputs, instead of learning unreferenced functions. We provide comprehensive empirical evidence showing that these residual networks are easier to optimize, and can gain accuracy from considerably increased depth. On the ImageNet dataset we evaluate residual nets with a depth of up to 152 layers—8× deeper than VGG nets [40] but still having lower complexity. An ensemble of these residual nets achieves 3.57% error on the ImageNet test set. This result won the 1st place on the ILSVRC 2015 classification task. We also present analysis on CIFAR-10 with 100 and 1000 layers. The depth of representations is of central importance for many visual recognition tasks. Solely due to our extremely deep representations, we obtain a 28% relative improvement on the COCO object detection dataset. Deep residual nets are foundations of our submissions to ILSVRC & COCO 2015 competitions1, where we also won the 1st places on the tasks of ImageNet detection, ImageNet localization, COCO detection, and COCO segmentation.

93,356 Citations

Open accessProceedings Article
Diederik P. Kingma1, Jimmy Ba2Institutions (2)
01 Jan 2015-
Abstract: We introduce Adam, an algorithm for first-order gradient-based optimization of stochastic objective functions, based on adaptive estimates of lower-order moments. The method is straightforward to implement, is computationally efficient, has little memory requirements, is invariant to diagonal rescaling of the gradients, and is well suited for problems that are large in terms of data and/or parameters. The method is also appropriate for non-stationary objectives and problems with very noisy and/or sparse gradients. The hyper-parameters have intuitive interpretations and typically require little tuning. Some connections to related algorithms, on which Adam was inspired, are discussed. We also analyze the theoretical convergence properties of the algorithm and provide a regret bound on the convergence rate that is comparable to the best known results under the online convex optimization framework. Empirical results demonstrate that Adam works well in practice and compares favorably to other stochastic optimization methods. Finally, we discuss AdaMax, a variant of Adam based on the infinity norm.

Topics: , Convex optimization (54%), Rate of convergence (52%) ... show more

78,539 Citations

Open accessProceedings Article
03 Dec 2012-
Abstract: We trained a large, deep convolutional neural network to classify the 1.2 million high-resolution images in the ImageNet LSVRC-2010 contest into the 1000 different classes. On the test data, we achieved top-1 and top-5 error rates of 37.5% and 17.0% which is considerably better than the previous state-of-the-art. The neural network, which has 60 million parameters and 650,000 neurons, consists of five convolutional layers, some of which are followed by max-pooling layers, and three fully-connected layers with a final 1000-way softmax. To make training faster, we used non-saturating neurons and a very efficient GPU implementation of the convolution operation. To reduce overriding in the fully-connected layers we employed a recently-developed regularization method called "dropout" that proved to be very effective. We also entered a variant of this model in the ILSVRC-2012 competition and achieved a winning top-5 test error rate of 15.3%, compared to 26.2% achieved by the second-best entry.

73,871 Citations

Open accessProceedings Article
Sergey Ioffe1, Christian Szegedy1Institutions (1)
06 Jul 2015-
Abstract: Training Deep Neural Networks is complicated by the fact that the distribution of each layer's inputs changes during training, as the parameters of the previous layers change. This slows down the training by requiring lower learning rates and careful parameter initialization, and makes it notoriously hard to train models with saturating nonlinearities. We refer to this phenomenon as internal covariate shift, and address the problem by normalizing layer inputs. Our method draws its strength from making normalization a part of the model architecture and performing the normalization for each training mini-batch. Batch Normalization allows us to use much higher learning rates and be less careful about initialization, and in some cases eliminates the need for Dropout. Applied to a state-of-the-art image classification model, Batch Normalization achieves the same accuracy with 14 times fewer training steps, and beats the original model by a significant margin. Using an ensemble of batch-normalized networks, we improve upon the best published result on ImageNet classification: reaching 4.82% top-5 test error, exceeding the accuracy of human raters.

Topics: , Initialization (52%)

23,723 Citations

Open accessProceedings Article
01 Jan 2019-
Abstract: Deep learning frameworks have often focused on either usability or speed, but not both. PyTorch is a machine learning library that shows that these two goals are in fact compatible: it was designed from first principles to support an imperative and Pythonic programming style that supports code as a model, makes debugging easy and is consistent with other popular scientific computing libraries, while remaining efficient and supporting hardware accelerators such as GPUs. In this paper, we detail the principles that drove the implementation of PyTorch and how they are reflected in its architecture. We emphasize that every aspect of PyTorch is a regular Python program under the full control of its user. We also explain how the careful and pragmatic implementation of the key components of its runtime enables them to work together to achieve compelling performance. We demonstrate the efficiency of individual subsystems, as well as the overall speed of PyTorch on several commonly used benchmarks.