The success of deep learning has revealed the application potential of neural networks across the sciences and opened up fundamental theoretical problems. In particular, the fact that learning algorithms based on simple variants of gradient methods are able to find near-optimal minima of highly nonconvex loss functions is an unexpected feature of neural networks. Moreover, such algorithms are able to fit the data even in the presence of noise, and yet they have excellent predictive capabilities. Several empirical results have shown a reproducible correlation between the so-called flatness of the minima achieved by the algorithms and the generalization performance. At the same time, statistical physics results have shown that in nonconvex networks a multitude of narrow minima may coexist with a much smaller number of wide flat minima, which generalize well. Here we show that wide flat minima arise as complex extensive structures, from the coalescence of minima around "high-margin" (i.e., locally robust) configurations. Despite being exponentially rare compared to zero-margin ones, high-margin minima tend to concentrate in particular regions. These minima are in turn surrounded by other solutions of smaller and smaller margin, leading to dense regions of solutions over long distances. Our analysis also provides an alternative analytical method for estimating when flat minima appear and when algorithms begin to find solutions, as the number of model parameters varies.
相關內容
Multi-layer perceptrons (MLP) have proven to be effective scene encoders when combined with higher-dimensional projections of the input, commonly referred to as \textit{positional encoding}. However, scenes with a wide frequency spectrum remain a challenge: choosing high frequencies for positional encoding introduces noise in low structure areas, while low frequencies result in poor fitting of detailed regions. To address this, we propose a progressive positional encoding, exposing a hierarchical MLP structure to incremental sets of frequency encodings. Our model accurately reconstructs scenes with wide frequency bands and learns a scene representation at progressive level of detail \textit{without explicit per-level supervision}. The architecture is modular: each level encodes a continuous implicit representation that can be leveraged separately for its respective resolution, meaning a smaller network for coarser reconstructions. Experiments on several 2D and 3D datasets show improvements in reconstruction accuracy, representational capacity and training speed compared to baselines.
The optimal design of magnetic devices becomes intractable using current computational methods when the number of design parameters is high. The emerging physics-informed deep learning framework has the potential to alleviate this curse of dimensionality. The objective of this paper is to investigate the ability of physics-informed neural networks to learn the magnetic field response as a function of design parameters in the context of a two-dimensional (2-D) magnetostatic problem. Our approach is as follows. We derive the variational principle for 2-D parametric magnetostatic problems, and prove the existence and uniqueness of the solution that satisfies the equations of the governing physics, i.e., Maxwell's equations. We use a deep neural network (DNN) to represent the magnetic field as a function of space and a total of ten parameters that describe geometric features and operating point conditions. We train the DNN by minimizing the physics-informed loss function using a variant of stochastic gradient descent. Subsequently, we conduct systematic numerical studies using a parametric EI-core electromagnet problem. In these studies, we vary the DNN architecture trying more than one hundred different possibilities. For each study, we evaluate the accuracy of the DNN by comparing its predictions to those of finite element analysis. In an exhaustive non-parametric study, we observe that sufficiently parameterized dense networks result in relative errors of less than 1%. Residual connections always improve relative errors for the same number of training iterations. Also, we observe that Fourier encoding features aligned with the device geometry do improve the rate of convergence, albeit higher-order harmonics are not necessary. Finally, we demonstrate our approach on a ten-dimensional problem with parameterized geometry.
Neural networks are powerful function estimators, leading to their status as a paradigm of choice for modeling structured data. However, unlike other structured representations that emphasize the modularity of the problem -- e.g., factor graphs -- neural networks are usually monolithic mappings from inputs to outputs, with a fixed computation order. This limitation prevents them from capturing different directions of computation and interaction between the modeled variables. In this paper, we combine the representational strengths of factor graphs and of neural networks, proposing undirected neural networks (UNNs): a flexible framework for specifying computations that can be performed in any order. For particular choices, our proposed models subsume and extend many existing architectures: feed-forward, recurrent, self-attention networks, auto-encoders, and networks with implicit layers. We demonstrate the effectiveness of undirected neural architectures, both unstructured and structured, on a range of tasks: tree-constrained dependency parsing, convolutional image classification, and sequence completion with attention. By varying the computation order, we show how a single UNN can be used both as a classifier and a prototype generator, and how it can fill in missing parts of an input sequence, making them a promising field for further research.
Knowledge distillation is a strategy of training a student network with guide of the soft output from a teacher network. It has been a successful method of model compression and knowledge transfer. However, currently knowledge distillation lacks a convincing theoretical understanding. On the other hand, recent finding on neural tangent kernel enables us to approximate a wide neural network with a linear model of the network's random features. In this paper, we theoretically analyze the knowledge distillation of a wide neural network. First we provide a transfer risk bound for the linearized model of the network. Then we propose a metric of the task's training difficulty, called data inefficiency. Based on this metric, we show that for a perfect teacher, a high ratio of teacher's soft labels can be beneficial. Finally, for the case of imperfect teacher, we find that hard labels can correct teacher's wrong prediction, which explains the practice of mixing hard and soft labels.
Neural Architecture Search (NAS) was first proposed to achieve state-of-the-art performance through the discovery of new architecture patterns, without human intervention. An over-reliance on expert knowledge in the search space design has however led to increased performance (local optima) without significant architectural breakthroughs, thus preventing truly novel solutions from being reached. In this work we 1) are the first to investigate casting NAS as a problem of finding the optimal network generator and 2) we propose a new, hierarchical and graph-based search space capable of representing an extremely large variety of network types, yet only requiring few continuous hyper-parameters. This greatly reduces the dimensionality of the problem, enabling the effective use of Bayesian Optimisation as a search strategy. At the same time, we expand the range of valid architectures, motivating a multi-objective learning approach. We demonstrate the effectiveness of this strategy on six benchmark datasets and show that our search space generates extremely lightweight yet highly competitive models.
Interpretation of Deep Neural Networks (DNNs) training as an optimal control problem with nonlinear dynamical systems has received considerable attention recently, yet the algorithmic development remains relatively limited. In this work, we make an attempt along this line by reformulating the training procedure from the trajectory optimization perspective. We first show that most widely-used algorithms for training DNNs can be linked to the Differential Dynamic Programming (DDP), a celebrated second-order trajectory optimization algorithm rooted in the Approximate Dynamic Programming. In this vein, we propose a new variant of DDP that can accept batch optimization for training feedforward networks, while integrating naturally with the recent progress in curvature approximation. The resulting algorithm features layer-wise feedback policies which improve convergence rate and reduce sensitivity to hyper-parameter over existing methods. We show that the algorithm is competitive against state-ofthe-art first and second order methods. Our work opens up new avenues for principled algorithmic design built upon the optimal control theory.
Sampling methods (e.g., node-wise, layer-wise, or subgraph) has become an indispensable strategy to speed up training large-scale Graph Neural Networks (GNNs). However, existing sampling methods are mostly based on the graph structural information and ignore the dynamicity of optimization, which leads to high variance in estimating the stochastic gradients. The high variance issue can be very pronounced in extremely large graphs, where it results in slow convergence and poor generalization. In this paper, we theoretically analyze the variance of sampling methods and show that, due to the composite structure of empirical risk, the variance of any sampling method can be decomposed into \textit{embedding approximation variance} in the forward stage and \textit{stochastic gradient variance} in the backward stage that necessities mitigating both types of variance to obtain faster convergence rate. We propose a decoupled variance reduction strategy that employs (approximate) gradient information to adaptively sample nodes with minimal variance, and explicitly reduces the variance introduced by embedding approximation. We show theoretically and empirically that the proposed method, even with smaller mini-batch sizes, enjoys a faster convergence rate and entails a better generalization compared to the existing methods.
When and why can a neural network be successfully trained? This article provides an overview of optimization algorithms and theory for training neural networks. First, we discuss the issue of gradient explosion/vanishing and the more general issue of undesirable spectrum, and then discuss practical solutions including careful initialization and normalization methods. Second, we review generic optimization methods used in training neural networks, such as SGD, adaptive gradient methods and distributed methods, and theoretical results for these algorithms. Third, we review existing research on the global issues of neural network training, including results on bad local minima, mode connectivity, lottery ticket hypothesis and infinite-width analysis.
Graph Convolutional Networks (GCNs) and their variants have experienced significant attention and have become the de facto methods for learning graph representations. GCNs derive inspiration primarily from recent deep learning approaches, and as a result, may inherit unnecessary complexity and redundant computation. In this paper, we reduce this excess complexity through successively removing nonlinearities and collapsing weight matrices between consecutive layers. We theoretically analyze the resulting linear model and show that it corresponds to a fixed low-pass filter followed by a linear classifier. Notably, our experimental evaluation demonstrates that these simplifications do not negatively impact accuracy in many downstream applications. Moreover, the resulting model scales to larger datasets, is naturally interpretable, and yields up to two orders of magnitude speedup over FastGCN.
Generative Adversarial Networks (GANs) are powerful generative models, but suffer from training instability. The recently proposed Wasserstein GAN (WGAN) makes progress toward stable training of GANs, but sometimes can still generate only low-quality samples or fail to converge. We find that these problems are often due to the use of weight clipping in WGAN to enforce a Lipschitz constraint on the critic, which can lead to undesired behavior. We propose an alternative to clipping weights: penalize the norm of gradient of the critic with respect to its input. Our proposed method performs better than standard WGAN and enables stable training of a wide variety of GAN architectures with almost no hyperparameter tuning, including 101-layer ResNets and language models over discrete data. We also achieve high quality generations on CIFAR-10 and LSUN bedrooms.
小貼士
登錄享
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191