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Register a new userGlobal Convergence and Geometric Characterization of Slow to Fast Weight Evolution in Neural Network Training for Classifying Linearly Non-Separable Data
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Ziang Long
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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In this paper, we study the dynamics of gradient descent in learning neural networks for classification problems. Unlike in existing works, we consider the linearly non-separable case where the training data of different classes lie in orthogonal subspaces. We show that when the network has sufficient (but not exceedingly large) number of neurons, (1) the corresponding minimization problem has a desirable landscape where all critical points are global minima with perfect classification; (2) gradient descent is guaranteed to converge to the global minima. Moreover, we discovered a geometric condition on the network weights so that when it is satisfied, the weight evolution transitions from a slow phase of weight direction spreading to a fast phase of weight convergence. The geometric condition says that the convex hull of the weights projected on the unit sphere contains the origin.
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Normalization is known to help the optimization of deep neural networks. Curiously, different architectures require specialized normalization methods. In this paper, we study what normalization is effective for Graph Neural Networks (GNNs). First, we adapt and evaluate the existing methods from other domains to GNNs. Faster convergence is achieved with InstanceNorm compared to BatchNorm and LayerNorm. We provide an explanation by showing that InstanceNorm serves as a preconditioner for GNNs, but such preconditioning effect is weaker with BatchNorm due to the heavy batch noise in graph datasets. Second, we show that the shift operation in InstanceNorm results in an expressiveness degradation of GNNs for highly regular graphs. We address this issue by proposing GraphNorm with a learnable shift. Empirically, GNNs with GraphNorm converge faster compared to GNNs using other normalization. GraphNorm also improves the generalization of GNNs, achieving better performance on graph classification benchmarks.
We investigate 1) the rate at which refined properties of the empirical risk---in particular, gradients---converge to their population counterparts in standard non-convex learning tasks, and 2) the consequences of this convergence for optimization. Our analysis follows the tradition of norm-based capacity control. We propose vector-valued Rademacher complexities as a simple, composable, and user-friendly tool to derive dimension-free uniform convergence bounds for gradients in non-convex learning problems. As an application of our techniques, we give a new analysis of batch gradient descent methods for non-convex generalized linear models and non-convex robust regression, showing how to use any algorithm that finds approximate stationary points to obtain optimal sample complexity, even when dimension is high or possibly infinite and multiple passes over the dataset are allowed. Moving to non-smooth models we show----in contrast to the smooth case---that even for a single ReLU it is not possible to obtain dimension-independent convergence rates for gradients in the worst case. On the positive side, it is still possible to obtain dimension-independent rates under a new type of distributional assumption.
Many optimizers have been proposed for training deep neural networks, and they often have multiple hyperparameters, which make it tricky to benchmark their performance. In this work, we propose a new benchmarking protocol to evaluate both end-to-end efficiency (training a model from scratch without knowing the best hyperparameter) and data-addition training efficiency (the previously selected hyperparameters are used for periodically re-training the model with newly collected data). For end-to-end efficiency, unlike previous work that assumes random hyperparameter tuning, which over-emphasizes the tuning time, we propose to evaluate with a bandit hyperparameter tuning strategy. A human study is conducted to show that our evaluation protocol matches human tuning behavior better than the random search. For data-addition training, we propose a new protocol for assessing the hyperparameter sensitivity to data shift. We then apply the proposed benchmarking framework to 7 optimizers and various tasks, including computer vision, natural language processing, reinforcement learning, and graph mining. Our results show that there is no clear winner across all the tasks.
This paper focuses on stochastic methods for solving smooth non-convex strongly-concave min-max problems, which have received increasing attention due to their potential applications in deep learning (e.g., deep AUC maximization). However, most of the existing algorithms are slow in practice, and their analysis revolves around the convergence to a nearly stationary point. We consider leveraging the Polyak-L ojasiewicz (PL) condition to design faster stochastic algorithms with stronger convergence guarantee. Although PL condition has been utilized for designing many stochastic minimization algorithms, their applications for non-convex min-max optimization remains rare. In this paper, we propose and analyze proximal epoch-based methods, and establish fast convergence in terms of both {bf the primal objective gap and the duality gap}. Our analysis is interesting in threefold: (i) it is based on a novel Lyapunov function that consists of the primal objective gap and the duality gap of a regularized function; (ii) it only requires a weaker PL condition for establishing the primal objective convergence than that required for the duality gap convergence; (iii) it yields the optimal dependence on the accuracy level $epsilon$, i.e., $O(1/epsilon)$. We also make explicit the dependence on the problem parameters and explore regions of weak convexity parameter that lead to improved dependence on condition numbers. Experiments on deep AUC maximization demonstrate the effectiveness of our methods. Our method (MaxAUC) achieved an AUC of 0.922 on private testing set on {bf CheXpert competition}.
We present the remote stochastic gradient (RSG) method, which computes the gradients at configurable remote observation points, in order to improve the convergence rate and suppress gradient noise at the same time for different curvatures. RSG is further combined with adaptive methods to construct ARSG for acceleration. The method is efficient in computation and memory, and is straightforward to implement. We analyze the convergence properties by modeling the training process as a dynamic system, which provides a guideline to select the configurable observation factor without grid search. ARSG yields $O(1/sqrt{T})$ convergence rate in non-convex settings, that can be further improved to $O(log(T)/T)$ in strongly convex settings. Numerical experiments demonstrate that ARSG achieves both faster convergence and better generalization, compared with popular adaptive methods, such as ADAM, NADAM, AMSGRAD, and RANGER for the tested problems. In particular, for training ResNet-50 on ImageNet, ARSG outperforms ADAM in convergence speed and meanwhile it surpasses SGD in generalization.
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