Block-Normalized Gradient Method: An Empirical Study for Training Deep Neural Network

TL;DR

The paper tackles the challenge of training deep neural networks under non-convex optimization by introducing a block-wise gradient normalization strategy that preserves descent directions while controlling update magnitudes. It formalizes a generic framework where per-layer gradients are normalized as $g_t^i = F_i'(x_t,\xi_t)/\|F_i'(x_t,\xi_t)\|_2$ and the update is $x_{t+1}=x_t-\tau_t\circ g_t$, with either constant or adaptive per-coordinate steps $\tau_t$. Through extensive experiments on MLPs, ResNets, ImageNet, LSTMs, and CNNs for NLP and sentiment tasks, the method consistently accelerates convergence and often improves generalization, with adaptive variants like AdamNG excelling on recurrent nets and constant-step variants like SGD+Momentum performing well on deep CNNs. A convex-case convergence analysis is provided in the appendix, while the primary emphasis remains on practical non-convex optimization benefits. Overall, Block-Normalized Gradient offers a simple, broadly applicable enhancement to first-order optimization in deep learning.

Abstract

In this paper, we propose a generic and simple strategy for utilizing stochastic gradient information in optimization. The technique essentially contains two consecutive steps in each iteration: 1) computing and normalizing each block (layer) of the mini-batch stochastic gradient; 2) selecting appropriate step size to update the decision variable (parameter) towards the negative of the block-normalized gradient. We conduct extensive empirical studies on various non-convex neural network optimization problems, including multi-layer perceptron, convolution neural networks and recurrent neural networks. The results indicate the block-normalized gradient can help accelerate the training of neural networks. In particular, we observe that the normalized gradient methods having constant step size with occasionally decay, such as SGD with momentum, have better performance in the deep convolution neural networks, while those with adaptive step sizes, such as Adam, perform better in recurrent neural networks. Besides, we also observe this line of methods can lead to solutions with better generalization properties, which is confirmed by the performance improvement over strong baselines.

Figures (3)

• Figure 1: The training and testing objective curves on MNIST dataset with multi layer perceptron. From left to right, the layer numbers are 6, 12 and 18 respectively. The first row is the training curve and the second is testing.
• Figure 2: The training and testing curves on CIFAR10 and CIFAR100 datasets with Resnet-110. Left: CIFAR10; Right: CIFAR100; Upper: SGD+Momentum; Lower: Adam. The thick curves are the training while the thin are testing.
• Figure 3: The training and testing objective curves on Penn Tree Bank dataset with LSTM recurrent neural networks. The first row is the training objective while the second is the testing. From left to right, the training sequence (BPTT) length are respectively 40, 400 and 1000. Dropout with 0.5 is imposed.

Theorems & Definitions (2)

• Theorem 1
• Remark 1