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Title: Effective and Efficient Batch Normalization Using Few Uncorrelated Data for Statistics' Estimation

Paper content:
Deep Neural Networks (DNNs) thrive in recent years in which Batch Normalization (BN) plays an indispensable role.
However, it has been observed that BN is costly due to the reduction operations.
In this paper, we propose alleviating the BN’s cost by using only a small fraction of data for mean & variance estimation at each iteration.
The key challenge to reach this goal is how to achieve a satisfactory balance between normalization effectiveness and execution efficiency.
We identify that the effectiveness expects less data correlation while the efficiency expects regular execution pattern.
To this end, we propose two categories of approach: sampling or creating few uncorrelated data for statistics’ estimation with certain strategy constraints.
The former includes “Batch Sampling (BS)” that randomly selects few samples from each batch and “Feature Sampling (FS)” that randomly selects a small patch from each feature map of all samples, and the latter is “Virtual Dataset Normalization (VDN)” that generates few synthetic random samples.
Accordingly, multi-way strategies are designed to reduce the data correlation for accurate estimation and optimize the execution pattern for running acceleration in the meantime.
All the proposed methods are comprehensively evaluated on various DNN models, where an overall training speedup by up to 21.7% on modern GPUs can be practically achieved without the support of any specialized libraries, and the loss of model accuracy and convergence rate are negligible.
Furthermore, our methods demonstrate powerful performance when solving the well-known “micro-batch normalization” problem in the case of tiny batch size.
Recent years, Deep Neural Networks (DNNs) have achieved remarkable success in a wide spectrum of domains such as computer vision BID16 and language modeling BID4 .
The success of DNNs largely relies on the capability of presentation benefit from the deep structure BID5 .
However, training a deep network is so difficult to converge that batch normalization (BN) has been proposed to solve it BID14 .
BN leverages the statistics (mean & variance) of mini-batches to standardize the activations.
It allows the network to go deeper without significant gradient explosion or vanishing BID23 BID14 .
Moreover, previous work has demonstrated that BN enables the use of higher learning rate and less awareness on the initialization BID14 , as well as produces mutual information across samples BID21 or introduces estimation noises BID2 for better generalization.
Despite BN's effectiveness, it is observed that BN introduces considerable training overhead due to the costly reduction operations.
The use of BN can lower the overall training speed (mini second per image) by >45% , especially in deep models.
To alleviate this problem, several methods were reported.
Range Batch Normalization (RBN) BID1 accelerated the forward pass by estimating the variance according to the data range of activations within each batch.
A similar approach, L 1 -norm BN (L1BN) , simplified both the forward and backward passes by replacing the L 2 -norm variance with its L 1 -norm version and re-derived the gradients for backpropagation (BP) training.
Different from the above two methods, Self-normalization BID15 provided another solution which totally eliminates the need of BN operation with an elaborate activation function called "scaled exponential linear unit" (SELU).
SELU can automatically force the activation towards zero mean and unit variance for better convergence.
Nevertheless, all of these methods are not sufficiently effective.
The strengths of L1BN & RBN are very limited since GPU has sufficient resources to optimize the execution speed of complex arithmetic operations such as root for the vanilla calculation of L 2 -norm variance.
Since the derivation of SELU is based on the plain convolutional network, currently it cannot handle other modern structures with skip paths like ResNet and DenseNet.In this paper, we propose mitigating BN's computational cost by just using few data to estimate the mean and variance at each iteration.
Whereas, the key challenge of this way lies at how to preserve the normalization effectiveness of the vanilla BN and improve the execution efficiency in the meantime, i.e. balance the effectiveness-efficiency trade-off.
We identify that the effectiveness preservation expects less data correlation and the efficiency improvement expects regular execution pattern.
This observation motivates us to propose two categories of approach to achieve the goal of effective and efficient BN: sampling or creating few uncorrelated data for statistics' estimation with certain strategy constraints.Sampling data includes "Batch Sampling (BS)" that randomly selects few samples from each batch and "Feature Sampling (FS)" that randomly selects a small patch from each feature map (FM) of all samples; creating data means "Virtual Dataset Normalization (VDN)" that generates few synthetic random samples, inspired by BID22 .
Consequently, multi-way strategies including intra-layer regularity, inter-layer randomness, and static execution graph during each epoch, are designed to reduce the data correlation for accurate estimation and optimize the execution pattern for running acceleration in the meantime.
All the proposed approaches with single-use or joint-use are comprehensively evaluated on various DNN models, where the loss of model accuracy and convergence rate is negligible.
We practically achieve an overall training speedup by up to 21.7% on modern GPUs.
Note that any support of specialized libraries is not needed in our work, which is not like the network pruning BID32 or quantization BID12 requiring extra library for sparse or low-precision computation, respectively.
Most previous acceleration works targeted inference which remained the training inefficient BID26 BID20 BID19 BID31 BID9 , and the rest works for training acceleration were orthogonal to our approach BID7 BID29 .
Additionally, our methods further shows powerful performance when solving the well-known "micro-batch normalization" problem in the case of tiny batch sizes.In summary, the major contributions of this work are summarized as follows.•
We propose a new way to alleviate BN's computational cost by using few data to estimate the mean and variance, in which we identify that the key challenge is to balance the normalization effectiveness via less data correlation and execution efficiency via regular execution pattern.•
We propose two categories of approach to achieve the above goal: sampling (BS/FS) or creating (VDN) few uncorrelated data for statistics' estimation, in which multi-way strategies are designed to reduce the data correlation for accurate estimation and optimize the execution pattern for running acceleration in the meantime. The
approaches can be used alone or jointly.• Various
benchmarks are evaluated, on which up to 21.7% practical acceleration is achieved for overall training on modern GPUs with negligible accuracy loss and without specialized library support.• Our methods
are also extended to the micro-BN problem and achieve advanced performance 1 .In order to make
this paper easier for understanding, we present the organization of the whole paper in FIG0 The activations in one layer for normalization can be described by a d-dimensional activation feature DISPLAYFORM0 , where for each feature we have DISPLAYFORM1 Note that in convolutional (Conv) layer, d is the number of FMs and m equals to the number of points in each FM across all the samples in one batch; while in fully-connected (FC) layer, d and m are the neuron number and batch size, respectively. BN uses the statistics
(mean E[ DISPLAYFORM2 of the intra-batch data for each feature to normalize activation by DISPLAYFORM3 where DISPLAYFORM4 are trainable parameters introduced to recover the representation capability, is a small constant to avoid numerical error, and DISPLAYFORM5 The detailed operations of a BN layer in the backward pass can be found in Appendix C. DISPLAYFORM6 Iter. per second. TAB3 ; (b)
usual optimization
of the
reduction operation using adder tree; (c) the computational graph of
BN in the forward pass (upper) and backward pass (lower); (d) the computation graph of BN
using few data for statistics' estimation in forward pass (upper) and backward pass (lower). x is neuronal activations, µ and
σ denote the mean and standard deviation of x within one batch, respectively, and is the summation operation.From FIG0 , we can see that adding BN will significantly slow down the training speed (iterations per second) by 32%-43% on ImageNet. The reason why BN is costly is that
it contains several "reduction operations", i.e. m j=1 . We offer more thorough data analysis
in Appendix E. If the reduction operations are not optimized, it's computational complexity should be O(m). With the optimized parallel algorithm
proposed in BID3 , the reduction operation is transformed to cascaded adders of depth of log(m) as shown in FIG0 . However, the computational cost is still
high since we usually have m larger than one million. As shown in FIG0 , the red " "s represent
operations that contain summations, which cause the BN inefficiency.
Motivated by the importance but high cost of BN layer, we propose using few data to estimate the mean and variance for training acceleration.
The key challenge towards this goal is how to balance the normalization effectiveness with much less data for statistics' estimation and the execution efficiency with irregular memory access.
To this end, we propose two categories of approach: sampling (BS/FS) or creating (VDN) few uncorrelated data, which can be used alone or jointly.
Specifically, BS randomly selects few samples from each batch, FS randomly selects a small patch from each FM of all samples, and VDN generates few synthetic random samples.
Then, multi-way strategies including intra-layer regularity, inter-layer randomness, and static execution graph are designed to reduce the data correlation and optimize the execution pattern in the meantime.
Comprehensive experiments evidence that the proposed approaches can achieve up to 21.7% overall training acceleration with negligible accuracy loss.
In addition, VDN can also be applied to the micro-BN scenario with advanced performance.
This paper preliminary proves the effectiveness and efficiency of BN using few data for statistics' estimation.
We emphasize that the training speedup is practically achieved on modern GPUs, and we do not need any support of specialized libraries making it easy-to-use.
Developing specialized kernel optimization deserves further investigation for more aggressive execution benefits.

assistant: text: We propose accelerating Batch Normalization (BN) through sampling less correlated data for reduction operations  with regular execution pattern, which achieves up to 2x and 20% speedup for BN itself and the overall training, respectively.