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2020-08-06 20:53:44 +00:00
Published as a conference paper at ICLR 2016
DEEP COMPRESSION : C OMPRESSING DEEP NEURAL
NETWORKS WITH PRUNING , T RAINED QUANTIZATION
AND HUFFMAN CODING
Song Han
Stanford University, Stanford, CA 94305, USA
songhan@stanford.edu
Huizi Mao
arXiv:1510.00149v5 [cs.CV] 15 Feb 2016 Tsinghua University, Beijing, 100084, China
mhz12@mails.tsinghua.edu.cn
William J. Dally
Stanford University, Stanford, CA 94305, USA
NVIDIA, Santa Clara, CA 95050, USA
dally@stanford.edu
ABSTRACT
Neural networks are both computationally intensive and memory intensive, making
them difficult to deploy on embedded systems with limited hardware resources. To
address this limitation, we introduce “deep compression”, a three stage pipeline:
pruning, trained quantization and Huffman coding, that work together to reduce
the storage requirement of neural networks by35to49without affecting their
accuracy. Our method first prunes the network by learning only the important
connections. Next, we quantize the weights to enforce weight sharing, finally, we
apply Huffman coding. After the first two steps we retrain the network to fine
tune the remaining connections and the quantized centroids. Pruning, reduces the
number of connections by9to13; Quantization then reduces the number of
bits that represent each connection from 32 to 5. On the ImageNet dataset, our
method reduced the storage required by AlexNet by35, from 240MB to 6.9MB,
without loss of accuracy. Our method reduced the size of VGG-16 by49from
552MB to 11.3MB, again with no loss of accuracy. This allows fitting the model
into on-chip SRAM cache rather than off-chip DRAM memory. Our compression
method also facilitates the use of complex neural networks in mobile applications
where application size and download bandwidth are constrained. Benchmarked on
CPU, GPU and mobile GPU, compressed network has3to4layerwise speedup
and3to7better energy efficiency.
1 I NTRODUCTION
Deep neural networks have evolved to the state-of-the-art technique for computer vision tasks
(Krizhevsky et al., 2012)(Simonyan & Zisserman, 2014). Though these neural networks are very
powerful, the large number of weights consumes considerable storage and memory bandwidth. For
example, the AlexNet Caffemodel is over 200MB, and the VGG-16 Caffemodel is over 500MB
(BVLC). This makes it difficult to deploy deep neural networks on mobile system.
First, for many mobile-first companies such as Baidu and Facebook, various apps are updated via
different app stores, and they are very sensitive to the size of the binary files. For example, App
Store has the restriction “apps above 100 MB will not download until you connect to Wi-Fi”. As a
result, a feature that increases the binary size by 100MB will receive much more scrutiny than one
that increases it by 10MB. Although having deep neural networks running on mobile has many great
1 Published as a conference paper at ICLR 2016
Quantization: less bits per weight
Pruning: less number of weights Huffman Encoding
Cluster the Weights
Train Connectivity Encode Weightsoriginal same same same
network accuracy Generate Code Book accuracy accuracy
Prune Connections
original 9x-13x Quantize the Weights 27x-31x Encode Index 35x-49x
size reduction with Code Book reduction reduction
Train Weights
Retrain Code Book
Figure 1: The three stage compression pipeline: pruning, quantization and Huffman coding. Pruning
reduces the number of weights by10, while quantization further improves the compression rate:
between27and31. Huffman coding gives more compression: between35and49. The
compression rate already included the meta-data for sparse representation. The compression scheme
doesn’t incur any accuracy loss.
features such as better privacy, less network bandwidth and real time processing, the large storage
overhead prevents deep neural networks from being incorporated into mobile apps.
The second issue is energy consumption. Running large neural networks require a lot of memory
bandwidth to fetch the weights and a lot of computation to do dot products— which in turn consumes
considerable energy. Mobile devices are battery constrained, making power hungry applications such
as deep neural networks hard to deploy.
Energy consumption is dominated by memory access. Under 45nm CMOS technology, a 32 bit
floating point add consumes0:9pJ, a 32bit SRAM cache access takes5pJ, while a 32bit DRAM
memory access takes640pJ, which is 3 orders of magnitude of an add operation. Large networks
do not fit in on-chip storage and hence require the more costly DRAM accesses. Running a 1 billion
connection neural network, for example, at 20fps would require(20Hz)(1G)(640pJ) = 12:8Wjust
for DRAM access - well beyond the power envelope of a typical mobile device.
Our goal is to reduce the storage and energy required to run inference on such large networks so they
can be deployed on mobile devices. To achieve this goal, we present “deep compression”: a three-
stage pipeline (Figure 1) to reduce the storage required by neural network in a manner that preserves
the original accuracy. First, we prune the networking by removing the redundant connections, keeping
only the most informative connections. Next, the weights are quantized so that multiple connections
share the same weight, thus only the codebook (effective weights) and the indices need to be stored.
Finally, we apply Huffman coding to take advantage of the biased distribution of effective weights.
Our main insight is that, pruning and trained quantization are able to compress the network without
interfering each other, thus lead to surprisingly high compression rate. It makes the required storage
so small (a few megabytes) that all weights can be cached on chip instead of going to off-chip DRAM
which is energy consuming. Based on “deep compression”, the EIE hardware accelerator Han et al.
(2016) was later proposed that works on the compressed model, achieving significant speedup and
energy efficiency improvement.
2 N ETWORK PRUNING
Network pruning has been widely studied to compress CNN models. In early work, network pruning
proved to be a valid way to reduce the network complexity and over-fitting (LeCun et al., 1989;
Hanson & Pratt, 1989; Hassibi et al., 1993; Strom, 1997). Recently Han et al. (2015) pruned state- ¨
of-the-art CNN models with no loss of accuracy. We build on top of that approach. As shown on
the left side of Figure 1, we start by learning the connectivity via normal network training. Next, we
prune the small-weight connections: all connections with weights below a threshold are removed
from the network. Finally, we retrain the network to learn the final weights for the remaining sparse
connections. Pruning reduced the number of parameters by9and13for AlexNet and VGG-16
model.
2 Published as a conference paper at ICLR 2016
Figure 2: Representing the matrix sparsity with relative index. Padding filler zero to prevent overflow.
weights cluster index fine-tuned
(32 bit float) (2 bit uint) centroids centroids
2.09-0.981.48 0.09 3 0 2 1 3:2.00 1.96
0.05-0.14-1.082.12 cluster 1 1 0 3 2:1.50 1.48 3 0 1 1
-0.911.92 0 -1.03 0 3 1 0 1:0.00 -0.04 1 1 0 3
0 3 1 0
1.87 0 1.53 1.49 3 1 2 2 0:-1.00 lr -0.97
3 1 2 2
gradient
-0.03-0.010.03 0.02 -0.030.12 0.02-0.07 0.04
-0.010.01-0.020.12group by0.03 0.01-0.02 reduce 0.02
-0.010.02 0.04 0.01 0.02-0.010.01 0.04-0.02 0.04
-0.07-0.020.01-0.02 -0.01-0.02-0.010.01 -0.03
Figure 3: Weight sharing by scalar quantization (top) and centroids fine-tuning (bottom).
We store the sparse structure that results from pruning using compressed sparse row (CSR) or
compressed sparse column (CSC) format, which requires2a+n+1numbers, whereais the number
of non-zero elements andnis the number of rows or columns.
To compress further, we store the index difference instead of the absolute position, and encode this
difference in 8 bits for conv layer and 5 bits for fc layer. When we need an index difference larger
than the bound, we the zero padding solution shown in Figure 2: in case when the difference exceeds
8, the largest 3-bit (as an example) unsigned number, we add a filler zero.
3 T RAINED QUANTIZATION AND WEIGHT SHARING
Network quantization and weight sharing further compresses the pruned network by reducing the
number of bits required to represent each weight. We limit the number of effective weights we need to
store by having multiple connections share the same weight, and then fine-tune those shared weights.
Weight sharing is illustrated in Figure 3. Suppose we have a layer that has 4 input neurons and 4
output neurons, the weight is a44matrix. On the top left is the44weight matrix, and on the
bottom left is the44gradient matrix. The weights are quantized to 4 bins (denoted with 4 colors),
all the weights in the same bin share the same value, thus for each weight, we then need to store only
a small index into a table of shared weights. During update, all the gradients are grouped by the color
and summed together, multiplied by the learning rate and subtracted from the shared centroids from
last iteration. For pruned AlexNet, we are able to quantize to 8-bits (256 shared weights) for each
CONV layers, and 5-bits (32 shared weights) for each FC layer without any loss of accuracy.
To calculate the compression rate, givenkclusters, we only needlog 2 (k)bits to encode the index. In
general, for a network withnconnections and each connection is represented withbbits, constraining
the connections to have onlykshared weights will result in a compression rate of:
nbr= (1)nlog 2 (k) +kb
For example, Figure 3 shows the weights of a single layer neural network with four input units and
four output units. There are44 = 16weights originally but there are only4shared weights: similar
weights are grouped together to share the same value. Originally we need to store 16 weights each
3 Published as a conference paper at ICLR 2016
1.0 20000
CDF linear quantization
PDF nonlinear quantization by
density initialization clustring and finetuning 0.8 linear initialization 15000
random initialization
cummulative distribution 0.6
10000
density 0.4
5000
0.2
0
0.0
0.10 0.05 0.00 0.05 0.10 0.04 0.02 0.00 0.02 0.04 0.06 weight value weight value
Figure 4: Left: Three different methods for centroids initialization. Right: Distribution of weights
(blue) and distribution of codebook before (green cross) and after fine-tuning (red dot).
has 32 bits, now we need to store only 4 effective weights (blue, green, red and orange), each has 32
bits, together with 16 2-bit indices giving a compression rate of1632=(432 + 216) = 3:2
3.1 W EIGHT SHARING
We use k-means clustering to identify the shared weights for each layer of a trained network, so that
all the weights that fall into the same cluster will share the same weight. Weights are not shared across
layers. We partitionnoriginal weightsW=fw1 ;w 2 ;:::;w n gintokclustersC=fc1 ;c 2 ;:::;c k g,
nk, so as to minimize the within-cluster sum of squares (WCSS):
Xk Xarg min jwci j2 (2)
C i=1w2ci
Different from HashNet (Chen et al., 2015) where weight sharing is determined by a hash function
before the networks sees any training data, our method determines weight sharing after a network is
fully trained, so that the shared weights approximate the original network.
3.2 I NITIALIZATION OF SHARED WEIGHTS
Centroid initialization impacts the quality of clustering and thus affects the network’s prediction
accuracy. We examine three initialization methods: Forgy(random), density-based, and linear
initialization. In Figure 4 we plotted the original weights’ distribution of conv3 layer in AlexNet
(CDF in blue, PDF in red). The weights forms a bimodal distribution after network pruning. On the
bottom it plots the effective weights (centroids) with 3 different initialization methods (shown in blue,
red and yellow). In this example, there are 13 clusters.
Forgy(random) initialization randomly chooses k observations from the data set and uses these as
the initial centroids. The initialized centroids are shown in yellow. Since there are two peaks in the
bimodal distribution, Forgy method tend to concentrate around those two peaks.
Density-basedinitialization linearly spaces the CDF of the weights in the y-axis, then finds the
horizontal intersection with the CDF, and finally finds the vertical intersection on the x-axis, which
becomes a centroid, as shown in blue dots. This method makes the centroids denser around the two
peaks, but more scatted than the Forgy method.
Linearinitialization linearly spaces the centroids between the [min, max] of the original weights.
This initialization method is invariant to the distribution of the weights and is the most scattered
compared with the former two methods.
Larger weights play a more important role than smaller weights (Han et al., 2015), but there are fewer
of these large weights. Thus for both Forgy initialization and density-based initialization, very few
centroids have large absolute value which results in poor representation of these few large weights.
Linear initialization does not suffer from this problem. The experiment section compares the accuracy
4 Published as a conference paper at ICLR 2016
100000 220000
75000 165000
Count
Count 50000 110000
25000 55000
0 0
1 3 5 7 91113151719212325272931 1 3 5 7 9 1113151719212325272931
Weight Index (32 Effective Weights) Sparse Matrix Location Index (Max Diff is 32)
Figure 5: Distribution for weight (Left) and index (Right). The distribution is biased.
of different initialization methods after clustering and fine-tuning, showing that linear initialization
works best.
3.3 F EED -FORWARD AND BACK -PROPAGATION
The centroids of the one-dimensional k-means clustering are the shared weights. There is one level
of indirection during feed forward phase and back-propagation phase looking up the weight table.
An index into the shared weight table is stored for each connection. During back-propagation, the
gradient for each shared weight is calculated and used to update the shared weight. This procedure is
shown in Figure 3.
We denote the loss byL, the weight in theith column andjth row byWij , the centroid index of
elementWi;j byIij , thekth centroid of the layer byCk . By using the indicator function1(:), the
gradient of the centroids is calculated as:
@L X @L @W X L= ij @= 1(I@C k @W @W ij =k) (3)
i;j ij @C k i;j ij
4 H UFFMAN CODING
A Huffman code is an optimal prefix code commonly used for lossless data compression(Van Leeuwen,
1976). It uses variable-length codewords to encode source symbols. The table is derived from the
occurrence probability for each symbol. More common symbols are represented with fewer bits.
Figure 5 shows the probability distribution of quantized weights and the sparse matrix index of the
last fully connected layer in AlexNet. Both distributions are biased: most of the quantized weights are
distributed around the two peaks; the sparse matrix index difference are rarely above 20. Experiments
show that Huffman coding these non-uniformly distributed values saves20%30%of network
storage.
5 E XPERIMENTS
We pruned, quantized, and Huffman encoded four networks: two on MNIST and two on ImageNet
data-sets. The network parameters and accuracy- 1 before and after pruning are shown in Table 1. The
compression pipeline saves network storage by35to49across different networks without loss
of accuracy. The total size of AlexNet decreased from 240MB to 6.9MB, which is small enough to
be put into on-chip SRAM, eliminating the need to store the model in energy-consuming DRAM
memory.
Training is performed with the Caffe framework (Jia et al., 2014). Pruning is implemented by adding
a mask to the blobs to mask out the update of the pruned connections. Quantization and weight
sharing are implemented by maintaining a codebook structure that stores the shared weight, and
group-by-index after calculating the gradient of each layer. Each shared weight is updated with all
the gradients that fall into that bucket. Huffman coding doesn’t require training and is implemented
offline after all the fine-tuning is finished.
5.1 L E NET -300-100 AND LE NET -5 ON MNIST
We first experimented on MNIST dataset with LeNet-300-100 and LeNet-5 network (LeCun et al.,
1998). LeNet-300-100 is a fully connected network with two hidden layers, with 300 and 100
1 Reference model is from Caffe model zoo, accuracy is measured without data augmentation
5 Published as a conference paper at ICLR 2016
Table 1: The compression pipeline can save35to49parameter storage with no loss of accuracy.
CompressNetwork Top-1 Error Top-5 Error Parameters Rate
LeNet-300-100 Ref 1.64% - 1070 KB
LeNet-300-100 Compressed 1.58% - 27 KB 40
LeNet-5 Ref 0.80% - 1720 KB
LeNet-5 Compressed 0.74% - 44 KB 39
AlexNet Ref 42.78% 19.73% 240 MB
AlexNet Compressed 42.78% 19.70% 6.9 MB 35
VGG-16 Ref 31.50% 11.32% 552 MB
VGG-16 Compressed 31.17% 10.91% 11.3 MB 49
Table 2: Compression statistics for LeNet-300-100. P: pruning, Q:quantization, H:Huffman coding.
Weight Weight Index Index Compress CompressWeights%Layer #Weights bits bits bits bits rate rate(P) (P+Q) (P+Q+H) (P+Q) (P+Q+H) (P+Q) (P+Q+H)
ip1 235K 8% 6 4.4 5 3.7 3.1% 2.32%
ip2 30K 9% 6 4.4 5 4.3 3.8% 3.04%
ip3 1K 26% 6 4.3 5 3.2 15.7% 12.70%
Total 266K 8%(12) 6 5.1 5 3.7 3.1% (32) 2.49% (40)
Table 3: Compression statistics for LeNet-5. P: pruning, Q:quantization, H:Huffman coding.
Weight Weight Index Index Compress CompressWeights%Layer #Weights bits bits bits bits rate rate(P) (P+Q) (P+Q+H) (P+Q) (P+Q+H) (P+Q) (P+Q+H)
conv1 0.5K 66% 8 7.2 5 1.5 78.5% 67.45%
conv2 25K 12% 8 7.2 5 3.9 6.0% 5.28%
ip1 400K 8% 5 4.5 5 4.5 2.7% 2.45%
ip2 5K 19% 5 5.2 5 3.7 6.9% 6.13%
Total 431K 8%(12) 5.3 4.1 5 4.4 3.05% (33) 2.55% (39)
neurons each, which achieves 1.6% error rate on Mnist. LeNet-5 is a convolutional network that
has two convolutional layers and two fully connected layers, which achieves 0.8% error rate on
Mnist. Table 2 and table 3 show the statistics of the compression pipeline. The compression rate
includes the overhead of the codebook and sparse indexes. Most of the saving comes from pruning
and quantization (compressed32), while Huffman coding gives a marginal gain (compressed40)
5.2 A LEX NET ON IMAGE NET
We further examine the performance of Deep Compression on the ImageNet ILSVRC-2012 dataset,
which has 1.2M training examples and 50k validation examples. We use the AlexNet Caffe model as
the reference model, which has 61 million parameters and achieved a top-1 accuracy of 57.2% and a
top-5 accuracy of 80.3%. Table 4 shows that AlexNet can be compressed to2:88%of its original size
without impacting accuracy. There are 256 shared weights in each CONV layer, which are encoded
with 8 bits, and 32 shared weights in each FC layer, which are encoded with only 5 bits. The relative
sparse index is encoded with 4 bits. Huffman coding compressed additional 22%, resulting in35
compression in total.
5.3 VGG-16 ON IMAGE NET
With promising results on AlexNet, we also looked at a larger, more recent network, VGG-16 (Si-
monyan & Zisserman, 2014), on the same ILSVRC-2012 dataset. VGG-16 has far more convolutional
layers but still only three fully-connected layers. Following a similar methodology, we aggressively
compressed both convolutional and fully-connected layers to realize a significant reduction in the
number of effective weights, shown in Table5.
The VGG16 network as a whole has been compressed by49. Weights in the CONV layers are
represented with 8 bits, and FC layers use 5 bits, which does not impact the accuracy. The two largest
fully-connected layers can each be pruned to less than 1.6% of their original size. This reduction
6 Published as a conference paper at ICLR 2016
Table 4: Compression statistics for AlexNet. P: pruning, Q: quantization, H:Huffman coding.
Weight Weight Index Index Compress CompressWeights%Layer #Weights bits bits bits bits rate rate(P) (P+Q) (P+Q+H) (P+Q) (P+Q+H) (P+Q) (P+Q+H)
conv1 35K 84% 8 6.3 4 1.2 32.6% 20.53%
conv2 307K 38% 8 5.5 4 2.3 14.5% 9.43%
conv3 885K 35% 8 5.1 4 2.6 13.1% 8.44%
conv4 663K 37% 8 5.2 4 2.5 14.1% 9.11%
conv5 442K 37% 8 5.6 4 2.5 14.0% 9.43%
fc6 38M 9% 5 3.9 4 3.2 3.0% 2.39%
fc7 17M 9% 5 3.6 4 3.7 3.0% 2.46%
fc8 4M 25% 5 4 4 3.2 7.3% 5.85%
Total 61M 11%(9) 5.4 4 4 3.2 3.7% (27) 2.88% (35)
Table 5: Compression statistics for VGG-16. P: pruning, Q:quantization, H:Huffman coding.
Weigh Weight Index Index Compress CompressWeights%Layer #Weights bits bits bits bits rate rate(P) (P+Q) (P+Q+H) (P+Q) (P+Q+H) (P+Q) (P+Q+H)
conv11 2K 58% 8 6.8 5 1.7 40.0% 29.97%
conv12 37K 22% 8 6.5 5 2.6 9.8% 6.99%
conv21 74K 34% 8 5.6 5 2.4 14.3% 8.91%
conv22 148K 36% 8 5.9 5 2.3 14.7% 9.31%
conv31 295K 53% 8 4.8 5 1.8 21.7% 11.15%
conv32 590K 24% 8 4.6 5 2.9 9.7% 5.67%
conv33 590K 42% 8 4.6 5 2.2 17.0% 8.96%
conv41 1M 32% 8 4.6 5 2.6 13.1% 7.29%
conv42 2M 27% 8 4.2 5 2.9 10.9% 5.93%
conv43 2M 34% 8 4.4 5 2.5 14.0% 7.47%
conv51 2M 35% 8 4.7 5 2.5 14.3% 8.00%
conv52 2M 29% 8 4.6 5 2.7 11.7% 6.52%
conv53 2M 36% 8 4.6 5 2.3 14.8% 7.79%
fc6 103M 4% 5 3.6 5 3.5 1.6% 1.10%
fc7 17M 4% 5 4 5 4.3 1.5% 1.25%
fc8 4M 23% 5 4 5 3.4 7.1% 5.24%
Total 138M 7.5%(13) 6.4 4.1 5 3.1 3.2% (31) 2.05% (49)
is critical for real time image processing, where there is little reuse of these layers across images
(unlike batch processing). This is also critical for fast object detection algorithms where one CONV
pass is used by many FC passes. The reduced layers will fit in an on-chip SRAM and have modest
bandwidth requirements. Without the reduction, the bandwidth requirements are prohibitive.
6 D ISCUSSIONS
6.1 P RUNING AND QUANTIZATION WORKING TOGETHER
Figure 6 shows the accuracy at different compression rates for pruning and quantization together
or individually. When working individually, as shown in the purple and yellow lines, accuracy of
pruned network begins to drop significantly when compressed below 8% of its original size; accuracy
of quantized network also begins to drop significantly when compressed below 8% of its original
size. But when combined, as shown in the red line, the network can be compressed to 3% of original
size with no loss of accuracy. On the far right side compared the result of SVD, which is inexpensive
but has a poor compression rate.
The three plots in Figure 7 show how accuracy drops with fewer bits per connection for CONV layers
(left), FC layers (middle) and all layers (right). Each plot reports both top-1 and top-5 accuracy.
Dashed lines only applied quantization but without pruning; solid lines did both quantization and
pruning. There is very little difference between the two. This shows that pruning works well with
quantization.
Quantization works well on pruned network because unpruned AlexNet has 60 million weights to
quantize, while pruned AlexNet has only 6.7 million weights to quantize. Given the same amount of
centroids, the latter has less error.
7 Published as a conference paper at ICLR 2016
Figure 6: Accuracy v.s. compression rate under different compression methods. Pruning and
quantization works best when combined.
Figure 7: Pruning doesn’t hurt quantization. Dashed: quantization on unpruned network. Solid:
quantization on pruned network; Accuracy begins to drop at the same number of quantization bits
whether or not the network has been pruned. Although pruning made the number of parameters less,
quantization still works well, or even better(3 bits case on the left figure) as in the unpruned network.
Figure 8: Accuracy of different initialization methods. Left: top-1 accuracy. Right: top-5 accuracy.
Linear initialization gives best result.
The first two plots in Figure 7 show that CONV layers require more bits of precision than FC layers.
For CONV layers, accuracy drops significantly below 4 bits, while FC layer is more robust: not until
2 bits did the accuracy drop significantly.
6.2 C ENTROID INITIALIZATION
Figure 8 compares the accuracy of the three different initialization methods with respect to top-1
accuracy (Left) and top-5 accuracy (Right). The network is quantized to28bits as shown on
x-axis. Linear initialization outperforms the density initialization and random initialization in all
cases except at 3 bits.
The initial centroids of linear initialization spread equally across the x-axis, from the min value to the
max value. That helps to maintain the large weights as the large weights play a more important role
than smaller ones, which is also shown in network pruning Han et al. (2015). Neither random nor
density-based initialization retains large centroids. With these initialization methods, large weights are
clustered to the small centroids because there are few large weights. In contrast, linear initialization
allows large weights a better chance to form a large centroid.
8 Published as a conference paper at ICLR 2016
Figure 9: Compared with the original network, pruned network layer achieved3speedup on CPU,
3:5on GPU and4:2on mobile GPU on average. Batch size = 1 targeting real time processing.
Performance number normalized to CPU.
Figure 10: Compared with the original network, pruned network layer takes7less energy on CPU,
3:3less on GPU and4:2less on mobile GPU on average. Batch size = 1 targeting real time
processing. Energy number normalized to CPU.
6.3 S PEEDUP AND ENERGY EFFICIENCY
Deep Compression is targeting extremely latency-focused applications running on mobile, which
requires real-time inference, such as pedestrian detection on an embedded processor inside an
autonomous vehicle. Waiting for a batch to assemble significantly adds latency. So when bench-
marking the performance and energy efficiency, we consider the case when batch size = 1. The cases
of batching are given in Appendix A.
Fully connected layer dominates the model size (more than90%) and got compressed the most by
Deep Compression (96%weights pruned in VGG-16). In state-of-the-art object detection algorithms
such as fast R-CNN (Girshick, 2015), upto38%computation time is consumed on FC layers on
uncompressed model. So it’s interesting to benchmark on FC layers, to see the effect of Deep
Compression on performance and energy. Thus we setup our benchmark on FC6, FC7, FC8 layers of
AlexNet and VGG-16. In the non-batched case, the activation matrix is a vector with just one column,
so the computation boils down to dense / sparse matrix-vector multiplication for original / pruned
model, respectively. Since current BLAS library on CPU and GPU doesn’t support indirect look-up
and relative indexing, we didn’t benchmark the quantized model.
We compare three different off-the-shelf hardware: the NVIDIA GeForce GTX Titan X and the Intel
Core i7 5930K as desktop processors (same package as NVIDIA Digits Dev Box) and NVIDIA Tegra
K1 as mobile processor. To run the benchmark on GPU, we used cuBLAS GEMV for the original
dense layer. For the pruned sparse layer, we stored the sparse matrix in in CSR format, and used
cuSPARSE CSRMV kernel, which is optimized for sparse matrix-vector multiplication on GPU. To
run the benchmark on CPU, we used MKL CBLAS GEMV for the original dense model and MKL
SPBLAS CSRMV for the pruned sparse model.
To compare power consumption between different systems, it is important to measure power at a
consistent manner (NVIDIA, b). For our analysis, we are comparing pre-regulation power of the
entire application processor (AP) / SOC and DRAM combined. On CPU, the benchmark is running on
single socket with a single Haswell-E class Core i7-5930K processor. CPU socket and DRAM power
are as reported by thepcm-powerutility provided by Intel. For GPU, we usednvidia-smi
utility to report the power of Titan X. For mobile GPU, we use a Jetson TK1 development board and
measured the total power consumption with a power-meter. We assume15%AC to DC conversion
loss,85%regulator efficiency and15%power consumed by peripheral components (NVIDIA, a) to
report the AP+DRAM power for Tegra K1.
9 Published as a conference paper at ICLR 2016
Table 6: Accuracy of AlexNet with different aggressiveness of weight sharing and quantization. 8/5
bit quantization has no loss of accuracy; 8/4 bit quantization, which is more hardware friendly, has
negligible loss of accuracy of 0.01%; To be really aggressive, 4/2 bit quantization resulted in 1.99%
and 2.60% loss of accuracy.
Top-1 Error Top-5 Error#CONV bits / #FC bits Top-1 Error Top-5 Error Increase Increase
32bits / 32bits 42.78% 19.73% - -
8 bits / 5 bits 42.78% 19.70% 0.00% -0.03%
8 bits / 4 bits 42.79% 19.73% 0.01% 0.00%
4 bits / 2 bits 44.77% 22.33% 1.99% 2.60%
The ratio of memory access over computation characteristic with and without batching is different.
When the input activations are batched to a matrix the computation becomes matrix-matrix multipli-
cation, where locality can be improved by blocking. Matrix could be blocked to fit in caches and
reused efficiently. In this case, the amount of memory access isO(n2 ), and that of computation is
O(n3 ), the ratio between memory access and computation is in the order of1=n.
In real time processing when batching is not allowed, the input activation is a single vector and the
computation is matrix-vector multiplication. In this case, the amount of memory access isO(n2 ), and
the computation isO(n2 ), memory access and computation are of the same magnitude (as opposed
to1=n). That indicates MV is more memory-bounded than MM. So reducing the memory footprint
is critical for the non-batching case.
Figure 9 illustrates the speedup of pruning on different hardware. There are 6 columns for each
benchmark, showing the computation time of CPU / GPU / TK1 on dense / pruned network. Time is
normalized to CPU. When batch size = 1, pruned network layer obtained3to4speedup over the
dense network on average because it has smaller memory footprint and alleviates the data transferring
overhead, especially for large matrices that are unable to fit into the caches. For example VGG16’s
FC6 layer, the largest layer in our experiment, contains2508840964Bytes400MBdata,
which is far from the capacity of L3 cache.
In those latency-tolerating applications , batching improves memory locality, where weights could
be blocked and reused in matrix-matrix multiplication. In this scenario, pruned network no longer
shows its advantage. We give detailed timing results in Appendix A.
Figure 10 illustrates the energy efficiency of pruning on different hardware. We multiply power
consumption with computation time to get energy consumption, then normalized to CPU to get
energy efficiency. When batch size = 1, pruned network layer consumes3to7less energy over
the dense network on average. Reported bynvidia-smi, GPU utilization is99%for both dense
and sparse cases.
6.4 R ATIO OF WEIGHTS , I NDEX AND CODEBOOK
Pruning makes the weight matrix sparse, so extra space is needed to store the indexes of non-zero
elements. Quantization adds storage for a codebook. The experiment section has already included
these two factors. Figure 11 shows the breakdown of three different components when quantizing
four networks. Since on average both the weights and the sparse indexes are encoded with 5 bits,
their storage is roughly half and half. The overhead of codebook is very small and often negligible.
Figure 11: Storage ratio of weight, index and codebook.
10 Published as a conference paper at ICLR 2016
Table 7: Comparison with other compression methods on AlexNet. (Collins & Kohli, 2014) reduced
the parameters by4and with inferior accuracy. Deep Fried Convnets(Yang et al., 2014) worked
on fully connected layers and reduced the parameters by less than4. SVD save parameters but
suffers from large accuracy loss as much as 2%. Network pruning (Han et al., 2015) reduced the
parameters by9, not including index overhead. On other networks similar to AlexNet, (Denton
et al., 2014) exploited linear structure of convnets and compressed the network by2:4to13:4
layer wise, with 0.9% accuracy loss on compressing a single layer. (Gong et al., 2014) experimented
with vector quantization and compressed the network by16to24, incurring 1% accuracy loss.
CompressNetwork Top-1 Error Top-5 Error Parameters Rate
Baseline Caffemodel (BVLC) 42.78% 19.73% 240MB 1
Fastfood-32-AD (Yang et al., 2014) 41.93% - 131MB 2
Fastfood-16-AD (Yang et al., 2014) 42.90% - 64MB 3:7
Collins & Kohli (Collins & Kohli, 2014) 44.40% - 61MB 4
SVD (Denton et al., 2014) 44.02% 20.56% 47.6MB 5
Pruning (Han et al., 2015) 42.77% 19.67% 27MB 9
Pruning+Quantization 42.78% 19.70% 8.9MB 27
Pruning+Quantization+Huffman 42.78% 19.70% 6.9MB 35
7 R ELATED WORK
Neural networks are typically over-parametrized, and there is significant redundancy for deep learning
models(Denil et al., 2013). This results in a waste of both computation and memory usage. There
have been various proposals to remove the redundancy: Vanhoucke et al. (2011) explored a fixed-
point implementation with 8-bit integer (vs 32-bit floating point) activations. Hwang & Sung
(2014) proposed an optimization method for the fixed-point network with ternary weights and 3-bit
activations. Anwar et al. (2015) quantized the neural network using L2 error minimization and
achieved better accuracy on MNIST and CIFAR-10 datasets.Denton et al. (2014) exploited the linear
structure of the neural network by finding an appropriate low-rank approximation of the parameters
and keeping the accuracy within 1% of the original model.
The empirical success in this paper is consistent with the theoretical study of random-like sparse
networks with +1/0/-1 weights (Arora et al., 2014), which have been proved to enjoy nice properties
(e.g. reversibility), and to allow a provably polynomial time algorithm for training.
Much work has been focused on binning the network parameters into buckets, and only the values in
the buckets need to be stored. HashedNets(Chen et al., 2015) reduce model sizes by using a hash
function to randomly group connection weights, so that all connections within the same hash bucket
share a single parameter value. In their method, the weight binning is pre-determined by the hash
function, instead of being learned through training, which doesn’t capture the nature of images. Gong
et al. (2014) compressed deep convnets using vector quantization, which resulted in 1% accuracy
loss. Both methods studied only the fully connected layer, ignoring the convolutional layers.
There have been other attempts to reduce the number of parameters of neural networks by replacing
the fully connected layer with global average pooling. The Network in Network architecture(Lin et al.,
2013) and GoogLenet(Szegedy et al., 2014) achieves state-of-the-art results on several benchmarks by
adopting this idea. However, transfer learning, i.e. reusing features learned on the ImageNet dataset
and applying them to new tasks by only fine-tuning the fully connected layers, is more difficult with
this approach. This problem is noted by Szegedy et al. (2014) and motivates them to add a linear
layer on the top of their networks to enable transfer learning.
Network pruning has been used both to reduce network complexity and to reduce over-fitting. An
early approach to pruning was biased weight decay (Hanson & Pratt, 1989). Optimal Brain Damage
(LeCun et al., 1989) and Optimal Brain Surgeon (Hassibi et al., 1993) prune networks to reduce
the number of connections based on the Hessian of the loss function and suggest that such pruning
is more accurate than magnitude-based pruning such as weight decay. A recent work (Han et al.,
2015) successfully pruned several state of the art large scale networks and showed that the number of
parameters could be reduce by an order of magnitude. There are also attempts to reduce the number
of activations for both compression and acceleration Van Nguyen et al. (2015).
11 Published as a conference paper at ICLR 2016
8 F UTURE WORK
While theprunednetwork has been benchmarked on various hardware, thequantizednetwork with
weight sharing has not, because off-the-shelf cuSPARSE or MKL SPBLAS library does not support
indirect matrix entry lookup, nor is the relative index in CSC or CSR format supported. So the full
advantage of Deep Compression that fit the model in cache is not fully unveiled. A software solution
is to write customized GPU kernels that support this. A hardware solution is to build custom ASIC
architecture specialized to traverse the sparse and quantized network structure, which also supports
customized quantization bit width. We expect this architecture to have energy dominated by on-chip
SRAM access instead of off-chip DRAM access.
9 C ONCLUSION
We have presented “Deep Compression” that compressed neural networks without affecting accuracy.
Our method operates by pruning the unimportant connections, quantizing the network using weight
sharing, and then applying Huffman coding. We highlight our experiments on AlexNet which
reduced the weight storage by35without loss of accuracy. We show similar results for VGG-16
and LeNet networks compressed by49and39without loss of accuracy. This leads to smaller
storage requirement of putting convnets into mobile app. After Deep Compression the size of these
networks fit into on-chip SRAM cache (5pJ/access) rather than requiring off-chip DRAM memory
(640pJ/access). This potentially makes deep neural networks more energy efficient to run on mobile.
Our compression method also facilitates the use of complex neural networks in mobile applications
where application size and download bandwidth are constrained.
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A A PPENDIX :DETAILED TIMING /POWER REPORTS OF DENSE &SPARSE
NETWORK LAYERS
Table 8: Average time on different layers. To avoid variance, we measured the time spent on each
layer for 4096 input samples, and averaged the time regarding each input sample. For GPU, the time
consumed bycudaMallocandcudaMemcpyis not counted. For batch size = 1,gemvis used;
For batch size = 64,gemmis used. For sparse case,csrmvandcsrmmis used, respectively.
AlexNet AlexNet AlexNet VGG16 VGG16 VGG16Time (us) FC6 FC7 FC8 FC6 FC7 FC8
dense (batch=1) 541.5 243.0 80.5 1467.8 243.0 80.5
sparse (batch=1) 134.8 65.8 54.6 167.0 39.8 48.0Titan X dense (batch=64) 19.8 8.9 5.9 53.6 8.9 5.9
sparse (batch=64) 94.6 51.5 23.2 121.5 24.4 22.0
dense (batch=1) 7516.2 6187.1 1134.9 35022.8 5372.8 774.2
Core sparse (batch=1) 3066.5 1282.1 890.5 3774.3 545.1 777.3
i7-5930k dense (batch=64) 318.4 188.9 45.8 1056.0 188.3 45.7
sparse (batch=64) 1417.6 682.1 407.7 1780.3 274.9 363.1
dense (batch=1) 12437.2 5765.0 2252.1 35427.0 5544.3 2243.1
sparse (batch=1) 2879.3 1256.5 837.0 4377.2 626.3 745.1Tegra K1 dense (batch=64) 1663.6 2056.8 298.0 2001.4 2050.7 483.9
sparse (batch=64) 4003.9 1372.8 576.7 8024.8 660.2 544.1
Table 9: Power consumption of different layers. We measured the Titan X GPU power with
nvidia-smi, Core i7-5930k CPU power withpcm-powerand Tegra K1 mobile GPU power with
an external power meter (scaled to AP+DRAM, see paper discussion). During power measurement,
we repeated each computation multiple times in order to get stable numbers. On CPU, dense matrix
multiplications consume2xenergy than sparse ones because it is accelerated with multi-threading.
AlexNet AlexNet AlexNet VGG16 VGG16 VGG16Power (Watts) FC6 FC7 FC8 FC6 FC7 FC8
dense (batch=1) 157 159 159 166 163 159
sparse (batch=1) 181 183 162 189 166 162TitanX dense (batch=64) 168 173 166 173 173 167
sparse (batch=64) 156 158 163 160 158 161
dense (batch=1) 83.5 72.8 77.6 70.6 74.6 77.0
Core sparse (batch=1) 42.3 37.4 36.5 38.0 37.4 36.0
i7-5930k dense (batch=64) 85.4 84.7 101.6 83.1 97.1 87.5
sparse (batch=64) 37.2 37.1 38 39.5 36.6 38.2
dense (batch=1) 5.1 5.1 5.4 5.3 5.3 5.4
sparse (batch=1) 5.9 6.1 5.8 5.6 6.3 5.8Tegra K1 dense (batch=64) 5.6 5.6 6.3 5.4 5.6 6.3
sparse (batch=64) 5.0 4.6 5.1 4.8 4.7 5.0
14