- 1Department of Mathematics, University of California, Irvine, Irvine, CA, United States
- 2Department of Mathematics and Computer Science, Whittier College, Whittier, CA, United States
Convolutional neural networks (CNN) have been hugely successful recently with superior accuracy and performance in various imaging applications, such as classification, object detection, and segmentation. However, a highly accurate CNN model requires millions of parameters to be trained and utilized. Even to increase its performance slightly would require significantly more parameters due to adding more layers and/or increasing the number of filters per layer. Apparently, many of these weight parameters turn out to be redundant and extraneous, so the original, dense model can be replaced by its compressed version attained by imposing inter- and intra-group sparsity onto the layer weights during training. In this paper, we propose a nonconvex family of sparse group lasso that blends nonconvex regularization (e.g., transformed
1 Introduction
Deep neural networks (DNNs) have proven to be advantageous for numerous modern computer vision tasks involving image or video data. In particular, convolutional neural networks (CNNs) yield highly accurate models with applications in image classification [28, 39, 77, 95], semantic segmentation [13, 49], and object detection [30, 72, 73]. These large models often contain millions of weight parameters that often exceed the number of training data. This is a double-edged sword since on one hand, large models allow for high accuracy, while on the other, they contain many redundant parameters that lead to overparametrization. Overparametrization is a well-known phenomenon in DNN models [6, 17] that results in overfitting, learning useless random patterns in data [96], and having inferior generalization. Additionally, these models also possess exorbitant computational and memory demands during both training and inference. Consequently, they may not be applicable for devices with low computational power and memory.
Resolving these problems requires compressing the networks through sparsification and pruning. Although removing weights might affect the accuracy and generalization of the models, previous works [25, 54, 66, 81] demonstrated that many networks can be substantially pruned with negligible effect on accuracy. There are many systematic approaches to achieving sparsity in DNNs, as discussed extensively in Refs. 14 and 15.
Han et al. [26] proposed to first train a dense network, prune it afterward by setting the weights to zeroes if below a fixed threshold, and retrain the network with the remaining weights. Jin et al. [32] extended this method by restoring the pruned weights, training the network again, and repeating the process. Rather than pruning by thresholding, Aghasi et al. [1, 2] proposed Net-Trim, which prunes an already trained network layer by layer using convex optimization in order to ensure that the layer inputs and outputs remain consistent with the original network. For CNNs in particular, filter or channel pruning is preferred because it significantly reduces the amount of weight parameters required compared to individual weight pruning. Li et al. [43] calculated the sums of absolute weights of the filters of each layer and pruned the ones with the smallest sums. Hu et al. [29] proposed a metric called average percentage of zeroes for channels to measure their redundancies and pruned those with highest values for each layer. Zhuang et al. [105] developed discrimination-aware channel pruning that selects channels that contribute to the network’s discriminative power.
An alternative approach to pruning a dense network is learning a compressed structure from scratch. A conventional approach is to optimize the loss function equipped with either the
In this paper, we propose a family of group regularization methods that balances both group lasso for group-wise sparsity and nonconvex regularization for element-wise sparsity. The family extends sparse group lasso by replacing the
2 Model and Algorithm
2.1 Preliminaries
Given a training dataset consisting of N input-output pairs
where
W is the set of weight parameters of the DNN.
The most common regularizer used for DNNs is
To determine which filters or channels are relevant in each layer, group sparsity using the group lasso penalty [93] is considered. The group lasso penalty has been utilized in various applications, such as microarray data analysis [62], machine learning [7, 65], and EEG data [46]. Suppose a DNN has L layers, so the set of weight parameters W is divided into L sets of weights:
where
As an alternative to group lasso that encourages feature sharing, exclusive sparsity [104] enforces the model weight parameters to compete for features, making the features discriminative for each class in the context of classification. The regularization for exclusive sparsity is
Now, within each group, sparsity is enforced. Because exclusivity cannot guarantee the optimal features since some features do need to be shared, exclusive sparsity can be combined with group sparsity to form combined group and exclusive sparsity (CGES) [92]. CGES is formulated as
where
To obtain an even sparser network, element-wise sparsity and group sparsity can be combined and applied together to the training of DNNs. One regularizer that combines these two types of sparsity is the sparse group lasso penalty [76], which is formulated as
where
Sparse group lasso simultaneously enforces group sparsity by having the regularizer
Figure 1 demonstrates the differences between lasso, group lasso, and sparse group lasso applied to a weight matrix connecting a 5-dimensional input layer to a 10-dimensional output layer. In white, the entries are zero’ed out; in gray; the entries are not. Unlike lasso, group lasso results in a more structured method of pruning since three of the five neurons can be zero’ed out. Combined with
FIGURE 1. Comparison between lasso, group lasso, and sparse group lasso applied to a weight matrix. Entries in white are zero’ed out or removed; entries in gray remain.
2.2 Nonconvex Sparse Group Lasso
We recall that the
where
when applied to the weight set
A continuous alternative to the
where
for
The transformed
where
In addition, it interpolates the
The transformed
Another Lipschitz continuous, nonconvex regularizer is the
where
Due to the advantages and recent successes of the aforementioned nonconvex regularizers, we propose to replace the
Using these regularizers, we expect to obtain a sparser and/or more accurate network than from using the original sparse group lasso. The
2.3 Notations and Definitions
Before discussing the algorithm, we summarize notations that we will use to save space. They are the following:
If
for
2.4 Numerical Optimization
We develop a general algorithm framework to solve
where
Assumption 1. The function
By introducing an auxiliary variable
The constraints can be relaxed by adding the quadratic penalty terms with
With β fixed, Eq. 16 can be solved by alternating minimization:
To solve Eq. 17a, we simultaneously update
where
To update V, we see that Eq. 17b can be rewritten as
The proximal operators for the considered regularizers are thresholding functions as their closed-form solutions, and as a result, the V update simplifies to thresholding W. The regularization functions and their corresponding proximal operators are summarized in Table 1.
Incorporating the algorithm that solves the quadratic penalty problem Eq. 16, we now develop a general algorithm to solve Eq. 14. We solve a sequence of quadratic penalty problems Eq. 16 with
An alternative algorithm to solve Eq. 14 is proximal gradient descent [70]. By this method, the update for
Using this algorithm results in weight parameters with some already zero’ed out.
However, the advantage of our proposed algorithm lies in Eq. 17a, written more specifically as
We see that this step performs exact weight decay or
2.5 Convergence Analysis
To establish convergence for the proposed algorithm, the results below state that the accumulation point of the sequence generated by Eqs 17a and 17b is a block-coordinate minimizer, and an accumulation point generated by Algorithm 1 is a sparse feasible solution to (15). Proofs are provided in Section 5. Unfortunately, the feasible solution generated may not be a local minimizer of Eq. 15 because the loss function
Theorem 2. Let
Theorem 3. Let
Remark: To safely ensure that
If
3 Numerical Experiments
3.1 Application to Deep Neural Networks
We compare the proposed nonconvex sparse group lasso against four other methods as baselines: group lasso, sparse group lasso (
3.1.1 MNIST Classification
MNIST is trained on Lenet-5-Caffe, which has four layers with 1,370 total neurons and 431,080 total weight parameters. All layers of the network are applied with strictly the same type of regularization. No other regularization methods (e.g., dropout and batch normalization) are used. The network is optimized using Adam [37] with initial learning rate 0.001. For every 40 epochs, the learning rate decays by a factor of 0.1. We set the regularization parameter to the following values:
Table 2 reports the mean results for test error, weight sparsity, and neuron sparsity across five runs of Lenet-5-Caffe trained after 200 epochs. We see that although CGES has the lowest test errors at
TABLE 2. Average test error, weight sparsity, and neuron sparsity of Lenet-5 models trained on MNIST after 200 epochs across 5 runs. Standard deviations are in parentheses.
Table 3 reports the mean results for test error, weight sparsity, and neuron sparsity of the Lenet-5-Caffe models with the lowest test errors from the five runs. According to the results, the best test errors are attained by
TABLE 3. Average test error, weight sparsity, and neuron sparsity of Lenet-5 models trained on MNIST with lowest test errors across 5 runs. Standard deviations are in parentheses.
MNIST is also trained on a 4-layer CNN with two convolutional layers with 32 and 64 channels, respectively, and an intermediate layer with 1000 neurons. Each convolutional layer has a
Table 4 reports the mean results for test error, weight sparsity, and neuron sparsity across five runs of the 4-layer CNN models trained after 200 epochs. Although CGES consistently has the highest weight sparsity, it does not yield the most accurate models until when
TABLE 4. Average test error, weight sparsity, and neuron sparsity of 4-layer CNN models trained on MNIST after 200 epochs across 5 runs. Standard deviations are in parentheses.
Table 5 reports the mean results for test error, weight sparsity, and neuron sparsity of the 4-layer CNN models with the lowest test errors from the five runs. At
TABLE 5. Average test error, weight sparsity, and neuron sparsity of 4-layer CNN models trained on MNIST with lowest test errors across 5 runs. Standard deviations are in parentheses.
3.1.2 CIFAR Classification
CIFAR 10/100 is trained on Resnet-40 and wide Resnet with depth 28 and width 10 (WRN-28-10). Resnet-40 has approximately 570,000 weight parameters and 1520 neurons while WRN-28-10 has approximately 36,500,000 weight parameters and 10,736 neurons. The networks are optimized using stochastic gradient descent with initial learning rate 0.1. After every 60 epochs, learning rate decays by a factor of 0.2. Strictly the same type of regularization is applied to the weights of the hidden layer where dropout is utilized in the residual block. We vary the regularization parameter
Table 6 reports mean test error, weight sparsity, and neuron sparsity across the Resnet-40 models trained on CIFAR 10 with the lowest test errors from the five runs. Group lasso has the lowest test errors for all α’s provided while CGES,
TABLE 6. Average test error, weight sparsity, and neuron sparsity of Resnet-40 models trained on CIFAR 10 with lowest test errors across 5 runs. Standard deviations are in parentheses.
Table 7 reports mean test error, weight sparsity, and neuron sparsity across the Resnet-40 models trained on CIFAR 100 with the lowest test errors from the five runs. Group lasso has the lowest test errors for
TABLE 7. Average test error, weight sparsity, and neuron sparsity of Resnet-40 models trained on CIFAR 100 with lowest test errors across 5 runs. Standard deviations are in parentheses.
Table 8 reports mean test error, weight sparsity, and neuron sparsity across the WRN-28-10 models trained on CIFAR 10 with the lowest test errors from the five runs. The best test errors are attained by
TABLE 8. Average test error, weight sparsity, and neuron sparsity of WRN-28-10 models trained on CIFAR 10 with lowest test errors across 5 runs. Standard deviations are in parentheses.
Table 9 reports mean test error, weight sparsity, and neuron sparsity across the WRN-28-10 models trained on CIFAR 100 with the lowest test errors from the five runs. According to the results, the best test errors are attained by CGES when
TABLE 9. Average test error, weight sparsity, and neuron sparsity of WRN-28-10 models trained on CIFAR 100 with lowest test errors across 5 runs. Standard deviations are in parentheses.
3.2 Algorithm Comparison
We compare the proposed Algorithm 1 with direct stochastic gradient descent, where the gradient of the regularizer is approximated by backpropagation, and proximal gradient descent, discussed in Section 2.4, by applying them to
TABLE 10. Average test error, weight sparsity, and neuron sparsity of
FIGURE 2. Mean results of algorithms applied to SGL1 for Lenet-5 models trained on MNIST for 200 epochs across 5 runs when varying the regularization parameter
TABLE 11. Average test error, weight sparsity, and neuron sparsity of
FIGURE 3. Mean results of algorithms applied to SGL1 for Lenet-5 models trained on MNIST with lowest test errors across 5 runs when varying the regularization parameter
4 Conclusion and Future Work
In this work, we propose nonconvex sparse group lasso, a nonconvex extension of sparse group lasso. The
According to the numerical results, there is no single sparse regularizer that outperforms all other on any CNN trained on a given dataset. One regularizer may perform well in one case while it may perform worse on a different case. Due to the myriad of sparse regularizers to select from and the various parameters to tune, especially for one CNN trained on a given dataset, one direction is to develop an automatic machine learning framework that efficiently selects the right regularizer and parameters. In recent works, automatic machine learning can be represented as a matrix completion problem [88] and a statistical learning problem [24]. These frameworks can be adapted for selecting the best sparse regularizer, thus saving time for users who are training sparse CNNs.
5 Proofs
We provide proofs for the results discussed in Section 2.5.
5.1 Proof of Theorem 2
By Eqs 17a and 17b, for each
for all W, and
for all V. By Eq. 23, we have
for each
for each
Taking the limit gives us
Since
Because
for
by continuity, so it follows that
For notational convenience, let
By Eq. 22, we have
Because
after applying Eq. 26 in the third inequality and by continuity in the last equality.
Taking the limit over the subsequence
by continuity. Adding
which follows that
5.2 Proof of Theorem 3
Because
As a result,
Taking the limit over
which follows that
Data Availability Statement
The datasets MNIST and CIFAR 10/100 for this study are available through the Pytorch package in Python. Codes for the numerical experiments in Section 3 are available at https://github.com/kbui1993/Official_Nonconvex_SGL.
Author Contributions
KB and FP performed the experiments and analysis. All authors contributed to the design, evaluation, discussions and production of the manuscript.
Funding
The work was partially supported by NSF grants IIS-1632935, DMS-1854434, DMS-1924548, DMS-1952644 and the Qualcomm Faculty Award.
Conflict of Interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Acknowledgments
The authors would like to thank Thu Dinh for helpful conversations. They also thank Christos Louizos for answering our questions we had regarding his work in [54]. Lastly, the authors thank AWS Cloud Credits for Research and Google Cloud Platform (GCP) for providing cloud based computational resources for this work.
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Keywords: deep learning, sparsity, nonconvex optimization, sparse group lasso, feature selection
Citation: Bui K, Park F, Zhang S, Qi Y and Xin J (2021) Structured Sparsity of Convolutional Neural Networks via Nonconvex Sparse Group Regularization. Front. Appl. Math. Stat. 6:529564. doi: 10.3389/fams.2020.529564
Received: 25 January 2020; Accepted: 16 October 2020;
Published: 24 February 2021.
Edited by:
Lucia Tabacu, Old Dominion University, United StatesCopyright © 2021 Bui, Park, Zhang, Qi and Xin. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Jack Xin, amFjay54aW5AdWNpLmVkdQ==