SCIENCE CHINA Information Sciences, Volume 63 , Issue 1 : 112102(2020) https://doi.org/10.1007/S11432-018-9944-X

Snapshot boosting: a fast ensemble framework for deep neural networks

• AcceptedApr 15, 2019
• PublishedDec 24, 2019
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Abstract

Boosting has been proven to be effective in improving the generalization of machine learning models in many fields. It is capable of getting high-diversity base learners and getting an accurate ensemble model by combining a sufficient number of weak learners. However, it is rarely used in deep learning due to the high training budget of the neural network. Another method named snapshot ensemble can significantly reduce the training budget, but it is hard to balance the tradeoff between training costs and diversity.Inspired by the ideas of snapshot ensemble and boosting, we propose a method named snapshot boosting.A series of operations are performed to get many base models with high diversity and accuracy,such as the use of the validation set, the boosting-based training framework, and the effectiveensemble strategy. Last, we evaluate our method on the computer vision (CV) and the natural languageprocessing (NLP) tasks, and the results show that snapshot boosting can get a more balanced trade-offbetween training expenses and ensemble accuracy than other well-known ensemble methods.

Acknowledgment

This work was supported by National Natural Science Foundation of China (Grant Nos. 61832001, 61702015, 61702016, 61572039), National Key Research and Development Program of China (Grant No. 2018YFB1004403), and PKU-Tencent Joint Research Lab.

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• Figure 1

(Color online) The learning rate schedule of a 40-layer DenseNet on CIFAR100 using snapshot ensemble and snapshot boosting. Note that SGDR is used in snapshot ensemble and its learning rate restarts every 50 epochs.

• Figure 2

(Color online) The procedure of snapshot boosting.

• Figure 3

(Color online) The test accuracy of a 40-layer DenseNet on CIFAR100 using snapshot ensemble and snapshot boosting. Note that SGDR is used in snapshot ensemble and its learning rate restarts every 50 epochs.

• Figure 4

(Color online) Test accuracy of ensembles on CIFAR-100 using ResNet-32 (a) and DenseNet-40 (b). For the single model, the test accuracy is directly calculated on the test set in the last epoch. For the ensemble method, the test accuracy is the ensemble accuracy which is calculated with the base models already trained.

• Figure 5

(Color online) Pairwise correlation of softmax outputs between base models for DenseNet-40 on CIFAR-100. (a) Snapshot ensemble; (b) snapshot boosting.

• Figure 6

(Color online) The test accuracy of snapshot boosting on CIFAR-100 using ResNet-32 with different resetting learning rate $r_2$ and split ratio $\alpha$.

• Table 1   Comparisons on CV tasks$^{\rm~a)}$
 Model Method Number of Best base model Ensemble Increased base model accuracy (%) accuracy (%) accuracy (%) 6*ResNet-32 Single model 1 69.66 – – Deep incremental boosting 7 67.78 71.34 3.56 Bagging 8 67.86 71.88 4.02 AdaBoost.M1 8 68.12 72.36 4.24 Snapshot ensemble 10 69.83 72.98 3.15 Snapshot boosting 16 68.75 74.16 5.41 6*DenseNet-40 Single model 1 72.07 – – Deep incremental boosting 7 71.22 73.45 2.23 Bagging 8 71.34 74.37 3.03 AdaBoost.M1 8 71.35 74.72 3.37 Snapshot ensemble 8 71.71 73.31 1.60 Snapshot boosting 14 71.55 74.97 3.42

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•

Algorithm 1 Snapshot boosting

Require:$D_0$: the original training set; $D_1$: the test set; $T$: the number of base models; $\alpha$: the split ratio; $r_1$: the initial learning rate for training the first base model; $r_2$: the initial learning rate for training other base models; $k$: the number of classes; $p_1$; $p_2$; $\delta$; $f$;

Output: The prediction on the test set, $R$;

$|D_2|~=|D_0|(1-\alpha)$; $|D_3|~=|D_0|\alpha$;

$m~=~|D_2|$; $n~=~|D_3|$; $t~=0$;

$W_0(i)=1/m$;

$h_0~\gets~~F(D_2,W_0,r_1,p_1,p_2,\delta,f)$;

for $t=1$ to $T$

Set the learning rate to $r_2$ and decay it according to the validation accuracy on $D_3$;

$h_t~\gets~~F(D_2,W_t,h_{t-1},r_2,p_1,p_2,\delta,f)$;

Save the best mode $h_{t}$ before the validation accuracy on $D_3$ is no significant increased;

Get the error $\epsilon_t$ on $D_3$, $\epsilon_t=\frac{1}{n}\sum_{i=1}^{n}~I(h_t(x_i)\ne~y_i)$;

$\beta_t=\frac{1}{2}{\rm~log}\frac{1-\epsilon_t}{\epsilon_t}+\frac{1}{10}{\rm~log}(k-1)$;

$W_{t+1}(i)=\frac{1/m}{Z_t}{\rm~e}^{-\beta_t~y_i~h_t(x_i)~},~i=1,2,\ldots,m$;

Where $Z_t$ is a normalization factor;

$V_t~\gets$ use $h_t$ to predict the softmax outputs of $D_3$;

$S_t~\gets$ use $h_t$ to predict the softmax outputs of $D_1$;

$t~=~t+1$;

end for

$V~\gets$ merge each base model's outputs $V_t$, $t=1,2,\ldots,T$;

$S~\gets$ merge each base model's outputs $S_t$, $t=1,2,\ldots,T$;

Use $V$ to do the model selection and feature engineering;

$V^{*}~\gets$ $V$'s remaining features;

$S^{*}~\gets$ $S$'s remaining features;

$M~\gets$ fit a meta-learner on $V^{*}$;

$R~\gets~~M(S^{*})$;

• Table 2   Ensemble accuracy on IMDB dataset$^{\rm~a)}$
 Model Method Accuracy (%) 5*LSTM Single model 89.02 Bagging 89.17 AdaBoost 90.13 Snapshot ensemble 90.71 Snapshot boosting (average) 91.17 Snapshot boosting 91.52

a

• Table 3   Ensemble accuracy on CIFAR-10 dataset$^{\rm~a)}$
 Model Method Accuracy (%) 6*ResNet-32 Single model 93.50 DIB 92.78 Bagging 93.24 AdaBoost.M1 93.47 Snapshot ensemble 94.26 Snapshot boosting (average) 94.87 Snapshot boosting 95.12 6*DenseNet-40 Single model 93.43 DIB 93.26 Bagging 93.74 AdaBoost.M1 93.12 Snapshot ensemble 93.57 Snapshot boosting (average) 93.96 Snapshot boosting 94.31

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