1 Introduction
Over the last decade, deep neural networks have become the machine learning method of choice in a variety of application domains, demonstrating outstanding, often close to humanlevel, performances in a variety of tasks. Much of this tremendous success should be attributed to the availability of resources; a massive amount of data and compute power, which in turn fueled the impressive and innovative algorithmic and modeling development. However, resources, although available, come with a price. Data in the big data era is available, but reliable labeled data is always a challenge, and so are the ETL (ExtractTransformLoad) processes, data transfer, and storage. With the introduction of GPUs, compute power is readily available, making the training of deep architectures feasible. However, the training phase, which to a large extent, relies on stochastic gradient descent methods, requires a large number of computational resources as well as a substantial amount of time. A closer look at the compute processes highlights the fact that there is a significant difference in the compute effort between the inference (forward pass) and the model update (backpropagation) where the latter being far more demanding. The implication is evidenced by the performance charts that hardware manufactures publish, where performance matrices such as throughput (e.g. image per second) are up to 10x better at inference vs. training for popular deep neural network architectures
(Jouppi et al., 2017).In this paper, we address the computing challenge. Specifically, we suggest a method to select for the backpropagation pass only those instances that accelerate the training convergence of the deep neural network, thus speeding up the entire training process. The selection process is continuously performed throughout the training process at each step and in every training epoch. Our selection criterion is based on computations that are calculated anyhow as an integral part of the forward pass, thus taking advantage of the ”cheaper” inference compute.
2 Previous Approaches
Accelerating the training process is a longstanding challenge that was already addressed by quite a few authors. A common approach is to increase the batch size, thus mitigating the inherent time load. This approach represents a delicate balance between available compute ingredients (e.g. memory size, bandwidth, and compute elements). Interestingly, increasing the batch size not only impacts the computational burden but may also impact the final accuracy of the model (Goyal et al., 2017; Jia et al., 2018; Ying et al., 2018).
Sample selection is another approach that has been suggested to accelerate the training. The most notable one is probably the hard negative mining
(Schroff et al., 2015) where samples are selected by their loss values. The underlying assumption is that samples with higher losses have a significant impact on the model. Most of the previous work that utilized this approach was mainly aimed at increasing the model accuracy, but the same approach can also be used to accelerate training. Recent works employ selection schemes that examine the importance of the samples (Alain et al., 2015; Loshchilov and Hutter, 2015). During the training, the samples are selected based on their gradient norm, which in turn leads to a variance reduction in the stochastic gradients. Inspired by the batch size approach, a recent work by Katharopoulos and Fleuret
(Katharopoulos and Fleuret, 2018) uses selective sampling to choose the training samples that reduce the gradient variance, rather than increasing the size of the batch.Our work is inspired by the active learning
paradigm that utilizes selective sampling to choose the most useful examples for training. In active learning, the goal is to reduce the cost of labeling the training data by querying the labels of only carefully selected examples. Thus, unlike the common supervised learning setting, where training data is randomly selected, in active learning, the learner is given the power to ask questions, e.g. to select the most valuable examples to query for their labels. Measuring the training value of examples is a subject of intensive research, and quite a few selection criteria have been proposed. The approach most related to our work is the
uncertainty sampling (()lewis1994sequential), where samples are selected based on the uncertainty of their predict labels. Two heavily used approaches to measure uncertainty are entropybased and marginbased (Settles, 2009). In the entropybased approach (Lewis and Catlett, 1994), uncertainty is measured by the entropy of the posterior probability distribution of the labels, given the sample. Thus, a higher entropy represents higher uncertainty with respect to the class label. This approach naturally handles both binary and multiclass classification settings, but it relies on an accurate estimate of the (predicted) posterior probabilities. In the marginbased approach
(Tong and Koller, 2001; Campbell et al., 2000), uncertainty is measured by the distance of the samples from the decision boundary. For linear classifiers, several works
(Dasgupta, 2006; Balcan et al., 2007) gave theoretical bounds for the exponential improvement in computational complexity by selecting as few labels as possible. The idea is to label samples that reduce the version space (a set of classifiers consistent with the samples labeled so far) to the point where it has a diameter at most (c.f (Dasgupta, 2011)). This approach was proven to be useful also in nonrealizable cases (Balcan et al., 2007). However, generalizing it to the multiclass setting is less obvious. Another challenge in adapting this approach for deep learning is how to measure the distance to the intractable decision boundary. Ducoffe and Precioso
(Ducoffe and Precioso, 2018) approximate the distance to the decision boundary using the distance to the nearest adversarial examples. The adversarial examples are generated using a DeepFool algorithm (MoosaviDezfooli et al., 2016). The suggested DeepFool Active Learning method (DFAL) labels both, the unlabeled samples and the adversarial counterparts, with the same label.Our selection method is also utilizing uncertainty sampling, where the selection criterion is the closeness to the decision boundary. We do, however, consider the decision boundaries at the (last) fullyconnected layer, i.e. a multiclass linear classification setting. To this aim, we introduce the minimal margin score (MMS), which measures the distance to the decision boundary of the two most competing predicted labels. This MMS serves us as a measure to score the assigned examples. A similar measure was suggested by Jiang et al. (Jiang et al., )
as a loss function and a measure to predict the generalization gap of the network. Jiang et al. used their measure in a supervised learning setting and applied it to all layers. In contrast, we apply this measure only at the last layer, taking advantage of the linearity of the decision boundaries. Moreover, we use it for selective sampling, based solely on the assigned scores, namely without the knowledge of the true labels. The MMS measure can also be viewed as an approximation measure for the amount of perturbation needed to cross the decision boundary. Unlike the DFAL algorithm, we are not generating additional (adversarial) examples to approximate this distance but rather calculate it based on the scores of the lastlayer.
Although our selective sampling method is founded by active learning principles, the objective is different. Rather than reducing the cost of labelling, our goal is to accelerate the training. Therefore, we are more aggressive in the selection of the examples to form a batch group at each learning step, at the cost of selecting many examples at the course of training.
The rest of the paper is organized as follows. In section 3, we present the MMS measure and describe our selective sampling algorithm and discuss its properties. In Section 4 we present the performances of our algorithm on the common datasets CIFAR10 and CIFAR100 (Krizhevsky et al., 2009) and compare results against the original training algorithm and hardnegative sampling. We demonstrate additional speedup when we adopt a more aggressive learningdrop regime. We conclude at Section 5 with a discussion and suggestions for further research.
3 Accelerating training using Minimal Margin Score selection
As mentioned above, our method is based on the evaluation of the minimal amount of displacement a training sample should undergo until its predicted classification is switched. We call this measure minimal margin score (MMS). This measure depends on the best and the 2nd best scores achieved for a sample. Our measure was inspired by the marginbased quantity suggested by Jiang et al. (Jiang et al., ) for predicting the generalization gap of a given network. However, in our scheme, we apply our measure to the output layer, and we calculate it linearly with respect to the input of the last layer. Additionally, unlike (Jiang et al., ), we do not care about the true label, and our measure is calculated based on the best and the 2nd best NN scores.
An illustrative example, demonstrating the proposed approach, is given in Figure 1. In this example, a multiclass classification problem is composed of three classes: Green, Red, and Blue along with three linear projections: and , respectively. The query point is marked by an empty black circle. The highest scores of the query point are and (assuming all biases are 0’s), where and is negative (not marked). Since the best two scores are for the Green and Red classes, the distance of the query point to the decision boundary between these two classes is . The magnitude of is the MMS of this query point.
Formally, let be a large set of samples and be input to the last layer of the neural network . Assume we have a classification problem with classes. At the last layer the classifier consists of linear functions: for where is a linear mapping . For sample , the classifier predicts its class label by the maximal score achieved: . Denote the sorted scores of by where and . The classifier gave the highest score and gave the second highest score. The decision boundary between class and class is defined as:
Using this definition, the confidence of the the predicted label of point is determined by the distance of to the decision boundary , namely the minimal distance of to :
It is easy to show (see Appendix A) that
The distance is the Minimal Margin Score (MMS) of point . The larger the , the more confident we are about the predicted label. Conversely, the smaller the , the less confident we are about the predicted label . Therefore, can serve as a confidence measure for the predicted labels. Accordingly, the best points to select for the backpropagation step are the points whose MMS are the smallest.
Our implementation consists of a generic but yet simple online selective sampling method as a preliminary part of the training optimization step. Specifically, at each training step, a selective batch of size is produced out of a larger batch before using it in the SGD optimization routine. At a specific iteration, we first apply a forward pass on a batch of size , producing the prediction scores. Secondly, we select samples () whose MMS measures are the smallest. This procedure is summarized in Algorithm 1.
4 Experiments
In this section^{1}^{1}1
All experiments were conducted using PyTorch framework, and the code is publicly available at
https://github.com/berryweinst/mmsselect., we empirically examined the performance of the proposed selection scheme. As a baseline, we compared our results against the original training algorithm using uniform sampling from the dataset. Additionally, we compared the MMS method against two popular schemes: hardnegative mining which prefers samples with low prediction scores, and entropybased uncertainty sampling. Our experimental workbench was composed of the common visual datasets CIFAR10 and CIFAR100 (Krizhevsky et al., 2009) that consist of color images in 10 or 100 classes and has 50,000 training examples and 10,000 test examples.Hard negative samples.
For this experiment, we implemented the ”NMsamples” procedure (Hoffer et al. ), similar to the classical methods for ”hardnegativemining” used by machinelearning practitioners over the years (Yu et al., 2018; Schroff et al., 2015). We used the crossentropy loss as a proxy for the selection, in which the highest loss samples were selected. We denote this approach as HNMsamples.
On both datasets we used ResNet44 (He et al., 2016) and WRN2810 (Zagoruyko and Komodakis, 2016) architectures, respectively. To compare our MMS selection scheme against the baseline and HNM, we applied the original hyperparameters and training regime using batchsize of 64. In addition, we used the original augmentation policy as described in He et al. (2016) for ResNet44, while adding cutout (DeVries and Taylor, 2017) and autoaugment (Cubuk et al., 2018) for WRN2810. Optimization was performed for 200 epochs (equivalent to iterations) after which baseline accuracy was obtained with no apparent improvement.
Cifar10.
For the CIFAR10 dataset, sampling with the MMS scheme obtained significantly lower error compared to the baseline and the HNMsamples throughout the entire training progress ( on average). The test results are depicted in Figure 1(b). Furthermore, the use of MMS provides a slight improvement of 0.1% in the final test accuracy as well as a clear indication of a faster generalization compared to the baseline and the HNM schemes.
Cifar100.
Inspired by the results on CIFAR10 using the MMS method, we continued to evaluate performance on a more challenging 100 classes dataset. The MMS method obtained a nonnegligible error decrease, particularly after the first learningrate drop ( on average) as can be seen in Figure 1(d). On the other hand, we did not observe similar behaviour using the HNM and the baseline schemes, similarly as in CIFAR10.
4.1 Mean MMS and training error
To estimate the MMS values of the selected samples during training, we defined the mean MMS in a training step as the average MMS of the first selected samples for the batch. This was compared to the mean MMS of the samples selected by the baseline and the HNM methods.
Figure 3 present the trace of the mean MMS that was recorded at the experiments presented in Figure 2 in the course of training. The mean MMS of the suggested scheme remains lower compared to the baseline in most of the training process. We argue that this behaviour stems from the nature of the uncertain classification with respect to the selected samples. This result suggests that there are ”better” samples to train the model on rather than selecting the batch randomly. Interestingly, the HNM method obtained a similar mean MMS at the early stages of training, On the other hand, the HNM method resulted in a similar mean MMS as our suggested method during the training, but it increases after the learningrate drop, and it deviates from the MMS scores obtained by our method. Lower mean MMS scores resemble a better (more informative) selected batch of samples. Hence, we may conclude that the batches selected by our method, provides a higher value for the training procedure vs. the HNM samples. Moreover, the mean MMS trace is monotonically increasing as the training progress, and it flattens when training converges.
All the selective sampling methods that we tested (HNM, entropybased, and our MMS method), yielded a significantly higher error throughout the training (Figures 1(a), 1(c), 4). This coincides with the main theme of selective sampling that strive to focus training on the more informative points. However, training loss can be a poor proxy to this notion. For example, the selection criterion of the HNM favours high loss scores, which obviously increases the training error, while our MMS approach select uncertain points, some of which might be correctly classified, others might be missclassified by a small margin (low absolute loss scores), but they are all close to the decision boundary, and hence useful for training. Evidently, the mean MMS provides a clearer perspective into the progress of training and usefulness of the selected samples.
4.2 Selective sampling using entropy measure
Additionally, we tested the entropybased selective sampling, which is a popular form of uncertainty sampling. We select the examples with the largest entropy, thus the examples with the most class overlap, forming a training batch of size 64 out of a larger batch. We compared performances with the vanilla training and the MMS selection method, using the same experimental setting.
This experiment shows (see Figure 4) that for a small problem as CIFAR10, this selection method is efficient as our MMS. However, as CIFAR100, inducing a more challenging task, this method fails. This entropy measure relies on the uncertainty of the posterior distribution with respect to the examples class. We consider this as an inferior method for selection. Also, as the ratio between the batch size and the number of classes increases, this measure becomes less accurate. Finally, as the number of classes grows, as in CIFAR100 compared to CIFAR10, the prediction scores signal has a longer tail with less information, which also diminishes its value. The last assumption is not valid for our method as we measure based on the two best predictions.
4.3 Additional speedup via an aggressive leaningrate drop regime
The experimental results have led us to conjecture that we may further accelerate training using the MMS selection, by applying an early learningrate drop. To this end, we designed a new, more aggressive leaningrate drop regime. Figure 5 present an empirical evidence to our conjecture that with the MMS selection method we can speed up training while preserving final model accuracy.
Network  Dataset  Steps  Accuracy  

Baseline  Ours  Baseline  Ours  
ResNet44  CIFAR10  156K  44K  93.24%  93% 
WRN2810  CIFAR100  156K  80K  82.26%  82.2% 
Cifar10.
For CIFAR10 and ResNet44 we used the original learning rates while decreasing them at steps equivalent to epochs with batch of size 64. As depicted in Figure 4(b), we can see that indeed our selection yields validation accuracy that is similar to the one obtained using the original training regime, in a much earlier step. As described in table 1, training with our selection scheme almost reached final model accuracy in considerably less training steps as originally suggested. Specifically, we reached accuracy after merely steps (a minor drop of compared to the baseline). We also apply the early drop regime to the baseline configuration as well as the HNMsamples. Both failed to reach the desired model accuracy while suffering from degradation of and for the baseline and HNMsamples, respectively.
Cifar100.
Similarly, we applied the early learningrate drop scheme for CIFAR100 and WRN2810, using and decreasing steps equivalent to epochs and batch of size 64. As depicted in Figure 4(d), accuracy reached with a drop of compared to baseline with almost halving the baseline required steps (from to 80K steps). On the other hand, the baseline and the HNMsamples configurations failed to reach the desired accuracy after applying a similar early drop regime similarly to CIFAR10. The degradation for the HNMsamples approach was the most significant, with a drop of compared to the final model accuracy, while the baseline drop was approximately of .
5 Discussion
We presented a selective sampling method designed to accelerate the training of deep neural networks. Specifically, we utilized uncertainty sampling, where the criterion for selection is the distance to the decision boundary. To this end, we introduced a novel measurement, the minimal margin score (MMS), which measure the minimal amount of displacement an input should take until its predicted classification is switched. For multiclass linear classification, the MMS measure is a natural generalization of the marginbased selection criterion, which was thoroughly studied in the binary classification setting. We demonstrate a substantial acceleration for training commonly used DNN architectures for popular image classification tasks. The efficiency of our method is compared against the standard training procedures, and against commonly used selective sampling methods: Hard negative mining selection, and Entropybased selection. Furthermore, we demonstrate an additional speedup when we adopt a more aggressive learningdrop regime.
Tracking the MMS measure throughout the training process provides an interesting insight into the training process. Figure 3 demonstrates that the MMS measure is monotonically increasing, even when training and validation errors are flattening. Subsequently, it flattens when training converges to the final model. This suggests that improvement in training can be obtained as long as there is uncertainty in the class labels. Furthermore, tracking the MMS measure may turn out to be useful for designing and monitoring new training regimes.
Our selection criterion was inspired by the Active Learning methods, but our goal, accelerate training, is different. Active learning mainly concerns about the labelling cost. Hence, it is common to keep on training till (almost) convergence, before turning to select additional examples to label. However, such an approach is less efficient when it comes to acceleration. In such a scenario, we can be more aggressive; since labelling cost is not a concern, we can reselect a new batch of examples in each training step.
An efficient implementation is also crucial for gaining speedup. Our scheme provides many opportunities for further acceleration. For example, finetuning the sample size used to select and fill up a new batch, to balance between the selection effort conducted at the end of the forward pass, and the compute resources and efforts required to conduct the backpropagation pass. This also opens an opportunity to design and use dedicated hardware for the selection. In the past few years, custom ASIC devices that accelerate the inference phase of neural networks were developed (9; 12; N. P. Jouppi, C. Young, N. Patil, D. Patterson, G. Agrawal, R. Bajwa, S. Bates, S. Bhatia, N. Boden, A. Borchers, et al. (2017)). Furthermore, in (Jacob et al., 2018), it was shown that using quantization for lowprecision computation induces little or no degradation in final model accuracy. This observation, together with the fast and efficient inference achieved by ASICs, make them appealing to be used as a supplement accelerator in the forward pass of our selection scheme.
The MMS measure doesn’t use the labels. Thus it can be used to select samples in an active learning setting as well. Moreover, similarly to (Jiang et al., ) the MMS measure can be implemented at other layers in the deep architecture. This enables to select examples that directly impact training at all levels. The additional compute associated with such calculating and selecting the right batch content, makes it less appealing for acceleration. However, for active learning, it may introduce an additional gain, since the selection criterion chooses examples which are more informative for various layers. The design of a novel Active Learning method is left for further study.
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Appendix A
Assume a given point and its DNN latest layer output . W.l.o.g let’s the largest and the second largest scores of the classifier be: and , respectively. We are looking for the smallest satisfying:
Rearranging terms we get:
The leastnorm solution of the above underdetermined equation is calculated using the right pseudoinverse of which gives:
and therefore the MMS of is:
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