Introduction
Deep convolutional neural networks (CNNs) trained with logistic or softmax losses (LGL and SML respectively for brevity), e.g., logistic or softmax layer followed by crossentropy loss, have achieved remarkable success in various visual recognition tasks
[17, 16, 12, 25, 27]. The success mainly accredits to CNN’s merit of highlevel feature learning and loss function’s differentiability and simplicity for optimization. When training data exhibit class imbalances, training CNNs with gradient descent is biased towards learning majority classes in the conventional (unweighted) loss, resulting in performance degradation for minority classes. To remedy this issue, the classwise reweighted loss is often used to emphasize the minority classes that can boost the predictive performance without introducing much additional difficulty in model training [6, 14, 20, 28]. A typical choice of weights for each class is the inverseclass frequency.A natural question then to ask is what roles are those classwise weights playing in CNN training using LGL or SML that lead to performance gain? Intuitively, those weights make tradeoffs on the predictive performance among different classes. In this paper, we answer this question quantitatively in a set of equations that tradeoffs are on the model predicted probabilities produced by the CNN models. Surprisingly, effectiveness of the reweighting mechanism for LGL is rather different from SML. Here, we view the conventional (e.g., no reweighting) LGL or SML as a special case where all classes are weighted equally.
As these tradeoffs are related to the logistic and softmax losses, answering the above question actually leads us to answering a more fundamental question about their learning behavior: what is the property that the decision boundary must satisfy when models are trained? To our best knowledge, this question has not been investigated systematically, despite logistic and softmax losses are extensively exploited in deep leaning community.
While SML can be viewed as a multiclass extension of LGL for binary classification, LGL is a different learning objective when used in multiclass classification [2]. From the perspective of learning structure of data manifold as pointed out in [1, 2, 7], SML treats all class labels equally and poses a competition between true and other class labels for each training sample, which may distort data manifold; for LGL, the onevs.all approach it takes avoids this limitation as it models each target class independently, which may better capture the inclass structure of data. Though LGL enjoys such merits, it is rarely adopted in existing CNN models. The property that LGL and SML decision boundaries must satisfy further reveals the difference between LGL and SML (see Eq. (9), (10) with analysis). If used for the multiclass classification problem, we can identify two issues for LGL. Compared with SML, LGL may introduce data imbalance, which can degrade model performance as sample size plays an important role in determining decision boundaries. More importantly, since the onevs.all approach in LGL treats all other classes as the negative class, which is of a multimodal distribution [19, 18], the averaging effect of the predicted probabilities of LGL can hinder learning discriminative feature representations to other classes that share some similarities with the target class.
Our contribution can be summarized as follows:

We provide a theoretical derivation on the relation among sample’s predicted probability (once CNN is trained), class weights in the loss function and sample size in a system of equations. Those equations explaining the reweighting mechanism are different in effect for LGL and SML.

We depict the learning property for LGL and SML for classification problems based on those probability equations. Under mild conditions, the expectation of model predicted probabilities must maintain a relation specified in Eq (9).

We identify that the multimodality neglect problem in LGL is the main obstacle for LGL in multiclass classification. To remedy this problem, we propose a novel learning objective, innegative class reweighted LGL, as a competitive alternative for LGL and SML.

We conduct experiments on several benchmark datasets to demonstrate the effectiveness of our method.
Related Work
With recent explosion in computational power and availability of large scale image datasets, deep learning models have repeatedly made breakthroughs in a wide spectrum of tasks in computer vision
[17, 9]. Those advancements include new CNN architectures for image classification[16, 12, 25, 27], objective detection and segmentation [23, 24], new loss functions [7, 30] and effective training techniques to improve CNN performance [26, 15].In those supervised learning problems, CNNs are mostly trained with loss functions such as LGL and SML. In practice, class imbalance naturally emerges in realworld data and training CNN models directly on those datasets may lead to poor performance. This phenomenon is referred as the imbalanced learning problem
[11]. To tackle this problem, costsensitive method [8, 31]is the widelyadopted approach in current training practices as they don’t introduce any obstacles in the backpropagation algorithm. One of the most popular methods is classwise reweighting loss function based on LGL and SML. For example,
[14, 28] reweight each class by its inverseclass frequency. In some longtailed datasets, a smoothed version of weights is adopted [20, 21], which emphasizes less on minority classes, such as the square root of inverseclass frequency. More recently, [6] proposed a weighting strategy based on the calculation of effective sample size. In the context of learning from noisy data, [30] provides analysis on the weighted SGL showing close connection to the mean absolute error (MAE) loss. However, what role classwise weights play in LGL and SML is not explained in previous works. In this paper, we provide a theoretical explication on how the weights control the tradeoffs among model predictions.If we decompose the multiclass classification as multiple binary classification subtasks, LGL can also be used as the objective function via onevs.all approach [10, 2], which is however rarely adopted in existing works of deep learning. Motivated to understand classwise reweighted LGL and SML, our analysis further leads us to a more profound discovery in the properties of decision boundaries for LGL and SML. Previous work in [7] showed that the learning objective using LGL is quite different from SML as each class is learned independently. They identified the negative class distraction (NCD) phenomenon that might be detrimental to model performance when using LGL in multiclass classification. From our analysis, the NCD problem can be partially explained that LGL treats the negative class (e.g., nontarget classes) as a single class and ignores its multimodality. If there exists one nontarget class that share some similarity with the target class, CNN trained with LGL may make less confident predictions for that nontarget class (e.g., probability of belonging to the negative class is small) as its predicted probabilities are averaged out due to other nontarget classes with confident predictions. Consequently, samples from that specific nontarget class can be misclassified into the target class, resulting in large predictive error.
Analysis on LGL and SML
In this section, we provide a theoretical explanation for the classwise weighting mechanism and depict the learning property of LGL and SML losses.
Notation Let be the set of training samples of size , where is the
dimensional feature vector and
is the true class label, and the subset of for the th class. The boldis used to represent the onehot encoding for
: if , otherwise. is used to represent sample size for the th class and hence . The maximum size is denoted as .Preliminaries
For classification problem, the probability for a sample belonging to one class is modeled by logistic (e.g., sigmoid) for binary classification
and by softmax for multiclass classification
where all
’s are the logits for
modeled by CNN with parameter vector . It is worth noting that softmax is equivalent to logistic in binary classification as can be seen fromHence, without loss of generality, we write classwise reweighted LGL () and SML () in a unified form as follows
(1) 
where each is the CNN predicted probability of sample belonging to the th class; s are weight parameters to control each class’s contribution in the loss. When all s are equal, is the conventional crossentropy loss and minimizing it is equivalent to maximizing likelihood. If the training data are imbalanced, a different setup of s is used, usually classes with smaller sizes are assigned with higher weights. Generally,
s are treated as hyperparameters and selected by crossvalidation.
We emphasize here that using logistic function for multiclass () is a different learning objective from softmax in this case as the classification problem is essentially reformulated as binary classification subproblems.
Key Equations for Weights s
Assume that CNN’s output layer, after convolutional layers, is a fully connected layer of neurons with bias terms, then the predicted probability for sample is given by the softmax activation:
(2) 
where is the feature representation of extracted from convolutional layers, and are parameters of the th neuron in the output layer. For notational simplicity, we have dropped in .
After CNN is trained, we assume that the reweighted SML is minimized to local optimum . By optimization theory, a necessary condition is that the gradient of is zero at ^{1}^{1}1More strictly, zero is in the subgradient of at . But this doesn’t affect the following analysis.:
(3) 
We specifically consider for the st class with respect to one component of
. Then with chain rule, the necessary condition above gives:
(4) 
where we use given by Eq. (2).
Let be the softmax function of with each component , its derivative is
(5) 
Since Eq. (4) holds valid for any component of , we specifically consider the case when . Therefore we have and . Then Eq. (6) becomes:
(7) 
With the same calculations, we can obtain other similar equations, each of which corresponds to one class. Remember is the probability of sample from the th class being predicted into the th class, and Eq. (8) reveals the quantitative relation between weights s, model predicted probabilities and training samples. Notice that CNN is often trained with regularization to prevent overfitting. If the bias term s are not penalized, Eq. (8) still holds valid. Another possible issue is that the calculation relies on the use of bias terms in the output layer. As using bias increases CNN’s flexibility and is not harmful to CNN performance, our analysis is still applicable to a wide range of CNN models trained with crossentropy loss.
We observe in Eq. (8), (approximately) represents the expected probability of CNN incorrectly predicting a sample of class and the expected probability of CNN misclassifying a sample of class into class . If we assume that the training data can well represent the true data distribution that testing data also follow, the learning property of trained CNN shown in Eq. (8) can be generalized to testing data.
More specifically, since the CNN model is a continuous mapping and the softmax output is bounded between 0 and 1, by the uniform law of large numbers
[22], we have the following system of equations once CNN is trained:(9) 
where for indices and , represents the expected probability of CNN predicting a sample from class into class :
where is the true data distribution for the th class.
Binary Case with LGL For binary classification problem (), Eq. (9) gives us the following relation about CNN predicted probabilities:
(10) 

In the conventional LGL where each class is weighted equally (), Eq. (10) becomes . If data exhibit severe imbalance, say , then we must have ()
If is the decision making threshold, this implies that the trained neural network can correctly predict a majority class (e.g., class 0) sample, confidently (at least) with probability 0.9, on average. However, for minority class, the predictive performance is more complex which depends on the trained model and data distribution. For example, if two classes can be well separated and the model made very confident predictions, say , then we must have for the minority class, implying a good predictive performance on class 1. If , then we have
. This means the predicted probability of a minority sample being minority is 0.2 on average. Hence, the classifier must misclassify most minority samples (
), resulting in very poor predictive accuracy for minority class. 
If LGL is reweighted using inverseclass frequencies, and , the equation above is equivalent to . Since predictions are made by and means , we can have a deterministic relation: if either class 0 or 1 can be well predicted (e.g., ), reweighting by class inverse frequencies can guarantee performance improvement for the minority class. However, the extent of “goodness” depends on the separability of the underlying data distributions of the two classes.
Simulations for Eq. (10) We conduct simulations under two settings for checking Eq. (10). The imbalance ratio is set to 10 in training data (), testing data size is ; both training and testing data follow the same data distribution. As the property only relies on the last fully connected hidden layer, we use the following setup:

Sim1: ,
. Logistic regression is fitted.
andrepresents normal and uniform distribution respectively.

Sim2: , , where , , , . A onehiddenlayer forward neural network of layer size with sigmoid activation.
Table 1 shows simulation results under three settings. We see from the Table that simulated values match with the theoretical values accurately, demonstrating the correctness of Equation (10).
RHS  10  1  0.5 

LHS (Sim1)  10.05  1.00  0.50 
(1.13)  (0.09)  (0.04)  
LHS (Sim2)  10.12  1.01  0.50 
(0.67)  (0.05)  (0.03) 
Simulation results (along with standard deviation) for Eq. (
10) over 100 runs, . RHS represents theoretical value on the righthand side of (10); LHS the simulated value on the left hand side.Multiclass Case with SML Because and Eq. (9) has variables with only equations, we can’t exactly solve it quantitatively for a relation among those ’s when . For the special case when weights are chosen as the inverseclass frequencies , considering for class 1, we have . Multiclass classification does not have a deterministic relation as in the binary case, as predictions are made by and we don’t have a decisive threshold for decision making (like the 0.5 in binary case). Our findings match the results in [31] in the sense that classwise reweighting for multiclass is indeterministic. However, our results are solely based on the mathematical property of the backpropagation algorithm from optimization theory whereas [31] is based on decision theory.
Learning property of LGL and SML As the classwise reweighting mechanism is explained in Eq. (9
), those equations also reveal the property of decision boundaries for LGL and SML. For comparison, the decision boundary of support vector machine (SVM)
[5] is determined by those support vectors that maximize the margin and those samples with larger margin have no effects on the position of decision boundary. On the contrary, all samples have their contribution to the decision boundary in LGL and SML so that their averaged probabilities that the model produces must satisfy Eq. (9). In particular for the binary case, we can see that if classes are balanced, the model must make correct predictions with equal confidence for the positive and negative classes, on average; whereas for imbalanced data, the decision boundary will be pushed towards the minority class in a position with Eq (10) always maintained. Another observation is that if the expectation of model predicted probabilities doesn’t match with its mode (e.g skewed distribution), the magnitude of tradeoff between performance of the majority and minority class depends on the direction of skewness. If the distribution of the majority class skews away from the decision boundary, upweighting minority class will boost model performance at a small cost of performance degradation for the majority class than if it skews towards the decision boundary. This implies that estimating the shape of data distribution in the latent feature space and choosing the weights accordingly would be very helpful to improve model overall performance.
Innegative Class Reweighted LGL
In this section, we focused on LGL for multiclass classification via onevs.all approach. In addition to the theoretical merits of LGL mentioned in the introduction section that LGL is capable of better capturing the structure of data manifold than SML, the guarantee of achieving good performance after properly reweighting (e.g., Eg.(10)) is also desirable as the onevs.all approach naturally introduces data imbalance issue.
Multimodality Neglect Problem In spite of those merits of LGL, it also introduces the multimodality neglect problem for multiclass classification. Since the expectation of model predicted probability must satisfy Eq (10) for LGL, the averaging effect might be harmful for model performance. In the onevs.all approach, the negative class consists of all the remaining nontarget classes, which follows a multimodal distribution (one modality for each nontarget class). LGL treats all nontarget classes equally in the learning process. If there is a hard nontarget class that shares nontrivial similarity with the target class, its contribution in LGL might be averaged out by other easy nontarget classes. In other words, those easy nontarget classes (e.g., correctly predicted as the negative class with high probabilities) would compensate the predicted probability of the hard nontarget class so that the probabilistic relation in Eq (10) is maintained. Consequently, model could incorrectly predict samples from the hard nontarget class into the target class, inducing large predictive error for that class. This phenomenon is not desirable as we want LGL to pay more attention on the separation of the targetclass with that hard class, meanwhile maintain the separation from the remaining easy nontarget classes.
To this end, we propose an improved version of LGL to reweight each nontarget class’s contribution within the negative class. Specifically, for the target class (e.g., positive class, labeled as ) and all nontarget classes (e.g., negative class, labeled as ), a twolevel reweighting mechanism is applied in LGL, which we term as innegativeclass reweighted LGL (LGLINR):
(11) 
where is the predicted probability of sample belonging to the positive class and is the weight for class as a subclass of the negative class.
The first reweighting is at the level of positive vs. negative class. If we require , using inversefrequencies will maintain the balance between the positive and negative class, as onevs.all is likely to introduce class imbalance. The second level of reweighting is within the negative class: we upweight the contribution of a hard subclass by assigning a larger , making LGLIGR focus more on the learning for that class.
Choice of s When there are a large number of classes, treating all s as hyperparameters and selecting the optimal values are not feasible in practice as we generally don’t have the prior knowledge about which classes are hard. Instead, we adopt a strategy that assigns the weights during the training process. For each nontarget class , let be the subset of in the minibatch, we use the mean predicted probability
as the classlevel hardness measurement. A larger implies class is harder to separate from the target class . We then transform those ’s using softmax to get :
where is the temperature that can smooth () or sharpen () each nontarget class’s contribution [3]. LGLINR adaptively shifts its learning focus to those hard classes, meanwhile keep attentive on those easy classes. Note that this strategy only introduces one extra parameter in LGLINR.
With the competition mechanism imposed by , LGLINR can be viewed as a smoothed learning objective between the onevs.one and onevs.all approach: when , , all nontarget classes are weighted equally, which is the innegativeclass balanced LGL using inverseclass frequencies; when is very large, concentrates on the hardest class (e.g., ) and LGLINR approximately performs onevs.one classification. We don’t specifically finetune the optimal value of and works well in our experiments.
Experiments
We evaluate LGLINR on several benchmark datasets for image classification. Note that in our experiments, applying LGL in multiclass classification naturally introduces data imbalance which is handled in our LGLINR formulation. Our primary goal here is to demonstrate that LGLINR can be used as a dropin replacement for LGL and SML with competitive or even better performance, rather than outperform the existing best models using extra training techniques. For fair comparison, all loss functions are evaluated in the same test setting. Code is made publicly available at https://github.com/Dichoto/LGLINR.
Experiment Setup
Dataset We perform experiments on four MNISTtype datasets, MNIST, FashionMNIST (FMNIST) [29], KuzushijiMNIST (KMNIST) [4] and CIFAR10. FMNIST and KMNIST are intended as dropin replacements for MNIST which are harder than MNIST. Both datasets are grayscale images consisting of 10 classes of clothing and Japanese character respectively. CIFAR10 consists of colored images of size from 10 objects.
Model  Architecture 

CNN2C  CV(C20K5S1)MP(K2S2) 
CV(C50K5S1)MP(K2S2)80010  
CNN5C  CV(C32K3S1)BNCV(C64K3S1)BN 
CV(C128K3S1)MP(K2S2)CV(C256K3S1)  
BNCV(C512K3S1)MP(K8S1)51210 
CNN architectures used for MNISTtype datasets. Cchannel represents number, Kkernel size, Sstride, BNbatch normalization and MPmax pooling
Model setup We test three loss functions on each dataset with different CNN architectures. For MNISTtype datasets, two CNNs with simple configurations are used. The first one (CNN2C) has two convolution layers and the other one (CNN5C) has 5 convolution layers with batch normalization [15]. For CIFAR10, we use MobilenetV2 [13] and Resnet18 [12] with publicly available implementations.
Implementation details
All models are trained with the standard stochastic gradient descent (SGD) algorithm. The training setups are as follows. For MNISTtype data, the learning rate is set to 0.01, the momentum is 0.5, batch size 64, number of epoch is 20. We don’t perform any data augmentation. For CIFAR data, we train the models with 100 epochs and set batch size to 64. The initial learning rate is set to 0.1, and divide it by 10 at 50th and 75th epoch. The weight decay is
and the momentum in SGD is 0.9. Data augmentation includes random crop and horizontal flip. We train all models without pretraining on largescale image data. Model performance is evaluated by the top1 accuracy rate and we report this metric on the testing data from the standard train/test split of those datasets for fair performance evaluation. For LGLINR, we report the results using .Predictive Results
Table 3 and Table 4 shows the classification accuracy using LGL, SML and LGLINR on the MNISTtype and CIFAR10 dataset respectively. From the table, we can observe that for all three loss functions, model with larger capacity yields higher accuracy. On MNISTtype data, LGL yields overall poorer performance than SML. This is because in those datasets, some classes are very similar to each other (like shirt vs. coat in FMNIST) and the negative class consists of 9 different subclasses. Hence the learning focus of LGL may get distracted from the hard subclasses due to the averaging behavior of LGL as shown in Eq (9). However, SML doesn’t suffer this problem as all negative subclasses are treated equally. On CIFAR10, LGL achieves better accuracy than SML. This is possibly due to the lack of very similar classes as in MNISTtype data. This observation demonstrates LGL’s potential as a competitive alternative to SML in some classification tasks.
Model  Loss  MNIST  FMNIST  KMNIST 

CNN2C  LGL  99.15  89.44  94.37 
SML  99.09  91.15  95.13  
LGLINR  99.29  91.15  96.43  
CNN5C  LGL  99.36  92.35  96.35 
SML  99.47  93.15  96.39  
LGLINR  99.63  93.54  97.46 
On the other hand, LGLINR adaptively pays more attention on the hard classes while keeps its separation from easy classes. This enables LGLIRN to outperform LGL and SML notably. Comparing LGLIRN with LGL, we see that the multimodality neglect problem deteriorates LGL’s ability of learning discriminative features representation, which can be relieved by the innegative class reweighing mechanism; comparing LGLIRN with SML, focusing on learning hard classes (not restricted to classes similar to the target class) is beneficial. Also, the adaptive weight assignment in the training process doesn’t require extra effort on the weight selection, making our method widely applicable.
Loss  MobilenetV2  Resnet18 

LGL  92.40  91.55 
SML  91.11  91.32 
LGLINR  93.34  93.68 
Further Analysis
We check the predictive behavior of LGLINR in detail by looking at the confusion matrix on testing data. Here, we use CNN2C and KMNIST dataset as an example. Fig. 1 show the results. We observe that for LGL, Class 1 and 2 have the lowest accuracy among 10 classes. By shifting LGL’s learning focus on hard classes, LGLINR significantly improves model performance on class 1 and 2. This is within our expectation backed by the theoretical depiction of LGL’s learning property. SML does not have the multimodality neglect problem as each class is treated equally in the learning process, yet it also does not pay more attention to the hard classes. This makes LGLINR advantageous: LGLINR outperforms SML on 9 classes out of 10. For example, class 0 have 18 samples misclassified into class 4 whereas only 6 are misclassified in LGLINR.
Figure 2 displays the training accuracy curve for LGL, SML and LGLINR on FMNIST and KMNIST. Under the same training protocol, LGLINR achieves slightly faster convergence rate than SML and LGL with comparative (FMNIST) or better (KMNIST) performance, implying that focusing on learning hard classes may facilitate model training process.
We also check the sensitivity of the temperature parameter in LGLINR weighting mechanism. Mathematically, a large or small value for is not desirable as the LGLINR is reduced to an approximate onevs.one or a classbalanced learning objective. We test on KMNIST. As shown in Table 5 and Fig. 2, model performance is not sensitive to in this range, making LGLINR a competitive alternative to LGL or SML without introducing much hyperparameter tuning.
1  2  4  

Accuracy  96.43  96.29  96.43 
Conclusion
In this paper, motivated to explain the classwise reweighting mechanism in LGL and SML, we theoretically deprived a system of probability equations that depicts the learning property of LGL and SML, as well as explains the roles of those classwise weights in the loss function. By examining the difference in the effects of the weight mechanism on LGL and SML, we identify the multimodality neglect problem is the major obstacle that can negatively affect LGL’s performance in multiclass classification. We remedy this shortcoming of LGL with a innegativeclass reweighting mechanism. The proposed method shows its effectiveness on several benchmark image datasets. For future works, we plan to incorporate the estimation of data distribution and use the reweighting mechanism of LGLINR at the sample level in the model training process to further improve the efficacy of the reweighting mechanism.
Acknowledgement
This work is supported by the National Science Foundation under grant no. IIS1724227.
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