# Review — CB Loss: Class-Balanced Loss Based on Effective Number of Samples (Image Classification)

## Using the Effective Number of Samples for Each Class to Re-Balance the Loss, Outperforms Focal Loss in RetinaNet

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In this paper, **Class-Balanced Loss Based on Effective Number of Samples**, (**CB Loss**), by Cornell University, Cornell Tech, Google Brain, and Alphabet Inc., is reviewed. In this paper:

- A re-weighting scheme is designed that
**uses the effective number of samples for each class to re-balance the loss**, called class-balanced loss.

This is a paper in **2019 CVPR **over **200** **citations**. (Sik-Ho Tsang @ Medium)

# Outline

**Class Imbalance Problem****Effective Number of Samples****Class-Balanced Loss (CB Loss)****Experimental Results**

**1. Class Imbalance Problem**

- Suppose there are classes that arm imbalanced as above.
**Head**: For the class with small indices, these classes have the larger number of samples.**Tail**: For the class with large index, these classes have the smaller number of samples.**Black Solid Line**: Models directly trained on these samples are**biased toward dominant classes.****Red Dashed Line**:**Re-weighting the loss by inverse class frequency**may yield poor performance on real-world data with high class imbalance.**Blue Dashed Line**: A class-balanced term is designed to**re-weight the loss by inverse effective number of samples.**

# 2. Effective Number of Samples

## 2.1. Definition

Intuitively, the more data, the better. However, since

there is information overlap among data, as the number of samples increases, the marginal benefit a model can extract from the data diminishes.

**Left**: Given a class, denote the set of all possible data in the**feature space of this class**as. Assume the volume of*S**S*is*N*and*N*≥ 1.**Middle**:**Each sample**in a subset of*S*has the**unit volume of 1**and**may overlap with other samples**.**Right**: Each subset is randomly sampled from*S*to cover the entire set of*S*. The more data is being sampled, the better the coverage of*S*is.- The expected total volume of sampled data increases as the number of samples increases and is bounded by
*N*.

Therefore, the effective number of samples is defined as the expected volume of samples.

**The idea is to capture the diminishing marginal benefits by using more data points of a class.**- Due to intrinsic similarities among real-world data, as the number of samples grows, it is highly possible that a newly added sample is a near-duplicate of existing samples.
- In addition, CNNs are trained with heavy data augmentations, all augmented examples are also considered as same with the original example.
- For a class,
can be viewed as*N***the number of unique prototypes**.

## 2.2. Mathematical Formulation

- Denote the
**effective number (expected volume) of samples**as.*En* - To simplify the problem, the situation of partial overlapping is not considered.
- That is,
**a newly sampled data point**can only interact with previously sampled data in two ways: either**entirely inside the set of previously sampled data**with the**probability of**or*p***entirely outside**with the**probability of 1-**.*p* **Proposition (Effective Number)**:*En*= (1−*β^n*)/(1−*β*), where*β*= (*N*− 1)/*N*. This proposition is proved by**mathematical induction**.- When
*E*1 = 1, there is no overlapping.*E*1 = (1−*β^*1)/(1−*β*) = 1 holds. - Consider there are previously sampled
*n*−1 examples and are about to sample the*n*th example. Now**the expected volume of previously sampled data is**and*En*−1**the newly sampled data point has the probability of**. Therefore,*p*=*E*(*n*−1)/*N*to be overlapped with previous samples**the expected volume after sampling***n*th example is:

- With:

- We got:

- The above proposition shows that
**the effective number of samples is an exponential function of***n*. **The hyperparameter***β*∈ [0, 1) controls how fast*En*grows as*n*increases.

**3. Class-Balanced Loss (CB Loss)**

**The class-balanced (CB) loss**is written as:

- where
*ny*is the number of samples in the ground-truth class*y*. *β*= 0 corresponds to no re-weighting and*β*→ 1 corresponds to re-weighing by inverse class frequency.

The proposed novel concept of effective number of samples enables us to use a

hyperparameterβto smoothly adjust the class-balanced term between no re-weighting and re-weighing by inverse class frequency.

- The proposed class-balanced term is
**model-agnostic**and**loss-agnostic**in the sense that it is independent to the choice of loss function*L*and predicted class probabilities*p*.

## 3.1. Class-Balanced Softmax Cross-Entropy Loss

- Given a sample with class label
*y*, the softmax cross-entropy (CE) loss for this sample is written as:

- Suppose class
*y*has*ny*training samples,**the class-balanced (CB) softmax cross-entropy loss**is:

## 3.2. Class-Balanced Sigmoid Cross-Entropy Loss

- When using sigmoid function for multi-class problem, each output ode of the network is performing a
**one-vs-all classification**to predict the probability of the target class over the rest of classes. - In this case,
**Sigmoid doesn’t assume the mutual exclusiveness among classes.** - Since each class is considered independent and has its own predictor, sigmoid unifies single-label classification with multi-label prediction. This is a nice property to have since real-world data often has more than one semantic label.
- The sigmoid cross-entropy (CE) loss can be written as:

**The class-balanced (CB) sigmoid cross-entropy loss**is:

## 3.3. Class-Balanced Focal Loss

- The focal loss (FL) proposed in RetinaNet, reduce the relative loss for well-classified samples and focus on difficult samples:

- (For more details, if interested, please feel free to read RetinaNet.)
**The class-balanced (CB)****Focal Loss**

# 4. Experimental Results

## 4.1. Datasets

- 5 long-tailed versions of both CIFAR-10 and CIFAR-100 with imbalance factors of 10, 20, 50, 100 and 200 respectively, are tried.
- iNaturalist and ILSVRC are class imbalance in nature.

- The above shows the number of images per class with different imbalance factors.

## 4.2. CIFAR Datasets

- The search space of hyperparameters is {softmax, sigmoid, focal} for loss type,
*β*∈ {0.9, 0.99, 0.999, 0.9999}, and*γ*∈ {0.5, 1.0, 2.0} for Focal Loss. **The best**unanimously.*β*is 0.9999 on CIFAR-10**But on CIFAR-100, datasets with different imbalance factors tend to have different and smaller optimal***β*.

- On
**CIFAR-10**, when re-weighting based on*β*= 0.9999, the effective number of samples is close to the number of samples. This means**the best re-weighting strategy on CIFAR-10 is similar with re-weighting by inverse class frequency.** - On
**CIFAR-100**, the poor performance of using larger*β*suggests that re-weighting by inverse class frequency is not a wise choice.**A smaller***β*is needed that has smoother weights across classes. - For example, the number of unique prototypes of a specific bird species should be smaller than the number of unique prototypes of a generic bird class.
**Since classes in CIFAR-100 are more fine-grained than CIFAR-10, CIFAR-100 have smaller***N*compared with CIFAR-10.

## 4.3. Large-Scale Datasets

**The class-balanced****Focal Loss**is used since it has more flexibility and it is found thatand*β*= 0.999*γ*= 0.5**good performance**on all datasets.- Notably,
**ResNet****-50 is able to achieve comparable performance with****ResNet****-152 on iNaturalist and****ResNet****-101 on ILSVRC 2012 when using class-balanced****Focal Loss**to replace softmax cross-entropy loss.

- The above figures show
**the class-balanced****Focal Loss****starts to show its advantage after 60 epochs**of training.

## Reference

[2019 CVPR] [CB Loss]

Class-Balanced Loss Based on Effective Number of Samples

## Image Classification

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