# Some Loss Functions and Their Intuitive Explanations

## Contents

Loss functions are frequently used in supervised machine learning to minimize the differences between the predicted output of the model and the ground truth labels. In other words, it is used to measure how good our model can predict the true class of a sample from the dataset. Here I would like to list some frequently-used loss functions and give my intuitive explanation.

# Cross Entropy Loss

Cross Entropy Loss is usually used in classification problems. In essence, it is a measure of difference between the desired probablity distribution and the predicted probablity distribution. Suppose the classification is binary classification problem, the label are \({0, 1}\). Then the loss function for a single sample in the dataset is expressed as:

\[-y \log(p)-(1-y) \log(1-p)\ ,\]

where \(y\) is the label of the sample, and \(p\) is the predicted probability of the sample belonging to class 1.

For \(K\)-class (\(K >2\) ) classification problems, the predicted probablity output for a single sample is a vector of length \(K\): \([p_0, p_1, \ldots, p_{K-1}]\). The Cross Entropy Loss is extended as: \[-log (p_k)\ ,\] where \(k\) is the ground truth label for the sample. If \(p_k\) equals 1, then there is no loss incurred for that sample.

We often see the term "Softmax Loss" in literature or blog post. In fact, there is no such thing as "Softmax Loss", as is discussed on this Quora post. Suppose the original prediction for a sample is \([x_0, x_1, \ldots, x_{K-1}]\), we can get the normalized probablity \([p_0, p_1, \ldots, p_{K-1}]\)， where we have:

\[p_k = \frac{\exp(x_k)}{\sum_{i=0}^{K-1}\exp(x_i)}\ .\]

So the Cross Entropy Loss really is:

\[-\log \frac{\exp(x_k)}{\sum_{i=0}^{K-1}\exp(x_i)}\ .\]

In brief, the so-called Softmax Loss is just Softmax function followed by Cross Entropy Loss.

# Contrastive Loss

Contrastive Loss is often used in image retrieval tasks to learn discriminative features for images. During training, an image pair is fed into the model with their ground truth relationship \(y\): \(y\) equals 1 if the two images are similar and 0 otherwise. The loss function for a single pair is:

\[yd^2+(1-y)\max (margin-d, 0)^2\ ,\]

where \(d\) is the Euclidean distance between the two image features (suppose their features are \(f_1\) and \(f_2\)): \(d = \Vert f_1 - f_2\Vert_2\). The \(margin\) term is used to "tighten" the constraint: if two images in a pair are dissimilar, then their distance shoud be at least \(margin\), or a loss will be incurred.

# Triplet Loss

Triplet Loss is another loss commonly used in CNN-based image retrieval. During training process, an image triplet \((I_a, I_n, I_p)\) is fed into the model as a single sample, where \(I_a\), \(I_n\) and \(I_p\) represent the anchor, postive and negative images respectively. The idea behind is that distance between anchor and positive images should be smaller than that between anchor and negative images. The formal definition is:

\[\max \left( {\Vert f_a- f_p \Vert}^2 - {\Vert f_a - f_n \Vert}^2 + m, 0\right)\ .\]

In the above equation, \(m\) is a margin term used to "stretch" the distance differences between similar and dissimilar pair in the triplet, \(f_a, f_p, f_n\) are the feature embeddings for the anchor, postive and negative images.