Yolo Loss function explanation I am trying to understand the Yolo v2 loss function:
\begin{align}
&\lambda_{coord} \sum_{i=0}^{S^2}\sum_{j=0}^B \mathbb{1}_{ij}^{obj}[(x_i-\hat{x}_i)^2 + (y_i-\hat{y}_i)^2 ] \\&+ \lambda_{coord} \sum_{i=0}^{S^2}\sum_{j=0}^B \mathbb{1}_{ij}^{obj}[(\sqrt{w_i}-\sqrt{\hat{w}_i})^2 +(\sqrt{h_i}-\sqrt{\hat{h}_i})^2 ]\\
&+ \sum_{i=0}^{S^2}\sum_{j=0}^B \mathbb{1}_{ij}^{obj}(C_i - \hat{C}_i)^2 + \lambda_{noobj}\sum_{i=0}^{S^2}\sum_{j=0}^B \mathbb{1}_{ij}^{noobj}(C_i - \hat{C}_i)^2 \\
&+ \sum_{i=0}^{S^2} \mathbb{1}_{i}^{obj}\sum_{c \in classes}(p_i(c) - \hat{p}_i(c))^2 \\
\end{align}
If any person can detail the function.
 A: Your loss function is for YOLO v1 and not YOLO v2. I was also confused with the difference in the two loss functions and seems like many people are: 
https://groups.google.com/forum/#!topic/darknet/TJ4dN9R4iJk
YOLOv2 paper explains the difference in architecture from YOLOv1 as follows:

We remove the fully connected layers from YOLO(v1) and use anchor
  boxes to predict bounding boxes... When we move to anchor boxes we
  also decouple the class prediction mechanism from the spatial location
  and instead predict class and objectness for every anchorbox.

This means that the confidence probability $p_i(c)$ above should depend not only on $i$ and $c$ but also an anchor box index, say $j$. Therefore, the loss needs to be different from above. Unfortunately, YOLOv2 paper does not explicitly state its loss function. 
I try to make a guess on the loss function of YOLOv2 and discuss it here: 
https://fairyonice.github.io/Part_4_Object_Detection_with_Yolo_using_VOC_2012_data_loss.html
A: The loss formula you wrote is of the original YOLO paper loss, not the v2, or v3 loss.
There are some major differences between versions. I suggest reading the papers, or checking the code implementations. Papers: v2, v3.
Some major differences I noticed:


*

*Class probability is calculated per bounding box (hence output is now S∗S∗B*(5+C) instead of SS(B*5 + C))

*Bounding box coordinates now have a different representation

*In v3 they use 3 boxes across 3 different "scales"
You can try getting into the nitty-gritty details of the loss, either by looking at the python/keras implementation v2, v3 (look for the function yolo_loss) or directly at the c implementation v3 (look for delta_yolo_box, and delta_yolo_class).
A: Explanation of the different terms : 


*

*The 3 $\lambda$ constants are just constants to take into account more one aspect of the loss function. In the article $\lambda_{coord}$ is the highest in order to have the more importance in the first term

*The prediction of YOLO is a $S*S*(B*5+C)$ vector : $B$ bbox predictions for each grid cells and $C$ class prediction for each grid cell (where $C$ is the number of classes). The 5 bbox outputs of the box j of cell i are coordinates of tte center of the bbox $x_{ij}$ $y_{ij}$ , height $h_{ij}$, width $w_{ij}$ and a confidence index $C_{ij}$

*I imagine that the values with a hat are the real one read from the label and the one without hat are the predicted ones. So what is the real value from the label for the confidence score for each bbox $\hat{C}_{ij}$ ? It is the intersection over union of the predicted bounding box with the one from the label.

*$\mathbb{1}_{i}^{obj}$ is $1$ when there is an object in cell $i$ and $0$ elsewhere

*$\mathbb{1}_{ij}^{obj}$ "denotes that the $j$th bounding box predictor in cell $i$ is responsible for that prediction". In other words, it is equal to $1$ if there is an object in cell $i$ and confidence of the $j$th predictors of this cell is the highest among all the predictors of this cell. $\mathbb{1}_{ij}^{noobj}$ is almost the same except it values 1 when there are NO objects in cell $i$


Note that I used two indexes $i$ and $j$ for each bbox predictions, this is not the case in the article because there is always a factor $\mathbb{1}_{ij}^{obj}$ or $\mathbb{1}_{ij}^{noobj}$ so there is no ambigous interpretation : the $j$ chosen is the one corresponding to the highest confidence score in that cell.
More general explanation of each term of the sum :


*

*this term penalize bad localization of center of cells

*this term penalize the bounding box with inacurate height and width. The square root is present so that erors in small bounding boxes are more penalizing than errors in big bounding boxes.

*this term tries to make the confidence score equal to the IOU between the object and the prediction when there is one object

*Tries to make confidence score close to $0$ when there are no object in the cell

*This is a simple classification loss (not explained in the article)

A: Here is my Study Note


*Loss function: sum-squared error
a. Reason: Easy to optimize
b. Problem: (1) Does not perfectly align with our goal of maximize average precision. (2) In every image, many grid cells do not contain any object. This pushes the confidence scores of those cells towards 0, often overpowering the gradient from cells that do contain an object.
c. Solution: increase loss from bounding box coordinate predictions and decrease the loss from confidence predictions from boxes that don't contain objects. We use two parameters $$\lambda_{coord} = 5$$ and $\lambda_{noobj}$ = 0.5
d. Sum-squared error also equally weights errors in large boxes and small boxes


*Only one bounding box should be responsible for each object. We assign one predictor to be responsible for predicting an object based on which prediction has the highest current IOU with the ground truth.
a. Loss from bound box coordinate (x, y) Note that the loss comes from one bounding box from one grid cell. Even if obj not in grid cell as ground truth.
$$
 \begin{cases}
  \lambda_{coord} \sum^{S^2}_{i=0} [(x_i - \hat{x}_i)^2 + (y_i - \hat{y_i})^2] &\text{responsible bounding box} \\
  0 &\text{ other} \\
 \end {cases} 
$$
b. Loss from width w and height h. Note that the loss comes from one bounding box from one grid cell, even if the object is not in the grid cell as ground truth.
$$
 \begin {cases}
  \lambda_{coord} \sum^{S^2}_{i=0} [(\sqrt{w_i} - \sqrt{\hat{w}_i})^2 + (\sqrt{h_i} - \sqrt{\hat{h}_i})^2] &\text{responsible bounding box} \\
  0 &\text{ other} \\
 \end {cases}
$$
c. Loss from the confidence in each bound box. Not that the loss comes from one bounding box from one grid cell, even if the object is not in the grid cell as ground truth.
$$
 \begin {cases}
  \sum^{S^2}_{i=0}(C_i - \hat{C}_i)^2 &\text{obj in grid cell and responsible bounding box} \\
  \lambda_{noobj} \sum^{S^2}_{i=0}(C_i - \hat{C}_i)^2 &\text{obj not in grid cell and responsible bounding box} \\
  0 &\text{other}
 \end {cases}
$$
d. Loss from the class probability of grid cell, only when object is in the grid cell as ground truth.
$$
 \begin {cases}
  \sum^{S^2}_{i=0} \sum_{c \in classes} (p_i(c) - \hat{p}_i(c))^2 &\text{obj in grid cell}\\
  0 &\text{other} \\
 \end {cases}
$$
Loss function only penalizes classification if obj is present in the grid cell.
It also penalize bounding box coordinate if that box is responsible for the ground box (highest IOU)
