The loss functions in PyTorch

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*My post explains optimizers in PyTorch.

A loss function is the function which can get the mean(average) of the sum of the losses(differences) between a model's predictions and true values(train or test data) to optimize a model during training or to evaluate how good a model is during testing. *Loss function is also called Cost Function or Error Function.

There are popular loss functions as shown below:

(1) L1 Loss:

• can compute the mean(average) of the sum of the absolute losses(differences) between a model's predictions and true values(train and test data).
• 's formula:
• is used for a regression model.
• is also called Mean Absolute Error(MAE).
• is L1Loss() in PyTorch.
• 's pros:
  • It's less sensitive to outliers and anomalies.
  • The losses can be easily compared because they are just made absolute so the range of them is not big.
• 's cons:
  • The absolute loss of |0| cannot be differentiable according to this post and this post.

(2) L2 Loss:

• can compute the mean(average) of the sum of the squared losses(differences) between a model's predictions and true values(train and test data).
• 's formula:
• is used for a regression model.
• is also called Mean Squared Error(MSE).
• is MSELoss() in PyTorch
• 's pros:
  • All squared losses can be differentiable.
• 's cons:
  • It's sensitive to outliers and anomalies.
  • The losses cannot be easily compared because they are squared so the range of them is big.

(3) Huber Loss:

• can do the similar computation of either L1 Loss or L2 Loss depending on the absolute losses(differences) between a model's predictions and true values(train and test data) compared with delta which you set. *Memos:
  • delta is 1.0 basically.
  • Be careful, the computation is not exactly same as L1 Loss or L2 Loss according to the formulas below.
• 's formula. *The 1st one is L2 Loss-like one and the 2nd one is L1 Loss-like one:
• is used for a regression model.
• is HuberLoss() in PyTorch.
• with delta of 1.0 is same as Smooth L1 Loss which is SmoothL1Loss() in PyTorch.
• 's pros:
  • It's less sensitive to outliers and anomalies.
  • All losses can be differentiable.
  • The losses can be more easily compared than L2 Loss because only small losses are squared so the range of them is smaller than L2 Loss.
• 's cons:
  • The computation is more than L1 Loss and L2 Loss because the formula is more complex than them.

(4) BCE(Binary Cross Entropy) Loss:

• can compute the mean(average) of the sum of the losses(differences) between a model's binary predictions and true binary values(train and test data).
• s' formula:
• is used for Binary Classification. *Binary Classification is the technology to classify data into two classes.
• is also called Binary Cross Entropy or Log(Logarithmic) Loss.
• is BCELoss() in PyTorch. *Memos:
  • There is also BCEWithLogitsLoss() which is the combination of BCE Loss and Sigmoid Activation Function in PyTorch.
  • Sigmoid Activation Function suits BCE Loss to be more stable.

(5) Cross Entropy Loss:

• can compute the mean(average) of the sum of the losses(differences) between a model's predictions and true values(train and test data). *A loss is between 0 and 1.
• s' formula:
• is used for Multiclass Classification and Computer Vision. *Memos:
  • Multiclass Classification is the technology to classify data into multiple classes.
  • Computer vision is the technology which enables a computer to understand objects.
• is CrossEntropyLoss() in PyTorch.
• s' code from scratch in PyTorch:

import torch

y_pred = torch.tensor([7.4, 2.8, -0.6])
y_true = torch.tensor([3.9, -5.1, 9.3])

def cross_entropy(y_pred, y_true):
    return -torch.sum(y_true * torch.log(y_pred))
print(cross_entropy(y_pred.softmax(dim=0), y_true.softmax(dim=0)))
# tensor(7.9744)

y_pred = torch.tensor([[7.4, 2.8, -0.6], [1.3, 0.0, 4.2]])
y_true = torch.tensor([[3.9, -5.1, 9.3], [-5.3, 7.2, -8.4]])

print(cross_entropy(y_pred.softmax(dim=1), y_true.softmax(dim=1)))
# tensor(12.2420)

• s' code with mean from scratch in PyTorch:

import torch

y_pred = torch.tensor([7.4, 2.8, -0.6])
y_true = torch.tensor([3.9, -5.1, 9.3])

def cross_entropy(y_pred, y_true):               # ↓ ↓ mean ↓ ↓
    return (-torch.sum(y_true * torch.log(y_pred))) / y_pred.ndim
print(cross_entropy(y_pred.softmax(dim=0), y_true.softmax(dim=0)))
# tensor(7.9744)

y_pred = torch.tensor([[7.4, 2.8, -0.6], [1.3, 0.0, 4.2]])
y_true = torch.tensor([[3.9, -5.1, 9.3], [-5.3, 7.2, -8.4]])

print(cross_entropy(y_pred.softmax(dim=1), y_true.softmax(dim=1)))
# tensor(6.1210)