跟 GPT 学 AI：Loss Function vs Cost Function

The loss function and the cost function are related concepts in machine learning, but they have different meanings and uses:

Loss Function

• The loss function measures the error for a single training example. It tells us how well the model is performing for one instance.

• It represents the difference between the predicted value (output by the model) and the actual value (ground truth).

• For example, in regression problems, a common loss function is the Mean Squared Error (MSE):

  $$\text{Loss}(y, \hat{y}) = (\hat{y} - y)^2$$

  where $\hat{y}$ is the predicted value and $y$ is the actual value.

• In classification problems, the loss function often used is the log loss (or cross-entropy loss).

Cost Function

• The cost function is the average of the loss function over the entire training dataset. It measures how well the model is performing overall.

• The cost function is used to find the optimal model parameters by minimizing the average error over all training examples.

• Mathematically, the cost function is often defined as:

  $$J(\theta) = \frac{1}{m} \sum_{i=1}^m \text{Loss}(y^{(i)}, \hat{y}^{(i)})$$

  where $m$ is the number of training examples, and $\theta$ represents the model parameters.

Key Differences:

• Scope: The loss function deals with individual samples, while the cost function is an aggregated measure over the entire training dataset.
• Purpose: The loss function helps understand the model's performance on a single data point, whereas the cost function is used to optimize the model parameters by minimizing the overall error.
• Terminology: In practice, "loss function" and "cost function" are sometimes used interchangeably, but the distinction is that the cost function typically refers to the average loss across all examples.

Example:

• For linear regression:
  • Loss function for a single data point: $(\hat{y} - y)^2$
  • Cost function (Mean Squared Error): $J(\theta) = \frac{1}{m} \sum_{i=1}^m (\hat{y}^{(i)} - y^{(i)})^2$

To summarize, the loss function is a measure of error for a single instance, while the cost function is the average loss across all instances and is the objective that we aim to minimize during model training.