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Approximation error in machine learning refers to the error introduced by approximating a real-world problem with a simplified model. In other words, it's the difference between the model's predictions and the actual values in the dataset, ε(h*) - ε:(g). Machine learning models aim to minimize this error to make accurate predictions.

There are different sources of approximation error:

1. Model Complexity: If a model is too simple, it may not capture the underlying patterns in the data, leading to high approximation error. On the other hand, an overly complex model might fit the training data too closely, resulting in poor generalization to new, unseen data.

2. Bias: Bias is the error introduced by approximating a real-world problem with a simplified model. High bias can lead to underfitting, where the model is too simplistic to capture the underlying patterns in the data.

3. Variance: Variance is the error introduced due to the model's sensitivity to fluctuations in the training data. High variance can lead to overfitting, where the model fits the training data too closely and fails to generalize well to new data.

4. Noise in the Data: If the dataset contains random or irrelevant information (noise), the model may try to fit these patterns, leading to an increase in approximation error.

Figure 3766a shows the expected risk (error), approximation error, and estimation error.

Figure 3766a. Expected risk (error), approximation error, and the estimation error. [1]

The expected risk (error) of a hypothesis h_s ∈H, which is selected based on the training dataset  S  from a hypothesis class H, can be decomposed into the approximation error, ε_app, and the estimation error, ε_est, as following,

          L_D(h_s) = ε_app + ε_est ------------------------------------- [3766]

Figure 3766b shows the relationship between these terms in Equation 3766. The red points are specific hypotheses.  The best hypothesis (the Bayes hypothesis) lies outside the chosen hypothesis class H. The distance between the risk of  h^ and the risk of h* is the estimation error, while the distance between  ℎ* and Bayes hypothesis is the approximation error.

Some properties are:

• The larger  H is, the smaller this error is, because it's more likely that a larger hypothesis class contains the actual hypothesis we are looking for. Therefore, if  H does not contain the actual hypothesis we are searching for, then this error could not be zero.

• This error does not depend on the training data since in Equation 3766, there's no S (the training dataset).

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[1] www.medium.com.