# Overfitting: L2 regularization

L2 regularization is a popular regularization metric, which uses the following formula:

$$L_2\text{ regularization } = {w_1^2 + w_2^2 + ... + w_n^2}$$

For example, the following table shows the calculation of L2 regularization for a model with six weights:

Value Squared value
w1 0.2 0.04
w2 -0.5 0.25
w3 5.0 25.0
w4 -1.2 1.44
w5 0.3 0.09
w6 -0.1 0.01
26.83 = total

Notice that weights close to zero don't affect L2 regularization much, but large weights can have a huge impact. For example, in the preceding calculation:

• A single weight (w3) contributes about 93% of the total complexity.
• The other five weights collectively contribute only about 7% of the total complexity.

L2 regularization encourages weights toward 0, but never pushes weights all the way to zero.

If you use L2 regularization while training a model, what will typically happen to the overall complexity of the model?
The overall complexity of the system will probably drop.
Since L2 regularization encourages weights towards 0, the overall complexity will probably drop.
The overall complexity of the model will probably stay constant.
This is very unlikely.
The overall complexity of the model will probably increase.
This is unlikely. Remember that L2 regularization encourages weights towards 0.
If you use L2 regularization while training a model, some features will be removed from the model.
True
Although L2 regularization may make some weights very small, it will never push any weights all the way to zero. Consequently, all features will still contribute something to the model.
False
L2 regularization never pushes weights all the way to zero.

## Regularization rate (lambda)

As noted, training attempts to minimize some combination of loss and complexity:

$$\text{minimize(loss} + \text{ complexity)}$$

Model developers tune the overall impact of complexity on model training by multiplying its value by a scalar called the regularization rate. The Greek character lambda typically symbolizes the regularization rate.

That is, model developers aim to do the following:

$$\text{minimize(loss} + \lambda \text{ complexity)}$$

A high regularization rate:

• Strengthens the influence of regularization, thereby reducing the chances of overfitting.
• Tends to produce a histogram of model weights having the following characteristics:
• a normal distribution
• a mean weight of 0.

A low regularization rate:

• Lowers the influence of regularization, thereby increasing the chances of overfitting.
• Tends to produce a histogram of model weights with a flat distribution.

For example, the histogram of model weights for a high regularization rate might look as shown in Figure 18.

In contrast, a low regularization rate tends to yield a flatter histogram, as shown in Figure 19.

### Picking the regularization rate

The ideal regularization rate produces a model that generalizes well to new, previously unseen data. Unfortunately, that ideal value is data-dependent, so you must do some tuning.

### Early stopping: an alternative to complexity-based regularization

Early stopping is a regularization method that doesn't involve a calculation of complexity. Instead, early stopping simply means ending training before the model fully converges. For example, you end training when the loss curve for the validation set starts to increase (slope becomes positive).

Although early stopping usually increases training loss, it can decrease test loss.

Early stopping is a quick, but rarely optimal, form of regularization. The resulting model is very unlikely to be as good as a model trained thoroughly on the ideal regularization rate.

### Finding equilibrium between learning rate and regularization rate

Learning rate and regularization rate tend to pull weights in opposite directions. A high learning rate often pulls weights away from zero; a high regularization rate pulls weights towards zero.

If the regularization rate is high with respect to the learning rate, the weak weights tend to produce a model that makes poor predictions. Conversely, if the learning rate is high with respect to the regularization rate, the strong weights tend to produce an overfit model.

Your goal is to find the equilibrium between learning rate and regularization rate. This can be challenging. Worst of all, once you find that elusive balance, you may have to ultimately change the learning rate. And, when you change the learning rate, you'll again have to find the ideal regularization rate.

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{ "lastModified": "Last updated 2024-08-13 UTC.", "confidential": False }