Regularization for Simplicity: L₂ Regularization

Consider the following generalization curve, which shows the loss for both the training set and validation set against the number of training iterations.

The loss function for the training set gradually declines. By contrast, the loss function for the validation set declines, but then starts to rise.

Figure 1. Loss on training set and validation set.

Figure 1 shows a model in which training loss gradually decreases, but validation loss eventually goes up. In other words, this generalization curve shows that the model is overfitting to the data in the training set. Channeling our inner Ockham, perhaps we could prevent overfitting by penalizing complex models, a principle called regularization.

In other words, instead of simply aiming to minimize loss (empirical risk minimization):

$$\text{minimize(Loss(Data|Model))}$$

we'll now minimize loss+complexity, which is called structural risk minimization:

$$\text{minimize(Loss(Data|Model) + complexity(Model))}$$

Our training optimization algorithm is now a function of two terms: the loss term, which measures how well the model fits the data, and the regularization term, which measures model complexity.

Machine Learning Crash Course focuses on two common (and somewhat related) ways to think of model complexity:

  • Model complexity as a function of the weights of all the features in the model.
  • Model complexity as a function of the total number of features with nonzero weights. (A later module covers this approach.)

If model complexity is a function of weights, a feature weight with a high absolute value is more complex than a feature weight with a low absolute value.

We can quantify complexity using the L2 regularization formula, which defines the regularization term as the sum of the squares of all the feature weights:

$$L_2\text{ regularization term} = ||\boldsymbol w||_2^2 = {w_1^2 + w_2^2 + ... + w_n^2}$$

In this formula, weights close to zero have little effect on model complexity, while outlier weights can have a huge impact.

For example, a linear model with the following weights:

$$\{w_1 = 0.2, w_2 = 0.5, w_3 = 5, w_4 = 1, w_5 = 0.25, w_6 = 0.75\}$$

Has an L2 regularization term of 26.915:

$$w_1^2 + w_2^2 + \boldsymbol{w_3^2} + w_4^2 + w_5^2 + w_6^2$$ $$= 0.2^2 + 0.5^2 + \boldsymbol{5^2} + 1^2 + 0.25^2 + 0.75^2$$ $$= 0.04 + 0.25 + \boldsymbol{25} + 1 + 0.0625 + 0.5625$$ $$= 26.915$$

But \(w_3\) (bolded above), with a squared value of 25, contributes nearly all the complexity. The sum of the squares of all five other weights adds just 1.915 to the L2 regularization term.

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