Regularization for Simplicity: Lambda

Model developers tune the overall impact of the regularization term by multiplying its value by a scalar known as lambda (also called the regularization rate). That is, model developers aim to do the following:

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

Performing L2 regularization has the following effect on a model

  • Encourages weight values toward 0 (but not exactly 0)
  • Encourages the mean of the weights toward 0, with a normal (bell-shaped or Gaussian) distribution.

Increasing the lambda value strengthens the regularization effect. For example, the histogram of weights for a high value of lambda might look as shown in Figure 2.

Histogram of a model's weights with a mean of zero and a normal distribution.

Figure 2. Histogram of weights.

Lowering the value of lambda tends to yield a flatter histogram, as shown in Figure 3.

Histogram of a model's weights with a mean of zero that is somewhere between a flat distribution and a normal distribution.

Figure 3. Histogram of weights produced by a lower lambda value.

When choosing a lambda value, the goal is to strike the right balance between simplicity and training-data fit:

  • If your lambda value is too high, your model will be simple, but you run the risk of underfitting your data. Your model won't learn enough about the training data to make useful predictions.

  • If your lambda value is too low, your model will be more complex, and you run the risk of overfitting your data. Your model will learn too much about the particularities of the training data, and won't be able to generalize to new data.

The ideal value of lambda produces a model that generalizes well to new, previously unseen data. Unfortunately, that ideal value of lambda is data-dependent, so you'll need to do some tuning.

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