University of Michigan - Ann Arbor

Weight decay, commonly known as $$\ell_2$$ regularization, is a widely used technique for regularizing parametric machine learning models. Instead of directly manipulating the number of parameters, weight decay operates by restricting the values that the parameters can take. The technique is motivated by the intuition that the simplest function is $$f = 0$$, and the complexity of a linear function, such as $$f(\mathbf{x}) = \mathbf{w}^	op \mathbf{x}$$, can be measured by the distance of its parameters from zero.

Weight Decay

In machine learning _Frobenius_ and _L2_ norms can be used interchangeably. Nevertheless there is slight difference between these two. _L2_ norm in essence is an Euclidean norm with special case, when p=2. More specifically for n-dimensional vector we will have:
   $$L_2 = \sqrt {\textstyle\sum_{i=1}^n x_{i}^{2}}$$.
The _Frobenius_ measures the same thing but in this case we have matrices. That's why in neural networks _L2_ norm is frequently referred as _Forbenius_ norm.

_Frobenius_ and _L2_

In Ridge regression the coefficients and bias are learned using the same least-square criterion, but it adds a penalty for large variations in coefficients; i.e., coefficients are found by minimizing a tuning parameter - which controls the strength of the penalty term.
Once the parameters are learned, the ridge regression prediction formula is the same as OLS.
Ridge regression uses L2 regularization that minimizes the sum of square of coefficients and the influence of the regularization term is controlled by the $\alpha$ parameter. Higher $\alpha$ means more regularization and simpler models.
Use Ridge regression when the number of predictor variables is greater than the number of observations. Below is the formula found in our textbook. 
$$RSS_{RIDGE}(\beta_{j}, \beta_{0}) =
\sum_{i=1}^{n} ({y}_{i}-{\beta}_{0}-\sum_{j=1}^{p}{\beta}_{j}{x}_{ij})^2 +\lambda\sum_{j=1}^{p}{\beta}_{j}^2=RSS+\lambda\sum_{j=1}^{p}{\beta}_{j}^2.
$$ 

Note: Ridge Regression is sensitive to scales of variable. Therefore, we usually standardize the predictors before applying Ridge Regression.

Ridge Regression

What is weight decay?

A normal distribution, also known as a Gaussian distribution, is a continuous probability distribution defined by its mean $$\mu$$ and variance $$\sigma^2$$ (where $$\sigma$$ is the standard deviation). The probability density function is given by the formula: $$p(x) = \frac{1}{\sqrt{2 \pi \sigma^2}} \exp\left(-\frac{1}{2 \sigma^2} (x - \mu)^2ight)$$. Changing the mean $$\mu$$ corresponds to a shift of the distribution along the $$x$$-axis, while increasing the variance $$\sigma^2$$ spreads the distribution out and lowers its peak.

Gaussian (Normal) Distribution

The trade-off between the standard prediction loss and the additive weight decay penalty is characterized by the regularization constant, $$\lambda$$. This nonnegative hyperparameter is fit using validation data and modifies the objective to $$L(\mathbf{w}, b) + \frac{\lambda}{2} \|\mathbf{w}\|^2$$. When $$\lambda = 0$$, the original loss function is recovered. For $$\lambda > 0$$, the size of the weights is restricted, with larger values of $$\lambda$$ constraining the weights more considerably. The penalty term is divided by $$2$$ by convention so that the constant cancels out gracefully when the derivative of the quadratic function is taken.

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