Why does ridge regression only have one hyperparameter $\lambda$?

Ridge Regression objective$$\underset{\beta}{\text{min}} \sum_{i=1}^n (y_i - \beta \cdot x_i)^2 + \lambda \|\beta\|_2^2$$

SVM primal problem:

\begin{align} \max_{\mathbf{\alpha}} \quad &\min_{\mathbf{w},b} \frac{\|\mathbf{w}\|}{2}+ C \sum_{i=1}^{N} \alpha^{(i)} \left(1-\mathbf{w^T}\phi \left(\mathbf{x}^{(i)}\right)+b)\right), \\ s.t. \quad&0 \leq \alpha^{(i)} \leq C, &\forall i \in \{1,\dots,N\} \end{align}

Why does SVM have a hyperparameter parameter C on the hinge-loss function while in Ridge Regression There's No C parameter infront of the quadratic loss function?

Likewise, why is there no $$\lambda$$ parameter behind $$||w||$$ in SVM so $$\lambda$$ is assumed 1/2 in SVM?

Why isn't ridge: $$\underset{\beta}{\text{min}} \ C\sum_{i=1}^n (y_i - \beta \cdot x_i)^2 + \lambda \|\beta\|_2^2$$

Why isn't the SVM:

\begin{align} \max_{\mathbf{\alpha}} \quad &\min_{\mathbf{w},b} \lambda\|\mathbf{w}\|+ C \sum_{i=1}^{N} \alpha^{(i)} \left(1-\mathbf{w^T}\phi \left(\mathbf{x}^{(i)}\right)+b)\right), \\ s.t. \quad&0 \leq \alpha^{(i)} \leq C, &\forall i \in \{1,\dots,N\} \end{align}?

To introduce both hyperparameters in one equation would be redundant.

If you have both $$C$$ and $$\lambda$$, then there are several equivalent optimization problems with the same ratio of $$C$$ to $$\lambda$$. For instance, in your proposed ridge regression equation, the solution for $$C=1, \lambda=1$$ is necessarily the same as for $$C=1000, \lambda=1000$$. (This is because you can just factor out the common factor.)

\begin{align} &\min_{x,y} 1x + 1 y \\ =&\min_{x,y} 1000x + 1000 y \end{align}

In other words, it looks like you have two hyperparameters ($$C$$ and $$\lambda$$), but you truly have only one: the ratio between $$C$$ and $$\lambda$$.

By clamping one of the two values, you avoid a bit of wasteful overparameterization. If $$C$$ is always $$1$$, then adjusting $$\lambda$$ is the only way to adjust the ratio. Same for keeping $$\lambda$$ always at $$1$$.

• How is Elastic Net/Generalizations of the Lasso with two+ hyperparameters in one equation non-redundant? May 14, 2021 at 4:13
• Lasso has two summands and one hyperparameter. ElasticNet has three summands and two hyperparameters. In general, you need $n-1$ hyperparameters for $n$ summands. The remaining one is clamped to 1. May 14, 2021 at 4:20
• There's no redundancy: it's perfectly possible to introduce multiple parameters into Ridge Regression. See my remarks in the last paragraph at stats.stackexchange.com/a/164546/919.
– whuber
May 14, 2021 at 12:08
• @whuber I think we’re talking past each other. There are perfectly valid opportunities to introduce multiple parameters. In this particular introduction, you gain no expressive power in your model. May 14, 2021 at 12:43