> For the complete documentation index, see [llms.txt](https://json007.gitbook.io/svm/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://json007.gitbook.io/svm/lr/l2_regularization.md).

# L2 regularization

$$
\frac {1} {N} \sum\_{n=1}^N \log (1+\exp(-y\_n W^T X\_n)) + \lambda \left | W \right |\_2^2
$$

## kernel logistic regression##\#

带L2正则的LR：

$$
\min\_W \frac {\lambda} {N} W^TW + \frac {1} {N} \sum\_{n=1}^N \log (1+\exp(-y\_n W^T X\_n)) \\
W = \sum\_{n=1}^N \beta\_n X\_n \\
\min\_{\beta} \frac {\lambda} {N} \sum\_{n=1}^N \sum\_{m=1}^n \beta\_n \beta\_m K(X\_n,X\_m) + \frac {1} {N} \sum\_{n=1}^N \log (1+\exp(-y\_n \sum\_{m=1}^N \beta\_m K(X\_m,X\_n) ))
$$

为什么W的最优解是X的线性组合？ ![](https://2270971654-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-M7DcNFhVrwIk3Tks_pB%2Fsync%2F5ad41911a09fe60c2b1610a24d2149acb4a9d6b2.png?generation=1589383940702543\&alt=media)

## 求解##\#

坐标下降法等