the problem of overfitting

underfitting or high bias—hypothesis function h maps poorly to the trend of the data.
usually caused by a function that is too simple or uses too few features.

overfitting or high variance—fits the available data but does not generalize well to predict new data.
usually caused by a complicated function that creates a lot of unnecessary curves and angles unrelated to the data.

to address it:
1) Reduce the number of features:
1. Manually select which features to keep.
2. Use a model selection algorithm .
2) Regularization
1. Keep all the features: but reduce the magnitude of parameters θj.
2. Regularization works well when we have a lot of slightly useful features.

Regularization:
1.regularized linear regression
Without actually getting rid of these features or changing the form of our hypothesis, we can instead modify our cost function:

$$min_\theta\frac{1}{2m}[\sum_{i=1}^{m} (h_\theta(x^{(i)})-y^{(i)})^2+\lambda\sum_{j=1}^{n} \theta_j^2]$$
The $\lambda$, is the regularization parameter.
If $\lambda$ is chosen to be too large, it may smooth out the function too much and cause underfitting.

As a result, we see that the new hypothesis (depicted by the pink curve) looks like a quadratic function but fits the data better due to the extra small terms θ

actually,$(1-\alpha\frac{\lambda}{m})<1$
so it shrink the parameter a little bit before do the same thing as previous.

Using regularization also takes care of any non-invertibility issues of the X transpose X matrix as well.

if m ≤ n, then $X^TX$ is non-invertible. However, when we add the term λ⋅L, then $X^TX + λ⋅L$ becomes invertible.

2.regularized logistic regression


the θ vector is indexed from 0 to n (holding n+1 values, $\theta_0$ through θn), and this sum explicitly skips $\theta_0$

b.t.w Because regularization causes J(θ) to no longer be convex, gradient descent may not always converge to the global minimum (when λ>0, and when using an appropriate learning rate α).