Logistic regression turns the linear regression framework into a classifier and various types of 'regularization', of which the Ridge and Lasso methods are most common, help avoid overfit in feature rich instances.

Is regularized regression?

This is a form of regression, that constrains/ regularizes or shrinks the coefficient estimates towards zero. In other words, this technique discourages learning a more complex or flexible model, so as to avoid the risk of overfitting. A simple relation for linear regression looks like this.

Is logistic regression with regularization convex?

Abstract We show that Logistic Regression and Softmax are convex.