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Both mean squared error (MSE) and mean absolute error (MAE) are used in predictive modeling. MSE has nice mathematical properties which makes it easier to compute the gradient. However, MAE requires more complicated tools such as linear programming to compute the gradient. Because of the square, large errors have relatively greater influence on MSE than do the smaller error. Therefore, MAE is more robust to outliers since it does not make use of square. On the other hand, MSE is more useful if we are concerned about large errors whose consequences are much bigger than equivalent smaller ones. MSE also correspons to maximizing the likelihood of Gaussian random variables.

AdaBoost is the shortcut for adaptive boosting. So what’s the differences between Adaptive boosting and Gradient boosting?Both are boosting algorithms which means that they convert a set of weak learners into a single strong learner. They both initialize a strong learner (usually a decision tree) and iteratively create a weak learner that is added to the strong learner. They differ on how they create the weak learners during the iterative process.At each iteration, adaptive boosting changes the sample distribution by modifying the weights attached to each of the instances. It increases the weights of the wrongly predicted instances and decreases the ones of the correctly predicted instances. The weak learner thus focuses more on the difficult instances. After being trained, the weak learner is added to the strong one according to his performance (so-called alpha weight). The higher it performs, the more it contributes to the strong learner.On the other hand, gradient boosting doesn’t modify the sample distribution. Instead of training on a newly sample distribution, the weak learner trains on the remaining errors (so-called pseudo-residuals) of the strong learner. It is another way to give more importance to the difficult instances. At each iteration, the pseudo-residuals are computed and a weak learner is fitted to these pseudo-residuals. Then, the contribution of the weak learner (so-called multiplier) to the strong one isn’t computed according to his performance on the newly distribution sample but using a gradient descent optimization process. The computed contribution is the one minimizing the overall error of the strong learner.For more information:AdaBoost - Wikipedia (Adaptive boosting)http://statweb.stanford.edu/~jhf/ftp/trebst.pdf (Gradient boosting)

When you fit a linear model to your data, you are trying to minimize the mean squared error (MSE) between the model and the data. The data is usually split into two parts (train and test data). You fit the model to your training data, and then evaluate the model on unseen samples, known as your test set. A way of measuring how good a model fits with the data is by calculating the MSE. The smaller the MSE, the better the model.One issue that occurs is that your model may fit very well to your training set (MSE is small), but when you try it on the test set it does not work well (MSE is high). This is known as overfitting.A way of controlling overfitting is by adding an extra term to the MSE that penalizes large coefficients in the model, known as regularization. There are two approaches to do this. One penalizes the sum of the squared coefficients, known as Ridge regularization. The other penalizes the sum of the absolute value of the coefficients, knows and Lasso regularization. As a result, a regularized linear regression with fine tunes parameters will not overfit, meaning that the MSE of both the train and test set will be similar.