Why is dimensionality reduction used if it almost always reduces the explained variation? Let's say I have $N$ covariates in my regression model, and they explain 95% of the variation of the target set, i.e. $r^2=0.95$. If there are multicollinearity between these covariates and PCA is performed to reduce the dimensionality. If the principal components explains, say 80% of the variation (as opposed to 95%), then I have incurred some loss in the accuracy of my model.
Effectively, if PCA solves the issue of multicollinearity at the cost of accuracy, is there any benefit to it, other than the fact it can speed up model training and can reduce collinear covariates into statistically independent and robust variables?


If the principal components explains, say 80% of the variation (as opposed to 95%), then I have incurred some loss in the accuracy of my model.

Performing PCA does not reduce the accuracy of the model. The principal components, when you use all of them, should also explain the 95%. It is the reduction of the dimensionality which reduces the explained variation.
So this is a matter of model selection and finding models with fewer parameters. The role of PCA is to do this model selection by redefining the parameter space in order to find a small number of components that explain a large amount of variation.
