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All right so where are we now last week we set up our framework for statistical learning input space is output space is action space is hypothesis spaces and we proposed a basically a method for doing machine learning empirical risk minimization this was finding a prick ssin function that minimizes the average loss on a training set and then we talked about the issues of overfitting if the if we're searching for the function that minimizes the empirical risk over all functions we over fit we said okay let's instead constrain the set of functions we're searching over constrain the hypothesis space to some set containing functions that we think have properties we like or or are simpler or it's just a smaller set and this could help control overfitting and today what we're gonna do is move more strongly into this issue of preventing overfitting with what's called a regularization so regularization has you know various definitions depending on the context but in machine learning these days I think people use it rather loosely to mean anything you do to the process of fitting your data that makes your prediction function fit the training data less well less well in the hopes that it's going to fit new data better so it's going to generalize better so very generically that's how we can think of regularization fitting data less well so that it generalizes with the intent to have it generalize better so today we're going to talk about the two most common forms of regularization l1 and l2 regularization and what's great is that with just these two forms of regularization we're gonna get what I think are the two most important methods of machine learning namely Ridge regression and lasso regression so we get them right off the bat the two biggest bang for your buck and now algorithms and we're gonna go pretty deep into these algorithms we're going to figure out how to actually implement them we're going to think about the properties from a math perspective with purely a practical I like what is this telling us about using these things in practice so we're talking about bigger and smaller positive spaces and often it's very convenient to think about nest that hypothesis spaces nested I pass the spaces are well we have one hypothesis space one set of friction functions completely contained in a larger one so an example polynomials give a good example we have all the polynomials of degree 2 and then polynomials of degree 3 contain the polynomials of degree 2 because we can just set the highest degree monomial to have a coefficient of zero and we can write that like this so here we have F 1 is completely contained in F 2 completely contained in F n etc so these are hypothesis spaces getting larger and larger each one containing the previous one so polynomial functions I mentioned is one example so one way to get these nested hypothesis spaces is to introduce a complexity measure on the individual prick ship functions the individual decision functions so a few...
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