This is problematic, because many popular econometric methods regularize fitted functions by counting the number of regressors (eg AIC, aR2). This paper shows that those classical approaches are misguided. Model complexity is not necessarily a function of the number of regressors
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I can represent anything using an infinitely long binary string too. I call it a database, you call it a single parameter. That said, the Wikipedia article on AIC is far too positive and uncritical. Someone needs to rewrite it to clarify its underlying assumptions!
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Indeed, although a very peculiar type of "database" that plenty of people in statistics inadvertently use to overfit data, particularly among the old-school researchers.
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And yet, techniques to counter overfitting are somewhat succesfull.
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When combined with cross-validation, yes. Unfortunately, most econometric textbooks appear to have been written for computer-deprived students.
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Well, you can fit any dataset with K - nearest neighbours. There is only a single parameter K. VC dimension of a function class need not depend upon the number of parameters.
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Indeed. And yet, many classical regularization methods rely on counting the number of variables.
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Very interesting paper. Congratz! But: what are the implications? Where is this function useful? (Compression?) How better is using alpha to represent a column of a dataset than using the real data values?
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Hi Luis. The paper illustrates that regularization by counting the number of variables (as most econometric methods do) is misguided. A simple 1-variable function can overfit.
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Surely if a single parameter can do it, more parameters increases the liklihood of it occurring. Most concerns of overfitting are its likelihood or how preventable it is, and not conditions of necessity or sufficiency.
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Andrew, that may be so in some cases, however the point of the paper is different: Measuring the risk of overfitting by counting the number of variables is pointless.
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