Learning Approach vs. Function Approximation
I'm trying to firm my sense of differences in the way functions are generally sought using the learning paradigm vs. the classical function approximation approaches. Ie, is it the same, or are there differences in thought/approach. :clueless:
As I see it, seeking linear models in supervised learning seems no different from similar methods in function approximation (variations in error measures and their minimization aside). In both instances we use inputoutput sets to accomplish the task, ie find an approximant. In learning, we call the underlying generative process the target function, and after learning, the resulting approximant, the hypothesis. Approximation theory has it's own lingo. I suspect there must be more to this than my superficial characterization. So, in general, do approaches adopted in supervised learning run in parallel to approaches in function approximation? Are there some philosophical differences? If someone would kindly suggest how & why I should view them as separate developments, I'd much appreciate it. Thank you. 
Re: Learning Approach vs. Function Approximation
This is a good question. The general conclusion you made is correct, that more or less the same problem with different lingo is addressed in function approximation arising from the statistics community and supervised learning arising in the learning community. But if you read a statistics book on function approximation, it will look very different from the text related to this forum. So while the problem is the same, (inputoutput examples to learn a funcction f), the approaches in these two fields are different.
Largely, the difference is in the assumptions made and the nature of the results. In the statistics approach one usually makes distributional assumptions on the nature of the data and then derives how a particular model like the linear model will behave. Function approximation in statistics typically only discusses regression problems. In learning, we make very mild assumptions and obtain different types of results, and we have a particular focus on classification. Section 1.2 gives a short discussion of different types of learning that may be helpful. Quote:

Re: Learning Approach vs. Function Approximation
Thank you, Sir. Your response helps in drawing the distinctions between developments in statistics & ML. As I read further, I think I need to keep an eye on: (a) core objective, (b) the formulation of the problem, and (c) the assumptions made. As you note, and I vaguely sensed that too, similarities between stats & ML approaches appear on the subject of regression, and then too, one needs to pay attention to assumptions made during problem formulation (eg. distributional assumptions on input data etc.). Your point on weaker assumptions in ML, generally speaking, is very helpful too. And finally, if one examines the various ML paradigms, for eg., as stated in Sec 1.2, the objectives may be dramatically different, and we get problems & formulations quite different from statistics.

Re: Learning Approach vs. Function Approximation
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Re: Learning Approach vs. Function Approximation
Yes, nonparametric methods make fewer asumptions and indeed there are many techniques imported from statistics into learning, including nonparametric methods. In learning the focus is more on classification, and hence there are several new concepts that one would not encounter in a traditional statistics setting, such as the VCdimension.
The way I would look at it is that what we want to do is learn from data. How we do it may differ from discipline to discipline. The traditional statistician typically looks for some rigorous mathematical model within which to deduce interesting statements. The machine learner uses computationally tractable algorithms to output a hypothesis and then asks what can one say about the performance of that final hypothesis. It is perfectly valid to be both a machine learner and a statistician; using their broadest definitions, some might argue that ML subsumes statistics, and others would say statistics subsumes ML. To me ML and statistics are how you do things, and there is indeed big overlap, but what we do stays the same: learning from data. Quote:

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