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- W4295343755 abstract "This paper studies the problem of learning an unknown function $f$ from given data about $f$. The learning problem is to give an approximation $hat f$ to $f$ that predicts the values of $f$ away from the data. There are numerous settings for this learning problem depending on (i) what additional information we have about $f$ (known as a model class assumption), (ii) how we measure the accuracy of how well $hat f$ predicts $f$, (iii) what is known about the data and data sites, (iv) whether the data observations are polluted by noise. A mathematical description of the optimal performance possible (the smallest possible error of recovery) is known in the presence of a model class assumption. Under standard model class assumptions, it is shown in this paper that a near optimal $hat f$ can be found by solving a certain discrete over-parameterized optimization problem with a penalty term. Here, near optimal means that the error is bounded by a fixed constant times the optimal error. This explains the advantage of over-parameterization which is commonly used in modern machine learning. The main results of this paper prove that over-parameterized learning with an appropriate loss function gives a near optimal approximation $hat f$ of the function $f$ from which the data is collected. Quantitative bounds are given for how much over-parameterization needs to be employed and how the penalization needs to be scaled in order to guarantee a near optimal recovery of $f$. An extension of these results to the case where the data is polluted by additive deterministic noise is also given." @default.
- W4295343755 created "2022-09-13" @default.
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- W4295343755 date "2022-03-29" @default.
- W4295343755 modified "2023-09-30" @default.
- W4295343755 title "Optimal Learning" @default.
- W4295343755 doi "https://doi.org/10.48550/arxiv.2203.15994" @default.
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