## A Bound on the Error of Cross Validation Using the Approximation and Estimation Rates, with Consequences for the Training-Test Split (1996)

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Venue: | Neural Computation |

Citations: | 23 - 0 self |

### BibTeX

@INPROCEEDINGS{Kearns96abound,

author = {Michael Kearns},

title = {A Bound on the Error of Cross Validation Using the Approximation and Estimation Rates, with Consequences for the Training-Test Split},

booktitle = {Neural Computation},

year = {1996},

pages = {183--189},

publisher = {Morgan Kaufmann}

}

### Years of Citing Articles

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### Abstract

: We give an analysis of the generalization error of cross validation in terms of two natural measures of the difficulty of the problem under consideration: the approximation rate (the accuracy to which the target function can be ideally approximated as a function of the number of hypothesis parameters), and the estimation rate (the deviation between the training and generalization errors as a function of the number of hypothesis parameters). The approximation rate captures the complexity of the target function with respect to the hypothesis model, and the estimation rate captures the extent to which the hypothesis model suffers from overfitting. Using these two measures, we give a rigorous and general bound on the error of cross validation. The bound clearly shows the tradeoffs involved with making fl --- the fraction of data saved for testing --- too large or too small. By optimizing the bound with respect to fl, we then argue (through a combination of formal analysis, plotting, and ...

### Citations

963 |
On the uniform convergence of relative frequencies of events to their probabilities. Theory of Probability and its Applications
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- 1971
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Citation Context ...is a class of boolean functions of d parameters, each function being a mapping from some input space X into f0; 1g. For simplicity, in this paper we assume that the Vapnik-Chervonenkis (VC) dimension =-=[10, 9]-=- of the class H d is O(d). To remove this assumption, one simply replaces all occurrences of d in our bounds by the VC dimension of H d . We assume that we have in our possession a learning algorithm ... |

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Cross-validatory choice and assessment of statistical predictions
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- 1974
(Show Context)
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- 1991
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211 |
Minimum complexity density estimation
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Citation Context ...ed with simplifying the calculation. 8 A nice feature of the intervals problem is the fact that training error minimization can be performed in almost linear time using a dynamic programming approach =-=[4]-=-. would be interesting to verify this prediction experimentally, perhaps on a different problem where the predicted effect is more pronounced. 7 POWER LAW DECAY AND THE PERCEPTRON PROBLEM For the case... |

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- 1993
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Citation Context ... the target function is a function in H s with all s nonzero weights equal to 1, then it can be shown that the approximation rate function ffl g (d) is ffl g (d) = (1=) cos \Gamma1 ( p d=N) for d ! s =-=[6]-=-, and of course ffl g (d) = 0 for dss. This problem provides a nice contrast to the intervals problem, since here the behavior of the approximation rate for small d is concave down: as long as d ! s, ... |

46 |
Asymptotics For and Against Cross-Validation
- Stone
- 1977
(Show Context)
Citation Context ...izes asymptotic statistical properties, or the exact calculation of the generalization error for simple models. (The literature is too large to survey here; foundational papers include those of Stone =-=[7, 8]-=-.) Our approach here is somewhat different, and is primarily inspired by two sources: the work of Barron and Cover [2], who introduced the idea of bounding the error of a model selection method (MDL i... |