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Strong minimax lower bounds for learning




Minimax lower bounds for concept learning state, for example, that for each sample size $n$ and learning rule $g_n$, there exists a distribution of the observation $X$ and a concept $C$ to be learnt such that the expected error of $g_n$ is at least a constant times $V/n$, where $V$ is the VC dimension of the concept class. However, these bounds do not tell anything about the rate of decrease of the error for a {\sl fixed} distribution--concept pair.\\ In this paper we investigate minimax lower bounds in such a--stronger--sense. We show that for several natural $k$--parameter concept classes, including the class of linear halfspaces, the class of balls, the class of polyhedra with a certain number of faces, and a class of neural networks, for any {\sl sequence} of learning rules $\{g_n\}$, there exists a fixed distribution of $X$ and a fixed concept $C$ such that the expected error is larger than a constant times $k/n$ for {\sl infinitely many n}. We also obtain such strong minimax lower bounds for the tail distribution of the probability of error, which extend the corresponding minimax lower bounds.

Suggested Citation

  • Andras Antos & Gábor Lugosi, 1997. "Strong minimax lower bounds for learning," Economics Working Papers 197, Department of Economics and Business, Universitat Pompeu Fabra.
  • Handle: RePEc:upf:upfgen:197

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    References listed on IDEAS

    1. Lugosi, Gábor, 1995. "Improved upper bounds for probabilities of uniform deviations," Statistics & Probability Letters, Elsevier, vol. 25(1), pages 71-77, October.
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    Cited by:

    1. Meister Alexander, 2008. "Uniform and individual convergence rates for convex density classes," Statistics & Risk Modeling, De Gruyter, vol. 26(1), pages 25-34, March.

    More about this item


    Estimation; hypothesis testing; statistical decision theory: operations research;

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C44 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Operations Research; Statistical Decision Theory

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