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Improved Inapproximability of VC Dimension and Littlestone's Dimension via (Unbalanced) Biclique

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arxiv 2211.01443 v1 pith:KXHJM52Z submitted 2022-11-02 cs.CC cs.LG

classification cs.CCcs.LG
keywords dimensioninapproximabilitylittlestoneapproximatingbicliquehypothesisresultstime
verification ladder T0 review T1 audit T2 compute T3 formal
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abstract

We study the complexity of computing (and approximating) VC Dimension and Littlestone's Dimension when we are given the concept class explicitly. We give a simple reduction from Maximum (Unbalanced) Biclique problem to approximating VC Dimension and Littlestone's Dimension. With this connection, we derive a range of hardness of approximation results and running time lower bounds. For example, under the (randomized) Gap-Exponential Time Hypothesis or the Strongish Planted Clique Hypothesis, we show a tight inapproximability result: both dimensions are hard to approximate to within a factor of $o(\log n)$ in polynomial-time. These improve upon constant-factor inapproximability results from [Manurangsi and Rubinstein, COLT 2017].

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