Tight bounds for maximum ℓ_1-margin classifiers
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Popular iterative algorithms such as boosting methods and coordinate descent on linear models converge to the maximum ℓ_1-margin classifier, a.k.a. sparse hard-margin SVM, in high dimensional regimes where the data is linearly separable. Previous works consistently show that many estimators relying on the ℓ_1-norm achieve improved statistical rates for hard sparse ground truths. We show that surprisingly, this adaptivity does not apply to the maximum ℓ_1-margin classifier for a standard discriminative setting. In particular, for the noiseless setting, we prove tight upper and lower bounds for the prediction error that match existing rates of order _1^2/3/n^1/3 for general ground truths. To complete the picture, we show that when interpolating noisy observations, the error vanishes at a rate of order 1/√(log(d/n)). We are therefore first to show benign overfitting for the maximum ℓ_1-margin classifier.
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