Under finite moments and a local envelope growth condition, the least squares estimator in nonparametric regression can achieve minimax rates with heavy-tailed, covariate-dependent errors.
`local' vs. `global' parameters -- breaking the gaussian complexity barrier
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We show that if $F$ is a convex class of functions that is $L$-subgaussian, the error rate of learning problems generated by independent noise is equivalent to a fixed point determined by `local' covering estimates of the class, rather than by the gaussian averages. To that end, we establish new sharp upper and lower estimates on the error rate for such problems.
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On Least Squares Estimation under Heteroscedastic and Heavy-Tailed Errors
Under finite moments and a local envelope growth condition, the least squares estimator in nonparametric regression can achieve minimax rates with heavy-tailed, covariate-dependent errors.