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$\mathscr{H}$-Consistency Estimation Error of Surrogate Loss Minimizers

We present a detailed study of estimation errors in terms of surrogate loss estimation errors. We refer to such guarantees as $\mathscr{H}$-consistency estimation error bounds, since they account for the hypothesis set $\mathscr{H}$ adopted. These guarantees are significantly stronger than $\mathscr{H}$-calibration or $\mathscr{H}$-consistency. They are also more informative than similar excess error bounds derived in the literature, when $\mathscr{H}$ is the family of all measurable functions. We prove general theorems providing such guarantees, for both the distribution-dependent and distribution-independent settings. We show that our bounds are tight, modulo a convexity assumption. We also show that previous excess error bounds can be recovered as special cases of our general results. We then present a series of explicit bounds in the case of the zero-one loss, with multiple choices of the surrogate loss and for both the family of linear functions and neural networks with one hidden-layer. We further prove more favorable distribution-dependent guarantees in that case. We also present a series of explicit bounds in the case of the adversarial loss, with surrogate losses based on the supremum of the $ρ$-margin, hinge or sigmoid loss and for the same two general hypothesis sets. Here too, we prove several enhancements of these guarantees under natural distributional assumptions. Finally, we report the results of simulations illustrating our bounds and their tightness.