lemmaroutine

Properties of the estimation loss

Loss L\mathcal L is comparable to ∣H1/s−H′1/s∣s|H^{1/s}-H'^{1/s}|^s and bounds clipped relative error
basic loss facts for mean comparisons
Lemma·lem:est-loss·§7.3 Mean comparisons and the cost of releasing a source

Statement (paper)

The loss in Equation  is comparable, by constants depending only on ss, to ∣H1/s−(H′)1/s∣s|H^{1/s}-(H')^{1/s}|^s. It contracts under sums and nonnegative mixtures. If H≤C0GH\le C_0G for a nonnegative GG, then for every a>0a>0 and H′≥0H'\ge0, amin⁡{1,∣H−H′∣a+G}s≤CL(H,H′), a\min\left\{1,\frac{|H-H'|}{a+G}\right\}^{s} \le C\mathcal L(H,H'), where CC depends only on s,C0s,C_0.

In plain English

The loss L(H,H′)\mathcal L(H,H') is comparable, up to constants depending only on ss, to ∣H1/s−(H′)1/s∣s|H^{1/s}-(H')^{1/s}|^s, and it contracts under sums and nonnegative mixtures. If H≤C0GH\le C_0G with G≥0G\ge0, then for every a>0a>0 and H′≥0H'\ge0, amin⁡{1,∣H−H′∣/(a+G)}s≤CL(H,H′)a\min\{1,|H-H'|/(a+G)\}^s \le C\mathcal L(H,H'), with CC depending only on s,C0s,C_0.

Why it's here

This collects the basic facts about the loss that later mean comparisons need: an equivalent form via ssth-root differences, stability under summing and mixing, and a clipped relative-error bound against the scale a+Ga+G. The mean-comparison lemma uses these to turn a loss bound on sample terms into bounds on clipped relative errors.
Classified routine:
Elementary inequalities for the s-th root difference |H^{1/s}-H'^{1/s}|^s (a quasi-metric on nonnegative reals) together with convexity/subadditivity arguments; standard calculations with no use of other results in the paper.

Uses

Nothing else in the paper

Used by

lemMean comparison scale factors and release costclaimCoarse bin scale factor b/a

Sources

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§7.3 Mean comparisons and the cost of releasing a source
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lem:est-loss
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sections/estimation.tex:195
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