Two-Term Asymptotic Expansion of Jensen–Shannon Leakage for Subsampled Gaussian Sum Release

We consider subsampling at rate q among n users and a Gaussian sum-release with noise variance sigma². For neighboring datasets D0= (0,. . . , 0) and D1= (mu, 0,. . . , 0), letting SNR=mu²/sigma², we prove JSD (P0, P1) = (q*SNR) / (8n) + ( (4-12q+7q²) /64) * (SNR²/n²) + O (n^-3). The remainder is uniform for q in eps, 1 (any fixed eps>0) and bounded SNR. The proof combines an exact mixture representation, a local Edgeworth expansion, uniform sixth-derivative control, and a parity argument. Numerical quadrature of the exact binomial–Gaussian mixture corroborates the uniform n^-3 behavior.