Shape-Optimized Private Gradient Noise

Integrate this at the noise-injection line of a private optimizer, initially for a scalar clipped gradient query or one coordinate of a clipped gradient. Clip each per-example gradient to norm C, so use scalar sensitivity Delta=C or 2C according to the neighboring-dataset convention; keep that convention identical for all baselines. For every candidate p on a grid such as {1.0,1.25,...,8.0}, solve for b(p) by bisection. At each b, evaluate the hockey-stick integral numerically by constructing f(x) and f(x-Delta), identifying where f(x)-exp(epsilon)f(x-Delta)>0, and integrating the positive difference with adaptive quadrature over an interval initially covering 12b+Delta. Expand the interval until the omitted tail is below 1e-10. Choose the smallest b whose divergence is at most delta minus a numerical safety margin, then calculate U_m(p) and select p*. Sample noise using inverse-CDF sampling: for u uniform on [-1/2,1/2], set z=b sign(u)[-log(1-2|u|)]^(1/p). Apply g_private=g_clipped+z before the optimizer update. The paper supplies the divergence, distribution family, and moment objective; quadrature and bisection are numerical implementations. First test logistic regression and a small MNIST MLP, comparing p=1, p=2, and selected p under equal epsilon, delta, steps, clipping C, parameter count, and training FLOPs. Pre-register: measured divergence must be within 1% of delta at the selected scale; multiplying Delta by 0.5, 1, and 2 must leave p* unchanged within one grid step while b* scales proportionally within 3%; and at least one privacy regime must produce a 5% lower U_2 or U_1 than both Laplace and Gaussian. Fixing p=2 is the ablation. The transfer is falsified if calibration is non-conservative, p* changes substantially under sensitivity rescaling, or no moment/task improvement appears across privacy regimes.