Lap2

Implements Lap2, a differentially private SGD that replaces the Gaussian mechanism with calibrated Laplace noise.

Lap2 follows the DP-SGD template: per-example gradients are clipped to bound their \(\ell_2\) sensitivity to \(C\), noise is added, and a plain SGD step is taken. Unlike standard DP-SGD, the perturbation is drawn from a multivariate Laplace distribution rather than a Gaussian, with scale \(b\) calibrated jointly with the clipping norm through a majorization-theoretic moments accountant. Privacy loss depends on the sensitivity-to-noise ratio \(\rho = C/b\), and the analysis yields a noise scale that grows with the clipping norm and the iteration count \(T\).

\[ \begin{aligned} \hat g_t(x) &= \frac{g_t(x)}{\max\!\left(1,\ \lVert g_t(x)\rVert_2 / C\right)} \\ \tilde g_t &= \frac{1}{L}\sum_{x} \left( \hat g_t(x) + w \right),\qquad w \sim \mathrm{Lap}(b)^{n} \\ b^{*} &\approx \frac{2\zeta}{\epsilon_{\mathrm{tar}}}\,\sqrt{T\,\log(1/\delta)}\; C \\ \theta_{t+1} &= \theta_t - \eta\, \tilde g_t \end{aligned} \]

where \(g_t(x)\) is the per-example gradient, \(C\) the clipping norm, \(L\) the lot size, \(w\) a length-\(n\) vector of i.i.d. Laplace samples with scale \(b\) and density \(\propto \exp(-\lVert w\rVert_1 / b)\), \(\eta\) the learning rate, \(T\) the number of steps, \((\epsilon_{\mathrm{tar}},\delta)\) the target privacy budget, \(\zeta\) the sampling rate, and \(\rho = C/b\) the sensitivity-to-noise ratio optimized by the accountant.

Reference: Meisam Mohammady, Qin Yang, Nicholas Stout, Ayesha Samreen, Han Wang, Christopher J. Quinn, Yuan Hong, "Lap2: Revisiting Laplace DP-SGD for High Dimensions via Majorization Theory", arXiv 2026. https://arxiv.org/abs/2602.23516

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