Sharp Sub-Gamma Laplace Regularization

Integrate this at a scalar perturbation point, initially the pre-logit vector of a classifier or one selected hidden activation channel. During a calibration pass, collect centered scalar residuals X from minibatches and estimate sigma and alpha by fitting the empirical MGF envelope over a grid of lambda values, using a safety multiplier such as 1.1. Compute c_star offline by constrained optimization over asymmetric two-point laws. Parameterize p in (0,1), set x_-=-r and x_+=pr/(1-p) to enforce zero mean, and minimize c subject to log[p exp(lambda x_-)+(1-p)exp(lambda x_+)] <= sigma^2 lambda^2/[2(1-alpha lambda)] and the analogous inequality at negative lambda for grid points satisfying |lambda|<1/alpha. For each candidate c, check the hinge inequalities over a threshold grid t_k using the displayed closed form for the Laplace hinge expectation, then binary-search c. In training, sample L~Laplace(0,1) and add delta=c_star*sigma*L to the chosen scalar or normalized channel before the loss computation. If the loss is not convex in the perturbation, use a local quadratic surrogate with gradient g and curvature h estimated by finite differences. Pseudocode: estimate (sigma,alpha); solve c_star; for each batch compute z; sample delta; evaluate loss(z+delta); backpropagate. Compare no noise, Gaussian noise with standard deviation sigma, and uncalibrated Laplace noise with scale sigma on CIFAR-10 using a two-layer MLP and a linear softmax head. Success means lower worst-seed validation loss or improved clean accuracy at equal noise level, reduced gradient-norm variance, and empirical confirmation that held-out convex hinge or quadratic losses under measured perturbations are below the scaled-Laplace estimate.