{
 "artifacts": null,
 "category": "regularization",
 "description": "Replace ad hoc Gaussian or Laplace noise injection with a Laplace majorant calibrated to the observed finite-range sub-Gamma parameters of a neural perturbation. For convex perturbation losses, the calibration guarantees that the expected loss under the scaled Laplace noise upper-bounds the expected loss under every centered random perturbation satisfying the same Bernstein-type MGF constraint.",
 "formulas_latex": [
  "$$\\mathbb{E}[f(X)]\\leq\\mathbb{E}[f(cG)]\\qquad\\text{for every convex }f\\colon\\mathbb{R}\\to\\mathbb{R}.$$",
  "$$\\mathbb{E}[e^{\\lambda X}]\\leq\\exp\\!\\left(\\frac{\\sigma^{2}\\lambda^{2}}{2(1-\\alpha|\\lambda|)}\\right),\\qquad |\\lambda|\u003c\\alpha^{-1}.$$",
  "$$c_\\star=\\inf\\left\\{c\u003e0:\\ \\sup_{X\\in\\mathcal{S}(\\sigma,\\alpha)}\\mathbb{E}[(X-t)_+]\\leq\\mathbb{E}[(cL-t)_+]\\ \\text{for every }t\\in\\mathbb{R}\\right\\}.$$",
  "$$\\mathbb{E}[(cL-t)_+]=\\begin{cases}\\frac{c}{2}e^{-t/c},\u0026t\\geq0,\\\\-t+\\frac{c}{2}e^{t/c},\u0026t\u003c0,\\end{cases}\\qquad \\mathbb{E}[e^{\\lambda cL}]=\\frac{1}{1-c^2\\lambda^2},\\ |\\lambda|\u003cc^{-1}.$$"
 ],
 "id": 3052,
 "implementation": "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_+)] \u003c= sigma^2 lambda^2/[2(1-alpha lambda)] and the analogous inequality at negative lambda for grid points satisfying |lambda|\u003c1/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.",
 "math_summary": "The paper studies convex order: a centered random variable X is dominated by cL if E[f(X)] \u003c= E[f(cL)] for every convex f, where L is a centered unit-scale Laplace random variable. The sub-Gamma constraint to implement is the Bernstein MGF condition E[exp(lambda X)] \u003c= exp(sigma^2 lambda^2/(2(1-alpha |lambda|))) for |lambda| \u003c 1/alpha, where sigma \u003e= 0 is the variance proxy and alpha \u003e= 0 is the scale parameter; apply the condition separately to positive and negative lambda. Define the sharp scale c_star(sigma, alpha) as the smallest c such that E[(X-t)_+] \u003c= E[(cL-t)_+] for every threshold t and every centered X satisfying the MGF constraint. The hinge-function condition is sufficient because convex functions on the real line are affine functions plus nonnegative mixtures of hinges. The paper states that c_star is strictly larger than sigma vee alpha for the sub-Gamma class and that the optimum is attained by an asymmetric two-point distribution, so c_star can be estimated by optimizing over X taking values x_- \u003c 0 and x_+ \u003e 0 with probabilities p and 1-p, subject to p x_- + (1-p)x_+ = 0 and the finite-range MGF inequalities. For L with density p_L(l)=0.5 exp(-|l|), E[exp(lambda cL)] = 1/(1-c^2 lambda^2) for |lambda| \u003c 1/c, and E[(cL-t)_+] has a closed form.",
 "math_tags": [
  "probability",
  "convex-analysis",
  "statistics",
  "optimization"
 ],
 "ml_areas": [
  "regularization",
  "loss",
  "training"
 ],
 "paper": {
  "arxiv_id": "2609.02398",
  "arxiv_url": "https://arxiv.org/abs/2609.02398",
  "summary_what_math_gives_to_ml": "The paper develops sharp convex-order majorants for centered random variables whose moment generating functions satisfy finite-range Bernstein/sub-Gamma bounds. Its transferable asset is not Laplace noise itself, which is standard, but a principled calibration constant that guarantees every convex test loss under an admissible perturbation is bounded by the corresponding loss under a scaled Laplace perturbation; the extremal law is an asymmetric two-point distribution. A practical neural-network use is certified scalar perturbation regularization: estimate local sub-Gamma parameters of activations, gradients, or logit noise, compute the sharp majorant scale numerically from the convex-order condition, and train with the resulting scaled Laplace perturbation as a distributionally robust surrogate.",
  "title": "Convex Order Comparisons for Sub-Gamma Random Variables",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 6,
  "novelty": 6,
  "usefulness": 5
 },
 "solves": [
  "stability",
  "generalization",
  "accuracy"
 ],
 "title": "Sharp Sub-Gamma Laplace Regularization",
 "url": "https://synthcore.org/idea/3052/sharp-sub-gamma-laplace-regularization",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)",
  "verification_axes": {
   "benchmark_mechanism": {
    "confirmed": null,
    "tested": false
   },
   "practical_benchmark": {
    "beats_baseline": null,
    "tested": false
   },
   "toy_mechanism_gate": {
    "confirmed": null,
    "tested": false
   }
  }
 }
}