{
 "artifacts": null,
 "category": "optimization",
 "description": "Replace soft-thresholding or an approximate smooth l_p penalty with the scalar proximal operator of lambda|x|^p for 0\u003cp\u003c1. Apply the exact threshold first, then evaluate the larger positive stationary root using the paper's Lagrange-inversion series when it is well-conditioned and safeguarded bisection near the threshold.",
 "formulas_latex": [
  "$$F(x;y)=\\frac{1}{2}(x-y)^2+\\lambda|x|^p,\\qquad \\operatorname{prox}_{\\lambda|\\cdot|^p}(y)\\in\\arg\\min_x F(x;y).$$",
  "$$h(u)=u+\\lambda p u^{p-1}=|y|,\\qquad u_0=[\\lambda p(1-p)]^{\\frac{1}{2-p}},\\qquad u_{\\mathrm{th}}=[2\\lambda(1-p)]^{\\frac{1}{2-p}},$$\n$$r_{\\mathrm{th}}=\\frac{2-p}{2(1-p)}u_{\\mathrm{th}},\\qquad \\operatorname{prox}_{\\lambda|\\cdot|^p}(y)=0\\ \\text{if }|y|\u003cr_{\\mathrm{th}}.$$",
  "$$s=\\lambda p|y|^{p-2},\\qquad u=|y|\\left[1-\\sum_{n=1}^{N}c_n(q)s^n\\right]+R_N,\\quad q=1-p,\\quad c_n(q)=\\frac{\\Gamma((q+1)n-1)}{n\\,\\Gamma(qn)\\Gamma(n)},\\quad c_1(q)=1.$$",
  "$$\\widehat{u}\\in[u_0,|y|],\\qquad |h(\\widehat{u})-|y||\\leq\\varepsilon_{\\mathrm{root}};\\qquad x^+=\\operatorname{sign}(y)\\widehat{u}.$$"
 ],
 "id": 290,
 "implementation": "Integrate this into a proximal-gradient optimizer for a selected parameter tensor, such as all weights of a small MLP or only the router weights of an MoE layer. At iteration t, compute the ordinary gradient step z=W_t-eta_t*grad_W L(W_t), then apply the scalar proximal map independently to every entry of z with lambda=eta_t*alpha, where alpha is the l_p regularization coefficient. For each z_i, let r=abs(z_i), compute u_th=[2*lambda*(1-p)]**(1/(2-p)) and r_th=((2-p)/(2*(1-p)))*u_th; if r\u003cr_th set W_new_i=0. Otherwise compute s=lambda*p*r**(p-2). If s is below a configurable series limit, initialize u=r*(1-s-sum_{n=2}^N c_n*s**n), evaluating c_n with log-Gamma functions, and accept it only when abs(h(u)-r)\u003c=epsilon. Otherwise set lo=u_0=[lambda*p*(1-p)]**(1/(2-p)) and hi=r, then perform 30-50 bisection iterations on h(u)-r for the larger-root branch; use Newton steps only when they stay inside the bracket. The mathematical quantities are the threshold, stationary-root equation, and series coefficients. Estimate truncation reliability by comparing orders N and 2N; certify root accuracy using residual and bracket width. Start with a 2-layer width-512 MLP on MNIST or CIFAR-10. Compare AdamW plus magnitude pruning, soft-threshold proximal l1, and direct subgradient l_p training at matched sparsity and FLOPs. Measure validation accuracy, exact zero fraction, proximal-call time, wall-clock time, and loss descent. Success means higher accuracy at the same zero fraction, or equal accuracy with fewer pruning operations, with no threshold oscillations and residuals below 1e-6.",
 "math_summary": "For a scalar input y, define P_{lambda,p}(y) as a global minimizer of F(x;y)=0.5(x-y)^2+lambda|x|^p, with lambda\u003e0 and 0\u003cp\u003c1. By symmetry it is enough to use r=|y| and return sign(y) times a nonnegative solution. Any nonzero stationary point u\u003e0 satisfies h(u)=u+lambda p u^{p-1}=r. The stationary equation has its minimum at u_0=[lambda p(1-p)]^{1/(2-p)}, and the global minimizer switches from zero at the threshold r_th=((2-p)/(2(1-p)))u_th, where u_th=[2lambda(1-p)]^{1/(2-p)}. Thus P_{lambda,p}(y)=0 for |y|\u003cr_th; at equality either zero or sign(y)u_th is globally optimal; above the threshold choose the larger root u\u003eu_0 of h(u)=|y|. For the larger root, set q=1-p and s=lambda p |y|^{p-2}. Lagrange-Burmann inversion gives u=|y|[1-s-sum_{n=2}^N c_n(q)s^n]+R_N, with c_n(q)=Gamma((q+1)n-1)/(n Gamma(qn)Gamma(n)); equivalently c_1(q)=1. The series is used only in its certified or empirically validated convergence region. Near threshold, bracket the larger root in [u_0, |y|] and use bisection or safeguarded Newton; the residual |h(u)-|y|| supplies an implementation certificate.",
 "math_tags": [
  "optimization",
  "approximation-theory",
  "numerical-analysis"
 ],
 "ml_areas": [
  "optimizer",
  "regularization",
  "pruning"
 ],
 "paper": {
  "arxiv_id": "2606.31951",
  "arxiv_url": "https://arxiv.org/abs/2606.31951",
  "summary_what_math_gives_to_ml": "The paper supplies a numerically careful scalar proximal map for the nonconvex sparsity penalty |x|^p, 0\u003cp\u003c1, including the discontinuous threshold at which the global minimizer leaves zero. This is transferable to neural-network optimization as an exact or certified inexact proximal step for inducing genuinely sparse weights, rather than using an ordinary differentiable approximation to an l_p penalty. The most useful engineering asset is the combination of a closed-form threshold, a rapidly evaluable Lagrange-inversion series away from the threshold, and a safeguarded root evaluator near the threshold. A practical first target is a proximal-gradient or alternating optimizer for sparse MLP weights, convolution kernels, or MoE router parameters.",
  "title": "Explicit Series and a Certified Hybrid Evaluator for the $\\ell_p$ Proximity Operator for $0\u003cp\u003c1$",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 4,
  "novelty": 6,
  "usefulness": 6
 },
 "solves": [
  "accuracy",
  "generalization",
  "memory"
 ],
 "title": "Certified nonconvex p-shrinkage",
 "url": "https://synthcore.org/idea/290/certified-nonconvex-p-shrinkage",
 "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
   }
  }
 }
}