logbook.json

{
  "schema_version": "1.0",
  "title": "Reproduction: Allocating Variance to Maximize Expectation",
  "emoji": "📊",
  "space_id": "snaykey/repro-variance-allocation",
  "paper": {
    "arxiv_id": "2502.18463",
    "openreview_id": "vqxprtjuKH"
  },
  "tags": [
    "icml2026-repro",
    "paper-vqxprtjuKH"
  ],
  "updated_at": "2026-07-29T12:00:00+00:00",
  "root": {
    "slug": "index",
    "title": "Reproduction: Allocating Variance to Maximize Expectation",
    "children": [
      {
        "slug": "executive-summary",
        "title": "Executive summary",
        "children": []
      },
      {
        "slug": "claim-1-theorem-1-1-independent-ptas",
        "title": "For the independent Gaussian variance allocation problem, the paper gives a PTAS achieving E[max_i X_i] ≥ OPT - ε in polynomial time (Theorem 1.1, Section 1.2).",
        "children": []
      },
      {
        "slug": "claim-2-theorem-1-2-correlated-ptas",
        "title": "For correlated Gaussian variables, a PTAS with the same additive ε guarantee is established (Theorem 1.2, Section 1.2).",
        "children": []
      },
      {
        "slug": "claim-3-theorem-1-3-graph-ologn",
        "title": "For the GraphVarAlloc problem with multiple constraint sets (general m>1), the paper gives an O(log n) multiplicative approximation guaranteeing Ω(1/log n)·OPT (Theorem 1.3, Section 1.2).",
        "children": []
      },
      {
        "slug": "claim-4-theorem-1-6-concentration",
        "title": "Theorem 1.6 proves that in the optimal allocation, only Θ(1/p) variables receive variance Ω(p), i.e., the allocation concentrates on a shrinking subset as the constraint parameter p grows (Theorem 1.6, Section 1.3).",
        "children": []
      },
      {
        "slug": "claim-5-lemma-2-1-small-variance",
        "title": "Lemma 2.1 bounds the contribution of small-variance variables by O(ε√ln(1/ε)), which is used to limit the number of high-variance variables to O(1/ε²) and underlies the PTAS construction (Lemma 2.1, Section 2.1).",
        "children": []
      },
      {
        "slug": "claim-6-monte-carlo-er-figures",
        "title": "Monte Carlo simulations on Erdős–Rényi random graphs with n=8 nodes and edge probabilities p ranging from 1/8 to 8/8 are used to illustrate the concentration and concavity results across independent, positively, and negatively correlated settings (Figures 1-2, Section 1.3).",
        "children": []
      },
      {
        "slug": "conclusion",
        "title": "Conclusion",
        "children": []
      }
    ]
  },
  "revision": 2
}