{
  "schema_version": "1.0",
  "title": "Reproduction: Multi-task Linear Regression without Eigenvalue Lower Bounds (Adaptivity, Robustness, and Safety)",
  "emoji": "📈",
  "space_id": "snaykey/repro-multitask-linreg",
  "paper": {
    "arxiv_id": "2605.17126",
    "openreview_id": "D5Ijcnz1L9"
  },
  "tags": [
    "icml2026-repro",
    "paper-D5Ijcnz1L9"
  ],
  "updated_at": "2026-07-28T14:00:00+00:00",
  "root": {
    "slug": "index",
    "title": "Reproduction: Multi-task Linear Regression without Eigenvalue Lower Bounds (Adaptivity, Robustness, and Safety)",
    "children": [
      {
        "slug": "executive-summary",
        "title": "Executive summary",
        "children": []
      },
      {
        "slug": "claim-1-theorem-2-safety",
        "title": "Theorem 2 establishes an in-sample MSE bound for each task j that guarantees safety, ℰⱼⁱⁿ(θ̂ⱼ) ≲ q²(d/n)ζ regardless of the balancedness constant B, outlier fraction ε, or heterogeneity δ (Section 5.1, Theorem 2).",
        "children": []
      },
      {
        "slug": "claim-2-theorem-2-transfer",
        "title": "Theorem 2 also shows a transfer guarantee for inlier tasks when B ≲ min(1/ε, m): ℰⱼⁱⁿ(θ̂ⱼ) ≲ (Bd/mn + min(Bδ², d/n) + B²ε²d/n)ζ, achieved without knowing ε, δ, or the inlier set S (Section 5.1, Theorem 2).",
        "children": []
      },
      {
        "slug": "claim-3-assumption-1-balancedness",
        "title": "Assumption 1 (Balancedness) replaces the classical Lower Boundedness of Second Moments condition ρI ⪯ Σⱼ with the one-sided condition Σⱼ ⪯ B·Σ_S, accommodating rank-deficient or decaying covariate spectra where prior eigenvalue-lower-bound approaches (e.g. Duan & Wang 2023, depending on 1/ρ²) fail (Section 4, Assumption 1).",
        "children": []
      },
      {
        "slug": "claim-4-theorem-3-population",
        "title": "Theorem 3 extends the in-sample MSE guarantees of Theorem 2 to population risk via an empirical-to-population comparability constant νⱼ, retaining an intrinsic-dimension fallback for the safety guarantee (Section 5.2, Theorem 3).",
        "children": []
      },
      {
        "slug": "claim-5-theorem-4-glm",
        "title": "Theorem 4 extends the same adaptive safety/transfer MSE guarantees to generalized linear models under bounded-domain assumptions (Section 6, Theorem 4).",
        "children": []
      },
      {
        "slug": "claim-6-algorithm-1-objective",
        "title": "Algorithm 1 solves a joint convex objective ℒ(Θ)=Σⱼ wⱼ(fⱼ(θⱼ)+λⱼ‖θⱼ-β‖_{Σⱼ}) that penalizes disagreement in prediction space (via task-specific norms) rather than raw parameter space (Section 3, Algorithm 1).",
        "children": []
      },
      {
        "slug": "conclusion",
        "title": "Conclusion",
        "children": []
      }
    ]
  },
  "revision": 1
}