{
  "schema_version": 1,
  "title": "Reproduction: All ERMs Can Fail in Stochastic Convex Optimization (Lower Bounds in Linear Dimension)",
  "emoji": "📉",
  "space_id": "snaykey/repro-flashoptim",
  "paper": {
    "openreview_id": "Lzwp2KXedc"
  },
  "tags": [
    "icml2026-repro",
    "paper-Lzwp2KXedc"
  ],
  "updated_at": "2026-07-30T00:00:00Z",
  "root": {
    "slug": "index",
    "title": "Reproduction: All ERMs Can Fail in Stochastic Convex Optimization (Lower Bounds in Linear Dimension)",
    "file": "pages/index.md",
    "children": [
      {
        "slug": "executive-summary",
        "title": "Executive summary",
        "file": "pages/executive-summary/page.md",
        "children": []
      },
      {
        "slug": "claim1-thm1-all-erms-fail-linear-dim",
        "title": "In dimension d = 6m, there exists a stochastic convex optimization instance where every ε-ERM solution with ε = Θ(m^{-3/2}) incurs constant excess risk and fails to generalize (Theorem 1).",
        "file": "pages/claim1-thm1-all-erms-fail-linear-dim/page.md",
        "children": []
      },
      {
        "slug": "claim2-thm2-strongly-convex-rate",
        "title": "For λ-strongly convex losses with m^{-3/2} ≤ λ ≤ m^{-1/2}, any ε-ERM solution w_S satisfies F(w_S) − min F(w) ≥ Ω(1/(λ m^{3/2})) (Theorem 2).",
        "file": "pages/claim2-thm2-strongly-convex-rate/page.md",
        "children": []
      },
      {
        "slug": "claim3-cor3-gd-constant-error",
        "title": "When ηT = Ω(m^{3/2}), gradient descent incurs constant generalization error with high probability, as a corollary of the ERM lower bound (Corollary 3).",
        "file": "pages/claim3-cor3-gd-constant-error/page.md",
        "children": []
      },
      {
        "slug": "claim4-thm4-gd-excess-risk-rate",
        "title": "Gradient descent's excess risk is lower-bounded by F(w_S^{GD}) − min F(w) = Ω(min{√(ηT/m^{3/2}), 1}) (Theorem 4).",
        "file": "pages/claim4-thm4-gd-excess-risk-rate/page.md",
        "children": []
      },
      {
        "slug": "claim5-thm4-narrowing-gap",
        "title": "This new generalization lower bound of Ω(ηT + √(ηT/m^{3/2})) for constrained gradient descent narrows the previously exponential gap to the best known upper bound of O(ηT + ηT/m) (Theorem 4).",
        "file": "pages/claim5-thm4-narrowing-gap/page.md",
        "children": []
      },
      {
        "slug": "claim6-construction-code-link-function",
        "title": "The construction uses Feldman's asymptotically good binary code combined with a novel 'link function' mapping sample information to bad ERM solutions while preserving convexity, giving the first proof that all approximate ERMs fail in linear dimension (Section on technical construction / proof of Theorem 1).",
        "file": "pages/claim6-construction-code-link-function/page.md",
        "children": []
      },
      {
        "slug": "conclusion",
        "title": "Conclusion",
        "file": "pages/conclusion/page.md",
        "children": []
      }
    ]
  }
}