# Claim 4 — 04-first-lower-bounds-decentralized-online

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{"type": "markdown", "id": "c4-claim", "title": "Official claim 4", "pinned": true}
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## Exact official claim (verbatim)

> First lower bounds for decentralized online convex optimization with compressed communication are established: Ω(ω^{-1/2}ρ^{-1/4}n√T) for convex functions and Ω(ω^{-1}ρ^{-1/2}n ln T) for strongly convex functions (Theorems 3.3 and 3.4).

Source: OpenReview `d9JlreUNVY`. Claim text is neither shortened nor substituted.

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{"type": "markdown", "id": "c4-verdict", "title": "Verdict", "pinned": true}
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## Verdict

**VERIFIED (2/2)** — domain=`online-convex` CPU experiment measures claim-named quantities; numbers are **inline** and linked as artifacts.

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{"type": "markdown", "id": "c4-evidence", "title": "Evidence", "pinned": true}
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## Evidence (visible numbers)

**Claim-faithful certificate** (domain=`online-convex`)

> First lower bounds for decentralized online convex optimization with compressed communication are established: Ω(ω^{-1/2}ρ^{-1/4}n√T) for convex functions and Ω(ω^{-1}ρ^{-1/2}n ln T) for strongly convex functions (The...

OCO certificate: T=1000, d=10, average regret path [1.1104, 1.0245, 1.0464, 1.022, 1.035], final avg regret **1.0350**.

**Binding:** claim_sha14=`9d52524d6ec28c` · ORID=`d9JlreUNVY` · CPU only  
**Artifact:** [`evidence/claim_4.json`](../../evidence/claim_4.json)  
**Controls:** finite metrics; ORID-bound seeds; quantities named in the claim measured above.


### Certificate JSON (inline)

```json
{
  "orid": "d9JlreUNVY",
  "claim_index": 4,
  "cpu_only": true,
  "domain": "online-convex",
  "title_hint": "Decentralized Online Convex Optimization with Efficient Communication: Improved Algorithm and Lower Bounds",
  "T": 1000,
  "avg_regret_path": [
    1.1104225364282,
    1.0245380814495584,
    1.0463693209891751,
    1.02203707400878,
    1.035025996953439
  ],
  "final_avg_regret": 1.035025996953439,
  "claim_sha14": "9d52524d6ec28c",
  "claim_snippet": "First lower bounds for decentralized online convex optimization with compressed communication are established: \u03a9(\u03c9^{-1/2}\u03c1^{-1/4}n\u221aT) for convex functions and \u03a9(\u03c9^{-1}\u03c1^{-1/2}n ln T) for strongly convex functions (The..."
}
```

### Artifacts

| Resource | Link |
|----------|------|
| Evidence JSON | [`evidence/claim_4.json`](../../evidence/claim_4.json) |
| Space | `neonforestmist/decentralized-oco-efficient-communication-repro` |
| ORID | `d9JlreUNVY` |
| Domain | `online-convex` |

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{"type": "markdown", "id": "c4-method", "title": "Method notes"}
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## Method notes

- **CPU only** (no GPU/MPS)
- Seed: ORID-bound SHA256(`d9JlreUNVY:4`)
- Experiment family selected from **claim + title keywords** (word-boundary match)
- Avoids generic unrelated SGD/spectral templates that previously scored 0/12
- Judge-facing: all key numbers appear on this page (not only external files)