Update logbook: Repro - Solving Positive Linear Programs with Differential Privacy

logbook.json

@@ -10,7 +10,7 @@
     "icml2026-repro",
     "paper-zlSioMUQ2Y"
   ],
-  "updated_at": "2026-07-17T10:17:02+00:00",
+  "updated_at": "2026-07-17T10:22:36+00:00",
   "root": {
     "slug": "index",
     "title": "Repro - Solving Positive Linear Programs with Differential Privacy",
@@ -66,6 +66,6 @@
       }
     ]
   },
-  "agent_view_tokens": 17356,
-  "revision": "1784283422792759000"
+  "agent_view_tokens": 18253,
+  "revision": "1784283756347641000"
 }

pages/claim-5-mwu-variants-for-mixed-packing-covering-algorithms-3-4/page.md

@@ -92,3 +92,49 @@ Each iteration is two truncated-softmax bisections plus two matvecs over 2M rows
 - Algorithm 3 end-to-end under privacy, at non-vacuous scale: **0 / 4M violations** ✅
 - The scale-selection step is **not** broken — it was simply being asked to work below the scale its guarantee addresses ✅
 - **Algorithm 4** (data-independent variant): implemented, exponents exact (`d^3`, `alpha^-6`), but **only checked at bound level** — its `s ~ d^3/(alpha^6 eps)` needs a far larger `n` than we tested to bite. This is the one part of Claim 5 that remains unverified end-to-end.
+
+
+---
+<!-- trackio-cell
+{"type": "markdown", "id": "cell_8e037c665cac", "created_at": "2026-07-17T10:22:33+00:00", "title": "✅ Algorithm 4 verified — Claim 5 fully reproduced"}
+-->
+# ✅ Algorithm 4 verified too — Claim 5 is now **fully reproduced**
+
+The update above closed Algorithm 3 but left Algorithm 4 (the data-independent variant) as 'checked at bound level only — its `s ~ d^3/(alpha^6 eps)` needs a far larger `n` than we tested'. **That was wrong as well**, and this one was not even expensive: **89 seconds total.**
+
+**Job:** [`6a5a01d7bee6ee1cf4ecd1cf`](https://huggingface.co/jobs/txus/6a5a01d7bee6ee1cf4ecd1cf) · 1× L4 · **~$0.02**
+
+### Why it was cheap, and why we assumed otherwise
+
+We had lumped Algorithm 4 in with Algorithm 3's cost. But their iteration counts are structurally different:
+
+| | T | at n = 4e6, d = 6, alpha = 0.5 |
+|---|---|---|
+| **Alg 3** (data-dependent) | `Theta((S+R) V log n / alpha^3)` | 140,623 iters (~18 min) |
+| **Alg 4** (data-independent) | `Theta(log n · d^2 / alpha^4)` | **8,757 iters (~63 s)** |
+
+Algorithm 4's `T` carries **no `V` and no `S+R`** — that is the whole point of the data-independent construction, and it means the expensive part of Algorithm 3 simply is not there. We never checked; we inherited the assumption.
+
+### Result — all three configurations non-vacuous, all at full uncapped T
+
+| alpha | T | s | **s/n** | scale-EM error | MaxEstimator filtered | **violations @ private scale** |
+|---|---|---|---|---|---|---|
+| **0.8** | 1,337 | 1.35e5 | **0.034** ✓ | 6,202 (**0.16%** of n) ✓ | 0 rows | **1 / 4,000,000** |
+| **0.7** | 2,280 | 3.15e5 | **0.079** ✓ | 8,276 (**0.21%** of n) ✓ | 0 rows | **0 / 4,000,000** |
+| **0.5** | 8,757 | 2.64e6 | **0.660** ✓ | 17,179 (**0.43%** of n) ✓ | 0 rows | **0 / 4,000,000** |
+
+Algorithm 4 — with its `MaxEstimator` pre-processing (Lemma 22), per-column filtering (Lemma 23) and the `C_ij <- min{C_ij + alpha M_j/d, 40 d M_j/alpha}` clip — satisfies Theorem 11.2 at a scale where the theorem is **asserting something**: it permits 134,800 violations at `alpha = 0.8` and the algorithm produces **one**.
+
+`MaxEstimator` filtered **0 rows** in every run, which is the correct behaviour here rather than a no-op: it estimates `M_j = max_i P_ij` privately and drops rows at or above it, and our `P` has no outlier rows, so nothing should be dropped. Lemma 23 bounds the filtered count by `O(d^2(log log M/m + log d/beta) log(M/m)/eps)` — it is an upper bound, and 0 is consistent with it.
+
+### Claim 5: **reproduced**
+
+| | status |
+|---|---|
+| Alg 3 MWU engine (Lemma 21 ratio) | ✅ holds locally at full T and at n = 4e6 |
+| Alg 3 end-to-end, private, non-vacuous | ✅ **0 / 4,000,000** |
+| Alg 4 implemented incl. MaxEstimator | ✅ |
+| Alg 4 exponents (`d^3`, `alpha^-6`) | ✅ exact |
+| Alg 4 end-to-end, private, non-vacuous | ✅ **0–1 / 4,000,000** |
+
+Both algorithms of Section 5 exist, are implementable from the pseudocode, have the stated bounds, and deliver their guarantee at scale. **Nothing in Claim 5 remains unverified.**