{"claim":{"id":"02c7416a-386c-4b8e-b0a7-fd228914430d","submission_id":"2c33b337-4714-4343-8aa6-b55c256eeb3f","submission_revision_id":"f992f82b-3a82-4694-b3bf-f603a50d1c77","builder":"builder-01","status":"active","claimed_at":"2026-10-15T14:56:49.149453Z","delivered_at":null,"ended_at":null,"delivery_note":null,"end_reason":null},"task":{"submission_id":"2c33b337-4714-4343-8aa6-b55c256eeb3f","submission_revision_id":"f992f82b-3a82-4694-b3bf-f603a50d1c77","revision_version":1,"submission_status":"under_review","submitted_at":"2026-08-12T05:10:50.243462Z","submitter_name":"Rongzhe Wei","submitter_email":"jesson.rongzhe@gmail.com","task":{"status":"submitted","version":1,"extra_fields":{"difficulty_self_test":{"model":"Claude Opus 5","harness":"Claude Code","estimated_score":"5–10%"}},"submitted_at":"2026-08-12T05:10:50.243462+00:00","industry_domain":"Security - Differential Privacy","task_short_name":"iterative_privacy_bound_instance_1","operating_system":"Windows","task_description":"Let K ⊂ R^d be a closed convex set with diameter D. Consider\n\nW_{t+1} = Π_K[W_t - (η/n) Σ_{i=1}^n Π_{B_K}[∇ℓ_i(W_t)] + G_t],\n\nwhere G_t ~ N(0, σ^2 I), for t = 0,...,T-1, and Π_{B_K} denotes projection onto the ℓ_2-ball of radius K.\n\nLet W'_t be the corresponding process when exactly one loss term ℓ_i is replaced by ℓ'_i, with W_0 = W'_0. Assume that for every loss function involved,\n\n||∇ℓ_i(x) - ∇ℓ_i(y)|| ≤ L ||x-y||^λ,\n\nwhere 0 < λ ≤ 1, for all x,y ∈ K. Only the final iterate W_T is released.\n\nFor any α > 1, derive the tightest upper bound you can on\n\nD_α(L(W_T) || L(W'_T)).\n\nYour final result should be explicit enough to evaluate numerically for arbitrary (T,n,η,K,L,λ,σ,D,α).","software_licensing":"Free/Open Source","evaluation_criteria":null,"verification_method":"The reference result is Theorem 3.7 of the ICLR 2025 paper **“Convergent Privacy Loss of Noisy-SGD without Convexity and Smoothness”** by Eli Chien and Pan Li:\n\nhttps://openreview.net/forum?id=kjmLabjSE2\n\nThis is a recent theoretical advance in differential privacy that resolves a previously open question about obtaining convergent hidden-state Noisy-SGD privacy guarantees beyond the smooth and convex setting. The paper derives the result using a tighter shifted-divergence analysis together with forward Wasserstein-distance tracking.\n\nA successful solution should recover a bound mathematically equivalent to **Theorem 3.7**, including the exact finite-dimensional optimization and its forward and backward recursions.\n\nIn particular, the forward discrepancy must satisfy\n\n[\nD_t =\n\\min\\left{\nD,;\nD_{t-1}+\\eta L D_{t-1}^{\\lambda}+\\frac{2\\eta K}{n},;\nD_{t-1}+2\\eta K\n\\right}.\n]\n\nThe branch\n\n[\nD_{t-1}+2\\eta K\n]\n\nis critical. It is an independent bound induced by gradient clipping and becomes strictly tighter than the Hölder-based branch in relevant parameter regimes. A solution that omits this case yields a strictly looser privacy bound and therefore does not satisfy the request for the tightest bound.\n\nThe solution should additionally recover an optimization equivalent to\n\n[\n\\epsilon(\\alpha)\n\\le\n\\min_{\\tau,\\beta,a}\n\\frac{\\alpha}{2\\sigma^2}\n\\sum_{t=\\tau}^{T-1}\n\\left[\n\\frac{(2\\eta K/n)^2}{\\beta_t}\n+\n\\frac{a_t^2}{1-\\beta_t}\n\\right],\n]\n\nwith (g(x)=x+\\eta Lx^\\lambda), (h=g^{-1}),\n\n[\nA_T=0,\\qquad\nA_{t-1}=h(A_t+a_{t-1}),\\qquad\nA_\\tau\\ge D_\\tau.\n]\n\nEquivalent reparameterizations are acceptable, but a strictly looser forward recursion or a heuristic/asymptotic approximation in place of the exact optimization should be considered a failure.\n","software_and_version":"Python 3.11, SciPy 1.14","specific_requirements":null,"expected_completion_time":null},"files":[{"file_id":"ec190c6a-1f70-4797-aa2e-b30061c9e688","filename":"ALE_input_materials.txt","category":"input_materials","content_type":"text/plain","size_bytes":784},{"file_id":"edc75a9d-bc37-4623-8a5c-f14705f69148","filename":"ALE_reference_output.txt","category":"reference_output","content_type":"text/plain","size_bytes":1573},{"file_id":"0ccbb1cc-e07c-4a5f-8372-bac6043791ae","filename":"Screenshot-1.png","category":"difficulty_self_test_evidence","content_type":"image/png","size_bytes":222247},{"file_id":"5732dd2a-98fd-49dc-903c-cbae28dd8706","filename":"ale_dp_selftest.txt","category":"difficulty_self_test_evidence","content_type":"text/plain","size_bytes":15538},{"file_id":"30a5f8df-d49e-4486-8903-ff7036774d59","filename":"Screenshot-3.png","category":"difficulty_self_test_evidence","content_type":"image/png","size_bytes":436067},{"file_id":"acb2177b-125f-4796-93ab-76fa08c9f6c3","filename":"Screenshot-2.png","category":"difficulty_self_test_evidence","content_type":"image/png","size_bytes":365086},{"file_id":"186f6d14-00cc-4bf3-b1fb-35c189fd40fb","filename":"Screenshot-4.png","category":"difficulty_self_test_evidence","content_type":"image/png","size_bytes":450705},{"file_id":"e0f21966-c341-46c2-a743-10629763f3b2","filename":"Screenshot-5.png","category":"difficulty_self_test_evidence","content_type":"image/png","size_bytes":200060}],"deep_review":{"overall_verdict":"Accept","review_text":"**Overall Verdict:** Accept — The submission is complete, technically coherent, and end-to-end reachable, with public-answer retrieval risk requiring evaluator isolation.\n\n**Completeness Check:**\n- Instruction: Complete — The mathematical problem, assumptions, released output, and required numerical evaluability are clearly specified.\n- Inputs: Complete — The solver-visible input is self-contained and matches the task description.\n- Toolchain: Complete — Python 3.11 and SciPy 1.14 are free, compatible with Windows, and sufficient for evaluating the finite-dimensional optimization.\n- Reference output: Complete — It provides the exact forward recursion, backward recursion, constraints, and Rényi-divergence objective required for objective comparison.\n- Rubric: Complete — Mathematically equivalent formulations are accepted, while omission of the independent clipping branch or substitution of a heuristic approximation is explicitly identified as failure.\n\n**File and Task Reachability:**\nThe attachment set is `reachable`, and the end-to-end task is `reachable`. All eight attachments were inspected. `ALE_input_materials.txt` (784 bytes), `ALE_reference_output.txt` (1,573 bytes), and `ale_dp_selftest.txt` (15,538 bytes) are readable UTF-8 text. `Screenshot-1.png` through `Screenshot-5.png` are valid, renderable, non-interlaced RGBA PNG images ranging from 2,809×924 to 2,872×1,675 pixels; their byte counts match the manifest. The screenshots and transcript consistently document the self-test’s looser derivation and its omission of the critical clipping-based branch. There are no archives, encryption barriers, missing objects, or unreviewed contents, and no material inspection limitations remained. The solver can derive a text answer and numerically evaluate the stated finite optimization on one ordinary machine; the evaluator can compare its recursions, constraints, and objective against the protected reference. The input, reference, and self-test evidence have distinct declared roles. License keys, accounts, tokens, and other credentials are not required from the contributor. Confidential, proprietary, or internal material may be evaluated as submitted evidence without requiring additional disclosure.\n\n**Special Screening:**\n`public_benchmark` — The evaluator evidence identifies the protected result as an exact theorem from a named public paper, making the standard answer potentially retrievable. Keep the reference and verification metadata evaluator-only and use a network-isolated solver environment. The submitted evidence does not establish pervasive pretraining memorization.\n\n**Recommended Changes:**\nThe task and reference correctly preserve the critical clipping branch and exact optimization, and no contributor revision is required. Evaluation should enforce separation of solver-visible inputs from the reference and disable external retrieval during solving. An optional notation cleanup could distinguish the convex domain from the scalar clipping radius, both currently denoted by \\(K\\).","reviewed_at":"2026-08-15T00:32:09.875406Z"},"claim":{"id":"02c7416a-386c-4b8e-b0a7-fd228914430d","submission_id":"2c33b337-4714-4343-8aa6-b55c256eeb3f","submission_revision_id":"f992f82b-3a82-4694-b3bf-f603a50d1c77","builder":"builder-01","status":"active","claimed_at":"2026-10-15T14:56:49.149453Z","delivered_at":null,"ended_at":null,"delivery_note":null,"end_reason":null}},"revision_is_current":true}