Upload folder using huggingface_hub
Browse files- icml26/5c3d6bff-e8ce-4d9f-840b-719084582491/appendix_chunks.jsonl +186 -0
- icml26/5c3d6bff-e8ce-4d9f-840b-719084582491/appendix_text_v3.txt +557 -0
- icml26/5c3d6bff-e8ce-4d9f-840b-719084582491/assets.json +13 -0
- icml26/5c3d6bff-e8ce-4d9f-840b-719084582491/assets/_page_1_Figure_1.jpeg +3 -0
- icml26/5c3d6bff-e8ce-4d9f-840b-719084582491/chunks_v3_anonymized.jsonl +0 -0
- icml26/5c3d6bff-e8ce-4d9f-840b-719084582491/dataset_meta.json +61 -0
- icml26/5c3d6bff-e8ce-4d9f-840b-719084582491/main_body_chunks.jsonl +107 -0
- icml26/5c3d6bff-e8ce-4d9f-840b-719084582491/marker_meta.json +1963 -0
- icml26/5c3d6bff-e8ce-4d9f-840b-719084582491/model_text_v3.txt +320 -0
- icml26/5c3d6bff-e8ce-4d9f-840b-719084582491/paper.blocks.json +0 -0
- icml26/5c3d6bff-e8ce-4d9f-840b-719084582491/paper.md +1244 -0
- icml26/5c3d6bff-e8ce-4d9f-840b-719084582491/parse_report.json +76 -0
- icml26/5c3d6bff-e8ce-4d9f-840b-719084582491/reference_chunks.jsonl +4 -0
- icml26/5c3d6bff-e8ce-4d9f-840b-719084582491/reference_text_v3.txt +11 -0
- icml26/5c3d6bff-e8ce-4d9f-840b-719084582491/sanitization_report.json +59 -0
- icml26/5c3d6bff-e8ce-4d9f-840b-719084582491/sanitized_v3.txt +611 -0
icml26/5c3d6bff-e8ce-4d9f-840b-719084582491/appendix_chunks.jsonl
ADDED
|
@@ -0,0 +1,186 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0111", "section": "A. Extended Problem Formulation", "page_start": 10, "page_end": 10, "type": "Text", "text": "We restate the assumptions from Section 2 for completeness. Let (\\mathcal{X}, d) be a compact metric space with diameter D = \\sup_{x,y \\in \\mathcal{X}} d(x,y) . We consider \\mathcal{X} = [0,1]^d with Euclidean metric, though our results extend to general metric spaces. Let f: \\mathcal{X} \\to [0,1] be an unknown function satisfying:", "source": "marker_v2", "marker_block_id": "/page/9/Text/14"}
|
| 2 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0112", "section": "A. Extended Problem Formulation", "page_start": 10, "page_end": 10, "type": "Text", "text": "Assumption 2.1 (Lipschitz continuity). There exists L > 0 such that for all x, y \\in \\mathcal{X} : |f(x) - f(y)| \\le L \\cdot d(x, y) .", "source": "marker_v2", "marker_block_id": "/page/9/Text/15"}
|
| 3 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0113", "section": "A. Extended Problem Formulation", "page_start": 10, "page_end": 10, "type": "Text", "text": "We observe f through noisy queries: querying x returns y = f(x) + \\epsilon , where:", "source": "marker_v2", "marker_block_id": "/page/9/Text/16"}
|
| 4 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0114", "section": "A. Extended Problem Formulation", "page_start": 10, "page_end": 10, "type": "Text", "text": "Assumption 2.2 (Sub-Gaussian noise). The noise \\epsilon is \\sigma -sub-Gaussian: \\mathbb{E}[e^{\\lambda\\epsilon}] \\leq e^{\\lambda^2\\sigma^2/2} for all \\lambda \\in \\mathbb{R} .", "source": "marker_v2", "marker_block_id": "/page/9/Text/17"}
|
| 5 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0115", "section": "A. Extended Problem Formulation", "page_start": 10, "page_end": 10, "type": "Text", "text": "After T samples, the algorithm outputs \\hat{x}_T \\in \\mathcal{X} . The goal is to minimize simple regret r_T = f(x^*) - f(\\hat{x}_T) , where x^* \\in \\arg\\max_{x \\in \\mathcal{X}} f(x) . We seek PAC-style guarantees: with probability at least 1 - \\delta , achieve r_T \\leq \\varepsilon .", "source": "marker_v2", "marker_block_id": "/page/9/Text/18"}
|
| 6 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0116", "section": "A. Extended Problem Formulation", "page_start": 10, "page_end": 10, "type": "Text", "text": "To obtain instance-dependent rates that improve upon worst case bounds, we assume margin structure:", "source": "marker_v2", "marker_block_id": "/page/9/Text/19"}
|
| 7 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0117", "section": "A. Extended Problem Formulation", "page_start": 10, "page_end": 10, "type": "Text", "text": "Assumption 2.3 (Margin / near-optimality dimension). There exist C>0 and \\alpha\\in[0,d] such that for all \\varepsilon>0 : \\operatorname{Vol}(\\{x\\in\\mathcal{X}:f(x)\\geq f^*-\\varepsilon\\})\\leq C\\,\\varepsilon^{d-\\alpha} .", "source": "marker_v2", "marker_block_id": "/page/9/Text/20"}
|
| 8 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0118", "section": "A. Extended Problem Formulation", "page_start": 10, "page_end": 10, "type": "Text", "text": "The parameter \\alpha is the near-optimality dimension: smaller \\alpha corresponds to a sharper optimum (easier), while larger \\alpha corresponds to a broader near-optimal region (harder). The worst case is \\alpha=d , which recovers the standard d-dimensional Lipschitz difficulty. This assumption is standard in bandit theory (Audibert & Bubeck, 2010; Bubeck et al., 2011; Lattimore & Szepesvári, 2020; Kaufmann et al., 2016; Garivier & Kaufmann, 2016).", "source": "marker_v2", "marker_block_id": "/page/9/Text/21"}
|
| 9 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0119", "section": "B.1. Score Maximization and Replication", "page_start": 10, "page_end": 10, "type": "Text", "text": "CGP optimizes \\operatorname{score}(x) = U_t(x) - \\lambda \\cdot \\min_i d(x, x_i) , which is piecewise linear. Since this is non-smooth, we use CMA-ES (Covariance Matrix Adaptation Evolution Strategy) with bounded domain and 10 random restarts within A_t . For d \\leq 3 , we additionally use Delaunay triangulation to identify candidate optima at Voronoi vertices.", "source": "marker_v2", "marker_block_id": "/page/9/Text/24"}
|
| 10 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0120", "section": "B.1. Score Maximization and Replication", "page_start": 10, "page_end": 10, "type": "Text", "text": "Membership in A_t is exact: given \\{x_i, \\hat{\\mu}_i, r_i\\} , checking U_t(x) \\geq \\ell_t requires O(N_t) time. Approximate maximization may slow convergence of \\eta_t but does not invalidate certificates: any x \\notin A_t remains certifiably suboptimal regardless of which x \\in A_t is queried.", "source": "marker_v2", "marker_block_id": "/page/9/Text/25"}
|
| 11 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0121", "section": "B.1. Score Maximization and Replication", "page_start": 10, "page_end": 10, "type": "Text", "text": "Replication Strategy. When an active point x_i has r_i(t) > \\beta_{\\text{target}}(t) , we allocate \\lceil (r_i(t)/\\beta_{\\text{target}}(t))^2 \\rceil additional samples to reduce its confidence radius. This ensures all active points have comparable confidence, preventing", "source": "marker_v2", "marker_block_id": "/page/9/Text/26"}
|
| 12 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0122", "section": "B.1. Score Maximization and Replication", "page_start": 11, "page_end": 11, "type": "Text", "text": "any single point from dominating the envelope.", "source": "marker_v2", "marker_block_id": "/page/10/Text/1"}
|
| 13 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0123", "section": "B.2. Active Set Computation", "page_start": 11, "page_end": 11, "type": "Text", "text": "For low dimensions (d \\le 5) , we compute A_t exactly using grid discretization with resolution \\eta = D/\\sqrt[d]{N_{\\rm grid}} where N_{\\rm grid} = 10^4 . For each grid point x, we evaluate U_t(x) in O(N_t) time and check if U_t(x) \\ge \\ell_t . The volume \\operatorname{Vol}(A_t) is estimated as the fraction of grid points in A_t .", "source": "marker_v2", "marker_block_id": "/page/10/Text/3"}
|
| 14 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0124", "section": "B.2. Active Set Computation", "page_start": 11, "page_end": 11, "type": "Text", "text": "For higher dimensions (d>5), uniform Monte Carlo becomes ineffective once \\operatorname{Vol}(A_t) is small. We therefore estimate \\operatorname{Vol}(A_t) via a nested-set ratio estimator (subset simulation): define thresholds \\ell_t - \\tau_0 < \\ell_t - \\tau_1 < \\cdots < \\ell_t inducing nested sets", "source": "marker_v2", "marker_block_id": "/page/10/Text/4"}
|
| 15 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0125", "section": "B.2. Active Set Computation", "page_start": 11, "page_end": 11, "type": "Equation", "text": "A_t^{(k)} = \\{x : U_t(x) \\ge \\ell_t - \\tau_k\\}, \\quad A_t^{(K)} = A_t.", "source": "marker_v2", "marker_block_id": "/page/10/Equation/5"}
|
| 16 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0126", "section": "B.2. Active Set Computation", "page_start": 11, "page_end": 11, "type": "Text", "text": "We estimate", "source": "marker_v2", "marker_block_id": "/page/10/Text/6"}
|
| 17 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0127", "section": "B.2. Active Set Computation", "page_start": 11, "page_end": 11, "type": "Equation", "text": "Vol(A_t) = Vol(A_t^{(0)}) \\prod_{k=1}^{K} \\mathbb{P}_{x \\sim Unif(A_t^{(k-1)})} [x \\in A_t^{(k)}],", "source": "marker_v2", "marker_block_id": "/page/10/Equation/7"}
|
| 18 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0128", "section": "B.2. Active Set Computation", "page_start": 11, "page_end": 11, "type": "Text", "text": "sampling approximately uniformly from A_t^{(k-1)} using a hit-and-run Markov chain with the membership oracle U_t(x) \\geq \\ell_t - \\tau_{k-1} . This yields stable estimates even when \\operatorname{Vol}(A_t) is very small. In all high-dimensional experiments, we report confidence intervals of \\log \\operatorname{Vol}(A_t) from repeated estimator runs.", "source": "marker_v2", "marker_block_id": "/page/10/Text/8"}
|
| 19 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0129", "section": "B.2. Active Set Computation", "page_start": 11, "page_end": 11, "type": "Text", "text": "Crucially, certificate validity (Remark 4.7) is independent of volume estimation accuracy: the set membership rule x \\in A_t \\Leftrightarrow U_t(x) \\ge \\ell_t is exact.", "source": "marker_v2", "marker_block_id": "/page/10/Text/9"}
|
| 20 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0130", "section": "B.2. Active Set Computation", "page_start": 11, "page_end": 11, "type": "Text", "text": "On volume-based stopping. The shrinkage bound (Theorem 4.6) provides an upper bound on \\operatorname{Vol}(A_t) as a function of the algorithmic gap proxy", "source": "marker_v2", "marker_block_id": "/page/10/Text/10"}
|
| 21 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0131", "section": "B.2. Active Set Computation", "page_start": 11, "page_end": 11, "type": "Equation", "text": "\\varepsilon_t := 2(\\beta_t + L\\eta_t) + \\gamma_t, \\quad \\gamma_t = f^* - \\ell_t,", "source": "marker_v2", "marker_block_id": "/page/10/Equation/11"}
|
| 22 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0132", "section": "B.2. Active Set Computation", "page_start": 11, "page_end": 11, "type": "Text", "text": "and therefore supports using \\operatorname{Vol}(A_t) as a practical progress diagnostic. However, \\operatorname{Vol}(A_t) alone does not yield an anytime upper bound on regret without additional lower-regularity assumptions linking volume back to function values. In our experiments we therefore use \\varepsilon_t as the primary certificate-based stopping criterion, and treat \\operatorname{Vol}(A_t) as a secondary monitoring signal.", "source": "marker_v2", "marker_block_id": "/page/10/Text/12"}
|
| 23 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0133", "section": "C.1. Proof to Lemma 4.1", "page_start": 11, "page_end": 11, "type": "Text", "text": "Proof. Define the good event \\mathcal{E} = \\bigcap_{t=1}^T \\bigcap_{i=1}^{N_t} \\{|\\hat{\\mu}_i(t) - f(x_i)| \\leq r_i(t)\\}.", "source": "marker_v2", "marker_block_id": "/page/10/Text/15"}
|
| 24 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0134", "section": "C.1. Proof to Lemma 4.1", "page_start": 11, "page_end": 11, "type": "Text", "text": "Step 1: Single-point concentration. By Hoeffding's inequality for \\sigma -sub-Gaussian random variables:", "source": "marker_v2", "marker_block_id": "/page/10/Text/16"}
|
| 25 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0135", "section": "C.1. Proof to Lemma 4.1", "page_start": 11, "page_end": 11, "type": "Equation", "text": "\\mathbb{P}\\big[|\\hat{\\mu}_i(t) - f(x_i)| > r\\big] \\le 2\\exp\\left(-\\frac{n_i r^2}{2\\sigma^2}\\right). \\tag{4}", "source": "marker_v2", "marker_block_id": "/page/10/Equation/17"}
|
| 26 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0136", "section": "C.1. Proof to Lemma 4.1", "page_start": 11, "page_end": 11, "type": "Text", "text": "Step 2: Calibration of confidence radius. Substituting r = r_i(t) = \\sigma \\sqrt{2 \\log(2N_t T/\\delta)/n_i} :", "source": "marker_v2", "marker_block_id": "/page/10/Text/18"}
|
| 27 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0137", "section": "C.1. Proof to Lemma 4.1", "page_start": 11, "page_end": 11, "type": "Equation", "text": "\\mathbb{P}[|\\hat{\\mu}_{i}(t) - f(x_{i})| > r_{i}(t)] \\leq 2 \\exp\\left(-\\frac{n_{i} \\cdot 2\\sigma^{2} \\log(2N_{t}T/\\delta)}{2\\sigma^{2} \\cdot n_{i}}\\right) = 2 \\exp\\left(-\\log(2N_{t}T/\\delta)\\right) = \\frac{\\delta}{N_{t}T}. (5)", "source": "marker_v2", "marker_block_id": "/page/10/Equation/19"}
|
| 28 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0138", "section": "C.1. Proof to Lemma 4.1", "page_start": 11, "page_end": 11, "type": "Text", "text": "Step 3: Union bound. Applying the union bound over all i \\in \\{1, ..., N_t\\} and t \\in \\{1, ..., T\\} :", "source": "marker_v2", "marker_block_id": "/page/10/Text/20"}
|
| 29 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0139", "section": "C.1. Proof to Lemma 4.1", "page_start": 11, "page_end": 11, "type": "Equation", "text": "\\mathbb{P}[\\mathcal{E}^c] \\le \\sum_{t=1}^T \\sum_{i=1}^{N_t} \\frac{\\delta}{N_t T} \\le \\sum_{t=1}^T \\frac{\\delta}{T} = \\delta. \\tag{6}", "source": "marker_v2", "marker_block_id": "/page/10/Equation/21"}
|
| 30 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0140", "section": "C.1. Proof to Lemma 4.1", "page_start": 11, "page_end": 11, "type": "Equation", "text": "Hence \\mathbb{P}[\\mathcal{E}] \\geq 1 - \\delta .", "source": "marker_v2", "marker_block_id": "/page/10/Equation/22"}
|
| 31 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0141", "section": "C.2. Proof to Lemma 4.2 (UCB envelope is valid)", "page_start": 11, "page_end": 11, "type": "Text", "text": "Proof. Fix any x \\in \\mathcal{X} . For any sampled point x_i , on the good event \\mathcal{E} :", "source": "marker_v2", "marker_block_id": "/page/10/Text/24"}
|
| 32 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0142", "section": "C.2. Proof to Lemma 4.2 (UCB envelope is valid)", "page_start": 11, "page_end": 11, "type": "Equation", "text": "f(x_i) \\le \\hat{\\mu}_i(t) + r_i(t) = UCB_i(t). \\tag{7}", "source": "marker_v2", "marker_block_id": "/page/10/Equation/25"}
|
| 33 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0143", "section": "C.2. Proof to Lemma 4.2 (UCB envelope is valid)", "page_start": 11, "page_end": 11, "type": "Text", "text": "By Lipschitz continuity of f:", "source": "marker_v2", "marker_block_id": "/page/10/Text/26"}
|
| 34 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0144", "section": "C.2. Proof to Lemma 4.2 (UCB envelope is valid)", "page_start": 11, "page_end": 11, "type": "Equation", "text": "f(x) \\le f(x_i) + L \\cdot d(x, x_i) \\le UCB_i(t) + L \\cdot d(x, x_i) . (8)", "source": "marker_v2", "marker_block_id": "/page/10/Equation/27"}
|
| 35 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0145", "section": "C.2. Proof to Lemma 4.2 (UCB envelope is valid)", "page_start": 11, "page_end": 11, "type": "Text", "text": "Since this holds for all sampled i , taking the minimum over i :", "source": "marker_v2", "marker_block_id": "/page/10/Text/28"}
|
| 36 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0146", "section": "C.2. Proof to Lemma 4.2 (UCB envelope is valid)", "page_start": 11, "page_end": 11, "type": "Equation", "text": "f(x) \\le \\min_{i \\le N_t} \\{ \\text{UCB}_i(t) + L \\cdot d(x, x_i) \\} = U_t(x). (9)", "source": "marker_v2", "marker_block_id": "/page/10/Equation/29"}
|
| 37 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0147", "section": "C.3. Proof to Lemma 4.3 (Envelope slack bound)", "page_start": 11, "page_end": 11, "type": "Text", "text": "Proof. Fix any x \\in \\mathcal{X} and any sampled point x_i .", "source": "marker_v2", "marker_block_id": "/page/10/Text/31"}
|
| 38 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0148", "section": "C.3. Proof to Lemma 4.3 (Envelope slack bound)", "page_start": 11, "page_end": 11, "type": "Text", "text": "Step 1: Upper confidence bound. On the good event \\mathcal{E} , we have \\hat{\\mu}_i(t) \\leq f(x_i) + r_i(t) . Therefore:", "source": "marker_v2", "marker_block_id": "/page/10/Text/32"}
|
| 39 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0149", "section": "C.3. Proof to Lemma 4.3 (Envelope slack bound)", "page_start": 11, "page_end": 11, "type": "Equation", "text": "UCB_{i}(t) = \\hat{\\mu}_{i}(t) + r_{i}(t) \\le f(x_{i}) + 2r_{i}(t). (10)", "source": "marker_v2", "marker_block_id": "/page/10/Equation/33"}
|
| 40 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0150", "section": "C.3. Proof to Lemma 4.3 (Envelope slack bound)", "page_start": 11, "page_end": 11, "type": "Text", "text": "Step 2: Lipschitz propagation. By Assumption 2.1 (Lipschitz continuity):", "source": "marker_v2", "marker_block_id": "/page/10/Text/34"}
|
| 41 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0151", "section": "C.3. Proof to Lemma 4.3 (Envelope slack bound)", "page_start": 11, "page_end": 11, "type": "Equation", "text": "f(x_i) \\le f(x) + L \\cdot d(x, x_i). \\tag{11}", "source": "marker_v2", "marker_block_id": "/page/10/Equation/35"}
|
| 42 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0152", "section": "C.3. Proof to Lemma 4.3 (Envelope slack bound)", "page_start": 11, "page_end": 11, "type": "Text", "text": "Step 3: Combining bounds. Substituting the Lipschitz bound into Step 1:", "source": "marker_v2", "marker_block_id": "/page/10/Text/36"}
|
| 43 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0153", "section": "C.3. Proof to Lemma 4.3 (Envelope slack bound)", "page_start": 11, "page_end": 11, "type": "Equation", "text": "UCB_{i}(t) + L d(x, x_{i}) \\leq f(x_{i}) + 2r_{i}(t) + L d(x, x_{i}) \\leq f(x) + L d(x, x_{i}) + 2r_{i}(t) + L d(x, x_{i}) = f(x) + 2(r_{i}(t) + L d(x, x_{i})). \\quad (12)", "source": "marker_v2", "marker_block_id": "/page/10/Equation/37"}
|
| 44 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0154", "section": "C.3. Proof to Lemma 4.3 (Envelope slack bound)", "page_start": 12, "page_end": 12, "type": "Text", "text": "Step 4: Taking the minimum. Since the above holds for all i, taking the minimum over i on both sides:", "source": "marker_v2", "marker_block_id": "/page/11/Text/1"}
|
| 45 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0155", "section": "C.3. Proof to Lemma 4.3 (Envelope slack bound)", "page_start": 12, "page_end": 12, "type": "Equation", "text": "U_{t}(x) = \\min_{i \\leq N_{t}} \\left\\{ \\text{UCB}_{i}(t) + L d(x, x_{i}) \\right\\} \\leq f(x) + 2 \\min_{i} \\left\\{ r_{i}(t) + L d(x, x_{i}) \\right\\} = f(x) + 2\\rho_{t}(x). (13)", "source": "marker_v2", "marker_block_id": "/page/11/Equation/2"}
|
| 46 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0156", "section": "C.4. Proof to Theorem 4.5", "page_start": 12, "page_end": 12, "type": "Text", "text": "Proof. We show that any point in the active set must have function value close to optimal.", "source": "marker_v2", "marker_block_id": "/page/11/Text/4"}
|
| 47 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0157", "section": "C.4. Proof to Theorem 4.5", "page_start": 12, "page_end": 12, "type": "Text", "text": "Step 1: Lower certificate validity. On E, for any sampled point x i :", "source": "marker_v2", "marker_block_id": "/page/11/Text/5"}
|
| 48 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0158", "section": "C.4. Proof to Theorem 4.5", "page_start": 12, "page_end": 12, "type": "Equation", "text": "LCB_i(t) = \\hat{\\mu}_i(t) - r_i(t) \\le f(x_i) \\le f^*. (14)", "source": "marker_v2", "marker_block_id": "/page/11/Equation/6"}
|
| 49 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0159", "section": "C.4. Proof to Theorem 4.5", "page_start": 12, "page_end": 12, "type": "Text", "text": "Taking the maximum over all i:", "source": "marker_v2", "marker_block_id": "/page/11/Text/7"}
|
| 50 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0160", "section": "C.4. Proof to Theorem 4.5", "page_start": 12, "page_end": 12, "type": "Equation", "text": "\\ell_t = \\max_{i \\le N_t} LCB_i(t) \\le f^*. \\tag{15}", "source": "marker_v2", "marker_block_id": "/page/11/Equation/8"}
|
| 51 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0161", "section": "C.4. Proof to Theorem 4.5", "page_start": 12, "page_end": 12, "type": "Text", "text": "Step 2: Active set membership implies high UCB. Let x ∈ At. By definition of the active set (2) :", "source": "marker_v2", "marker_block_id": "/page/11/Text/9"}
|
| 52 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0162", "section": "C.4. Proof to Theorem 4.5", "page_start": 12, "page_end": 12, "type": "Equation", "text": "U_t(x) \\ge \\ell_t. \\tag{16}", "source": "marker_v2", "marker_block_id": "/page/11/Equation/10"}
|
| 53 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0163", "section": "C.4. Proof to Theorem 4.5", "page_start": 12, "page_end": 12, "type": "Text", "text": "Step 3: Applying the envelope bound. By Lemma 4.3:", "source": "marker_v2", "marker_block_id": "/page/11/Text/11"}
|
| 54 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0164", "section": "C.4. Proof to Theorem 4.5", "page_start": 12, "page_end": 12, "type": "Equation", "text": "f(x) + 2\\rho_t(x) \\ge U_t(x) \\ge \\ell_t. \\tag{17}", "source": "marker_v2", "marker_block_id": "/page/11/Equation/12"}
|
| 55 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0165", "section": "C.4. Proof to Theorem 4.5", "page_start": 12, "page_end": 12, "type": "Text", "text": "Step 4: Rearranging to obtain the containment. Solving for f(x):", "source": "marker_v2", "marker_block_id": "/page/11/Text/13"}
|
| 56 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0166", "section": "C.4. Proof to Theorem 4.5", "page_start": 12, "page_end": 12, "type": "Equation", "text": "f(x) \\ge \\ell_t - 2\\rho_t(x) = f^* - (f^* - \\ell_t) - 2\\rho_t(x) \\ge f^* - (f^* - \\ell_t) - 2 \\sup_{x' \\in A_t} \\rho_t(x') = f^* - 2\\Delta_t. (18)", "source": "marker_v2", "marker_block_id": "/page/11/Equation/14"}
|
| 57 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0167", "section": "C.4. Proof to Theorem 4.5", "page_start": 12, "page_end": 12, "type": "Equation", "text": "Hence A_t \\subseteq \\{x : f(x) \\ge f^* - 2\\Delta_t\\}.", "source": "marker_v2", "marker_block_id": "/page/11/Equation/15"}
|
| 58 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0168", "section": "C.5. Proof to Theorem 4.6", "page_start": 12, "page_end": 12, "type": "Text", "text": "Proof. We connect the active set volume to the margin condition via the containment theorem.", "source": "marker_v2", "marker_block_id": "/page/11/Text/17"}
|
| 59 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0169", "section": "C.5. Proof to Theorem 4.6", "page_start": 12, "page_end": 12, "type": "Text", "text": "Step 1: Bounding ∆t . From Theorem 4.5, A t ⊆ {x : f(x) ≥ f ∗ − 2∆t} where:", "source": "marker_v2", "marker_block_id": "/page/11/Text/18"}
|
| 60 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0170", "section": "C.5. Proof to Theorem 4.6", "page_start": 12, "page_end": 12, "type": "Equation", "text": "\\Delta_t = \\sup_{x \\in A_t} \\rho_t(x) + (f^* - \\ell_t). (19)", "source": "marker_v2", "marker_block_id": "/page/11/Equation/19"}
|
| 61 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0171", "section": "C.5. Proof to Theorem 4.6", "page_start": 12, "page_end": 12, "type": "Text", "text": "For any x ∈ At:", "source": "marker_v2", "marker_block_id": "/page/11/Text/20"}
|
| 62 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0172", "section": "C.5. Proof to Theorem 4.6", "page_start": 12, "page_end": 12, "type": "Equation", "text": "\\rho_t(x) = \\min_{i} \\left\\{ r_i(t) + L \\cdot d(x, x_i) \\right\\} \\leq \\max_{i:x_i \\text{ active}} r_i(t) + L \\cdot \\sup_{x \\in A_t} \\min_{i} d(x, x_i) = \\beta_t + L\\eta_t. \\tag{20}", "source": "marker_v2", "marker_block_id": "/page/11/Equation/21"}
|
| 63 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0173", "section": "C.5. Proof to Theorem 4.6", "page_start": 12, "page_end": 12, "type": "Text", "text": "Therefore:", "source": "marker_v2", "marker_block_id": "/page/11/Text/22"}
|
| 64 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0174", "section": "C.5. Proof to Theorem 4.6", "page_start": 12, "page_end": 12, "type": "Equation", "text": "\\Delta_t \\le \\beta_t + L\\eta_t + \\frac{\\gamma_t}{2}.\\tag{21}", "source": "marker_v2", "marker_block_id": "/page/11/Equation/23"}
|
| 65 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0175", "section": "C.5. Proof to Theorem 4.6", "page_start": 12, "page_end": 12, "type": "Text", "text": "Step 2: Applying the margin condition. By Assumption 2.3, for any ε > 0:", "source": "marker_v2", "marker_block_id": "/page/11/Text/24"}
|
| 66 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0176", "section": "C.5. Proof to Theorem 4.6", "page_start": 12, "page_end": 12, "type": "Equation", "text": "\\operatorname{Vol}(\\{x: f(x) \\ge f^* - \\varepsilon\\}) \\le C \\cdot \\varepsilon^{d-\\alpha}. \\tag{22}", "source": "marker_v2", "marker_block_id": "/page/11/Equation/25"}
|
| 67 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0177", "section": "C.5. Proof to Theorem 4.6", "page_start": 12, "page_end": 12, "type": "Text", "text": "Step 3: Combining the bounds. Setting ε = 2∆ t ≤ 2(β t + Lηt) + γt:", "source": "marker_v2", "marker_block_id": "/page/11/Text/26"}
|
| 68 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0178", "section": "C.5. Proof to Theorem 4.6", "page_start": 12, "page_end": 12, "type": "Equation", "text": "\\operatorname{Vol}(A_t) \\leq \\operatorname{Vol}(\\{x : f(x) \\geq f^* - 2\\Delta_t\\}) \\leq C \\cdot (2\\Delta_t)^{d-\\alpha} \\leq C \\cdot (2(\\beta_t + L\\eta_t) + \\gamma_t)^{d-\\alpha}. (23)", "source": "marker_v2", "marker_block_id": "/page/11/Equation/27"}
|
| 69 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0179", "section": "C.6. Proof to Theorem 4.8", "page_start": 12, "page_end": 12, "type": "Text", "text": "Proof. We derive the sample complexity by analyzing the requirements for ε-optimality.", "source": "marker_v2", "marker_block_id": "/page/11/Text/29"}
|
| 70 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0180", "section": "C.6. Proof to Theorem 4.8", "page_start": 12, "page_end": 12, "type": "Text", "text": "Step 1: Optimality condition. To achieve simple regret r T ≤ ε, it suffices to ensure:", "source": "marker_v2", "marker_block_id": "/page/11/Text/30"}
|
| 71 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0181", "section": "C.6. Proof to Theorem 4.8", "page_start": 12, "page_end": 12, "type": "Equation", "text": "2(\\beta_t + L\\eta_t) + \\gamma_t \\le \\varepsilon. \\tag{24}", "source": "marker_v2", "marker_block_id": "/page/11/Equation/31"}
|
| 72 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0182", "section": "C.6. Proof to Theorem 4.8", "page_start": 12, "page_end": 12, "type": "Text", "text": "This requires β t ≤ ε/6, η t ≤ ε/(6L), and γ t ≤ ε/3.", "source": "marker_v2", "marker_block_id": "/page/11/Text/32"}
|
| 73 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0183", "section": "C.6. Proof to Theorem 4.8", "page_start": 12, "page_end": 12, "type": "Text", "text": "Step 2: Covering the active set. By the shrinkage theorem (Theorem 4.6) , once 2(β t + Lηt) + γ t ≤ ε we have:", "source": "marker_v2", "marker_block_id": "/page/11/Text/33"}
|
| 74 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0184", "section": "C.6. Proof to Theorem 4.8", "page_start": 12, "page_end": 12, "type": "Equation", "text": "Vol(A_t) \\le C \\cdot \\varepsilon^{d-\\alpha}. (25)", "source": "marker_v2", "marker_block_id": "/page/11/Equation/34"}
|
| 75 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0185", "section": "C.6. Proof to Theorem 4.8", "page_start": 12, "page_end": 12, "type": "Text", "text": "To achieve covering radius η = ε/(6L) over a region of volume Cε d − α , we need:", "source": "marker_v2", "marker_block_id": "/page/11/Text/35"}
|
| 76 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0186", "section": "C.6. Proof to Theorem 4.8", "page_start": 12, "page_end": 12, "type": "Equation", "text": "N_{\\text{cover}} = O\\left(\\frac{\\text{Vol}(A_t)}{\\eta^d}\\right) = O\\left(\\frac{C\\varepsilon^{d-\\alpha}}{(\\varepsilon/(6L))^d}\\right) = O\\left(L^d\\varepsilon^{-\\alpha}\\right) (26)", "source": "marker_v2", "marker_block_id": "/page/11/Equation/36"}
|
| 77 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0187", "section": "C.6. Proof to Theorem 4.8", "page_start": 12, "page_end": 12, "type": "Text", "text": "distinct sample locations.", "source": "marker_v2", "marker_block_id": "/page/11/Text/37"}
|
| 78 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0188", "section": "C.6. Proof to Theorem 4.8", "page_start": 12, "page_end": 12, "type": "Text", "text": "Step 3: Samples per location. To achieve confidence radius β t ≤ ε/6 at each location, we need:", "source": "marker_v2", "marker_block_id": "/page/11/Text/38"}
|
| 79 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0189", "section": "C.6. Proof to Theorem 4.8", "page_start": 12, "page_end": 12, "type": "Equation", "text": "\\sigma\\sqrt{\\frac{2\\log(2N_tT/\\delta)}{n_i}} \\le \\frac{\\varepsilon}{6}. (27)", "source": "marker_v2", "marker_block_id": "/page/11/Equation/39"}
|
| 80 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0190", "section": "C.6. Proof to Theorem 4.8", "page_start": 12, "page_end": 12, "type": "Text", "text": "Solving for n i :", "source": "marker_v2", "marker_block_id": "/page/11/Text/40"}
|
| 81 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0191", "section": "C.6. Proof to Theorem 4.8", "page_start": 12, "page_end": 12, "type": "Equation", "text": "n_i = O\\left(\\frac{\\sigma^2 \\log(T/\\delta)}{\\varepsilon^2}\\right). (28)", "source": "marker_v2", "marker_block_id": "/page/11/Equation/41"}
|
| 82 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0192", "section": "C.6. Proof to Theorem 4.8", "page_start": 13, "page_end": 13, "type": "Text", "text": "Step 4: Total sample complexity. Combining Steps 2 and 3:", "source": "marker_v2", "marker_block_id": "/page/12/Text/1"}
|
| 83 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0193", "section": "C.6. Proof to Theorem 4.8", "page_start": 13, "page_end": 13, "type": "Equation", "text": "T = N_{\\text{cover}} \\cdot n_{i} = O(L^{d} \\varepsilon^{-\\alpha}) \\cdot O\\left(\\frac{\\sigma^{2} \\log(T/\\delta)}{\\varepsilon^{2}}\\right) = \\tilde{O}(L^{d} \\varepsilon^{-(2+\\alpha)}). \\tag{29}", "source": "marker_v2", "marker_block_id": "/page/12/Equation/2"}
|
| 84 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0194", "section": "C.7. Proof to Theorem 4.9", "page_start": 13, "page_end": 13, "type": "Text", "text": "Proof. We construct a hard instance via a randomized reduction.", "source": "marker_v2", "marker_block_id": "/page/12/Text/4"}
|
| 85 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0195", "section": "C.7. Proof to Theorem 4.9", "page_start": 13, "page_end": 13, "type": "Text", "text": "Step 1: Hard instance construction (continuous bump). We use a standard \"one bump among M locations\" construction that ensures global Lipschitz continuity. Partition \\mathcal{X} = [0,1]^d into M = \\varepsilon^{-\\alpha} disjoint cells with centers \\{c_1,\\ldots,c_M\\} , each cell having diameter \\Theta(\\varepsilon^{\\alpha/d}) . Select one cell i^* uniformly at random to contain the optimum, and place x^* = c_{i^*} . Define:", "source": "marker_v2", "marker_block_id": "/page/12/Text/5"}
|
| 86 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0196", "section": "C.7. Proof to Theorem 4.9", "page_start": 13, "page_end": 13, "type": "Equation", "text": "f(x) = (1 - \\varepsilon) + \\varepsilon \\cdot \\max \\left\\{ 0, 1 - \\frac{L \\|x - x^*\\|}{\\varepsilon} \\right\\}. \\quad (30)", "source": "marker_v2", "marker_block_id": "/page/12/Equation/6"}
|
| 87 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0197", "section": "C.7. Proof to Theorem 4.9", "page_start": 13, "page_end": 13, "type": "Text", "text": "This is a cone/bump centered at x^* that peaks at f(x^*)=1 and decreases linearly with slope L until it reaches the baseline value 1-\\varepsilon at radius \\varepsilon/L from x^* . Outside this radius, f(x)\\equiv 1-\\varepsilon . The function is globally L-Lipschitz: within the bump, the gradient has magnitude L; outside, the function is constant; and at the boundary \\|x-x^*\\|=\\varepsilon/L , both pieces match at value 1-\\varepsilon .", "source": "marker_v2", "marker_block_id": "/page/12/Text/7"}
|
| 88 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0198", "section": "C.7. Proof to Theorem 4.9", "page_start": 13, "page_end": 13, "type": "Text", "text": "Verification of Assumption 2.3. The \\varepsilon -near-optimal set \\{x:f(x)\\geq 1-\\varepsilon\\} is exactly the ball of radius \\varepsilon/L around x^* , which has volume \\operatorname{Vol}(B_{\\varepsilon/L})=O((\\varepsilon/L)^d)=O(\\varepsilon^d) . With M=\\varepsilon^{-\\alpha} candidate locations, only one contains the bump, so the near-optimal fraction of the domain is O(\\varepsilon^d) . Since the domain has unit volume, \\operatorname{Vol}(\\{f\\geq f^*-\\varepsilon\\})=O(\\varepsilon^d)\\leq C\\varepsilon^{d-\\alpha} for \\alpha\\geq 0 , satisfying Assumption 2.3.", "source": "marker_v2", "marker_block_id": "/page/12/Text/8"}
|
| 89 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0199", "section": "C.7. Proof to Theorem 4.9", "page_start": 13, "page_end": 13, "type": "Text", "text": "Step 2: Information-theoretic lower bound. To identify the correct cell with probability \\geq 2/3 , the algorithm must distinguish between M hypotheses. By Fano's inequality, this requires:", "source": "marker_v2", "marker_block_id": "/page/12/Text/9"}
|
| 90 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0200", "section": "C.7. Proof to Theorem 4.9", "page_start": 13, "page_end": 13, "type": "Equation", "text": "\\sum_{i=1}^{M} n_i \\cdot \\text{KL}(P_i || P_0) \\ge \\log(M/3), \\tag{31}", "source": "marker_v2", "marker_block_id": "/page/12/Equation/10"}
|
| 91 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0201", "section": "C.7. Proof to Theorem 4.9", "page_start": 13, "page_end": 13, "type": "Text", "text": "where n_i is the number of samples in cell i, and \\mathrm{KL}(P_i || P_0) is the KL divergence between observations under hypothesis i versus the null.", "source": "marker_v2", "marker_block_id": "/page/12/Text/11"}
|
| 92 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0202", "section": "C.7. Proof to Theorem 4.9", "page_start": 13, "page_end": 13, "type": "Text", "text": "Step 3: Per-cell sample requirement. For \\sigma -sub-Gaussian noise, distinguishing a cell with optimum from one without requires:", "source": "marker_v2", "marker_block_id": "/page/12/Text/12"}
|
| 93 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0203", "section": "C.7. Proof to Theorem 4.9", "page_start": 13, "page_end": 13, "type": "Equation", "text": "n_i = \\Omega\\left(\\frac{\\sigma^2}{\\varepsilon^2}\\right) \\tag{32}", "source": "marker_v2", "marker_block_id": "/page/12/Equation/13"}
|
| 94 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0204", "section": "C.7. Proof to Theorem 4.9", "page_start": 13, "page_end": 13, "type": "Text", "text": "samples per cell to detect the \\varepsilon gap with constant probability. Step 4: Total sample complexity. Summing over all M cells:", "source": "marker_v2", "marker_block_id": "/page/12/Text/14"}
|
| 95 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0205", "section": "C.7. Proof to Theorem 4.9", "page_start": 13, "page_end": 13, "type": "Equation", "text": "T = \\Omega \\left( M \\cdot \\frac{\\sigma^2}{\\varepsilon^2} \\right) = \\Omega \\left( \\varepsilon^{-\\alpha} \\cdot \\varepsilon^{-2} \\right) = \\Omega \\left( \\varepsilon^{-(2+\\alpha)} \\right). \\tag{33}", "source": "marker_v2", "marker_block_id": "/page/12/Equation/15"}
|
| 96 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0206", "section": "C.8. Proof to Theorem 5.1", "page_start": 13, "page_end": 13, "type": "Text", "text": "Proof. We prove each claim separately.", "source": "marker_v2", "marker_block_id": "/page/12/Text/17"}
|
| 97 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0207", "section": "C.8. Proof to Theorem 5.1", "page_start": 13, "page_end": 13, "type": "Text", "text": "Proof of (1): Bounded doubling events. Each doubling event multiplies \\hat{L} by 2. Starting from \\hat{L}_0 \\leq L^* :", "source": "marker_v2", "marker_block_id": "/page/12/Text/18"}
|
| 98 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0208", "section": "C.8. Proof to Theorem 5.1", "page_start": 13, "page_end": 13, "type": "Equation", "text": "\\hat{L}_k = 2^k \\hat{L}_0 after k doublings. (34)", "source": "marker_v2", "marker_block_id": "/page/12/Equation/19"}
|
| 99 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0209", "section": "C.8. Proof to Theorem 5.1", "page_start": 13, "page_end": 13, "type": "Text", "text": "The algorithm stops doubling when \\hat{L} \\geq L^* , which requires:", "source": "marker_v2", "marker_block_id": "/page/12/Text/20"}
|
| 100 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0210", "section": "C.8. Proof to Theorem 5.1", "page_start": 13, "page_end": 13, "type": "Equation", "text": "2^K \\hat{L}_0 \\ge L^* \\implies K \\ge \\log_2(L^*/\\hat{L}_0). (35)", "source": "marker_v2", "marker_block_id": "/page/12/Equation/21"}
|
| 101 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0211", "section": "C.8. Proof to Theorem 5.1", "page_start": 13, "page_end": 13, "type": "Text", "text": "Hence K \\leq \\lceil \\log_2(L^*/\\hat{L}_0) \\rceil .", "source": "marker_v2", "marker_block_id": "/page/12/Text/22"}
|
| 102 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0212", "section": "C.8. Proof to Theorem 5.1", "page_start": 13, "page_end": 13, "type": "Text", "text": "Proof of (2): Final estimate accuracy. A violation is detected when:", "source": "marker_v2", "marker_block_id": "/page/12/Text/23"}
|
| 103 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0213", "section": "C.8. Proof to Theorem 5.1", "page_start": 13, "page_end": 13, "type": "Equation", "text": "|\\hat{\\mu}_i - \\hat{\\mu}_j| - 2(r_i + r_j) > \\hat{L} \\cdot d(x_i, x_j). (36)", "source": "marker_v2", "marker_block_id": "/page/12/Equation/24"}
|
| 104 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0214", "section": "C.8. Proof to Theorem 5.1", "page_start": 13, "page_end": 13, "type": "Text", "text": "On the good event \\mathcal{E} :", "source": "marker_v2", "marker_block_id": "/page/12/Text/25"}
|
| 105 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0215", "section": "C.8. Proof to Theorem 5.1", "page_start": 13, "page_end": 13, "type": "Equation", "text": "|f(x_i) - f(x_j)| \\le |\\hat{\\mu}_i - \\hat{\\mu}_j| + 2(r_i + r_j). (37)", "source": "marker_v2", "marker_block_id": "/page/12/Equation/26"}
|
| 106 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0216", "section": "C.8. Proof to Theorem 5.1", "page_start": 13, "page_end": 13, "type": "Text", "text": "If \\hat{L} \\geq L^* , then by Lipschitz continuity:", "source": "marker_v2", "marker_block_id": "/page/12/Text/27"}
|
| 107 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0217", "section": "C.8. Proof to Theorem 5.1", "page_start": 13, "page_end": 13, "type": "Equation", "text": "|f(x_i) - f(x_i)| \\le L^* \\cdot d(x_i, x_i) \\le \\hat{L} \\cdot d(x_i, x_i), \\quad (38)", "source": "marker_v2", "marker_block_id": "/page/12/Equation/28"}
|
| 108 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0218", "section": "C.8. Proof to Theorem 5.1", "page_start": 13, "page_end": 13, "type": "Text", "text": "so no violation can occur. Thus violations only occur when \\hat{L} < L^* , and after all doublings complete, \\hat{L} \\geq L^* . Since we double (rather than increase by smaller factors), \\hat{L} \\leq 2L^* .", "source": "marker_v2", "marker_block_id": "/page/12/Text/29"}
|
| 109 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0219", "section": "C.8. Proof to Theorem 5.1", "page_start": 13, "page_end": 13, "type": "Text", "text": "Proof of (3): Sample complexity overhead. Between doublings, CGP runs with either:", "source": "marker_v2", "marker_block_id": "/page/12/Text/30"}
|
| 110 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0220", "section": "C.8. Proof to Theorem 5.1", "page_start": 13, "page_end": 13, "type": "ListGroup", "text": "Invalid \\hat{L} < L^* (before sufficient doublings): certificates may be incorrect, but each such phase has at most O(T/K) samples before a violation triggers doubling. Valid \\hat{L} \\geq L^* (after final doubling): CGP achieves \\tilde{O}(\\varepsilon^{-(2+\\alpha)}) complexity by Theorem 4.8.", "source": "marker_v2", "marker_block_id": "/page/12/ListGroup/482"}
|
| 111 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0221", "section": "C.8. Proof to Theorem 5.1", "page_start": 13, "page_end": 13, "type": "Text", "text": "There are at most K invalid phases, each contributing O(T/K) samples. The final valid phase dominates, giving total complexity T = \\tilde{O}(\\varepsilon^{-(2+\\alpha)} \\cdot K) = \\tilde{O}(\\varepsilon^{-(2+\\alpha)} \\cdot \\log(L^*/\\hat{L}_0)) .", "source": "marker_v2", "marker_block_id": "/page/12/Text/33"}
|
| 112 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0222", "section": "C.9. Certificate Validity Under Adaptive Lipschitz Estimation", "page_start": 14, "page_end": 14, "type": "Text", "text": "Lemma C.1 (Certificate validity once \\hat{L} \\geq L^* ). Assume the good event \\mathcal{E} holds. Fix any time t at which the current estimate satisfies \\hat{L} \\geq L^* . Then the envelope constructed with \\hat{L} satisfies f(x) \\leq U_t(x) for all x \\in \\mathcal{X} , and consequently the active set", "source": "marker_v2", "marker_block_id": "/page/13/Text/2"}
|
| 113 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0223", "section": "C.9. Certificate Validity Under Adaptive Lipschitz Estimation", "page_start": 14, "page_end": 14, "type": "Equation", "text": "A_t = \\{ x \\in \\mathcal{X} : U_t(x) \\ge \\ell_t \\}", "source": "marker_v2", "marker_block_id": "/page/13/Equation/3"}
|
| 114 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0224", "section": "C.9. Certificate Validity Under Adaptive Lipschitz Estimation", "page_start": 14, "page_end": 14, "type": "Text", "text": "contains x^* and certifies that any x \\notin A_t is suboptimal (in the sense U_t(x) < \\ell_t \\le f^* ).", "source": "marker_v2", "marker_block_id": "/page/13/Text/4"}
|
| 115 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0225", "section": "C.9. Certificate Validity Under Adaptive Lipschitz Estimation", "page_start": 14, "page_end": 14, "type": "Text", "text": "Proof. If \\hat{L} \\geq L^* , then for all x, y \\in \\mathcal{X} we have |f(x) - f(y)| \\leq L^* d(x, y) \\leq \\hat{L} d(x, y) , i.e., f is \\hat{L} -Lipschitz. On \\mathcal{E} , for each sampled point x_i , f(x_i) \\leq \\mathrm{UCB}_i(t) . Therefore for all x,", "source": "marker_v2", "marker_block_id": "/page/13/Text/5"}
|
| 116 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0226", "section": "C.9. Certificate Validity Under Adaptive Lipschitz Estimation", "page_start": 14, "page_end": 14, "type": "Equation", "text": "f(x) \\le f(x_i) + \\hat{L} d(x, x_i) \\le UCB_i(t) + \\hat{L} d(x, x_i).", "source": "marker_v2", "marker_block_id": "/page/13/Equation/6"}
|
| 117 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0227", "section": "C.9. Certificate Validity Under Adaptive Lipschitz Estimation", "page_start": 14, "page_end": 14, "type": "Text", "text": "Taking the minimum over i yields f(x) \\leq U_t(x) for all x. In particular, U_t(x^*) \\geq f^* . Also \\ell_t = \\max_i \\mathrm{LCB}_i(t) \\leq f^* on \\mathcal{E} , hence U_t(x^*) \\geq \\ell_t and so x^* \\in A_t . Finally, if x \\notin A_t , then U_t(x) < \\ell_t \\leq f^* , certifying x cannot be optimal under \\mathcal{E} .", "source": "marker_v2", "marker_block_id": "/page/13/Text/7"}
|
| 118 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0228", "section": "C.9. Certificate Validity Under Adaptive Lipschitz Estimation", "page_start": 14, "page_end": 14, "type": "Text", "text": "Remark C.2 (Why pre-final certificates need not be valid). If \\hat{L} < L^* , then the envelope may fail to upper-bound f globally, and the rule U_t(x) \\geq \\ell_t can (in principle) exclude near-optimal points. CGP-Adaptive therefore guarantees certificate validity only after the final doubling event ensures \\hat{L} \\geq L^* .", "source": "marker_v2", "marker_block_id": "/page/13/Text/8"}
|
| 119 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0229", "section": "C.10. Global Safety of Certified Restarts (No False Elimination)", "page_start": 14, "page_end": 14, "type": "Text", "text": "We restate and prove the key safety property underlying certified restarts in CGP-TR.", "source": "marker_v2", "marker_block_id": "/page/13/Text/10"}
|
| 120 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0230", "section": "C.10. Global Safety of Certified Restarts (No False Elimination)", "page_start": 14, "page_end": 14, "type": "Text", "text": "Lemma C.3 (No false certified restart for the region containing x^* ). Fix any trust region \\mathcal{T}^* such that x^* \\in \\mathcal{T}^* . On the good event \\mathcal{E} , the certified restart condition", "source": "marker_v2", "marker_block_id": "/page/13/Text/11"}
|
| 121 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0231", "section": "C.10. Global Safety of Certified Restarts (No False Elimination)", "page_start": 14, "page_end": 14, "type": "Equation", "text": "u_t^{(\\mathcal{T}^*)} := \\max_{x \\in \\mathcal{T}^*} U_t(x) < \\ell_t", "source": "marker_v2", "marker_block_id": "/page/13/Equation/12"}
|
| 122 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0232", "section": "C.10. Global Safety of Certified Restarts (No False Elimination)", "page_start": 14, "page_end": 14, "type": "Text", "text": "never holds. Hence, \\mathcal{T}^* is never restarted by the certified rule.", "source": "marker_v2", "marker_block_id": "/page/13/Text/13"}
|
| 123 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0233", "section": "C.10. Global Safety of Certified Restarts (No False Elimination)", "page_start": 14, "page_end": 14, "type": "Text", "text": "Proof. On the good event \\mathcal{E} , Lemma 4.2 implies f(x) \\leq U_t(x) for all x \\in \\mathcal{X} . Since x^* \\in \\mathcal{T}^* ,", "source": "marker_v2", "marker_block_id": "/page/13/Text/14"}
|
| 124 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0234", "section": "C.10. Global Safety of Certified Restarts (No False Elimination)", "page_start": 14, "page_end": 14, "type": "Equation", "text": "u_t^{(\\mathcal{T}^*)} = \\max_{x \\in \\mathcal{T}^*} U_t(x) \\ge U_t(x^*) \\ge f(x^*) = f^*.", "source": "marker_v2", "marker_block_id": "/page/13/Equation/15"}
|
| 125 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0235", "section": "C.10. Global Safety of Certified Restarts (No False Elimination)", "page_start": 14, "page_end": 14, "type": "Text", "text": "Also, by definition \\ell_t = \\max_i \\mathrm{LCB}_i(t) . On \\mathcal{E} , \\mathrm{LCB}_i(t) \\leq f(x_i) \\leq f^* for every sampled point x_i , hence \\ell_t \\leq f^* . Therefore,", "source": "marker_v2", "marker_block_id": "/page/13/Text/16"}
|
| 126 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0236", "section": "C.10. Global Safety of Certified Restarts (No False Elimination)", "page_start": 14, "page_end": 14, "type": "Equation", "text": "u_t^{(\\mathcal{T}^*)} \\geq f^* \\geq \\ell_t,", "source": "marker_v2", "marker_block_id": "/page/13/Equation/17"}
|
| 127 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0237", "section": "C.10. Global Safety of Certified Restarts (No False Elimination)", "page_start": 14, "page_end": 14, "type": "Text", "text": "so the strict inequality u_t^{(\\mathcal{T}^*)} < \\ell_t cannot occur on \\mathcal{E} .", "source": "marker_v2", "marker_block_id": "/page/13/Text/18"}
|
| 128 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0238", "section": "C.11. Proof to Theorem 6.1", "page_start": 14, "page_end": 14, "type": "Text", "text": "We first establish a key lemma showing that certified elimination is safe.", "source": "marker_v2", "marker_block_id": "/page/13/Text/20"}
|
| 129 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0239", "section": "C.11. Proof to Theorem 6.1", "page_start": 14, "page_end": 14, "type": "Text", "text": "Lemma C.4 (Certified elimination). On the good event \\mathcal{E} , for any trust region \\mathcal{T}_j ,", "source": "marker_v2", "marker_block_id": "/page/13/Text/21"}
|
| 130 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0240", "section": "C.11. Proof to Theorem 6.1", "page_start": 14, "page_end": 14, "type": "Equation", "text": "u_t^{(j)} := \\max_{x \\in \\mathcal{T}_j} U_t(x) < \\ell_t \\quad \\Longrightarrow \\quad \\sup_{x \\in \\mathcal{T}_i} f(x) < f(x^*).", "source": "marker_v2", "marker_block_id": "/page/13/Equation/22"}
|
| 131 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0241", "section": "C.11. Proof to Theorem 6.1", "page_start": 14, "page_end": 14, "type": "Text", "text": "Proof. On \\mathcal{E} , by Lemma 4.2, f(x) \\leq U_t(x) for all x \\in \\mathcal{X} . Also \\ell_t = \\max_i \\mathrm{LCB}_i(t) \\leq f(x^*) on \\mathcal{E} since each \\mathrm{LCB}_i(t) \\leq f(x_i) \\leq f(x^*) . Therefore", "source": "marker_v2", "marker_block_id": "/page/13/Text/23"}
|
| 132 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0242", "section": "C.11. Proof to Theorem 6.1", "page_start": 14, "page_end": 14, "type": "Equation", "text": "\\sup_{x \\in \\mathcal{T}_j} f(x) \\le \\sup_{x \\in \\mathcal{T}_j} U_t(x) = u_t^{(j)} < \\ell_t \\le f(x^*),", "source": "marker_v2", "marker_block_id": "/page/13/Equation/24"}
|
| 133 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0243", "section": "C.11. Proof to Theorem 6.1", "page_start": 14, "page_end": 14, "type": "Text", "text": "which proves the claim.", "source": "marker_v2", "marker_block_id": "/page/13/Text/25"}
|
| 134 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0244", "section": "C.11. Proof to Theorem 6.1", "page_start": 14, "page_end": 14, "type": "Text", "text": "Proof of Theorem 6.1(1): No false restarts. If x^* \\in \\mathcal{T}^* , then u_t^{(\\mathcal{T}^*)} = \\max_{x \\in \\mathcal{T}^*} U_t(x) \\geq U_t(x^*) \\geq f(x^*) on \\mathcal{E} . Also \\ell_t \\leq f(x^*) on \\mathcal{E} . Hence u_t^{(\\mathcal{T}^*)} \\geq \\ell_t and the restart condition u_t^{(\\mathcal{T}^*)} < \\ell_t never triggers.", "source": "marker_v2", "marker_block_id": "/page/13/Text/26"}
|
| 135 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0245", "section": "C.11. Proof to Theorem 6.1", "page_start": 14, "page_end": 14, "type": "Text", "text": "Proof of Theorem 6.1(2): Local certificate. Conditioned on T^* evaluations within \\mathcal{T}^* , the CGP analysis applies verbatim on the restricted domain \\mathcal{T}^* : the good event \\mathcal{E} implies all within-region confidence bounds hold; Lipschitz continuity holds on \\mathcal{T}^* ; and Assumption 2.3 holds restricted to \\mathcal{T}^* . Therefore the containment and shrinkage results follow with \\mathcal{X} replaced by \\mathcal{T}^* , yielding \\operatorname{Vol}(A_T^{(\\mathcal{T}^*)}) \\leq C\\varepsilon^{d-\\alpha} after T^* = \\tilde{O}(\\varepsilon^{-(2+\\alpha)}) within-region samples.", "source": "marker_v2", "marker_block_id": "/page/13/Text/27"}
|
| 136 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0246", "section": "C.11. Proof to Theorem 6.1", "page_start": 14, "page_end": 14, "type": "Text", "text": "Proof of Theorem 6.1(3): Allocation bound. Fix a sub-optimal region j with gap \\Delta_j > 0 . On \\mathcal{E} , using the envelope bound from Lemma 4.3, we have for all x:", "source": "marker_v2", "marker_block_id": "/page/13/Text/28"}
|
| 137 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0247", "section": "C.11. Proof to Theorem 6.1", "page_start": 14, "page_end": 14, "type": "Equation", "text": "U_t(x) < f(x) + 2\\rho_t(x),", "source": "marker_v2", "marker_block_id": "/page/13/Equation/29"}
|
| 138 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0248", "section": "C.11. Proof to Theorem 6.1", "page_start": 14, "page_end": 14, "type": "Text", "text": "where \\rho_t(x) = \\min_i \\{r_i(t) + Ld(x, x_i)\\} . Restricting to points sampled in \\mathcal{T}_j and using the within-region replication schedule, we bound \\rho_t(x) \\leq \\beta_j(n) + L \\cdot \\operatorname{diam}(\\mathcal{T}_j) , yielding", "source": "marker_v2", "marker_block_id": "/page/13/Text/30"}
|
| 139 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0249", "section": "C.11. Proof to Theorem 6.1", "page_start": 14, "page_end": 14, "type": "Equation", "text": "u_t^{(j)} = \\max_{x \\in \\mathcal{T}_j} U_t(x) \\le \\sup_{x \\in \\mathcal{T}_j} f(x) + 2\\beta_j(n) + 2L \\cdot \\operatorname{diam}(\\mathcal{T}_j),", "source": "marker_v2", "marker_block_id": "/page/13/Equation/31"}
|
| 140 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0250", "section": "C.11. Proof to Theorem 6.1", "page_start": 14, "page_end": 14, "type": "Text", "text": "after n within-region samples. By the diameter condition L \\operatorname{diam}(\\mathcal{T}_j) \\leq \\Delta_j/8 and by requiring \\beta_j(n) \\leq \\Delta_j/8 , we obtain", "source": "marker_v2", "marker_block_id": "/page/13/Text/32"}
|
| 141 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0251", "section": "C.11. Proof to Theorem 6.1", "page_start": 14, "page_end": 14, "type": "Equation", "text": "u_t^{(j)} \\le \\sup_{\\mathcal{T}_j} f + \\Delta_j/4 + \\Delta_j/4 = f^* - \\Delta_j/2.", "source": "marker_v2", "marker_block_id": "/page/13/Equation/33"}
|
| 142 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0252", "section": "C.11. Proof to Theorem 6.1", "page_start": 14, "page_end": 14, "type": "Text", "text": "Meanwhile, on \\mathcal{E} we have \\ell_t \\leq f^* always, and once the region containing x^* has been sampled sufficiently (which occurs because it is never restarted and is favored by UCB", "source": "marker_v2", "marker_block_id": "/page/13/Text/34"}
|
| 143 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0253", "section": "C.11. Proof to Theorem 6.1", "page_start": 15, "page_end": 15, "type": "Text", "text": "selection), \\ell_t becomes at least f^* - \\Delta_j/4 . Hence eventually u_t^{(j)} < \\ell_t , triggering certified restart/elimination.", "source": "marker_v2", "marker_block_id": "/page/14/Text/1"}
|
| 144 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0254", "section": "C.11. Proof to Theorem 6.1", "page_start": 15, "page_end": 15, "type": "Text", "text": "Solving \\beta_j(n) \\leq \\Delta_j/8 under \\beta_j(n) \\leq c_\\sigma \\sqrt{\\log(c_T/\\delta)/n} gives", "source": "marker_v2", "marker_block_id": "/page/14/Text/2"}
|
| 145 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0255", "section": "C.11. Proof to Theorem 6.1", "page_start": 15, "page_end": 15, "type": "Equation", "text": "n \\ge \\frac{64c_{\\sigma}^2}{\\Delta_i^2} \\log \\left(\\frac{c_T}{\\delta}\\right),\\,", "source": "marker_v2", "marker_block_id": "/page/14/Equation/3"}
|
| 146 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0256", "section": "C.11. Proof to Theorem 6.1", "page_start": 15, "page_end": 15, "type": "Text", "text": "yielding the stated bound N_j \\leq \\frac{64c_\\sigma^2}{\\Delta_j^2}\\log(c_T/\\delta) + 1 .", "source": "marker_v2", "marker_block_id": "/page/14/Text/4"}
|
| 147 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0257", "section": "C.12. Proof to Proposition 7.1", "page_start": 15, "page_end": 15, "type": "Text", "text": "Proof. We analyze each case and the certificate validity separately.", "source": "marker_v2", "marker_block_id": "/page/14/Text/6"}
|
| 148 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0258", "section": "C.12. Proof to Proposition 7.1", "page_start": 15, "page_end": 15, "type": "Text", "text": "Proof of (1): High smoothness ratio case. When \\rho \\geq 0.5 , CGP-Hybrid continues with CGP in Phase 2. The algorithm is identical to vanilla CGP, so by Theorem 4.8:", "source": "marker_v2", "marker_block_id": "/page/14/Text/7"}
|
| 149 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0259", "section": "C.12. Proof to Proposition 7.1", "page_start": 15, "page_end": 15, "type": "Equation", "text": "T = \\tilde{O}(\\varepsilon^{-(2+\\alpha)}). \\tag{39}", "source": "marker_v2", "marker_block_id": "/page/14/Equation/8"}
|
| 150 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0260", "section": "C.12. Proof to Proposition 7.1", "page_start": 15, "page_end": 15, "type": "Text", "text": "Proof of (2): Low smoothness ratio case. When \\rho < 0.5 , the function is significantly smoother near the optimum. After Phase 1, CGP has reduced the active set to volume V < 0.1 \\cdot \\mathrm{Vol}(\\mathcal{X}) . In Phase 2, GP-UCB operates within A_t , which has:", "source": "marker_v2", "marker_block_id": "/page/14/Text/9"}
|
| 151 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0261", "section": "C.12. Proof to Proposition 7.1", "page_start": 15, "page_end": 15, "type": "ListGroup", "text": "Effective diameter diam (A_t) = O(V^{1/d}) . Local Lipschitz constant L_{local} = \\rho \\cdot L < 0.5L .", "source": "marker_v2", "marker_block_id": "/page/14/ListGroup/480"}
|
| 152 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0262", "section": "C.12. Proof to Proposition 7.1", "page_start": 15, "page_end": 15, "type": "Text", "text": "The sample complexity of GP-UCB on this restricted domain depends on the kernel's maximum information gain \\gamma_T over A_t (Srinivas et al., 2009). For commonly used kernels (Matérn, SE), \\gamma_T scales polylogarithmically with T when the domain is bounded. The reduced diameter O(V^{1/d}) and local smoothness \\rho < 0.5 empirically yield faster convergence than continuing CGP; we validate this empirically in Section 9 rather than claiming a specific rate.", "source": "marker_v2", "marker_block_id": "/page/14/Text/12"}
|
| 153 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0263", "section": "C.12. Proof to Proposition 7.1", "page_start": 15, "page_end": 15, "type": "Text", "text": "Proof of (3): Certificate validity. The certificate A_T is computed in Phase 1 using CGP's Lipschitz envelope construction. By Theorem 4.5, on the good event \\mathcal{E} :", "source": "marker_v2", "marker_block_id": "/page/14/Text/13"}
|
| 154 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0264", "section": "C.12. Proof to Proposition 7.1", "page_start": 15, "page_end": 15, "type": "Equation", "text": "x^* \\in A_T with probability \\geq 1 - \\delta . (40)", "source": "marker_v2", "marker_block_id": "/page/14/Equation/14"}
|
| 155 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0265", "section": "C.12. Proof to Proposition 7.1", "page_start": 15, "page_end": 15, "type": "Text", "text": "This guarantee depends only on the Lipschitz assumption and confidence bounds, not on the Phase 2 optimization method. Therefore, switching to GP-UCB in Phase 2 does not invalidate the certificate: A_T still contains x^* with high probability, and any point outside A_T remains certifiably suboptimal.", "source": "marker_v2", "marker_block_id": "/page/14/Text/15"}
|
| 156 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0266", "section": "D.1. Adaptive Lipschitz Estimation", "page_start": 15, "page_end": 15, "type": "Text", "text": "CGP-Adaptive removes the requirement for known L with O(\\log T) overhead. The doubling scheme is conservative", "source": "marker_v2", "marker_block_id": "/page/14/Text/18"}
|
| 157 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0267", "section": "D.1. Adaptive Lipschitz Estimation", "page_start": 15, "page_end": 15, "type": "Text", "text": "but provably correct; more aggressive schemes (e.g., multiplicative updates with factor 1.5) may reduce overhead but risk certificate invalidation.", "source": "marker_v2", "marker_block_id": "/page/14/Text/19"}
|
| 158 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0268", "section": "D.1. Adaptive Lipschitz Estimation", "page_start": 15, "page_end": 15, "type": "Text", "text": "We recommend initializing \\hat{L}_0 from finite differences on initial Sobol samples:", "source": "marker_v2", "marker_block_id": "/page/14/Text/20"}
|
| 159 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0269", "section": "D.1. Adaptive Lipschitz Estimation", "page_start": 15, "page_end": 15, "type": "Equation", "text": "\\hat{L}_0 = \\max_{i \\neq j} \\frac{|y_i - y_j|}{d(x_i, x_j)}", "source": "marker_v2", "marker_block_id": "/page/14/Equation/21"}
|
| 160 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0270", "section": "D.1. Adaptive Lipschitz Estimation", "page_start": 15, "page_end": 15, "type": "Text", "text": "over the first 10 samples. This typically underestimates L by a factor of 2 to 10, requiring 1 to 4 doublings to reach \\hat{L} \\geq L^* .", "source": "marker_v2", "marker_block_id": "/page/14/Text/22"}
|
| 161 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0271", "section": "D.2. Trust Region Configuration", "page_start": 15, "page_end": 15, "type": "Text", "text": "CGP-TR trades global certificates for scalability. The local certificates within trust regions still enable principled stopping and progress assessment, but do not guarantee global optimality.", "source": "marker_v2", "marker_block_id": "/page/14/Text/24"}
|
| 162 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0272", "section": "D.2. Trust Region Configuration", "page_start": 15, "page_end": 15, "type": "Text", "text": "Recommended settings:", "source": "marker_v2", "marker_block_id": "/page/14/Text/25"}
|
| 163 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0273", "section": "D.2. Trust Region Configuration", "page_start": 15, "page_end": 15, "type": "ListGroup", "text": "Number of trust regions: n_{\\text{trust}} = 5 (balances exploration vs. overhead) Initial radius: r_0 = 0.2 (covers 20% of domain diameter per region) Minimum radius: r_{\\min} = 0.01 (prevents over-contraction) Failure threshold: \\tau_{\\text{fail}} = 10 (triggers contraction after 10 non-improving samples)", "source": "marker_v2", "marker_block_id": "/page/14/ListGroup/481"}
|
| 164 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0274", "section": "D.2. Trust Region Configuration", "page_start": 15, "page_end": 15, "type": "Text", "text": "For applications requiring global certificates in high dimensions, combining CGP-TR with random embeddings (Wang et al., 2016) is promising: project to a low-dimensional subspace, run CGP with global certificates, then lift back.", "source": "marker_v2", "marker_block_id": "/page/14/Text/30"}
|
| 165 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0275", "section": "D.3. Smoothness Detection", "page_start": 15, "page_end": 15, "type": "Text", "text": "CGP-Hybrid's smoothness detection via \\rho = L_{\\rm local}/L_{\\rm global} is heuristic but effective. We estimate L_{\\rm local} from points within A_t using the same finite difference approach as L_{\\rm global} .", "source": "marker_v2", "marker_block_id": "/page/14/Text/32"}
|
| 166 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0276", "section": "D.3. Smoothness Detection", "page_start": 15, "page_end": 15, "type": "Text", "text": "The threshold \\rho_{\\rm thresh}=0.5 was selected via cross-validation on held-out benchmarks. More sophisticated detection could use local GP posterior variance or curvature estimates. The key insight is that CGP's certificate remains valid regardless of Phase 2 method, so switching is always safe.", "source": "marker_v2", "marker_block_id": "/page/14/Text/33"}
|
| 167 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0277", "section": "D.4. When to Use CGP", "page_start": 15, "page_end": 15, "type": "Text", "text": "CGP is well suited when:", "source": "marker_v2", "marker_block_id": "/page/14/Text/35"}
|
| 168 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0278", "section": "D.4. When to Use CGP", "page_start": 15, "page_end": 15, "type": "Text", "text": "Evaluations are expensive and interpretable progress is valued", "source": "marker_v2", "marker_block_id": "/page/14/Text/36"}
|
| 169 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0279", "section": "D.4. When to Use CGP", "page_start": 16, "page_end": 16, "type": "Text", "text": "851852", "source": "marker_v2", "marker_block_id": "/page/15/Text/36"}
|
| 170 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0280", "section": "D.4. When to Use CGP", "page_start": 16, "page_end": 16, "type": "Text", "text": "854855", "source": "marker_v2", "marker_block_id": "/page/15/Text/38"}
|
| 171 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0281", "section": "D.4. When to Use CGP", "page_start": 16, "page_end": 16, "type": "Text", "text": "858859", "source": "marker_v2", "marker_block_id": "/page/15/Text/41"}
|
| 172 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0282", "section": "D.4. When to Use CGP", "page_start": 16, "page_end": 16, "type": "Text", "text": "860861", "source": "marker_v2", "marker_block_id": "/page/15/Text/42"}
|
| 173 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0283", "section": "D.4. When to Use CGP", "page_start": 16, "page_end": 16, "type": "Text", "text": "862863", "source": "marker_v2", "marker_block_id": "/page/15/Text/43"}
|
| 174 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0284", "section": "D.4. When to Use CGP", "page_start": 16, "page_end": 16, "type": "Text", "text": "865866", "source": "marker_v2", "marker_block_id": "/page/15/Text/45"}
|
| 175 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0285", "section": "D.4. When to Use CGP", "page_start": 16, "page_end": 16, "type": "Text", "text": "867868", "source": "marker_v2", "marker_block_id": "/page/15/Text/46"}
|
| 176 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0286", "section": "D.4. When to Use CGP", "page_start": 16, "page_end": 16, "type": "Text", "text": "869870", "source": "marker_v2", "marker_block_id": "/page/15/Text/47"}
|
| 177 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0287", "section": "D.4. When to Use CGP", "page_start": 16, "page_end": 16, "type": "Text", "text": "872873", "source": "marker_v2", "marker_block_id": "/page/15/Text/49"}
|
| 178 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0288", "section": "D.4. When to Use CGP", "page_start": 16, "page_end": 16, "type": "Text", "text": "875876", "source": "marker_v2", "marker_block_id": "/page/15/Text/51"}
|
| 179 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0289", "section": "D.4. When to Use CGP", "page_start": 16, "page_end": 16, "type": "Text", "text": "877878", "source": "marker_v2", "marker_block_id": "/page/15/Text/52"}
|
| 180 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0290", "section": "D.4. When to Use CGP", "page_start": 16, "page_end": 16, "type": "ListGroup", "text": "2. The objective has margin structure (sharp peak rather than wide plateau) 3. Lipschitz continuity is a reasonable assumption 4. Dimension is moderate ( d \\le 15 for vanilla CGP, d \\le 100 for CGP-TR) 5. Anytime stopping decisions are needed", "source": "marker_v2", "marker_block_id": "/page/15/ListGroup/326"}
|
| 181 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0291", "section": "D.4. When to Use CGP", "page_start": 16, "page_end": 16, "type": "Text", "text": "For very high-dimensional problems (d>100), trust region methods like TuRBO may scale better. For smooth problems with cheap evaluations, GP-based methods may be more sample efficient due to their ability to exploit higher-order smoothness.", "source": "marker_v2", "marker_block_id": "/page/15/Text/6"}
|
| 182 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0292", "section": "E.1. Baseline Configurations", "page_start": 16, "page_end": 16, "type": "ListGroup", "text": "Random Search: Sobol sequences for quasi-random sampling GP-UCB : Matérn-5/2 kernel via BoTorch, \\beta_t = 2\\log(t^2\\pi^2/6\\delta) TuRBO : Default settings from Eriksson et al. (2019), 1 trust region HEBO: Heteroscedastic GP with input warping, default settings BORE : Tree-Parzen estimator with density ratio, default settings HOO : Binary tree with \\nu_1 = 1 , \\rho = 0.5 StoSOO: k = 3 children per node, h_{\\text{max}} = 20 LIPO: Pure Lipschitz optimization, L estimated online SAASBO: Sparse axis-aligned GP, 10 active dimensions", "source": "marker_v2", "marker_block_id": "/page/15/ListGroup/327"}
|
| 183 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0293", "section": "Low-dimensional.", "page_start": 16, "page_end": 16, "type": "ListGroup", "text": "Needle-2D: f(x) = 1 \\|x x^*\\|^{1/\\alpha} with \\alpha = 2 , sharp peak Branin: Standard 2D benchmark with 3 global optima Hartmann-6: 6D benchmark with narrow global basin Levy-5: 5D benchmark with global structure Rosenbrock-4: 4D benchmark with curved valley", "source": "marker_v2", "marker_block_id": "/page/15/ListGroup/328"}
|
| 184 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0294", "section": "Medium-dimensional.", "page_start": 16, "page_end": 16, "type": "ListGroup", "text": "Ackley-10: 10D benchmark with many local optima SVM-RBF-6 : Real hyperparameter tuning (C, \\gamma, 4) preprocessing) on MNIST LunarLander-12: RL reward optimization with 12 policy parameters", "source": "marker_v2", "marker_block_id": "/page/15/ListGroup/329"}
|
| 185 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0295", "section": "High-dimensional.", "page_start": 16, "page_end": 16, "type": "ListGroup", "text": "Rover-60 : Mars rover trajectory with 60 waypoint parameters (Wang et al., 2016) NAS-36: Neural architecture search on CIFAR-10, 36 continuous encodings Ant-100: MuJoCo Ant locomotion, 100 morphology and control parameters", "source": "marker_v2", "marker_block_id": "/page/15/ListGroup/330"}
|
| 186 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0296", "section": "E.3. Computational Resources", "page_start": 16, "page_end": 16, "type": "Text", "text": "All experiments run on AMD EPYC 7763 with 256GB RAM. CGP variants use NumPy/SciPy; GP baselines use BoTorch/GPyTorch with GPU acceleration (NVIDIA A100) where available.", "source": "marker_v2", "marker_block_id": "/page/15/Text/34"}
|
icml26/5c3d6bff-e8ce-4d9f-840b-719084582491/appendix_text_v3.txt
ADDED
|
@@ -0,0 +1,557 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
[p. 10 | section: A. Extended Problem Formulation | type: Text]
|
| 2 |
+
We restate the assumptions from Section 2 for completeness. Let (\mathcal{X}, d) be a compact metric space with diameter D = \sup_{x,y \in \mathcal{X}} d(x,y) . We consider \mathcal{X} = [0,1]^d with Euclidean metric, though our results extend to general metric spaces. Let f: \mathcal{X} \to [0,1] be an unknown function satisfying:
|
| 3 |
+
|
| 4 |
+
[p. 10 | section: A. Extended Problem Formulation | type: Text]
|
| 5 |
+
Assumption 2.1 (Lipschitz continuity). There exists L > 0 such that for all x, y \in \mathcal{X} : |f(x) - f(y)| \le L \cdot d(x, y) .
|
| 6 |
+
|
| 7 |
+
[p. 10 | section: A. Extended Problem Formulation | type: Text]
|
| 8 |
+
We observe f through noisy queries: querying x returns y = f(x) + \epsilon , where:
|
| 9 |
+
|
| 10 |
+
[p. 10 | section: A. Extended Problem Formulation | type: Text]
|
| 11 |
+
Assumption 2.2 (Sub-Gaussian noise). The noise \epsilon is \sigma -sub-Gaussian: \mathbb{E}[e^{\lambda\epsilon}] \leq e^{\lambda^2\sigma^2/2} for all \lambda \in \mathbb{R} .
|
| 12 |
+
|
| 13 |
+
[p. 10 | section: A. Extended Problem Formulation | type: Text]
|
| 14 |
+
After T samples, the algorithm outputs \hat{x}_T \in \mathcal{X} . The goal is to minimize simple regret r_T = f(x^*) - f(\hat{x}_T) , where x^* \in \arg\max_{x \in \mathcal{X}} f(x) . We seek PAC-style guarantees: with probability at least 1 - \delta , achieve r_T \leq \varepsilon .
|
| 15 |
+
|
| 16 |
+
[p. 10 | section: A. Extended Problem Formulation | type: Text]
|
| 17 |
+
To obtain instance-dependent rates that improve upon worst case bounds, we assume margin structure:
|
| 18 |
+
|
| 19 |
+
[p. 10 | section: A. Extended Problem Formulation | type: Text]
|
| 20 |
+
Assumption 2.3 (Margin / near-optimality dimension). There exist C>0 and \alpha\in[0,d] such that for all \varepsilon>0 : \operatorname{Vol}(\{x\in\mathcal{X}:f(x)\geq f^*-\varepsilon\})\leq C\,\varepsilon^{d-\alpha} .
|
| 21 |
+
|
| 22 |
+
[p. 10 | section: A. Extended Problem Formulation | type: Text]
|
| 23 |
+
The parameter \alpha is the near-optimality dimension: smaller \alpha corresponds to a sharper optimum (easier), while larger \alpha corresponds to a broader near-optimal region (harder). The worst case is \alpha=d , which recovers the standard d-dimensional Lipschitz difficulty. This assumption is standard in bandit theory (Audibert & Bubeck, 2010; Bubeck et al., 2011; Lattimore & Szepesvári, 2020; Kaufmann et al., 2016; Garivier & Kaufmann, 2016).
|
| 24 |
+
|
| 25 |
+
[p. 10 | section: B.1. Score Maximization and Replication | type: Text]
|
| 26 |
+
CGP optimizes \operatorname{score}(x) = U_t(x) - \lambda \cdot \min_i d(x, x_i) , which is piecewise linear. Since this is non-smooth, we use CMA-ES (Covariance Matrix Adaptation Evolution Strategy) with bounded domain and 10 random restarts within A_t . For d \leq 3 , we additionally use Delaunay triangulation to identify candidate optima at Voronoi vertices.
|
| 27 |
+
|
| 28 |
+
[p. 10 | section: B.1. Score Maximization and Replication | type: Text]
|
| 29 |
+
Membership in A_t is exact: given \{x_i, \hat{\mu}_i, r_i\} , checking U_t(x) \geq \ell_t requires O(N_t) time. Approximate maximization may slow convergence of \eta_t but does not invalidate certificates: any x \notin A_t remains certifiably suboptimal regardless of which x \in A_t is queried.
|
| 30 |
+
|
| 31 |
+
[p. 10 | section: B.1. Score Maximization and Replication | type: Text]
|
| 32 |
+
Replication Strategy. When an active point x_i has r_i(t) > \beta_{\text{target}}(t) , we allocate \lceil (r_i(t)/\beta_{\text{target}}(t))^2 \rceil additional samples to reduce its confidence radius. This ensures all active points have comparable confidence, preventing
|
| 33 |
+
|
| 34 |
+
[p. 11 | section: B.1. Score Maximization and Replication | type: Text]
|
| 35 |
+
any single point from dominating the envelope.
|
| 36 |
+
|
| 37 |
+
[p. 11 | section: B.2. Active Set Computation | type: Text]
|
| 38 |
+
For low dimensions (d \le 5) , we compute A_t exactly using grid discretization with resolution \eta = D/\sqrt[d]{N_{\rm grid}} where N_{\rm grid} = 10^4 . For each grid point x, we evaluate U_t(x) in O(N_t) time and check if U_t(x) \ge \ell_t . The volume \operatorname{Vol}(A_t) is estimated as the fraction of grid points in A_t .
|
| 39 |
+
|
| 40 |
+
[p. 11 | section: B.2. Active Set Computation | type: Text]
|
| 41 |
+
For higher dimensions (d>5), uniform Monte Carlo becomes ineffective once \operatorname{Vol}(A_t) is small. We therefore estimate \operatorname{Vol}(A_t) via a nested-set ratio estimator (subset simulation): define thresholds \ell_t - \tau_0 < \ell_t - \tau_1 < \cdots < \ell_t inducing nested sets
|
| 42 |
+
|
| 43 |
+
[p. 11 | section: B.2. Active Set Computation | type: Equation]
|
| 44 |
+
A_t^{(k)} = \{x : U_t(x) \ge \ell_t - \tau_k\}, \quad A_t^{(K)} = A_t.
|
| 45 |
+
|
| 46 |
+
[p. 11 | section: B.2. Active Set Computation | type: Text]
|
| 47 |
+
We estimate
|
| 48 |
+
|
| 49 |
+
[p. 11 | section: B.2. Active Set Computation | type: Equation]
|
| 50 |
+
Vol(A_t) = Vol(A_t^{(0)}) \prod_{k=1}^{K} \mathbb{P}_{x \sim Unif(A_t^{(k-1)})} [x \in A_t^{(k)}],
|
| 51 |
+
|
| 52 |
+
[p. 11 | section: B.2. Active Set Computation | type: Text]
|
| 53 |
+
sampling approximately uniformly from A_t^{(k-1)} using a hit-and-run Markov chain with the membership oracle U_t(x) \geq \ell_t - \tau_{k-1} . This yields stable estimates even when \operatorname{Vol}(A_t) is very small. In all high-dimensional experiments, we report confidence intervals of \log \operatorname{Vol}(A_t) from repeated estimator runs.
|
| 54 |
+
|
| 55 |
+
[p. 11 | section: B.2. Active Set Computation | type: Text]
|
| 56 |
+
Crucially, certificate validity (Remark 4.7) is independent of volume estimation accuracy: the set membership rule x \in A_t \Leftrightarrow U_t(x) \ge \ell_t is exact.
|
| 57 |
+
|
| 58 |
+
[p. 11 | section: B.2. Active Set Computation | type: Text]
|
| 59 |
+
On volume-based stopping. The shrinkage bound (Theorem 4.6) provides an upper bound on \operatorname{Vol}(A_t) as a function of the algorithmic gap proxy
|
| 60 |
+
|
| 61 |
+
[p. 11 | section: B.2. Active Set Computation | type: Equation]
|
| 62 |
+
\varepsilon_t := 2(\beta_t + L\eta_t) + \gamma_t, \quad \gamma_t = f^* - \ell_t,
|
| 63 |
+
|
| 64 |
+
[p. 11 | section: B.2. Active Set Computation | type: Text]
|
| 65 |
+
and therefore supports using \operatorname{Vol}(A_t) as a practical progress diagnostic. However, \operatorname{Vol}(A_t) alone does not yield an anytime upper bound on regret without additional lower-regularity assumptions linking volume back to function values. In our experiments we therefore use \varepsilon_t as the primary certificate-based stopping criterion, and treat \operatorname{Vol}(A_t) as a secondary monitoring signal.
|
| 66 |
+
|
| 67 |
+
[p. 11 | section: C.1. Proof to Lemma 4.1 | type: Text]
|
| 68 |
+
Proof. Define the good event \mathcal{E} = \bigcap_{t=1}^T \bigcap_{i=1}^{N_t} \{|\hat{\mu}_i(t) - f(x_i)| \leq r_i(t)\}.
|
| 69 |
+
|
| 70 |
+
[p. 11 | section: C.1. Proof to Lemma 4.1 | type: Text]
|
| 71 |
+
Step 1: Single-point concentration. By Hoeffding's inequality for \sigma -sub-Gaussian random variables:
|
| 72 |
+
|
| 73 |
+
[p. 11 | section: C.1. Proof to Lemma 4.1 | type: Equation]
|
| 74 |
+
\mathbb{P}\big[|\hat{\mu}_i(t) - f(x_i)| > r\big] \le 2\exp\left(-\frac{n_i r^2}{2\sigma^2}\right). \tag{4}
|
| 75 |
+
|
| 76 |
+
[p. 11 | section: C.1. Proof to Lemma 4.1 | type: Text]
|
| 77 |
+
Step 2: Calibration of confidence radius. Substituting r = r_i(t) = \sigma \sqrt{2 \log(2N_t T/\delta)/n_i} :
|
| 78 |
+
|
| 79 |
+
[p. 11 | section: C.1. Proof to Lemma 4.1 | type: Equation]
|
| 80 |
+
\mathbb{P}[|\hat{\mu}_{i}(t) - f(x_{i})| > r_{i}(t)] \leq 2 \exp\left(-\frac{n_{i} \cdot 2\sigma^{2} \log(2N_{t}T/\delta)}{2\sigma^{2} \cdot n_{i}}\right) = 2 \exp\left(-\log(2N_{t}T/\delta)\right) = \frac{\delta}{N_{t}T}. (5)
|
| 81 |
+
|
| 82 |
+
[p. 11 | section: C.1. Proof to Lemma 4.1 | type: Text]
|
| 83 |
+
Step 3: Union bound. Applying the union bound over all i \in \{1, ..., N_t\} and t \in \{1, ..., T\} :
|
| 84 |
+
|
| 85 |
+
[p. 11 | section: C.1. Proof to Lemma 4.1 | type: Equation]
|
| 86 |
+
\mathbb{P}[\mathcal{E}^c] \le \sum_{t=1}^T \sum_{i=1}^{N_t} \frac{\delta}{N_t T} \le \sum_{t=1}^T \frac{\delta}{T} = \delta. \tag{6}
|
| 87 |
+
|
| 88 |
+
[p. 11 | section: C.1. Proof to Lemma 4.1 | type: Equation]
|
| 89 |
+
Hence \mathbb{P}[\mathcal{E}] \geq 1 - \delta .
|
| 90 |
+
|
| 91 |
+
[p. 11 | section: C.2. Proof to Lemma 4.2 (UCB envelope is valid) | type: Text]
|
| 92 |
+
Proof. Fix any x \in \mathcal{X} . For any sampled point x_i , on the good event \mathcal{E} :
|
| 93 |
+
|
| 94 |
+
[p. 11 | section: C.2. Proof to Lemma 4.2 (UCB envelope is valid) | type: Equation]
|
| 95 |
+
f(x_i) \le \hat{\mu}_i(t) + r_i(t) = UCB_i(t). \tag{7}
|
| 96 |
+
|
| 97 |
+
[p. 11 | section: C.2. Proof to Lemma 4.2 (UCB envelope is valid) | type: Text]
|
| 98 |
+
By Lipschitz continuity of f:
|
| 99 |
+
|
| 100 |
+
[p. 11 | section: C.2. Proof to Lemma 4.2 (UCB envelope is valid) | type: Equation]
|
| 101 |
+
f(x) \le f(x_i) + L \cdot d(x, x_i) \le UCB_i(t) + L \cdot d(x, x_i) . (8)
|
| 102 |
+
|
| 103 |
+
[p. 11 | section: C.2. Proof to Lemma 4.2 (UCB envelope is valid) | type: Text]
|
| 104 |
+
Since this holds for all sampled i , taking the minimum over i :
|
| 105 |
+
|
| 106 |
+
[p. 11 | section: C.2. Proof to Lemma 4.2 (UCB envelope is valid) | type: Equation]
|
| 107 |
+
f(x) \le \min_{i \le N_t} \{ \text{UCB}_i(t) + L \cdot d(x, x_i) \} = U_t(x). (9)
|
| 108 |
+
|
| 109 |
+
[p. 11 | section: C.3. Proof to Lemma 4.3 (Envelope slack bound) | type: Text]
|
| 110 |
+
Proof. Fix any x \in \mathcal{X} and any sampled point x_i .
|
| 111 |
+
|
| 112 |
+
[p. 11 | section: C.3. Proof to Lemma 4.3 (Envelope slack bound) | type: Text]
|
| 113 |
+
Step 1: Upper confidence bound. On the good event \mathcal{E} , we have \hat{\mu}_i(t) \leq f(x_i) + r_i(t) . Therefore:
|
| 114 |
+
|
| 115 |
+
[p. 11 | section: C.3. Proof to Lemma 4.3 (Envelope slack bound) | type: Equation]
|
| 116 |
+
UCB_{i}(t) = \hat{\mu}_{i}(t) + r_{i}(t) \le f(x_{i}) + 2r_{i}(t). (10)
|
| 117 |
+
|
| 118 |
+
[p. 11 | section: C.3. Proof to Lemma 4.3 (Envelope slack bound) | type: Text]
|
| 119 |
+
Step 2: Lipschitz propagation. By Assumption 2.1 (Lipschitz continuity):
|
| 120 |
+
|
| 121 |
+
[p. 11 | section: C.3. Proof to Lemma 4.3 (Envelope slack bound) | type: Equation]
|
| 122 |
+
f(x_i) \le f(x) + L \cdot d(x, x_i). \tag{11}
|
| 123 |
+
|
| 124 |
+
[p. 11 | section: C.3. Proof to Lemma 4.3 (Envelope slack bound) | type: Text]
|
| 125 |
+
Step 3: Combining bounds. Substituting the Lipschitz bound into Step 1:
|
| 126 |
+
|
| 127 |
+
[p. 11 | section: C.3. Proof to Lemma 4.3 (Envelope slack bound) | type: Equation]
|
| 128 |
+
UCB_{i}(t) + L d(x, x_{i}) \leq f(x_{i}) + 2r_{i}(t) + L d(x, x_{i}) \leq f(x) + L d(x, x_{i}) + 2r_{i}(t) + L d(x, x_{i}) = f(x) + 2(r_{i}(t) + L d(x, x_{i})). \quad (12)
|
| 129 |
+
|
| 130 |
+
[p. 12 | section: C.3. Proof to Lemma 4.3 (Envelope slack bound) | type: Text]
|
| 131 |
+
Step 4: Taking the minimum. Since the above holds for all i, taking the minimum over i on both sides:
|
| 132 |
+
|
| 133 |
+
[p. 12 | section: C.3. Proof to Lemma 4.3 (Envelope slack bound) | type: Equation]
|
| 134 |
+
U_{t}(x) = \min_{i \leq N_{t}} \left\{ \text{UCB}_{i}(t) + L d(x, x_{i}) \right\} \leq f(x) + 2 \min_{i} \left\{ r_{i}(t) + L d(x, x_{i}) \right\} = f(x) + 2\rho_{t}(x). (13)
|
| 135 |
+
|
| 136 |
+
[p. 12 | section: C.4. Proof to Theorem 4.5 | type: Text]
|
| 137 |
+
Proof. We show that any point in the active set must have function value close to optimal.
|
| 138 |
+
|
| 139 |
+
[p. 12 | section: C.4. Proof to Theorem 4.5 | type: Text]
|
| 140 |
+
Step 1: Lower certificate validity. On E, for any sampled point x i :
|
| 141 |
+
|
| 142 |
+
[p. 12 | section: C.4. Proof to Theorem 4.5 | type: Equation]
|
| 143 |
+
LCB_i(t) = \hat{\mu}_i(t) - r_i(t) \le f(x_i) \le f^*. (14)
|
| 144 |
+
|
| 145 |
+
[p. 12 | section: C.4. Proof to Theorem 4.5 | type: Text]
|
| 146 |
+
Taking the maximum over all i:
|
| 147 |
+
|
| 148 |
+
[p. 12 | section: C.4. Proof to Theorem 4.5 | type: Equation]
|
| 149 |
+
\ell_t = \max_{i \le N_t} LCB_i(t) \le f^*. \tag{15}
|
| 150 |
+
|
| 151 |
+
[p. 12 | section: C.4. Proof to Theorem 4.5 | type: Text]
|
| 152 |
+
Step 2: Active set membership implies high UCB. Let x ∈ At. By definition of the active set (2) :
|
| 153 |
+
|
| 154 |
+
[p. 12 | section: C.4. Proof to Theorem 4.5 | type: Equation]
|
| 155 |
+
U_t(x) \ge \ell_t. \tag{16}
|
| 156 |
+
|
| 157 |
+
[p. 12 | section: C.4. Proof to Theorem 4.5 | type: Text]
|
| 158 |
+
Step 3: Applying the envelope bound. By Lemma 4.3:
|
| 159 |
+
|
| 160 |
+
[p. 12 | section: C.4. Proof to Theorem 4.5 | type: Equation]
|
| 161 |
+
f(x) + 2\rho_t(x) \ge U_t(x) \ge \ell_t. \tag{17}
|
| 162 |
+
|
| 163 |
+
[p. 12 | section: C.4. Proof to Theorem 4.5 | type: Text]
|
| 164 |
+
Step 4: Rearranging to obtain the containment. Solving for f(x):
|
| 165 |
+
|
| 166 |
+
[p. 12 | section: C.4. Proof to Theorem 4.5 | type: Equation]
|
| 167 |
+
f(x) \ge \ell_t - 2\rho_t(x) = f^* - (f^* - \ell_t) - 2\rho_t(x) \ge f^* - (f^* - \ell_t) - 2 \sup_{x' \in A_t} \rho_t(x') = f^* - 2\Delta_t. (18)
|
| 168 |
+
|
| 169 |
+
[p. 12 | section: C.4. Proof to Theorem 4.5 | type: Equation]
|
| 170 |
+
Hence A_t \subseteq \{x : f(x) \ge f^* - 2\Delta_t\}.
|
| 171 |
+
|
| 172 |
+
[p. 12 | section: C.5. Proof to Theorem 4.6 | type: Text]
|
| 173 |
+
Proof. We connect the active set volume to the margin condition via the containment theorem.
|
| 174 |
+
|
| 175 |
+
[p. 12 | section: C.5. Proof to Theorem 4.6 | type: Text]
|
| 176 |
+
Step 1: Bounding ∆t . From Theorem 4.5, A t ⊆ {x : f(x) ≥ f ∗ − 2∆t} where:
|
| 177 |
+
|
| 178 |
+
[p. 12 | section: C.5. Proof to Theorem 4.6 | type: Equation]
|
| 179 |
+
\Delta_t = \sup_{x \in A_t} \rho_t(x) + (f^* - \ell_t). (19)
|
| 180 |
+
|
| 181 |
+
[p. 12 | section: C.5. Proof to Theorem 4.6 | type: Text]
|
| 182 |
+
For any x ∈ At:
|
| 183 |
+
|
| 184 |
+
[p. 12 | section: C.5. Proof to Theorem 4.6 | type: Equation]
|
| 185 |
+
\rho_t(x) = \min_{i} \left\{ r_i(t) + L \cdot d(x, x_i) \right\} \leq \max_{i:x_i \text{ active}} r_i(t) + L \cdot \sup_{x \in A_t} \min_{i} d(x, x_i) = \beta_t + L\eta_t. \tag{20}
|
| 186 |
+
|
| 187 |
+
[p. 12 | section: C.5. Proof to Theorem 4.6 | type: Text]
|
| 188 |
+
Therefore:
|
| 189 |
+
|
| 190 |
+
[p. 12 | section: C.5. Proof to Theorem 4.6 | type: Equation]
|
| 191 |
+
\Delta_t \le \beta_t + L\eta_t + \frac{\gamma_t}{2}.\tag{21}
|
| 192 |
+
|
| 193 |
+
[p. 12 | section: C.5. Proof to Theorem 4.6 | type: Text]
|
| 194 |
+
Step 2: Applying the margin condition. By Assumption 2.3, for any ε > 0:
|
| 195 |
+
|
| 196 |
+
[p. 12 | section: C.5. Proof to Theorem 4.6 | type: Equation]
|
| 197 |
+
\operatorname{Vol}(\{x: f(x) \ge f^* - \varepsilon\}) \le C \cdot \varepsilon^{d-\alpha}. \tag{22}
|
| 198 |
+
|
| 199 |
+
[p. 12 | section: C.5. Proof to Theorem 4.6 | type: Text]
|
| 200 |
+
Step 3: Combining the bounds. Setting ε = 2∆ t ≤ 2(β t + Lηt) + γt:
|
| 201 |
+
|
| 202 |
+
[p. 12 | section: C.5. Proof to Theorem 4.6 | type: Equation]
|
| 203 |
+
\operatorname{Vol}(A_t) \leq \operatorname{Vol}(\{x : f(x) \geq f^* - 2\Delta_t\}) \leq C \cdot (2\Delta_t)^{d-\alpha} \leq C \cdot (2(\beta_t + L\eta_t) + \gamma_t)^{d-\alpha}. (23)
|
| 204 |
+
|
| 205 |
+
[p. 12 | section: C.6. Proof to Theorem 4.8 | type: Text]
|
| 206 |
+
Proof. We derive the sample complexity by analyzing the requirements for ε-optimality.
|
| 207 |
+
|
| 208 |
+
[p. 12 | section: C.6. Proof to Theorem 4.8 | type: Text]
|
| 209 |
+
Step 1: Optimality condition. To achieve simple regret r T ≤ ε, it suffices to ensure:
|
| 210 |
+
|
| 211 |
+
[p. 12 | section: C.6. Proof to Theorem 4.8 | type: Equation]
|
| 212 |
+
2(\beta_t + L\eta_t) + \gamma_t \le \varepsilon. \tag{24}
|
| 213 |
+
|
| 214 |
+
[p. 12 | section: C.6. Proof to Theorem 4.8 | type: Text]
|
| 215 |
+
This requires β t ≤ ε/6, η t ≤ ε/(6L), and γ t ≤ ε/3.
|
| 216 |
+
|
| 217 |
+
[p. 12 | section: C.6. Proof to Theorem 4.8 | type: Text]
|
| 218 |
+
Step 2: Covering the active set. By the shrinkage theorem (Theorem 4.6) , once 2(β t + Lηt) + γ t ≤ ε we have:
|
| 219 |
+
|
| 220 |
+
[p. 12 | section: C.6. Proof to Theorem 4.8 | type: Equation]
|
| 221 |
+
Vol(A_t) \le C \cdot \varepsilon^{d-\alpha}. (25)
|
| 222 |
+
|
| 223 |
+
[p. 12 | section: C.6. Proof to Theorem 4.8 | type: Text]
|
| 224 |
+
To achieve covering radius η = ε/(6L) over a region of volume Cε d − α , we need:
|
| 225 |
+
|
| 226 |
+
[p. 12 | section: C.6. Proof to Theorem 4.8 | type: Equation]
|
| 227 |
+
N_{\text{cover}} = O\left(\frac{\text{Vol}(A_t)}{\eta^d}\right) = O\left(\frac{C\varepsilon^{d-\alpha}}{(\varepsilon/(6L))^d}\right) = O\left(L^d\varepsilon^{-\alpha}\right) (26)
|
| 228 |
+
|
| 229 |
+
[p. 12 | section: C.6. Proof to Theorem 4.8 | type: Text]
|
| 230 |
+
distinct sample locations.
|
| 231 |
+
|
| 232 |
+
[p. 12 | section: C.6. Proof to Theorem 4.8 | type: Text]
|
| 233 |
+
Step 3: Samples per location. To achieve confidence radius β t ≤ ε/6 at each location, we need:
|
| 234 |
+
|
| 235 |
+
[p. 12 | section: C.6. Proof to Theorem 4.8 | type: Equation]
|
| 236 |
+
\sigma\sqrt{\frac{2\log(2N_tT/\delta)}{n_i}} \le \frac{\varepsilon}{6}. (27)
|
| 237 |
+
|
| 238 |
+
[p. 12 | section: C.6. Proof to Theorem 4.8 | type: Text]
|
| 239 |
+
Solving for n i :
|
| 240 |
+
|
| 241 |
+
[p. 12 | section: C.6. Proof to Theorem 4.8 | type: Equation]
|
| 242 |
+
n_i = O\left(\frac{\sigma^2 \log(T/\delta)}{\varepsilon^2}\right). (28)
|
| 243 |
+
|
| 244 |
+
[p. 13 | section: C.6. Proof to Theorem 4.8 | type: Text]
|
| 245 |
+
Step 4: Total sample complexity. Combining Steps 2 and 3:
|
| 246 |
+
|
| 247 |
+
[p. 13 | section: C.6. Proof to Theorem 4.8 | type: Equation]
|
| 248 |
+
T = N_{\text{cover}} \cdot n_{i} = O(L^{d} \varepsilon^{-\alpha}) \cdot O\left(\frac{\sigma^{2} \log(T/\delta)}{\varepsilon^{2}}\right) = \tilde{O}(L^{d} \varepsilon^{-(2+\alpha)}). \tag{29}
|
| 249 |
+
|
| 250 |
+
[p. 13 | section: C.7. Proof to Theorem 4.9 | type: Text]
|
| 251 |
+
Proof. We construct a hard instance via a randomized reduction.
|
| 252 |
+
|
| 253 |
+
[p. 13 | section: C.7. Proof to Theorem 4.9 | type: Text]
|
| 254 |
+
Step 1: Hard instance construction (continuous bump). We use a standard "one bump among M locations" construction that ensures global Lipschitz continuity. Partition \mathcal{X} = [0,1]^d into M = \varepsilon^{-\alpha} disjoint cells with centers \{c_1,\ldots,c_M\} , each cell having diameter \Theta(\varepsilon^{\alpha/d}) . Select one cell i^* uniformly at random to contain the optimum, and place x^* = c_{i^*} . Define:
|
| 255 |
+
|
| 256 |
+
[p. 13 | section: C.7. Proof to Theorem 4.9 | type: Equation]
|
| 257 |
+
f(x) = (1 - \varepsilon) + \varepsilon \cdot \max \left\{ 0, 1 - \frac{L \|x - x^*\|}{\varepsilon} \right\}. \quad (30)
|
| 258 |
+
|
| 259 |
+
[p. 13 | section: C.7. Proof to Theorem 4.9 | type: Text]
|
| 260 |
+
This is a cone/bump centered at x^* that peaks at f(x^*)=1 and decreases linearly with slope L until it reaches the baseline value 1-\varepsilon at radius \varepsilon/L from x^* . Outside this radius, f(x)\equiv 1-\varepsilon . The function is globally L-Lipschitz: within the bump, the gradient has magnitude L; outside, the function is constant; and at the boundary \|x-x^*\|=\varepsilon/L , both pieces match at value 1-\varepsilon .
|
| 261 |
+
|
| 262 |
+
[p. 13 | section: C.7. Proof to Theorem 4.9 | type: Text]
|
| 263 |
+
Verification of Assumption 2.3. The \varepsilon -near-optimal set \{x:f(x)\geq 1-\varepsilon\} is exactly the ball of radius \varepsilon/L around x^* , which has volume \operatorname{Vol}(B_{\varepsilon/L})=O((\varepsilon/L)^d)=O(\varepsilon^d) . With M=\varepsilon^{-\alpha} candidate locations, only one contains the bump, so the near-optimal fraction of the domain is O(\varepsilon^d) . Since the domain has unit volume, \operatorname{Vol}(\{f\geq f^*-\varepsilon\})=O(\varepsilon^d)\leq C\varepsilon^{d-\alpha} for \alpha\geq 0 , satisfying Assumption 2.3.
|
| 264 |
+
|
| 265 |
+
[p. 13 | section: C.7. Proof to Theorem 4.9 | type: Text]
|
| 266 |
+
Step 2: Information-theoretic lower bound. To identify the correct cell with probability \geq 2/3 , the algorithm must distinguish between M hypotheses. By Fano's inequality, this requires:
|
| 267 |
+
|
| 268 |
+
[p. 13 | section: C.7. Proof to Theorem 4.9 | type: Equation]
|
| 269 |
+
\sum_{i=1}^{M} n_i \cdot \text{KL}(P_i || P_0) \ge \log(M/3), \tag{31}
|
| 270 |
+
|
| 271 |
+
[p. 13 | section: C.7. Proof to Theorem 4.9 | type: Text]
|
| 272 |
+
where n_i is the number of samples in cell i, and \mathrm{KL}(P_i || P_0) is the KL divergence between observations under hypothesis i versus the null.
|
| 273 |
+
|
| 274 |
+
[p. 13 | section: C.7. Proof to Theorem 4.9 | type: Text]
|
| 275 |
+
Step 3: Per-cell sample requirement. For \sigma -sub-Gaussian noise, distinguishing a cell with optimum from one without requires:
|
| 276 |
+
|
| 277 |
+
[p. 13 | section: C.7. Proof to Theorem 4.9 | type: Equation]
|
| 278 |
+
n_i = \Omega\left(\frac{\sigma^2}{\varepsilon^2}\right) \tag{32}
|
| 279 |
+
|
| 280 |
+
[p. 13 | section: C.7. Proof to Theorem 4.9 | type: Text]
|
| 281 |
+
samples per cell to detect the \varepsilon gap with constant probability. Step 4: Total sample complexity. Summing over all M cells:
|
| 282 |
+
|
| 283 |
+
[p. 13 | section: C.7. Proof to Theorem 4.9 | type: Equation]
|
| 284 |
+
T = \Omega \left( M \cdot \frac{\sigma^2}{\varepsilon^2} \right) = \Omega \left( \varepsilon^{-\alpha} \cdot \varepsilon^{-2} \right) = \Omega \left( \varepsilon^{-(2+\alpha)} \right). \tag{33}
|
| 285 |
+
|
| 286 |
+
[p. 13 | section: C.8. Proof to Theorem 5.1 | type: Text]
|
| 287 |
+
Proof. We prove each claim separately.
|
| 288 |
+
|
| 289 |
+
[p. 13 | section: C.8. Proof to Theorem 5.1 | type: Text]
|
| 290 |
+
Proof of (1): Bounded doubling events. Each doubling event multiplies \hat{L} by 2. Starting from \hat{L}_0 \leq L^* :
|
| 291 |
+
|
| 292 |
+
[p. 13 | section: C.8. Proof to Theorem 5.1 | type: Equation]
|
| 293 |
+
\hat{L}_k = 2^k \hat{L}_0 after k doublings. (34)
|
| 294 |
+
|
| 295 |
+
[p. 13 | section: C.8. Proof to Theorem 5.1 | type: Text]
|
| 296 |
+
The algorithm stops doubling when \hat{L} \geq L^* , which requires:
|
| 297 |
+
|
| 298 |
+
[p. 13 | section: C.8. Proof to Theorem 5.1 | type: Equation]
|
| 299 |
+
2^K \hat{L}_0 \ge L^* \implies K \ge \log_2(L^*/\hat{L}_0). (35)
|
| 300 |
+
|
| 301 |
+
[p. 13 | section: C.8. Proof to Theorem 5.1 | type: Text]
|
| 302 |
+
Hence K \leq \lceil \log_2(L^*/\hat{L}_0) \rceil .
|
| 303 |
+
|
| 304 |
+
[p. 13 | section: C.8. Proof to Theorem 5.1 | type: Text]
|
| 305 |
+
Proof of (2): Final estimate accuracy. A violation is detected when:
|
| 306 |
+
|
| 307 |
+
[p. 13 | section: C.8. Proof to Theorem 5.1 | type: Equation]
|
| 308 |
+
|\hat{\mu}_i - \hat{\mu}_j| - 2(r_i + r_j) > \hat{L} \cdot d(x_i, x_j). (36)
|
| 309 |
+
|
| 310 |
+
[p. 13 | section: C.8. Proof to Theorem 5.1 | type: Text]
|
| 311 |
+
On the good event \mathcal{E} :
|
| 312 |
+
|
| 313 |
+
[p. 13 | section: C.8. Proof to Theorem 5.1 | type: Equation]
|
| 314 |
+
|f(x_i) - f(x_j)| \le |\hat{\mu}_i - \hat{\mu}_j| + 2(r_i + r_j). (37)
|
| 315 |
+
|
| 316 |
+
[p. 13 | section: C.8. Proof to Theorem 5.1 | type: Text]
|
| 317 |
+
If \hat{L} \geq L^* , then by Lipschitz continuity:
|
| 318 |
+
|
| 319 |
+
[p. 13 | section: C.8. Proof to Theorem 5.1 | type: Equation]
|
| 320 |
+
|f(x_i) - f(x_i)| \le L^* \cdot d(x_i, x_i) \le \hat{L} \cdot d(x_i, x_i), \quad (38)
|
| 321 |
+
|
| 322 |
+
[p. 13 | section: C.8. Proof to Theorem 5.1 | type: Text]
|
| 323 |
+
so no violation can occur. Thus violations only occur when \hat{L} < L^* , and after all doublings complete, \hat{L} \geq L^* . Since we double (rather than increase by smaller factors), \hat{L} \leq 2L^* .
|
| 324 |
+
|
| 325 |
+
[p. 13 | section: C.8. Proof to Theorem 5.1 | type: Text]
|
| 326 |
+
Proof of (3): Sample complexity overhead. Between doublings, CGP runs with either:
|
| 327 |
+
|
| 328 |
+
[p. 13 | section: C.8. Proof to Theorem 5.1 | type: ListGroup]
|
| 329 |
+
Invalid \hat{L} < L^* (before sufficient doublings): certificates may be incorrect, but each such phase has at most O(T/K) samples before a violation triggers doubling. Valid \hat{L} \geq L^* (after final doubling): CGP achieves \tilde{O}(\varepsilon^{-(2+\alpha)}) complexity by Theorem 4.8.
|
| 330 |
+
|
| 331 |
+
[p. 13 | section: C.8. Proof to Theorem 5.1 | type: Text]
|
| 332 |
+
There are at most K invalid phases, each contributing O(T/K) samples. The final valid phase dominates, giving total complexity T = \tilde{O}(\varepsilon^{-(2+\alpha)} \cdot K) = \tilde{O}(\varepsilon^{-(2+\alpha)} \cdot \log(L^*/\hat{L}_0)) .
|
| 333 |
+
|
| 334 |
+
[p. 14 | section: C.9. Certificate Validity Under Adaptive Lipschitz Estimation | type: Text]
|
| 335 |
+
Lemma C.1 (Certificate validity once \hat{L} \geq L^* ). Assume the good event \mathcal{E} holds. Fix any time t at which the current estimate satisfies \hat{L} \geq L^* . Then the envelope constructed with \hat{L} satisfies f(x) \leq U_t(x) for all x \in \mathcal{X} , and consequently the active set
|
| 336 |
+
|
| 337 |
+
[p. 14 | section: C.9. Certificate Validity Under Adaptive Lipschitz Estimation | type: Equation]
|
| 338 |
+
A_t = \{ x \in \mathcal{X} : U_t(x) \ge \ell_t \}
|
| 339 |
+
|
| 340 |
+
[p. 14 | section: C.9. Certificate Validity Under Adaptive Lipschitz Estimation | type: Text]
|
| 341 |
+
contains x^* and certifies that any x \notin A_t is suboptimal (in the sense U_t(x) < \ell_t \le f^* ).
|
| 342 |
+
|
| 343 |
+
[p. 14 | section: C.9. Certificate Validity Under Adaptive Lipschitz Estimation | type: Text]
|
| 344 |
+
Proof. If \hat{L} \geq L^* , then for all x, y \in \mathcal{X} we have |f(x) - f(y)| \leq L^* d(x, y) \leq \hat{L} d(x, y) , i.e., f is \hat{L} -Lipschitz. On \mathcal{E} , for each sampled point x_i , f(x_i) \leq \mathrm{UCB}_i(t) . Therefore for all x,
|
| 345 |
+
|
| 346 |
+
[p. 14 | section: C.9. Certificate Validity Under Adaptive Lipschitz Estimation | type: Equation]
|
| 347 |
+
f(x) \le f(x_i) + \hat{L} d(x, x_i) \le UCB_i(t) + \hat{L} d(x, x_i).
|
| 348 |
+
|
| 349 |
+
[p. 14 | section: C.9. Certificate Validity Under Adaptive Lipschitz Estimation | type: Text]
|
| 350 |
+
Taking the minimum over i yields f(x) \leq U_t(x) for all x. In particular, U_t(x^*) \geq f^* . Also \ell_t = \max_i \mathrm{LCB}_i(t) \leq f^* on \mathcal{E} , hence U_t(x^*) \geq \ell_t and so x^* \in A_t . Finally, if x \notin A_t , then U_t(x) < \ell_t \leq f^* , certifying x cannot be optimal under \mathcal{E} .
|
| 351 |
+
|
| 352 |
+
[p. 14 | section: C.9. Certificate Validity Under Adaptive Lipschitz Estimation | type: Text]
|
| 353 |
+
Remark C.2 (Why pre-final certificates need not be valid). If \hat{L} < L^* , then the envelope may fail to upper-bound f globally, and the rule U_t(x) \geq \ell_t can (in principle) exclude near-optimal points. CGP-Adaptive therefore guarantees certificate validity only after the final doubling event ensures \hat{L} \geq L^* .
|
| 354 |
+
|
| 355 |
+
[p. 14 | section: C.10. Global Safety of Certified Restarts (No False Elimination) | type: Text]
|
| 356 |
+
We restate and prove the key safety property underlying certified restarts in CGP-TR.
|
| 357 |
+
|
| 358 |
+
[p. 14 | section: C.10. Global Safety of Certified Restarts (No False Elimination) | type: Text]
|
| 359 |
+
Lemma C.3 (No false certified restart for the region containing x^* ). Fix any trust region \mathcal{T}^* such that x^* \in \mathcal{T}^* . On the good event \mathcal{E} , the certified restart condition
|
| 360 |
+
|
| 361 |
+
[p. 14 | section: C.10. Global Safety of Certified Restarts (No False Elimination) | type: Equation]
|
| 362 |
+
u_t^{(\mathcal{T}^*)} := \max_{x \in \mathcal{T}^*} U_t(x) < \ell_t
|
| 363 |
+
|
| 364 |
+
[p. 14 | section: C.10. Global Safety of Certified Restarts (No False Elimination) | type: Text]
|
| 365 |
+
never holds. Hence, \mathcal{T}^* is never restarted by the certified rule.
|
| 366 |
+
|
| 367 |
+
[p. 14 | section: C.10. Global Safety of Certified Restarts (No False Elimination) | type: Text]
|
| 368 |
+
Proof. On the good event \mathcal{E} , Lemma 4.2 implies f(x) \leq U_t(x) for all x \in \mathcal{X} . Since x^* \in \mathcal{T}^* ,
|
| 369 |
+
|
| 370 |
+
[p. 14 | section: C.10. Global Safety of Certified Restarts (No False Elimination) | type: Equation]
|
| 371 |
+
u_t^{(\mathcal{T}^*)} = \max_{x \in \mathcal{T}^*} U_t(x) \ge U_t(x^*) \ge f(x^*) = f^*.
|
| 372 |
+
|
| 373 |
+
[p. 14 | section: C.10. Global Safety of Certified Restarts (No False Elimination) | type: Text]
|
| 374 |
+
Also, by definition \ell_t = \max_i \mathrm{LCB}_i(t) . On \mathcal{E} , \mathrm{LCB}_i(t) \leq f(x_i) \leq f^* for every sampled point x_i , hence \ell_t \leq f^* . Therefore,
|
| 375 |
+
|
| 376 |
+
[p. 14 | section: C.10. Global Safety of Certified Restarts (No False Elimination) | type: Equation]
|
| 377 |
+
u_t^{(\mathcal{T}^*)} \geq f^* \geq \ell_t,
|
| 378 |
+
|
| 379 |
+
[p. 14 | section: C.10. Global Safety of Certified Restarts (No False Elimination) | type: Text]
|
| 380 |
+
so the strict inequality u_t^{(\mathcal{T}^*)} < \ell_t cannot occur on \mathcal{E} .
|
| 381 |
+
|
| 382 |
+
[p. 14 | section: C.11. Proof to Theorem 6.1 | type: Text]
|
| 383 |
+
We first establish a key lemma showing that certified elimination is safe.
|
| 384 |
+
|
| 385 |
+
[p. 14 | section: C.11. Proof to Theorem 6.1 | type: Text]
|
| 386 |
+
Lemma C.4 (Certified elimination). On the good event \mathcal{E} , for any trust region \mathcal{T}_j ,
|
| 387 |
+
|
| 388 |
+
[p. 14 | section: C.11. Proof to Theorem 6.1 | type: Equation]
|
| 389 |
+
u_t^{(j)} := \max_{x \in \mathcal{T}_j} U_t(x) < \ell_t \quad \Longrightarrow \quad \sup_{x \in \mathcal{T}_i} f(x) < f(x^*).
|
| 390 |
+
|
| 391 |
+
[p. 14 | section: C.11. Proof to Theorem 6.1 | type: Text]
|
| 392 |
+
Proof. On \mathcal{E} , by Lemma 4.2, f(x) \leq U_t(x) for all x \in \mathcal{X} . Also \ell_t = \max_i \mathrm{LCB}_i(t) \leq f(x^*) on \mathcal{E} since each \mathrm{LCB}_i(t) \leq f(x_i) \leq f(x^*) . Therefore
|
| 393 |
+
|
| 394 |
+
[p. 14 | section: C.11. Proof to Theorem 6.1 | type: Equation]
|
| 395 |
+
\sup_{x \in \mathcal{T}_j} f(x) \le \sup_{x \in \mathcal{T}_j} U_t(x) = u_t^{(j)} < \ell_t \le f(x^*),
|
| 396 |
+
|
| 397 |
+
[p. 14 | section: C.11. Proof to Theorem 6.1 | type: Text]
|
| 398 |
+
which proves the claim.
|
| 399 |
+
|
| 400 |
+
[p. 14 | section: C.11. Proof to Theorem 6.1 | type: Text]
|
| 401 |
+
Proof of Theorem 6.1(1): No false restarts. If x^* \in \mathcal{T}^* , then u_t^{(\mathcal{T}^*)} = \max_{x \in \mathcal{T}^*} U_t(x) \geq U_t(x^*) \geq f(x^*) on \mathcal{E} . Also \ell_t \leq f(x^*) on \mathcal{E} . Hence u_t^{(\mathcal{T}^*)} \geq \ell_t and the restart condition u_t^{(\mathcal{T}^*)} < \ell_t never triggers.
|
| 402 |
+
|
| 403 |
+
[p. 14 | section: C.11. Proof to Theorem 6.1 | type: Text]
|
| 404 |
+
Proof of Theorem 6.1(2): Local certificate. Conditioned on T^* evaluations within \mathcal{T}^* , the CGP analysis applies verbatim on the restricted domain \mathcal{T}^* : the good event \mathcal{E} implies all within-region confidence bounds hold; Lipschitz continuity holds on \mathcal{T}^* ; and Assumption 2.3 holds restricted to \mathcal{T}^* . Therefore the containment and shrinkage results follow with \mathcal{X} replaced by \mathcal{T}^* , yielding \operatorname{Vol}(A_T^{(\mathcal{T}^*)}) \leq C\varepsilon^{d-\alpha} after T^* = \tilde{O}(\varepsilon^{-(2+\alpha)}) within-region samples.
|
| 405 |
+
|
| 406 |
+
[p. 14 | section: C.11. Proof to Theorem 6.1 | type: Text]
|
| 407 |
+
Proof of Theorem 6.1(3): Allocation bound. Fix a sub-optimal region j with gap \Delta_j > 0 . On \mathcal{E} , using the envelope bound from Lemma 4.3, we have for all x:
|
| 408 |
+
|
| 409 |
+
[p. 14 | section: C.11. Proof to Theorem 6.1 | type: Equation]
|
| 410 |
+
U_t(x) < f(x) + 2\rho_t(x),
|
| 411 |
+
|
| 412 |
+
[p. 14 | section: C.11. Proof to Theorem 6.1 | type: Text]
|
| 413 |
+
where \rho_t(x) = \min_i \{r_i(t) + Ld(x, x_i)\} . Restricting to points sampled in \mathcal{T}_j and using the within-region replication schedule, we bound \rho_t(x) \leq \beta_j(n) + L \cdot \operatorname{diam}(\mathcal{T}_j) , yielding
|
| 414 |
+
|
| 415 |
+
[p. 14 | section: C.11. Proof to Theorem 6.1 | type: Equation]
|
| 416 |
+
u_t^{(j)} = \max_{x \in \mathcal{T}_j} U_t(x) \le \sup_{x \in \mathcal{T}_j} f(x) + 2\beta_j(n) + 2L \cdot \operatorname{diam}(\mathcal{T}_j),
|
| 417 |
+
|
| 418 |
+
[p. 14 | section: C.11. Proof to Theorem 6.1 | type: Text]
|
| 419 |
+
after n within-region samples. By the diameter condition L \operatorname{diam}(\mathcal{T}_j) \leq \Delta_j/8 and by requiring \beta_j(n) \leq \Delta_j/8 , we obtain
|
| 420 |
+
|
| 421 |
+
[p. 14 | section: C.11. Proof to Theorem 6.1 | type: Equation]
|
| 422 |
+
u_t^{(j)} \le \sup_{\mathcal{T}_j} f + \Delta_j/4 + \Delta_j/4 = f^* - \Delta_j/2.
|
| 423 |
+
|
| 424 |
+
[p. 14 | section: C.11. Proof to Theorem 6.1 | type: Text]
|
| 425 |
+
Meanwhile, on \mathcal{E} we have \ell_t \leq f^* always, and once the region containing x^* has been sampled sufficiently (which occurs because it is never restarted and is favored by UCB
|
| 426 |
+
|
| 427 |
+
[p. 15 | section: C.11. Proof to Theorem 6.1 | type: Text]
|
| 428 |
+
selection), \ell_t becomes at least f^* - \Delta_j/4 . Hence eventually u_t^{(j)} < \ell_t , triggering certified restart/elimination.
|
| 429 |
+
|
| 430 |
+
[p. 15 | section: C.11. Proof to Theorem 6.1 | type: Text]
|
| 431 |
+
Solving \beta_j(n) \leq \Delta_j/8 under \beta_j(n) \leq c_\sigma \sqrt{\log(c_T/\delta)/n} gives
|
| 432 |
+
|
| 433 |
+
[p. 15 | section: C.11. Proof to Theorem 6.1 | type: Equation]
|
| 434 |
+
n \ge \frac{64c_{\sigma}^2}{\Delta_i^2} \log \left(\frac{c_T}{\delta}\right),\,
|
| 435 |
+
|
| 436 |
+
[p. 15 | section: C.11. Proof to Theorem 6.1 | type: Text]
|
| 437 |
+
yielding the stated bound N_j \leq \frac{64c_\sigma^2}{\Delta_j^2}\log(c_T/\delta) + 1 .
|
| 438 |
+
|
| 439 |
+
[p. 15 | section: C.12. Proof to Proposition 7.1 | type: Text]
|
| 440 |
+
Proof. We analyze each case and the certificate validity separately.
|
| 441 |
+
|
| 442 |
+
[p. 15 | section: C.12. Proof to Proposition 7.1 | type: Text]
|
| 443 |
+
Proof of (1): High smoothness ratio case. When \rho \geq 0.5 , CGP-Hybrid continues with CGP in Phase 2. The algorithm is identical to vanilla CGP, so by Theorem 4.8:
|
| 444 |
+
|
| 445 |
+
[p. 15 | section: C.12. Proof to Proposition 7.1 | type: Equation]
|
| 446 |
+
T = \tilde{O}(\varepsilon^{-(2+\alpha)}). \tag{39}
|
| 447 |
+
|
| 448 |
+
[p. 15 | section: C.12. Proof to Proposition 7.1 | type: Text]
|
| 449 |
+
Proof of (2): Low smoothness ratio case. When \rho < 0.5 , the function is significantly smoother near the optimum. After Phase 1, CGP has reduced the active set to volume V < 0.1 \cdot \mathrm{Vol}(\mathcal{X}) . In Phase 2, GP-UCB operates within A_t , which has:
|
| 450 |
+
|
| 451 |
+
[p. 15 | section: C.12. Proof to Proposition 7.1 | type: ListGroup]
|
| 452 |
+
Effective diameter diam (A_t) = O(V^{1/d}) . Local Lipschitz constant L_{local} = \rho \cdot L < 0.5L .
|
| 453 |
+
|
| 454 |
+
[p. 15 | section: C.12. Proof to Proposition 7.1 | type: Text]
|
| 455 |
+
The sample complexity of GP-UCB on this restricted domain depends on the kernel's maximum information gain \gamma_T over A_t (Srinivas et al., 2009). For commonly used kernels (Matérn, SE), \gamma_T scales polylogarithmically with T when the domain is bounded. The reduced diameter O(V^{1/d}) and local smoothness \rho < 0.5 empirically yield faster convergence than continuing CGP; we validate this empirically in Section 9 rather than claiming a specific rate.
|
| 456 |
+
|
| 457 |
+
[p. 15 | section: C.12. Proof to Proposition 7.1 | type: Text]
|
| 458 |
+
Proof of (3): Certificate validity. The certificate A_T is computed in Phase 1 using CGP's Lipschitz envelope construction. By Theorem 4.5, on the good event \mathcal{E} :
|
| 459 |
+
|
| 460 |
+
[p. 15 | section: C.12. Proof to Proposition 7.1 | type: Equation]
|
| 461 |
+
x^* \in A_T with probability \geq 1 - \delta . (40)
|
| 462 |
+
|
| 463 |
+
[p. 15 | section: C.12. Proof to Proposition 7.1 | type: Text]
|
| 464 |
+
This guarantee depends only on the Lipschitz assumption and confidence bounds, not on the Phase 2 optimization method. Therefore, switching to GP-UCB in Phase 2 does not invalidate the certificate: A_T still contains x^* with high probability, and any point outside A_T remains certifiably suboptimal.
|
| 465 |
+
|
| 466 |
+
[p. 15 | section: D.1. Adaptive Lipschitz Estimation | type: Text]
|
| 467 |
+
CGP-Adaptive removes the requirement for known L with O(\log T) overhead. The doubling scheme is conservative
|
| 468 |
+
|
| 469 |
+
[p. 15 | section: D.1. Adaptive Lipschitz Estimation | type: Text]
|
| 470 |
+
but provably correct; more aggressive schemes (e.g., multiplicative updates with factor 1.5) may reduce overhead but risk certificate invalidation.
|
| 471 |
+
|
| 472 |
+
[p. 15 | section: D.1. Adaptive Lipschitz Estimation | type: Text]
|
| 473 |
+
We recommend initializing \hat{L}_0 from finite differences on initial Sobol samples:
|
| 474 |
+
|
| 475 |
+
[p. 15 | section: D.1. Adaptive Lipschitz Estimation | type: Equation]
|
| 476 |
+
\hat{L}_0 = \max_{i \neq j} \frac{|y_i - y_j|}{d(x_i, x_j)}
|
| 477 |
+
|
| 478 |
+
[p. 15 | section: D.1. Adaptive Lipschitz Estimation | type: Text]
|
| 479 |
+
over the first 10 samples. This typically underestimates L by a factor of 2 to 10, requiring 1 to 4 doublings to reach \hat{L} \geq L^* .
|
| 480 |
+
|
| 481 |
+
[p. 15 | section: D.2. Trust Region Configuration | type: Text]
|
| 482 |
+
CGP-TR trades global certificates for scalability. The local certificates within trust regions still enable principled stopping and progress assessment, but do not guarantee global optimality.
|
| 483 |
+
|
| 484 |
+
[p. 15 | section: D.2. Trust Region Configuration | type: Text]
|
| 485 |
+
Recommended settings:
|
| 486 |
+
|
| 487 |
+
[p. 15 | section: D.2. Trust Region Configuration | type: ListGroup]
|
| 488 |
+
Number of trust regions: n_{\text{trust}} = 5 (balances exploration vs. overhead) Initial radius: r_0 = 0.2 (covers 20% of domain diameter per region) Minimum radius: r_{\min} = 0.01 (prevents over-contraction) Failure threshold: \tau_{\text{fail}} = 10 (triggers contraction after 10 non-improving samples)
|
| 489 |
+
|
| 490 |
+
[p. 15 | section: D.2. Trust Region Configuration | type: Text]
|
| 491 |
+
For applications requiring global certificates in high dimensions, combining CGP-TR with random embeddings (Wang et al., 2016) is promising: project to a low-dimensional subspace, run CGP with global certificates, then lift back.
|
| 492 |
+
|
| 493 |
+
[p. 15 | section: D.3. Smoothness Detection | type: Text]
|
| 494 |
+
CGP-Hybrid's smoothness detection via \rho = L_{\rm local}/L_{\rm global} is heuristic but effective. We estimate L_{\rm local} from points within A_t using the same finite difference approach as L_{\rm global} .
|
| 495 |
+
|
| 496 |
+
[p. 15 | section: D.3. Smoothness Detection | type: Text]
|
| 497 |
+
The threshold \rho_{\rm thresh}=0.5 was selected via cross-validation on held-out benchmarks. More sophisticated detection could use local GP posterior variance or curvature estimates. The key insight is that CGP's certificate remains valid regardless of Phase 2 method, so switching is always safe.
|
| 498 |
+
|
| 499 |
+
[p. 15 | section: D.4. When to Use CGP | type: Text]
|
| 500 |
+
CGP is well suited when:
|
| 501 |
+
|
| 502 |
+
[p. 15 | section: D.4. When to Use CGP | type: Text]
|
| 503 |
+
Evaluations are expensive and interpretable progress is valued
|
| 504 |
+
|
| 505 |
+
[p. 16 | section: D.4. When to Use CGP | type: Text]
|
| 506 |
+
851852
|
| 507 |
+
|
| 508 |
+
[p. 16 | section: D.4. When to Use CGP | type: Text]
|
| 509 |
+
854855
|
| 510 |
+
|
| 511 |
+
[p. 16 | section: D.4. When to Use CGP | type: Text]
|
| 512 |
+
858859
|
| 513 |
+
|
| 514 |
+
[p. 16 | section: D.4. When to Use CGP | type: Text]
|
| 515 |
+
860861
|
| 516 |
+
|
| 517 |
+
[p. 16 | section: D.4. When to Use CGP | type: Text]
|
| 518 |
+
862863
|
| 519 |
+
|
| 520 |
+
[p. 16 | section: D.4. When to Use CGP | type: Text]
|
| 521 |
+
865866
|
| 522 |
+
|
| 523 |
+
[p. 16 | section: D.4. When to Use CGP | type: Text]
|
| 524 |
+
867868
|
| 525 |
+
|
| 526 |
+
[p. 16 | section: D.4. When to Use CGP | type: Text]
|
| 527 |
+
869870
|
| 528 |
+
|
| 529 |
+
[p. 16 | section: D.4. When to Use CGP | type: Text]
|
| 530 |
+
872873
|
| 531 |
+
|
| 532 |
+
[p. 16 | section: D.4. When to Use CGP | type: Text]
|
| 533 |
+
875876
|
| 534 |
+
|
| 535 |
+
[p. 16 | section: D.4. When to Use CGP | type: Text]
|
| 536 |
+
877878
|
| 537 |
+
|
| 538 |
+
[p. 16 | section: D.4. When to Use CGP | type: ListGroup]
|
| 539 |
+
2. The objective has margin structure (sharp peak rather than wide plateau) 3. Lipschitz continuity is a reasonable assumption 4. Dimension is moderate ( d \le 15 for vanilla CGP, d \le 100 for CGP-TR) 5. Anytime stopping decisions are needed
|
| 540 |
+
|
| 541 |
+
[p. 16 | section: D.4. When to Use CGP | type: Text]
|
| 542 |
+
For very high-dimensional problems (d>100), trust region methods like TuRBO may scale better. For smooth problems with cheap evaluations, GP-based methods may be more sample efficient due to their ability to exploit higher-order smoothness.
|
| 543 |
+
|
| 544 |
+
[p. 16 | section: E.1. Baseline Configurations | type: ListGroup]
|
| 545 |
+
Random Search: Sobol sequences for quasi-random sampling GP-UCB : Matérn-5/2 kernel via BoTorch, \beta_t = 2\log(t^2\pi^2/6\delta) TuRBO : Default settings from Eriksson et al. (2019), 1 trust region HEBO: Heteroscedastic GP with input warping, default settings BORE : Tree-Parzen estimator with density ratio, default settings HOO : Binary tree with \nu_1 = 1 , \rho = 0.5 StoSOO: k = 3 children per node, h_{\text{max}} = 20 LIPO: Pure Lipschitz optimization, L estimated online SAASBO: Sparse axis-aligned GP, 10 active dimensions
|
| 546 |
+
|
| 547 |
+
[p. 16 | section: Low-dimensional. | type: ListGroup]
|
| 548 |
+
Needle-2D: f(x) = 1 \|x x^*\|^{1/\alpha} with \alpha = 2 , sharp peak Branin: Standard 2D benchmark with 3 global optima Hartmann-6: 6D benchmark with narrow global basin Levy-5: 5D benchmark with global structure Rosenbrock-4: 4D benchmark with curved valley
|
| 549 |
+
|
| 550 |
+
[p. 16 | section: Medium-dimensional. | type: ListGroup]
|
| 551 |
+
Ackley-10: 10D benchmark with many local optima SVM-RBF-6 : Real hyperparameter tuning (C, \gamma, 4) preprocessing) on MNIST LunarLander-12: RL reward optimization with 12 policy parameters
|
| 552 |
+
|
| 553 |
+
[p. 16 | section: High-dimensional. | type: ListGroup]
|
| 554 |
+
Rover-60 : Mars rover trajectory with 60 waypoint parameters (Wang et al., 2016) NAS-36: Neural architecture search on CIFAR-10, 36 continuous encodings Ant-100: MuJoCo Ant locomotion, 100 morphology and control parameters
|
| 555 |
+
|
| 556 |
+
[p. 16 | section: E.3. Computational Resources | type: Text]
|
| 557 |
+
All experiments run on AMD EPYC 7763 with 256GB RAM. CGP variants use NumPy/SciPy; GP baselines use BoTorch/GPyTorch with GPU acceleration (NVIDIA A100) where available.
|
icml26/5c3d6bff-e8ce-4d9f-840b-719084582491/assets.json
ADDED
|
@@ -0,0 +1,13 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
[
|
| 2 |
+
{
|
| 3 |
+
"filename": "_page_1_Figure_1.jpeg",
|
| 4 |
+
"path": "data/processed_papers/icml26_20260429_1952_duequeue/marker_raw/5c3d6bff-e8ce-4d9f-840b-719084582491/marker_markdown/5c3d6bff-e8ce-4d9f-840b-719084582491/_page_1_Figure_1.jpeg",
|
| 5 |
+
"bytes": 18058,
|
| 6 |
+
"width": 585,
|
| 7 |
+
"height": 449,
|
| 8 |
+
"aspect_ratio": 1.3028953229398663,
|
| 9 |
+
"keep": true,
|
| 10 |
+
"reject_reason": null,
|
| 11 |
+
"model_path": "assets/_page_1_Figure_1.jpeg"
|
| 12 |
+
}
|
| 13 |
+
]
|
icml26/5c3d6bff-e8ce-4d9f-840b-719084582491/assets/_page_1_Figure_1.jpeg
ADDED
|
Git LFS Details
|
icml26/5c3d6bff-e8ce-4d9f-840b-719084582491/chunks_v3_anonymized.jsonl
ADDED
|
The diff for this file is too large to render.
See raw diff
|
|
|
icml26/5c3d6bff-e8ce-4d9f-840b-719084582491/dataset_meta.json
ADDED
|
@@ -0,0 +1,61 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
{
|
| 2 |
+
"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491",
|
| 3 |
+
"pipeline": "Paper2Markdown-V3",
|
| 4 |
+
"ok": true,
|
| 5 |
+
"page_count": 16,
|
| 6 |
+
"chunk_count": 297,
|
| 7 |
+
"main_body_chunk_count": 107,
|
| 8 |
+
"appendix_chunk_count": 186,
|
| 9 |
+
"reference_chunk_count": 4,
|
| 10 |
+
"model_text_chars": 43413,
|
| 11 |
+
"raw_markdown_chars": 83365,
|
| 12 |
+
"sanitized_chars": 76947,
|
| 13 |
+
"page_provenance": {
|
| 14 |
+
"min_page": 1,
|
| 15 |
+
"max_page": 16,
|
| 16 |
+
"invalid_count": 0
|
| 17 |
+
},
|
| 18 |
+
"marker_block_type_counts": {
|
| 19 |
+
"Caption": 1,
|
| 20 |
+
"Code": 2,
|
| 21 |
+
"Equation": 58,
|
| 22 |
+
"Figure": 1,
|
| 23 |
+
"Footnote": 1,
|
| 24 |
+
"ListGroup": 18,
|
| 25 |
+
"PageFooter": 16,
|
| 26 |
+
"PageHeader": 18,
|
| 27 |
+
"SectionHeader": 48,
|
| 28 |
+
"Table": 2,
|
| 29 |
+
"TableGroup": 8,
|
| 30 |
+
"Text": 696
|
| 31 |
+
},
|
| 32 |
+
"asset_count_raw": 1,
|
| 33 |
+
"asset_count_model_kept": 1,
|
| 34 |
+
"asset_count_rejected": 0,
|
| 35 |
+
"asset_reject_reasons": {
|
| 36 |
+
"kept": 1
|
| 37 |
+
},
|
| 38 |
+
"artifact_leak_audit": {
|
| 39 |
+
"ok": true,
|
| 40 |
+
"hits": {
|
| 41 |
+
"Anonymous Authors": [],
|
| 42 |
+
"ACKNOWLEDGMENT": [],
|
| 43 |
+
"OpenReview": [],
|
| 44 |
+
"\"accept_label\"": [],
|
| 45 |
+
"\"decision\"": [],
|
| 46 |
+
"\"decision_tier\"": [],
|
| 47 |
+
"\"source_status\"": [],
|
| 48 |
+
"Meta-review": [],
|
| 49 |
+
"Official Review": [],
|
| 50 |
+
"official_reviews": [],
|
| 51 |
+
"meta_reviews": [],
|
| 52 |
+
"suggested_verdict_score": []
|
| 53 |
+
},
|
| 54 |
+
"artifact_count": 2
|
| 55 |
+
},
|
| 56 |
+
"default_model_input": "model_text_v3.txt",
|
| 57 |
+
"appendix_input": "appendix_text_v3.txt",
|
| 58 |
+
"reference_input": "reference_text_v3.txt",
|
| 59 |
+
"source": "koala_icml26_due_queue",
|
| 60 |
+
"run_name": "icml26_20260429_1952_duequeue"
|
| 61 |
+
}
|
icml26/5c3d6bff-e8ce-4d9f-840b-719084582491/main_body_chunks.jsonl
ADDED
|
@@ -0,0 +1,107 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0000", "section": "Abstract", "page_start": 1, "page_end": 1, "type": "Text", "text": "We study black-box optimization of Lipschitz functions under noisy evaluations. Existing adaptive discretization methods implicitly avoid suboptimal regions but do not provide explicit certificates of optimality or measurable progress guarantees. We introduce Certificate-Guided Pruning (CGP), which maintains an explicit active set A_t of potentially optimal points via confidence-adjusted Lipschitz envelopes. Any point outside A_t is certifiably suboptimal with high probability, and under a margin condition with near-optimality dimension \\alpha , we prove Vol(A_t) shrinks at a controlled rate yielding sample complexity \\tilde{O}(\\varepsilon^{-(2+\\alpha)}) . We develop three extensions: CGP-Adaptive learns L online with O(\\log T) overhead; CGP-TR scales to d > 50via trust regions with local certificates; and CGP-Hybrid switches to GP refinement when local smoothness is detected. Experiments on 12 benchmarks (d \\in [2, 100]) show CGP variants match or exceed strong baselines while providing principled stopping criteria via certificate volume.", "source": "marker_v2", "marker_block_id": "/page/0/Text/4"}
|
| 2 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0001", "section": "1. Introduction", "page_start": 1, "page_end": 1, "type": "Text", "text": "Black-box optimization, the task of finding the maximum of a function f:\\mathcal{X}\\to\\mathbb{R} accessible only through noisy point evaluations, is fundamental to machine learning, with applications spanning hyperparameter tuning (Snoek et al., 2012; Bergstra & Bengio, 2012), neural architecture search (Zoph & Le, 2017), and simulation-based optimization (Fu, 2015; Brochu et al., 2010). In many such settings, evaluations are expensive: training a neural network or running a physical simulation may cost hours or dollars per query. We call these \"precious calls\" where each evaluation must count, motivating the need for methods that provide explicit progress guarantees.", "source": "marker_v2", "marker_block_id": "/page/0/Text/6"}
|
| 3 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0002", "section": "1. Introduction", "page_start": 1, "page_end": 1, "type": "Text", "text": "The Lipschitz continuity assumption provides a natural framework for addressing this challenge. If |f(x)-f(y)| \\le L \\cdot d(x,y) for a known constant L, then observations at sampled points constrain f globally, enabling pruning of provably suboptimal regions. Classical methods exploiting this structure include DIRECT (Jones et al., 1993; 1998), Lipschitz bandits (Kleinberg et al., 2008; Auer et al., 2002), and adaptive discretization algorithms (Bubeck et al., 2011; Valko et al., 2013; Munos, 2014). However, existing methods implicitly avoid suboptimal regions via tree-based refinement without exposing two properties that matter for precious call optimization: (1) explicit certificates identifying which regions are provably suboptimal at any time t, and (2) measurable progress indicating how much of the domain remains plausibly optimal.", "source": "marker_v2", "marker_block_id": "/page/0/Text/9"}
|
| 4 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0003", "section": "1. Introduction", "page_start": 1, "page_end": 1, "type": "Text", "text": "To address these limitations, we introduce Certificate-Guided Pruning (CGP), which maintains an explicit active set A_t \\subseteq \\mathcal{X} of potentially optimal points. This set is defined via a Lipschitz UCB envelope U_t(x) that upper bounds f(x)with high probability, a global lower certificate \\ell_t that lower bounds f(x^*) , and the active set A_t = \\{x : U_t(x) \\ge \\ell_t\\} . Points outside A_t are certifiably suboptimal, and as sampling proceeds, A_t shrinks, providing anytime valid progress certificates. Unlike prior work that uses similar mathematical tools implicitly, CGP exposes the pruning mechanism as a first-class algorithmic object: the certificate is computable in closed form, the shrinkage rate is provably controlled, and the certificate provides valid optimality bounds even when stopped early. Figure 1 illustrates this mechanism. To understand how CGP differs from existing approaches, consider zooming algorithms (Kleinberg et al., 2008; Bubeck et al., 2011). While zooming maintains a tree of \"active arms\" and expands nodes with high UCB, the implicit pruning is an analysis artifact not exposed to the user (Shihab et al., 2026). Table 1 makes this distinction precise: CGP provides explicit certificates, computable progress metrics, and principled stopping rules that zooming-based methods lack. Similarly, Thompson sampling (Thompson, 1933; Russo & Van Roy, 2014; Daniel et al., 2018) and information-directed methods (Hennig & Schuler, 2012; Hernández-Lobato et al., 2014) maintain implicit uncertainty without providing explicit geometric certificates.", "source": "marker_v2", "marker_block_id": "/page/0/Text/10"}
|
| 5 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0004", "section": "1. Introduction", "page_start": 1, "page_end": 1, "type": "Text", "text": "Building on this foundation, our contributions are fourfold. First, we present CGP with explicit active set maintenance", "source": "marker_v2", "marker_block_id": "/page/0/Text/11"}
|
| 6 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0005", "section": "1. Introduction", "page_start": 2, "page_end": 2, "type": "Caption", "text": "Figure 1. The active set A_t (shaded) consists of points where the Lipschitz envelope U_t(x) (red) exceeds the global lower bound \\ell_t (green dashed). Regions where U_t(x) < \\ell_t are certifiably suboptimal and pruned, causing A_t to shrink as sampling proceeds.", "source": "marker_v2", "marker_block_id": "/page/1/Caption/2"}
|
| 7 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0006", "section": "1. Introduction", "page_start": 2, "page_end": 2, "type": "TableGroup", "text": "Table 1. Comparison with zooming-based bandits. CGP uniquely exports explicit certificates (active set A_t , gap bound \\varepsilon_t , and volume \\operatorname{Vol}(A_t) ), enabling principled stopping criteria that implicit zooming methods lack. Property CGP Zooming HOO/StoSOO Explicit active set A_t ✓ _ _ Computable Vol(A_t) ✓ _ - Anytime optimality bound ✓ _ _ Principled stopping rule ✓ _ _ Certificate export ✓ _ _ Sample complexity \\tilde{O}(\\varepsilon^{-(2+\\alpha)}) \\tilde{O}(\\varepsilon^{-(2+\\alpha)}) \\tilde{O}(\\varepsilon^{-(2+d)}) O(\\log N_t) Per-iteration cost O(N_t) O(\\log N_t) O(\\log N_t) Adaptive L √(CGP-A) _ _ High-dim scaling √(CGP-TR) -", "source": "marker_v2", "marker_block_id": "/page/1/TableGroup/393"}
|
| 8 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0007", "section": "1. Introduction", "page_start": 2, "page_end": 2, "type": "Text", "text": "and prove a shrinkage theorem: under a margin condition with near-optimality dimension \\alpha (i.e., Vol( \\{x: f(x) \\geq f^* - \\varepsilon ) \\leq C\\varepsilon^{d-\\alpha} , we show Vol(A_t) \\leq C \\cdot (2(\\beta_t + \\beta_t)) (L\\eta_t)^{d-\\alpha} , yielding sample complexity T = \\tilde{O}(\\varepsilon^{-(2+\\alpha)}) that improves on the worst case \\tilde{O}(\\varepsilon^{-(2+d)}) when \\alpha < d (Section 4). Second, we develop CGP-Adaptive (Section 5), which learns L online via a doubling scheme, proving that unknown L adds only O(\\log T) multiplicative overhead, the first such guarantee for Lipschitz optimization with certificates. Third, we introduce CGP-TR (Section 6), a trust region variant that scales to d > 50 by maintaining local certificates within adaptively sized regions, enabling high-dimensional applications previously intractable for Lipschitz methods. Fourth, we propose CGP-Hybrid (Section 7), which detects local smoothness via the ratio \\rho = L_{\\rm local}/L_{\\rm global} and switches to GP refinement when \\rho < 0.5 , achieving best of both worlds performance across diverse function classes.", "source": "marker_v2", "marker_block_id": "/page/1/Text/5"}
|
| 9 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0008", "section": "1. Introduction", "page_start": 2, "page_end": 2, "type": "Text", "text": "These theoretical contributions translate to strong empirical performance. Experiments (Section 9) demonstrate that CGP variants are competitive with strong baselines on 12 benchmarks spanning d \\in [2, 100] , including Rover tra-", "source": "marker_v2", "marker_block_id": "/page/1/Text/6"}
|
| 10 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0009", "section": "1. Introduction", "page_start": 2, "page_end": 2, "type": "TableGroup", "text": "Table 2. Summary of Notation Symbol Description f^* Global maximum value of the objective function L Lipschitz constant (or global upper bound) A_t Active set at time t (contains potential optimizers) U_t(x) Lipschitz Upper Confidence Bound envelope \\ell_t Global lower certificate (\\max_i LCB_i) \\alpha Near-optimality dimension (problem hardness) \\beta_t Active confidence radius (uncertainty in A_t ) \\eta_t Covering radius (resolution of A_t ) \\gamma_t Gap to optimum proxy (f^* - \\ell_t) ρ Local smoothness ratio ( L_{\\rm local}/L_{\\rm global} )", "source": "marker_v2", "marker_block_id": "/page/1/TableGroup/394"}
|
| 11 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0010", "section": "1. Introduction", "page_start": 2, "page_end": 2, "type": "Text", "text": "jectory optimization (d=60), neural architecture search (d=36), and safe robotics where certificates enable stopping with guaranteed bounds (Shihab et al., 2025a). CGP-Hybrid performs best among tested methods on all 12 benchmarks under matched budgets, including Branin and Rosenbrock where vanilla CGP previously lost to GP-based methods.", "source": "marker_v2", "marker_block_id": "/page/1/Text/9"}
|
| 12 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0011", "section": "2. Problem Formulation", "page_start": 2, "page_end": 2, "type": "Text", "text": "Let (\\mathcal{X}, d) be a compact metric space with diameter D = \\sup_{x,y} d(x,y) . We consider \\mathcal{X} = [0,1]^d with Euclidean metric. Let f: \\mathcal{X} \\to [0,1] satisfy:", "source": "marker_v2", "marker_block_id": "/page/1/Text/11"}
|
| 13 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0012", "section": "2. Problem Formulation", "page_start": 2, "page_end": 2, "type": "Text", "text": "Assumption 2.1 (Lipschitz continuity). There exists L > 0 such that for all x, y \\in \\mathcal{X} : |f(x) - f(y)| \\le L \\cdot d(x, y) .", "source": "marker_v2", "marker_block_id": "/page/1/Text/12"}
|
| 14 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0013", "section": "2. Problem Formulation", "page_start": 2, "page_end": 2, "type": "Text", "text": "We observe f through noisy queries: querying x returns y = f(x) + \\epsilon , where:", "source": "marker_v2", "marker_block_id": "/page/1/Text/13"}
|
| 15 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0014", "section": "2. Problem Formulation", "page_start": 2, "page_end": 2, "type": "Text", "text": "Assumption 2.2 (Sub-Gaussian noise). The noise \\epsilon is \\sigma -sub-Gaussian: \\mathbb{E}[e^{\\lambda\\epsilon}] \\leq e^{\\lambda^2 \\sigma^2/2} for all \\lambda \\in \\mathbb{R} .", "source": "marker_v2", "marker_block_id": "/page/1/Text/14"}
|
| 16 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0015", "section": "2. Problem Formulation", "page_start": 2, "page_end": 2, "type": "Text", "text": "After T samples, the algorithm outputs \\hat{x}_T \\in \\mathcal{X} . The goal is to minimize simple regret r_T = f(x^*) - f(\\hat{x}_T) . We seek PAC guarantees: r_T \\leq \\varepsilon with probability \\geq 1 - \\delta .", "source": "marker_v2", "marker_block_id": "/page/1/Text/15"}
|
| 17 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0016", "section": "2. Problem Formulation", "page_start": 2, "page_end": 2, "type": "Text", "text": "Assumption 2.3 (Margin / near-optimality dimension). There exist C>0 and \\alpha\\in[0,d] such that for all \\varepsilon>0 : \\operatorname{Vol}(\\{x\\in\\mathcal{X}:f(x)\\geq f^*-\\varepsilon\\})\\leq C\\varepsilon^{d-\\alpha} .", "source": "marker_v2", "marker_block_id": "/page/1/Text/16"}
|
| 18 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0017", "section": "2. Problem Formulation", "page_start": 2, "page_end": 2, "type": "Text", "text": "The parameter \\alpha is the near-optimality dimension: smaller \\alpha means a sharper optimum (easier), \\alpha=d is worst case. For isolated maxima with nondegenerate Hessian, \\alpha=d/2 ; for f(x)\\approx f^*-c\\|x-x^*\\|^p , we have \\alpha=d/p . This assumption is standard in bandit theory (Audibert & Bubeck, 2010; Bubeck et al., 2011; Lattimore & Szepesvári, 2020); see Appendix A for extended discussion.", "source": "marker_v2", "marker_block_id": "/page/1/Text/17"}
|
| 19 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0018", "section": "3. Algorithm: Certificate-Guided Pruning", "page_start": 2, "page_end": 2, "type": "Text", "text": "With the problem formalized, we now describe the CGP algorithm (Algorithm 1). CGP maintains sampled points", "source": "marker_v2", "marker_block_id": "/page/1/Text/19"}
|
| 20 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0019", "section": "Algorithm 1 Certificate-Guided Pruning (CGP)", "page_start": 3, "page_end": 3, "type": "Text", "text": "Require: Domain \\mathcal{X} , Lipschitz constant L, noise \\sigma , budget T, confidence \\delta", "source": "marker_v2", "marker_block_id": "/page/2/Text/2"}
|
| 21 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0020", "section": "Algorithm 1 Certificate-Guided Pruning (CGP)", "page_start": 3, "page_end": 3, "type": "ListGroup", "text": "1: Initialize: Sample x_1 uniformly, observe y_1 2: for t = 1, ..., T 1 do 3: Compute r_i(t) , \\ell_t = \\max_i LCB_i(t) , A_t = \\{x : U_t(x) \\ge \\ell_t\\} 4: x_{t+1} \\leftarrow \\arg\\max_{x \\in A_t} [U_t(x) L \\cdot \\min_i d(x, x_i)] 5: Query x_{t+1} , observe y_{t+1} , update statistics 6: Replicate active points with r_i(t) > \\beta_{\\text{target}}(t) 7: end for", "source": "marker_v2", "marker_block_id": "/page/2/ListGroup/570"}
|
| 22 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0021", "section": "Algorithm 1 Certificate-Guided Pruning (CGP)", "page_start": 3, "page_end": 3, "type": "Text", "text": "8: Output: \\hat{x}_T = \\arg \\max_i \\hat{\\mu}_i(T) , certificate A_T", "source": "marker_v2", "marker_block_id": "/page/2/Text/571"}
|
| 23 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0022", "section": "Algorithm 1 Certificate-Guided Pruning (CGP)", "page_start": 3, "page_end": 3, "type": "Text", "text": "with empirical estimates, confidence intervals, and the active set. At time t, let \\{x_1,\\ldots,x_{N_t}\\} be distinct points sampled, with n_i observations at x_i . Define empirical mean \\hat{\\mu}_i(t) = \\frac{1}{n_i} \\sum_{j=1}^{n_i} y_{i,j} and confidence radius r_i(t) = \\sigma \\sqrt{2\\log(2N_tT/\\delta)}/n_i , ensuring |f(x_i) - \\hat{\\mu}_i(t)| \\leq r_i(t) with high probability.", "source": "marker_v2", "marker_block_id": "/page/2/Text/11"}
|
| 24 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0023", "section": "Algorithm 1 Certificate-Guided Pruning (CGP)", "page_start": 3, "page_end": 3, "type": "Text", "text": "The upper and lower confidence bounds are UCB_i(t) = \\hat{\\mu}_i(t) + r_i(t) and LCB_i(t) = \\hat{\\mu}_i(t) - r_i(t) . The global lower certificate \\ell_t = \\max_{i \\leq N_t} LCB_i(t) satisfies \\ell_t \\leq f(x^*) under the good event. The Lipschitz UCB envelope propagates uncertainty:", "source": "marker_v2", "marker_block_id": "/page/2/Text/12"}
|
| 25 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0024", "section": "Algorithm 1 Certificate-Guided Pruning (CGP)", "page_start": 3, "page_end": 3, "type": "Equation", "text": "U_t(x) = \\min_{i \\le N_t} \\left\\{ \\text{UCB}_i(t) + L \\cdot d(x, x_i) \\right\\}, \\tag{1}", "source": "marker_v2", "marker_block_id": "/page/2/Equation/13"}
|
| 26 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0025", "section": "Algorithm 1 Certificate-Guided Pruning (CGP)", "page_start": 3, "page_end": 3, "type": "Text", "text": "which upper-bounds f(x) everywhere. The active set is", "source": "marker_v2", "marker_block_id": "/page/2/Text/14"}
|
| 27 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0026", "section": "Algorithm 1 Certificate-Guided Pruning (CGP)", "page_start": 3, "page_end": 3, "type": "Equation", "text": "A_t = \\left\\{ x \\in \\mathcal{X} : U_t(x) \\ge \\ell_t \\right\\},\\tag{2}", "source": "marker_v2", "marker_block_id": "/page/2/Equation/15"}
|
| 28 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0027", "section": "Algorithm 1 Certificate-Guided Pruning (CGP)", "page_start": 3, "page_end": 3, "type": "Text", "text": "and points outside A_t are certifiably suboptimal: their upper bound is below the lower bound on f^* .", "source": "marker_v2", "marker_block_id": "/page/2/Text/16"}
|
| 29 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0028", "section": "Algorithm 1 Certificate-Guided Pruning (CGP)", "page_start": 3, "page_end": 3, "type": "Text", "text": "The algorithm selects queries via \\mathrm{score}(x) = U_t(x) - \\lambda \\cdot \\min_{i \\leq N_t} d(x, x_i) where \\lambda = L , selecting x_{t+1} = \\arg\\max_{x \\in A_t} \\mathrm{score}(x) . The first term favors high UCB regions while the second encourages coverage. CGP allocates additional samples to active points with r_i(t) > \\beta_{\\mathrm{target}}(t) to reduce confidence radii. The target confidence radius follows a schedule \\beta_{\\mathrm{target}}(t) = \\sigma \\sqrt{2\\log(2T^2/\\delta)/t} , ensuring that confidence radii decrease at rate O(1/\\sqrt{t}) .", "source": "marker_v2", "marker_block_id": "/page/2/Text/17"}
|
| 30 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0029", "section": "Algorithm 1 Certificate-Guided Pruning (CGP)", "page_start": 3, "page_end": 3, "type": "Text", "text": "We compute A_t via discretization for low dimensions and Monte Carlo sampling for d>5 (details in Appendix B.2). Theoretically, CGP assumes oracle access to \\arg\\max_{x\\in A_t}\\mathrm{score}(x) ; practically, we use CMA-ES with 10 random restarts within A_t (see Appendix B for details). For d\\leq 3 , we additionally use Delaunay triangulation to identify candidate optima at Voronoi vertices. Membership in A_t is exact: checking U_t(x)\\geq \\ell_t requires O(N_t) time. Approximate maximization may slow convergence of \\eta_t but does not invalidate certificates: any x\\notin A_t remains certifiably suboptimal regardless of which x\\in A_t is queried.", "source": "marker_v2", "marker_block_id": "/page/2/Text/18"}
|
| 31 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0030", "section": "Algorithm 1 Certificate-Guided Pruning (CGP)", "page_start": 3, "page_end": 3, "type": "Text", "text": "Replication Strategy. When an active point x_i has r_i(t) > \\beta_{\\text{target}}(t) , we allocate \\lceil (r_i(t)/\\beta_{\\text{target}}(t))^2 \\rceil additional samples to reduce its confidence radius. This ensures all active points have comparable confidence, preventing any single point from dominating the envelope.", "source": "marker_v2", "marker_block_id": "/page/2/Text/19"}
|
| 32 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0031", "section": "4. Theoretical Analysis", "page_start": 3, "page_end": 3, "type": "Text", "text": "Having described the algorithm, we now establish its theoretical guarantees. We show that the active set is contained in the near-optimal set, its volume shrinks at a controlled rate, and this yields instance-dependent sample complexity. All results hold on the good event \\mathcal E where |f(x_i) - \\hat{\\mu}_i(t)| \\leq r_i(t) for all t,i. All proofs are deferred to Appendix C.", "source": "marker_v2", "marker_block_id": "/page/2/Text/21"}
|
| 33 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0032", "section": "4. Theoretical Analysis", "page_start": 3, "page_end": 3, "type": "Text", "text": "Lemma 4.1 (Good event). With r_i(t) = \\sigma \\sqrt{2 \\log(2N_t T/\\delta)/n_i} , we have \\mathbb{P}[\\mathcal{E}] \\geq 1 - \\delta .", "source": "marker_v2", "marker_block_id": "/page/2/Text/22"}
|
| 34 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0033", "section": "4. Theoretical Analysis", "page_start": 3, "page_end": 3, "type": "Text", "text": "Lemma 4.2 (UCB envelope is valid). On \\mathcal{E} , for all x \\in \\mathcal{X} : f(x) \\leq U_t(x) .", "source": "marker_v2", "marker_block_id": "/page/2/Text/23"}
|
| 35 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0034", "section": "4. Theoretical Analysis", "page_start": 3, "page_end": 3, "type": "Text", "text": "Proof sketch. For any sampled x_i , on \\mathcal{E} : f(x_i) \\leq \\mathrm{UCB}_i(t) . By Lipschitz continuity: f(x) \\leq f(x_i) + L \\cdot d(x, x_i) \\leq \\mathrm{UCB}_i(t) + L \\cdot d(x, x_i) . Taking min over i gives f(x) \\leq U_t(x) .", "source": "marker_v2", "marker_block_id": "/page/2/Text/24"}
|
| 36 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0035", "section": "4. Theoretical Analysis", "page_start": 3, "page_end": 3, "type": "Text", "text": "Lemma 4.3 (Envelope slack bound). On \\mathcal{E} , for all x \\in \\mathcal{X} : U_t(x) \\leq f(x) + 2\\rho_t(x) , where \\rho_t(x) = \\min_i \\{r_i(t) + L \\cdot d(x, x_i)\\} .", "source": "marker_v2", "marker_block_id": "/page/2/Text/25"}
|
| 37 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0036", "section": "4. Theoretical Analysis", "page_start": 3, "page_end": 3, "type": "Text", "text": "Remark 4.4 (Slack in envelope bound). The factor of 2 arises from applying Lipschitz continuity (f(x_i) \\leq f(x) + L \\cdot d(x, x_i)) after bounding UCB_i \\leq f(x_i) + 2r_i . A tighter bound separating the confidence and distance terms is possible but complicates notation without affecting rate dependencies. Constants throughout are not optimized.", "source": "marker_v2", "marker_block_id": "/page/2/Text/26"}
|
| 38 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0037", "section": "4. Theoretical Analysis", "page_start": 3, "page_end": 3, "type": "Text", "text": "Theorem 4.5 (Active set containment). On \\mathcal{E} , A_t \\subseteq \\{x : f(x) \\ge f^* - 2\\Delta_t\\} where \\Delta_t = \\sup_{x \\in A_t} \\rho_t(x) + (f^* - \\ell_t) .", "source": "marker_v2", "marker_block_id": "/page/2/Text/27"}
|
| 39 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0038", "section": "4. Theoretical Analysis", "page_start": 3, "page_end": 3, "type": "Text", "text": "Proof sketch. On \\mathcal{E} , \\ell_t = \\max_i \\mathrm{LCB}_i(t) \\leq f^* since each \\mathrm{LCB}_i(t) \\leq f(x_i) \\leq f^* . For x \\in A_t , by definition U_t(x) \\geq \\ell_t . Applying Lemma 4.3: f(x) + 2\\rho_t(x) \\geq U_t(x) \\geq \\ell_t . Rearranging: f(x) \\geq \\ell_t - 2\\rho_t(x) = f^* - (f^* - \\ell_t) - 2\\rho_t(x) \\geq f^* - 2\\Delta_t .", "source": "marker_v2", "marker_block_id": "/page/2/Text/28"}
|
| 40 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0039", "section": "4. Theoretical Analysis", "page_start": 3, "page_end": 3, "type": "Text", "text": "The containment theorem bounds how far active points can be from optimal. To translate this into a volume bound, we introduce two key quantities: the covering radius \\eta_t = \\sup_{x \\in A_t} \\min_i d(x, x_i) measuring how well samples cover A_t , and the active confidence radius \\beta_t = \\max_{i:x_i \\text{ active }} r_i(t) measuring confidence precision.", "source": "marker_v2", "marker_block_id": "/page/2/Text/29"}
|
| 41 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0040", "section": "4. Theoretical Analysis", "page_start": 3, "page_end": 3, "type": "Text", "text": "Theorem 4.6 (Shrinkage theorem). Under Assumptions 2.1–2.3, on \\mathcal{E} :", "source": "marker_v2", "marker_block_id": "/page/2/Text/30"}
|
| 42 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0041", "section": "4. Theoretical Analysis", "page_start": 3, "page_end": 3, "type": "Equation", "text": "Vol(A_t) \\le C \\cdot \\left(2(\\beta_t + L\\eta_t) + \\gamma_t\\right)^{d-\\alpha},\\tag{3}", "source": "marker_v2", "marker_block_id": "/page/2/Equation/31"}
|
| 43 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0042", "section": "4. Theoretical Analysis", "page_start": 4, "page_end": 4, "type": "Code", "text": "where \\gamma_t = f^* - \\ell_t.", "source": "marker_v2", "marker_block_id": "/page/3/Code/1"}
|
| 44 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0043", "section": "4. Theoretical Analysis", "page_start": 4, "page_end": 4, "type": "Text", "text": "205206", "source": "marker_v2", "marker_block_id": "/page/3/Text/62"}
|
| 45 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0044", "section": "4. Theoretical Analysis", "page_start": 4, "page_end": 4, "type": "Text", "text": "Proof sketch. From Theorem 4.5, A_t \\subseteq \\{x: f(x) \\ge f^* - 2\\Delta_t\\} . For x \\in A_t , \\rho_t(x) \\le \\beta_t + L\\eta_t (the worst-case slack from active-point confidence plus covering distance), so \\Delta_t \\le \\beta_t + L\\eta_t + \\gamma_t/2 . Applying Assumption 2.3 with \\varepsilon = 2\\Delta_t : \\operatorname{Vol}(A_t) \\le C(2\\Delta_t)^{d-\\alpha} \\le C(2(\\beta_t + L\\eta_t) + \\gamma_t)^{d-\\alpha} .", "source": "marker_v2", "marker_block_id": "/page/3/Text/2"}
|
| 46 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0045", "section": "4. Theoretical Analysis", "page_start": 4, "page_end": 4, "type": "Text", "text": "This makes pruning measurable: \\beta_t is controlled by replication, \\eta_t by the query rule, \\gamma_t by best-point improvement. All three quantities can be computed during the run, enabling practitioners to monitor progress.", "source": "marker_v2", "marker_block_id": "/page/3/Text/3"}
|
| 47 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0046", "section": "4. Theoretical Analysis", "page_start": 4, "page_end": 4, "type": "Text", "text": "Remark 4.7 (Certificate validity vs. progress estimation). The certificate itself is the set membership rule x \\in A_t \\Leftrightarrow U_t(x) \\geq \\ell_t , which is exact given (\\hat{\\mu}_i, r_i) and does not depend on any volume estimator. Approximations (grid/Monte Carlo) are used only to estimate \\operatorname{Vol}(A_t) for monitoring and optional stopping heuristics; certificate validity is unaffected by volume estimation errors.", "source": "marker_v2", "marker_block_id": "/page/3/Text/4"}
|
| 48 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0047", "section": "4. Theoretical Analysis", "page_start": 4, "page_end": 4, "type": "Text", "text": "The shrinkage theorem directly yields sample complexity by bounding how many samples are needed to drive \\beta_t , \\eta_t , and \\gamma_t below \\varepsilon .", "source": "marker_v2", "marker_block_id": "/page/3/Text/5"}
|
| 49 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0048", "section": "4. Theoretical Analysis", "page_start": 4, "page_end": 4, "type": "Text", "text": "Theorem 4.8 (Sample complexity). Under Assumptions 2.1–2.3, CGP achieves r_T \\le \\varepsilon with probability \\ge 1 - \\delta using T = \\tilde{O}(L^d \\varepsilon^{-(2+\\alpha)} \\log(1/\\delta)) samples. When \\alpha < d , this improves upon the worst-case \\tilde{O}(\\varepsilon^{-(2+d)}) rate.", "source": "marker_v2", "marker_block_id": "/page/3/Text/6"}
|
| 50 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0049", "section": "4. Theoretical Analysis", "page_start": 4, "page_end": 4, "type": "Text", "text": "The following lower bound shows that our sample complexity is optimal up to logarithmic factors.", "source": "marker_v2", "marker_block_id": "/page/3/Text/7"}
|
| 51 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0050", "section": "4. Theoretical Analysis", "page_start": 4, "page_end": 4, "type": "Text", "text": "Theorem 4.9 (Lower bound). For any algorithm and \\alpha \\in (0, d] , there exists f satisfying Assumption 2.3 requiring T = \\Omega(\\varepsilon^{-(2+\\alpha)}) samples for \\varepsilon -optimality with probability \\geq 2/3 .", "source": "marker_v2", "marker_block_id": "/page/3/Text/8"}
|
| 52 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0051", "section": "4. Theoretical Analysis", "page_start": 4, "page_end": 4, "type": "Text", "text": "This establishes CGP is minimax optimal up to logarithmic factors. A key property is anytime validity: at any t, any x \\notin A_t satisfies f(x) < f^* - \\varepsilon_t for computable \\varepsilon_t > 0 . Full proofs are in Appendix C.", "source": "marker_v2", "marker_block_id": "/page/3/Text/9"}
|
| 53 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0052", "section": "5. CGP-Adaptive: Learning L Online", "page_start": 4, "page_end": 4, "type": "Text", "text": "The theoretical results above assume known L, which is often unavailable in practice. Underestimating L invalidates certificates, while overestimating is safe but conservative. We develop CGP-Adaptive (Algorithm 2), which learns L online via a doubling scheme with provable guarantees.", "source": "marker_v2", "marker_block_id": "/page/3/Text/11"}
|
| 54 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0053", "section": "5. CGP-Adaptive: Learning L Online", "page_start": 4, "page_end": 4, "type": "Text", "text": "The key insight is that Lipschitz violations are detectable. If |\\hat{\\mu}_i - \\hat{\\mu}_j| - 2(r_i + r_j) > \\hat{L} \\cdot d(x_i, x_j) , then \\hat{L} underestimates L with high probability. CGP-Adaptive uses a doubling scheme: start with conservative \\hat{L}_0 , and upon detecting a violation, double \\hat{L} .", "source": "marker_v2", "marker_block_id": "/page/3/Text/12"}
|
| 55 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0054", "section": "Algorithm 2 CGP-Adaptive", "page_start": 4, "page_end": 4, "type": "Code", "text": "Require: Domain \\mathcal{X}, initial estimate \\hat{L}_0, noise \\sigma, budget T, confidence \\delta 1: \\hat{L} \\leftarrow \\hat{L}_0, k \\leftarrow 0 (doubling counter) 2: for t = 1, ..., T do Run CGP iteration with current \\hat{L} for all pairs (i, j) with n_i, n_j \\ge \\log(T/\\delta) do 4: 5: if |\\hat{\\mu}_i - \\hat{\\mu}_j| - 2(r_i + r_j) > \\hat{L} \\cdot d(x_i, x_j) then \\hat{L} \\leftarrow 2\\hat{L}, k \\leftarrow k + 1 {Doubling event} 6: Recompute A_t with new \\hat{L} 7: 8: end if end for 10: end for", "source": "marker_v2", "marker_block_id": "/page/3/Code/14"}
|
| 56 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0055", "section": "Algorithm 2 CGP-Adaptive", "page_start": 4, "page_end": 4, "type": "Text", "text": "Theorem 5.1 (Adaptive L guarantee: learning regime). Let L^* = \\sup_{x \\neq y} |f(x) - f(y)|/d(x,y) be the true Lipschitz constant. CGP-Adaptive with initial \\hat{L}_0 \\leq L^* (learning from underestimation) satisfies:", "source": "marker_v2", "marker_block_id": "/page/3/Text/15"}
|
| 57 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0056", "section": "Algorithm 2 CGP-Adaptive", "page_start": 4, "page_end": 4, "type": "ListGroup", "text": "1. The number of doubling events is at most K = \\lceil \\log_2(L^*/\\hat{L}_0) \\rceil . 2. After all doublings, \\hat{L} \\in [L^*, 2L^*] with probability \\geq 1 \\delta . 3. The total sample complexity is T = \\tilde{O}(\\varepsilon^{-(2+\\alpha)} \\cdot K) , i.e., O(\\log(L^*/\\hat{L}_0)) multiplicative overhead. 4. Certificate validity: Certificates are valid only after the final doubling (when \\hat{L} \\geq L^* ). Before this, certificates may falsely exclude near-optimal points.", "source": "marker_v2", "marker_block_id": "/page/3/ListGroup/546"}
|
| 58 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0057", "section": "Algorithm 2 CGP-Adaptive", "page_start": 4, "page_end": 4, "type": "Text", "text": "Remark 5.2 (Anytime-valid certificates). For applications requiring certificates valid at all times, use \\hat{L}_0 \\geq L^* (conservative overestimate). This ensures \\hat{L} \\geq L^* throughout, so all certificates are valid, but may be overly conservative. One can optionally decrease \\hat{L} when evidence suggests overestimation, but this requires different analysis than the doubling scheme above.", "source": "marker_v2", "marker_block_id": "/page/3/Text/20"}
|
| 59 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0058", "section": "Algorithm 2 CGP-Adaptive", "page_start": 4, "page_end": 4, "type": "Text", "text": "This is the first provably correct adaptive L estimation for Lipschitz optimization with certificates. Prior work (Malherbe & Vayatis, 2017) estimates L but without guarantees on certificate validity. Table 5 shows CGP-Adaptive matches oracle performance (known L) within 8% while being robust to 100\\times underestimation of initial \\hat{L}_0 .", "source": "marker_v2", "marker_block_id": "/page/3/Text/21"}
|
| 60 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0059", "section": "6. CGP-TR: Trust Regions for High Dimensions", "page_start": 4, "page_end": 4, "type": "Text", "text": "CGP-Adaptive addresses the unknown L problem, but another challenge remains: scalability. The covering number of A_t grows as O(\\eta^{-d}) , making CGP intractable for d > 15.", "source": "marker_v2", "marker_block_id": "/page/3/Text/23"}
|
| 61 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0060", "section": "6. CGP-TR: Trust Regions for High Dimensions", "page_start": 5, "page_end": 5, "type": "TableGroup", "text": "Table 3. Simple regret ( \\times 10^{-2} ) at T = 200. Bold: best; †: significant vs second-best. Method Needle Branin Hartmann Ackley Levy Rosen. SVM Random 8.2 12.1 15.3 22.4 18.7 14.2 11.2 GP-UCB 2.1 1.8 4.2 12.3 5.1 4.8 3.9 TuRBO 1.8 2.1 3.1 9.8 4.3 3.9 3.2 HEBO 1.9 1.6 3.3 9.4 4.1 3.7 3.1 BORE 2.0 1.9 3.5 10.1 4.5 4.1 3.4 HOO 3.4 5.2 8.7 14.2 9.8 8.1 7.1 CGP 1.2 2.0 2.9 8.1 3.8 3.8 2.8 CGP-A 1.3 2.1 3.0 8.3 3.9 3.9 2.9 CGP-H 1.1 ^{\\dagger} 1.4 ^{\\dagger} 2.7 ^{\\dagger} 7.8 ^{\\dagger} 3.5 ^{\\dagger} 3.4 ^{\\dagger} 2.6 ^{\\dagger}", "source": "marker_v2", "marker_block_id": "/page/4/TableGroup/410"}
|
| 62 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0061", "section": "6. CGP-TR: Trust Regions for High Dimensions", "page_start": 5, "page_end": 5, "type": "Text", "text": "To enable high-dimensional optimization, we develop CGP-TR (Algorithm 3), which maintains local certificates within trust regions that adapt based on observed progress.", "source": "marker_v2", "marker_block_id": "/page/4/Text/4"}
|
| 63 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0062", "section": "6. CGP-TR: Trust Regions for High Dimensions", "page_start": 5, "page_end": 5, "type": "Text", "text": "The key insight is that certificates need not be global. A local certificate A_t^{\\mathcal{T}} within trust region \\mathcal{T} \\subset \\mathcal{X} still provides valid bounds for \\arg\\max_{x \\in \\mathcal{T}} f(x) . CGP-TR maintains multiple trust regions \\{\\mathcal{T}_1, \\dots, \\mathcal{T}_m\\} centered at promising points, with radii that expand on success and contract on failure (following TuRBO (Eriksson et al., 2019)).", "source": "marker_v2", "marker_block_id": "/page/4/Text/5"}
|
| 64 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0063", "section": "6. CGP-TR: Trust Regions for High Dimensions", "page_start": 5, "page_end": 5, "type": "Text", "text": "Certified restarts. We restart a trust region only when it is certifiably suboptimal: if u_t^{(j)} := \\max_{x \\in \\mathcal{T}_j} U_t(x) satisfies u_t^{(j)} < \\ell_t where \\ell_t := \\max_i \\mathrm{LCB}_i(t) , then with high probability \\sup_{x \\in \\mathcal{T}_j} f(x) < f(x^*) , so \\mathcal{T}_j cannot contain x^* and can be safely restarted. This certified restart rule ensures that regions containing x^* are never falsely eliminated. In our implementation, contraction is lower-bounded by r_{\\min} and centers are fixed, so a region that contains x^* cannot be contracted to exclude it; restarts occur only via the certified condition u_t^{(j)} < \\ell_t .", "source": "marker_v2", "marker_block_id": "/page/4/Text/6"}
|
| 65 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0064", "section": "6. CGP-TR: Trust Regions for High Dimensions", "page_start": 5, "page_end": 5, "type": "Text", "text": "Theorem 6.1 (CGP-TR with certified restarts: correctness and allocation). Assume the good event \\mathcal{E} holds for the confidence bounds used to construct U_t and \\ell_t . CGP-TR uses certified restarts: restart \\mathcal{T}_j only if u_t^{(j)} := \\max_{x \\in \\mathcal{T}_j} U_t(x) < \\ell_t .", "source": "marker_v2", "marker_block_id": "/page/4/Text/7"}
|
| 66 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0065", "section": "6. CGP-TR: Trust Regions for High Dimensions", "page_start": 5, "page_end": 5, "type": "Text", "text": "Let \\mathcal{T}^* be a trust region that contains x^* at some time and is not contracted to exclude x^* (e.g., contraction is lower-bounded by r_{\\min} and the center remains fixed). Then:", "source": "marker_v2", "marker_block_id": "/page/4/Text/8"}
|
| 67 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0066", "section": "6. CGP-TR: Trust Regions for High Dimensions", "page_start": 5, "page_end": 5, "type": "ListGroup", "text": "1. (No false restarts) On \\mathcal{E} , \\mathcal{T}^* is never restarted by the certified rule. 2. (Local certificate) Conditioned on receiving T^* evaluations inside T^* , the local active set A_t^{(T^*)} = \\{x \\in T^* : U_t(x) \\geq \\ell_t^{(T^*)}\\} satisfies the same containment/shrinkage/sample-complexity bounds as CGP on the restricted domain T^* . 3. (Allocation bound) Define the region gap \\Delta_i :=", "source": "marker_v2", "marker_block_id": "/page/4/ListGroup/411"}
|
| 68 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0067", "section": "Algorithm 3 CGP-TR (Trust Region with Certified Restarts)", "page_start": 5, "page_end": 5, "type": "Text", "text": "Require: Domain \\mathcal{X}, L, \\sigma , budget T, initial radius r_0, n_{\\text{trust}} regions", "source": "marker_v2", "marker_block_id": "/page/4/Text/13"}
|
| 69 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0068", "section": "Algorithm 3 CGP-TR (Trust Region with Certified Restarts)", "page_start": 5, "page_end": 5, "type": "ListGroup", "text": "1: Initialize n_{\\text{trust}} trust regions at Sobol points with radius r_0 2: for t = 1, ..., T do 3: Compute \\ell_t := \\max_{i \\leq N_t} LCB_i(t) (global lower certificate) 4: Select trust region \\mathcal{T}_j with highest u_t^{(j)} := \\max_{x \\in \\mathcal{T}_i} U_t(x) 5: Run CGP within \\mathcal{T}_j : compute local A_t^{(j)} = \\{x \\in \\mathcal{T}_j : U_t(x) \\ge \\ell_t^{(j)}\\} 6: Query x_{t+1} \\in A_t^{(j)} , observe y_{t+1} 7: if u_t^{(j)} < \\ell_t then 8: Certified restart: restart \\mathcal{T}_j at a new Sobol point with radius r_0 9: else if improvement in \\mathcal{T}_i then 10: Expand: r_j \\leftarrow \\min(2r_j, D/2) 11: else if no improvement for \\tau_{\\text{fail}} iterations then 12: Contract: r_j \\leftarrow \\max(r_j/2, r_{\\min}) 13: end if 14: end for 15: Output: Best point across all regions, local certificate A_T^{(j^*)}", "source": "marker_v2", "marker_block_id": "/page/4/ListGroup/412"}
|
| 70 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0069", "section": "Algorithm 3 CGP-TR (Trust Region with Certified Restarts)", "page_start": 5, "page_end": 5, "type": "Text", "text": "f^* - \\sup_{x \\in \\mathcal{T}_j} f(x) (with \\Delta_j > 0 for suboptimal regions). Assume each region runs CGP with replication ensuring its maximal active-point confidence radius after n within-region samples satisfies \\beta_j(n) \\le c_\\sigma \\sqrt{\\log(c_T/\\delta)/n} for constants c_\\sigma, c_T matching the paper's confidence schedule. If the region radii are eventually bounded so that L \\cdot \\operatorname{diam}(\\mathcal{T}_j) \\le \\Delta_j/8 , then any suboptimal region j is selected at most", "source": "marker_v2", "marker_block_id": "/page/4/Text/29"}
|
| 71 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0070", "section": "Algorithm 3 CGP-TR (Trust Region with Certified Restarts)", "page_start": 5, "page_end": 5, "type": "Equation", "text": "N_j \\le \\frac{64c_\\sigma^2}{\\Delta_j^2} \\log\\left(\\frac{c_T}{\\delta}\\right) + 1", "source": "marker_v2", "marker_block_id": "/page/4/Equation/30"}
|
| 72 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0071", "section": "Algorithm 3 CGP-TR (Trust Region with Certified Restarts)", "page_start": 5, "page_end": 5, "type": "Text", "text": "times before it is eliminated by the certified restart rule.", "source": "marker_v2", "marker_block_id": "/page/4/Text/31"}
|
| 73 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0072", "section": "Algorithm 4 CGP-Hybrid", "page_start": 6, "page_end": 6, "type": "Text", "text": "Require: Domain X , L, σ, budget T, switch threshold ρthresh = 0.5", "source": "marker_v2", "marker_block_id": "/page/5/Text/2"}
|
| 74 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0073", "section": "Algorithm 4 CGP-Hybrid", "page_start": 6, "page_end": 6, "type": "ListGroup", "text": "1: Phase 1: Run CGP until Vol(At) < 0.1 · Vol(X ) or t > T /3 2: Estimate ρ t = Lˆ local(t)/Lˆ global 3: if ρ t < ρthresh then 4: Phase 2: Switch to GP-UCB within A t (GP refinement) 5: Fit GP to points in At, continue with GP-UCB acquisition 6: else", "source": "marker_v2", "marker_block_id": "/page/5/ListGroup/1046"}
|
| 75 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0074", "section": "Algorithm 4 CGP-Hybrid", "page_start": 6, "page_end": 6, "type": "ListGroup", "text": "7: Phase 2: Continue CGP within A t 8: end if 9: Output: Best point, certificate A T (from CGP phase)", "source": "marker_v2", "marker_block_id": "/page/5/ListGroup/1047"}
|
| 76 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0075", "section": "Algorithm 4 CGP-Hybrid", "page_start": 6, "page_end": 6, "type": "Text", "text": "The key advantage is that covering T j requires O((rj/η) d ) points, and since r j ≪ D, this is tractable even for large d. With ntrust = O(log T) regions, CGP-TR explores globally while maintaining local certificates. The allocation bound (Theorem 6.1, item 3) ensures that suboptimal regions receive only O(log T /∆ 2 j ) evaluations before certified elimination, preventing wasted samples.", "source": "marker_v2", "marker_block_id": "/page/5/Text/12"}
|
| 77 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0076", "section": "Algorithm 4 CGP-Hybrid", "page_start": 6, "page_end": 6, "type": "Text", "text": "CGP-TR provides local rather than global certificates, but the certified restart rule guarantees that the region containing x ∗ is never falsely eliminated. This enables scaling to d = 50 to 100 where global Lipschitz methods fail entirely.", "source": "marker_v2", "marker_block_id": "/page/5/Text/13"}
|
| 78 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0077", "section": "7. CGP-Hybrid: Best of Both Worlds", "page_start": 6, "page_end": 6, "type": "Text", "text": "While CGP-TR addresses scalability, some functions exhibit local smoothness that GPs can exploit more effectively than Lipschitz methods. CGP-Hybrid (Algorithm 4) preserves CGP's anytime certificates while allowing any optimizer to refine within the certified active set. The key point is modularity: Phase 1 constructs a certificate At; Phase 2 performs additional optimization restricted to A t without affecting certificate validity. We instantiate Phase 2 with GP-UCB when local smoothness is detected, but other optimizers can be used. This design captures the best of both worlds: CGP's explicit pruning guarantees and GP's ability to exploit local smoothness when present.", "source": "marker_v2", "marker_block_id": "/page/5/Text/15"}
|
| 79 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0078", "section": "7. CGP-Hybrid: Best of Both Worlds", "page_start": 6, "page_end": 6, "type": "Text", "text": "Define the effective smoothness ratio ρ t = Lˆ local(t)/Lˆ global, where Lˆ local(t) is estimated from points within At. When ρ t < 0.5, the function is significantly smoother near the optimum, and GP refinement is beneficial.", "source": "marker_v2", "marker_block_id": "/page/5/Text/16"}
|
| 80 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0079", "section": "7. CGP-Hybrid: Best of Both Worlds", "page_start": 6, "page_end": 6, "type": "Text", "text": "Proposition 7.1 (Hybrid guarantee). CGP-Hybrid achieves:", "source": "marker_v2", "marker_block_id": "/page/5/Text/17"}
|
| 81 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0080", "section": "7. CGP-Hybrid: Best of Both Worlds", "page_start": 6, "page_end": 6, "type": "ListGroup", "text": "1. If ρ ≥ 0.5 : same guarantee as CGP, T = O˜(ε −(2+α) ) . 2. If ρ < 0.5 : after CGP reduces A t to volume V , GP- UCB operates within a restricted domain of effective", "source": "marker_v2", "marker_block_id": "/page/5/ListGroup/1048"}
|
| 82 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0081", "section": "7. CGP-Hybrid: Best of Both Worlds", "page_start": 6, "page_end": 6, "type": "Text", "text": "diameter O(V 1 /d) . The additional sample complexity depends on the GP kernel's information gain γ T over At ; empirically, this yields faster convergence than continuing CGP when the function is locally smooth.", "source": "marker_v2", "marker_block_id": "/page/5/Text/20"}
|
| 83 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0082", "section": "7. CGP-Hybrid: Best of Both Worlds", "page_start": 6, "page_end": 6, "type": "Text", "text": "3. The certificate A T from Phase 1 remains valid regard less of Phase 2 method.", "source": "marker_v2", "marker_block_id": "/page/5/Text/21"}
|
| 84 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0083", "section": "7. CGP-Hybrid: Best of Both Worlds", "page_start": 6, "page_end": 6, "type": "Text", "text": "Proposition 7.2 (Certificate invariance under Phase 2). The certificate A t computed by CGP in Phase 1 remains valid regardless of the Phase-2 optimizer, since validity depends only on the confidence bounds and Lipschitz envelope used to define U t and ℓt . Specifically, any point x /∈ A t sat isfies f(x) < f ∗ − ε t with high probability, where ε t is computable from Phase 1 quantities alone.", "source": "marker_v2", "marker_block_id": "/page/5/Text/22"}
|
| 85 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0084", "section": "7. CGP-Hybrid: Best of Both Worlds", "page_start": 6, "page_end": 6, "type": "Text", "text": "The key insight is that CGP's certificate remains valid even when switching to GP: A t still contains x ∗ with high probability, so GP refinement within A t is safe. This provides the best of both worlds: CGP's certificates and pruning efficiency when ρ ≥ 0.5, and GP's smoothness exploitation when ρ < 0.5.", "source": "marker_v2", "marker_block_id": "/page/5/Text/23"}
|
| 86 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0085", "section": "7. CGP-Hybrid: Best of Both Worlds", "page_start": 6, "page_end": 6, "type": "Text", "text": "Table 6 shows CGP-Hybrid wins on all 12 benchmarks, including Branin (ρ = 0.31) and Rosenbrock (ρ = 0.28) where it detects low ρ and switches to GP refinement, and Needle (ρ = 0.98) where it stays with CGP.", "source": "marker_v2", "marker_block_id": "/page/5/Text/24"}
|
| 87 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0086", "section": "7. CGP-Hybrid: Best of Both Worlds", "page_start": 6, "page_end": 6, "type": "TableGroup", "text": "Table 4. High-dimensional benchmarks (d > 20) at T = 500. Method Rover-60 NAS-36 Ant-100 Random 42.1 ± 1.2 38.4 ± 0.9 51.2 ± 1.4 TuRBO 12.4 ± 0.4 11.2 ± 0.3 18.7 ± 0.6 HEBO 14.1 ± 0.5 12.8 ± 0.4 21.3 ± 0.7 CMA-ES 15.8 ± 0.6 14.1 ± 0.5 19.4 ± 0.6 CGP (intractable for d > 15) CGP-TR † 11.2 ± 0.3 † 10.4 ± 0.3 † 17.1 ± 0.5 † CGP-TR additionally provides local optimality certificates.", "source": "marker_v2", "marker_block_id": "/page/5/TableGroup/1045"}
|
| 88 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0087", "section": "8. Related Work", "page_start": 6, "page_end": 6, "type": "Text", "text": "Our work is primarily grounded in the literature on Lipschitz bandits and global optimization. Foundational approaches, such as the continuum-armed bandits of Kleinberg et al. (2008) and the X-armed bandit framework of Bubeck et al. (2011) , utilize zooming mechanisms to achieve regret bounds depending on the near-optimality dimension. These concepts were refined for deterministic and stochastic settings via tree-based algorithms like DOO/SOO (Munos, 2011) and StoSOO (Valko et al., 2013) . However, a key distinction is that zooming algorithms maintain pruning implicitly as an analysis artifact, whereas CGP exposes the active set A t as a computable object with measurable volume. This explicit geometric approach also relates to partitionbased global optimization methods like DIRECT (Jones", "source": "marker_v2", "marker_block_id": "/page/5/Text/29"}
|
| 89 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0088", "section": "8. Related Work", "page_start": 7, "page_end": 7, "type": "TableGroup", "text": "Table 5. CGP-Adaptive with varying initial Lˆ0. Robust to 100× underestimation. Initial Lˆ 0 Doublings Final L/L ˆ ∗ Regret (×10−2 ) Overhead ∗ L (oracle) 0 1.0 2.9 ± 0.1 1.0× ∗/2 L 1 1.0 3.0 ± 0.1 1.03× ∗/10 L 4 1.6 3.1 ± 0.1 1.07× ∗/100 L 7 1.3 3.2 ± 0.2 1.12× LIPO (adaptive) – – 6.2 ± 0.3 – Table 6. CGP-Hybrid smoothness detection. ρ < 0.5 triggers GP refinement.", "source": "marker_v2", "marker_block_id": "/page/6/TableGroup/804"}
|
| 90 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0089", "section": "8. Related Work", "page_start": 7, "page_end": 7, "type": "Table", "text": "Benchmark ρˆ Phase 2 CGP-H Regret Best Baseline Needle-2D 0.98 CGP 1.1 ± 0.1 1.8 (TuRBO) Branin 0.31 GP 1.4 ± 0.1 1.6 (HEBO) Hartmann-6 0.72 CGP 2.7 ± 0.1 3.1 (TuRBO) Rosenbrock 0.28 GP 3.4 ± 0.1 3.7 (HEBO) Ackley-10 0.85 CGP 7.8 ± 0.3 9.4 (HEBO) Levy-5 0.67 CGP 3.5 ± 0.2 4.1 (HEBO) SVM-RBF 0.74 CGP 2.6 ± 0.1 3.1 (HEBO) LunarLander 0.81 CGP 6.1 ± 0.3 7.0 (HEBO)", "source": "marker_v2", "marker_block_id": "/page/6/Table/5"}
|
| 91 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0090", "section": "8. Related Work", "page_start": 7, "page_end": 7, "type": "Text", "text": "et al., 1993) and LIPO (Malherbe & Vayatis, 2017) . While LIPO addresses the unknown Lipschitz constant, it lacks the certificate guarantees provided by our CGP-Adaptive doubling scheme (Shihab et al., 2025c) . Our theoretical analysis further draws on sample complexity results under margin conditions from the finite-arm setting (Audibert & Bubeck, 2010; Jamieson & Nowak, 2014; Shihab et al., 2025b) , adapting confidence bounds from the UCB frame w ork (Auer et al., 2002) to continuous spaces with explicit uncertainty representation similar to safe optimization levelsets (Sui et al., 2015) .", "source": "marker_v2", "marker_block_id": "/page/6/Text/6"}
|
| 92 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0091", "section": "8. Related Work", "page_start": 7, "page_end": 7, "type": "Text", "text": "In the broader context of black-box optimization, Bayesian methods provide the standard alternative for uncertainty quantification. Classic approaches like GP-UCB (Srinivas et al., 2009) , Entropy Search (Hennig & Schuler, 2012; Hernandez-Lobato et al. ´ , 2014) , and Thompson Sampling (Thompson, 1933) offer strong performance but scale cubically with observations. To address high-dimensional scaling, recent work has introduced trust regions (TuRBO) (Eriksson et al., 2019) and nonstationary priors (HEBO) (Cowen-Rivers et al., 2022) . More recently, advanced heuristics such as Bounce (Papenmeier et al., 2024) have improved geometric adaptation for mixed spaces, while Prior-Fitted Networks (Hollmann et al., 2023) and generative diffusion models like Diff-BBO (Wu et al., 2024) exploit massive pretraining to minimize regret rapidly. While these emerging methods achieve impressive empirical results, they remain fundamentally heuristic, lacking the computable stopping criteria or active set containment guarantees that are central to CGP. We therefore focus our comparison on established", "source": "marker_v2", "marker_block_id": "/page/6/Text/7"}
|
| 93 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0092", "section": "8. Related Work", "page_start": 7, "page_end": 7, "type": "Text", "text": "baselines to isolate the specific utility of our certification mechanism, distinguishing our approach from heuristic hyperparameter tuners like Hyperband (Li et al., 2018) by prioritizing provable safety over raw speed.", "source": "marker_v2", "marker_block_id": "/page/6/Text/8"}
|
| 94 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0093", "section": "9. Experiments", "page_start": 7, "page_end": 7, "type": "Text", "text": "We evaluate CGP variants on 12 benchmarks spanning d ∈ [2, 100], measuring simple regret, certificate utility, and scalability. Code will be available upon acceptance.", "source": "marker_v2", "marker_block_id": "/page/6/Text/10"}
|
| 95 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0094", "section": "9. Experiments", "page_start": 7, "page_end": 7, "type": "Text", "text": "Setup. We compare against 9 baselines: Random Search, GP-UCB, TuRBO, HEBO, BORE, HOO, StoSOO, LIPO, and SAASBO (see Appendix E for configurations). We evaluate on 12 benchmarks spanning d ∈ [2, 100]: lowdimensional (Needle-2D, Branin, Hartmann-6, Levy-5, Rosenbrock-4), medium-dimensional (Ackley-10, SVM-RBF-6, LunarLander-12), and high-dimensional (Rover-60, NAS-36, MuJoCo-Ant-100). All experiments use 30 runs with σ = 0.1 noise; we report mean ± SE with Bonferronicorrected t-tests (p < 0.05).", "source": "marker_v2", "marker_block_id": "/page/6/Text/11"}
|
| 96 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0095", "section": "9. Experiments", "page_start": 7, "page_end": 7, "type": "Text", "text": "Table 3 shows CGP-Hybrid performs best among tested methods on all 7 low and medium-dimensional benchmarks. On Branin and Rosenbrock where vanilla CGP lost to HEBO, CGP-Hybrid detects ρ < 0.5 and switches to GP refinement, achieving 12% and 8% improvement over HEBO respectively. On Needle where ρ ≈ 1, CGP-Hybrid stays with CGP and matches vanilla CGP performance. For high-dimensional problems, Table 4 demonstrates CGP-TR scales to d = 100 while outperforming TuRBO by", "source": "marker_v2", "marker_block_id": "/page/6/Text/12"}
|
| 97 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0096", "section": "9. Experiments", "page_start": 8, "page_end": 8, "type": "Text", "text": "9 to 12%. Critically, CGP-TR provides local certificates within trust regions, enabling principled stopping—a capability TuRBO lacks. Regarding adaptive estimation, Table 5 shows CGP-Adaptive is robust to initial underestimation: even with \\hat{L}_0=L^*/100 , performance degrades only 10% with 7 doublings, validating Theorem 5.1's O(\\log(L^*/\\hat{L}_0)) overhead. Finally, Table 6 confirms CGP-Hybrid correctly identifies when to switch: Branin ( \\rho=0.31 ) and Rosenbrock ( \\rho=0.28 ) trigger GP refinement, achieving 12% and 8% improvement over HEBO. Benchmarks with \\rho>0.5 stay with CGP, maintaining certificate validity.", "source": "marker_v2", "marker_block_id": "/page/7/Text/1"}
|
| 98 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0097", "section": "9. Experiments", "page_start": 8, "page_end": 8, "type": "Text", "text": "Shrinkage validation. Across all benchmarks, we observe Vol(A_t) shrinks to < 5% by T = 100, with empirical decay rates closely matching the theoretical bound from Theorem 4.6. This confirms our analysis is tight and the margin condition captures the true problem difficulty.", "source": "marker_v2", "marker_block_id": "/page/7/Text/2"}
|
| 99 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0098", "section": "9. Experiments", "page_start": 8, "page_end": 8, "type": "TableGroup", "text": "Table 7. Certificate-enabled early stopping on Hartmann-6. CGP uniquely provides actionable stopping criteria. Stopping Rule Samples Regret ( \\times 10^{-2} ) Savings Fixed T = 200 200 2.9 \\pm 0.1 _ Vol(A_t) < 10\\% 82 \\pm 9 3.8 \\pm 0.2 59% Vol(A_t) < 5\\% 118 \\pm 14 3.2 \\pm 0.2 41% {\\rm Gap\\ bound} < 0.05 134 \\pm 12 3.0 \\pm 0.1 33%", "source": "marker_v2", "marker_block_id": "/page/7/TableGroup/395"}
|
| 100 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0099", "section": "9. Experiments", "page_start": 8, "page_end": 8, "type": "Text", "text": "Table 7 demonstrates certificate utility: stopping at Vol(A_t) < 10\\% saves 59% of samples with only 31% regret increase. No baseline provides such principled stopping rules. In d>20, we use the computable gap proxy \\varepsilon_t:=2(\\beta_t+L\\eta_t)+\\gamma_t as the primary criterion. Beyond sample efficiency, CGP also offers computational advantages. Table 8 shows CGP variants are 6 to 8 times faster than GP-based methods due to their O(n) per-iteration cost versus GP's O(n^3) . Finally, we ablate CGP's components to understand their individual contributions.", "source": "marker_v2", "marker_block_id": "/page/7/Text/5"}
|
| 101 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0100", "section": "9. Experiments", "page_start": 8, "page_end": 8, "type": "TableGroup", "text": "Table 8. Wall-clock time (seconds) for T=200 on Hartmann-6. Method Time (s) Regret ( \\times 10^{-2} ) Speedup CGP 58 2.9 8× CGP-Adaptive 64 3.0 7.5 \\times CGP-Hybrid 72 2.7 6.7 \\times GP-UCB 480 4.2 1 \\times TuRBO 620 3.1 0.8 \\times HEBO 890 3.3 0.5 \\times", "source": "marker_v2", "marker_block_id": "/page/7/TableGroup/396"}
|
| 102 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0101", "section": "9. Experiments", "page_start": 8, "page_end": 8, "type": "Text", "text": "Table 9 shows all components contribute: removing pruning certificates increases regret 78%, coverage penalty 44%, replication 33%. GP refinement provides 7% improvement on Hartmann-6 where \\rho=0.72 is borderline. Table 10 validates that \\hat{\\alpha} < d across all benchmarks, confirming the margin condition holds and our complexity bounds apply.", "source": "marker_v2", "marker_block_id": "/page/7/Text/8"}
|
| 103 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0102", "section": "9. Experiments", "page_start": 8, "page_end": 8, "type": "TableGroup", "text": "Table 9. Ablation study on Hartmann-6. All components contribute. Variant Regret ( \\times 10^{-2} ) Vol(A_{200}) CGP-Hybrid (full) \\textbf{2.7} \\pm \\textbf{0.1} 2.1% GP refinement 2.9 \\pm 0.1 2.1% pruning certificate 4.8 \\pm 0.2 _ coverage penalty 3.9 \\pm 0.2 3.8% replication 3.6 \\pm 0.2 2.9% CGP-TR (d=6) 2.8 \\pm 0.1 2.4% (local) CGP-Adaptive 3.0 \\pm 0.1 2.4% Table 10. Empirical \\hat{\\alpha} estimates from shrinkage trajectories.", "source": "marker_v2", "marker_block_id": "/page/7/TableGroup/397"}
|
| 104 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0103", "section": "9. Experiments", "page_start": 8, "page_end": 8, "type": "Table", "text": "Benchmark d â (95% CI) True \\alpha \\hat{\\alpha} < d ? Needle-2D 2 1.8 \\pm 0.2 2.0 √ Branin 2 1.2 \\pm 0.1 _ \\checkmark Hartmann-6 6 2.4 \\pm 0.3 _ \\checkmark Ackley-10 10 3.2 \\pm 0.4 _ \\checkmark Rover-60 60 8.4 \\pm 1.2 _ \\checkmark", "source": "marker_v2", "marker_block_id": "/page/7/Table/12"}
|
| 105 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0104", "section": "10. Conclusion", "page_start": 8, "page_end": 8, "type": "Text", "text": "We introduced Certificate-Guided Pruning (CGP), an algorithm for stochastic Lipschitz optimization that maintains explicit active sets with provable shrinkage guarantees. Under a margin condition with near-optimality dimension \\alpha , we prove Vol(A_t) \\leq C \\cdot (2(\\beta_t + L\\eta_t) + \\gamma_t)^{d-\\alpha} , yielding sample complexity \\tilde{O}(\\varepsilon^{-(2+\\alpha)}) with anytime valid certificates. Three extensions broaden applicability: CGP-Adaptive learns L online with O(\\log T) overhead, CGP-TR scales to d > 50 via trust regions, and CGP-Hybrid switches to GP refinement when local smoothness is detected. The margin condition holds broadly for isolated maxima with nondegenerate Hessian \\alpha = d/2 , for polynomial decay \\alpha = d/p and can be estimated online from shrinkage trajectories. Limitations include requiring Lipschitz continuity and dimension constraints ( d \\le 15 for vanilla CGP, d \\le 100 for CGP-TR); practical guidance is in Appendix D. Future directions include safe optimization using A_t for safety certificates and CGP-TR with random embeddings for global high-dimensional certificates.", "source": "marker_v2", "marker_block_id": "/page/7/Text/14"}
|
| 106 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0105", "section": "Impact Statement", "page_start": 8, "page_end": 8, "type": "Text", "text": "This paper introduces Certificate-Guided Pruning (CGP), a method designed to improve the sample efficiency of black-box optimization in resource-constrained settings. By providing explicit optimality certificates and principled stopping criteria, our approach significantly reduces the computational budget required for expensive tasks such as neural architecture search and simulation-based engineering, directly contributing to lower energy consumption and carbon footprints. Furthermore, the ability to certify suboptimal regions enhances reliability in safety-critical applications like", "source": "marker_v2", "marker_block_id": "/page/7/Text/16"}
|
| 107 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0106", "section": "Impact Statement", "page_start": 9, "page_end": 9, "type": "Text", "text": "robotics. However, practitioners must ensure the validity of the Lipschitz assumption, as violations could lead to the incorrect pruning of optimal solutions.", "source": "marker_v2", "marker_block_id": "/page/8/Text/1"}
|
icml26/5c3d6bff-e8ce-4d9f-840b-719084582491/marker_meta.json
ADDED
|
@@ -0,0 +1,1963 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
{
|
| 2 |
+
"table_of_contents": [
|
| 3 |
+
{
|
| 4 |
+
"title": "Certificate-Guided Pruning for Stochastic Lipschitz Optimization",
|
| 5 |
+
"heading_level": null,
|
| 6 |
+
"page_id": 0,
|
| 7 |
+
"polygon": [
|
| 8 |
+
[
|
| 9 |
+
96.75,
|
| 10 |
+
89.25
|
| 11 |
+
],
|
| 12 |
+
[
|
| 13 |
+
500.25,
|
| 14 |
+
89.25
|
| 15 |
+
],
|
| 16 |
+
[
|
| 17 |
+
500.25,
|
| 18 |
+
103.25390625
|
| 19 |
+
],
|
| 20 |
+
[
|
| 21 |
+
96.75,
|
| 22 |
+
103.25390625
|
| 23 |
+
]
|
| 24 |
+
]
|
| 25 |
+
},
|
| 26 |
+
{
|
| 27 |
+
"title": "Anonymous Authors1",
|
| 28 |
+
"heading_level": null,
|
| 29 |
+
"page_id": 0,
|
| 30 |
+
"polygon": [
|
| 31 |
+
[
|
| 32 |
+
246.75,
|
| 33 |
+
141.15234375
|
| 34 |
+
],
|
| 35 |
+
[
|
| 36 |
+
340.06640625,
|
| 37 |
+
141.15234375
|
| 38 |
+
],
|
| 39 |
+
[
|
| 40 |
+
340.06640625,
|
| 41 |
+
151.5
|
| 42 |
+
],
|
| 43 |
+
[
|
| 44 |
+
246.75,
|
| 45 |
+
151.5
|
| 46 |
+
]
|
| 47 |
+
]
|
| 48 |
+
},
|
| 49 |
+
{
|
| 50 |
+
"title": "Abstract",
|
| 51 |
+
"heading_level": null,
|
| 52 |
+
"page_id": 0,
|
| 53 |
+
"polygon": [
|
| 54 |
+
[
|
| 55 |
+
148.5,
|
| 56 |
+
175.5
|
| 57 |
+
],
|
| 58 |
+
[
|
| 59 |
+
195.75,
|
| 60 |
+
175.5
|
| 61 |
+
],
|
| 62 |
+
[
|
| 63 |
+
195.75,
|
| 64 |
+
186.0
|
| 65 |
+
],
|
| 66 |
+
[
|
| 67 |
+
148.5,
|
| 68 |
+
186.0
|
| 69 |
+
]
|
| 70 |
+
]
|
| 71 |
+
},
|
| 72 |
+
{
|
| 73 |
+
"title": "1. Introduction",
|
| 74 |
+
"heading_level": null,
|
| 75 |
+
"page_id": 0,
|
| 76 |
+
"polygon": [
|
| 77 |
+
[
|
| 78 |
+
53.490234375,
|
| 79 |
+
486.0
|
| 80 |
+
],
|
| 81 |
+
[
|
| 82 |
+
132.75,
|
| 83 |
+
486.0
|
| 84 |
+
],
|
| 85 |
+
[
|
| 86 |
+
132.75,
|
| 87 |
+
496.16015625
|
| 88 |
+
],
|
| 89 |
+
[
|
| 90 |
+
53.490234375,
|
| 91 |
+
496.16015625
|
| 92 |
+
]
|
| 93 |
+
]
|
| 94 |
+
},
|
| 95 |
+
{
|
| 96 |
+
"title": "2. Problem Formulation",
|
| 97 |
+
"heading_level": null,
|
| 98 |
+
"page_id": 1,
|
| 99 |
+
"polygon": [
|
| 100 |
+
[
|
| 101 |
+
303.609375,
|
| 102 |
+
333.75
|
| 103 |
+
],
|
| 104 |
+
[
|
| 105 |
+
430.5,
|
| 106 |
+
333.75
|
| 107 |
+
],
|
| 108 |
+
[
|
| 109 |
+
430.5,
|
| 110 |
+
343.5
|
| 111 |
+
],
|
| 112 |
+
[
|
| 113 |
+
303.609375,
|
| 114 |
+
343.5
|
| 115 |
+
]
|
| 116 |
+
]
|
| 117 |
+
},
|
| 118 |
+
{
|
| 119 |
+
"title": "3. Algorithm: Certificate-Guided Pruning",
|
| 120 |
+
"heading_level": null,
|
| 121 |
+
"page_id": 1,
|
| 122 |
+
"polygon": [
|
| 123 |
+
[
|
| 124 |
+
306.0,
|
| 125 |
+
675.0
|
| 126 |
+
],
|
| 127 |
+
[
|
| 128 |
+
521.25,
|
| 129 |
+
675.0
|
| 130 |
+
],
|
| 131 |
+
[
|
| 132 |
+
521.25,
|
| 133 |
+
685.265625
|
| 134 |
+
],
|
| 135 |
+
[
|
| 136 |
+
306.0,
|
| 137 |
+
685.265625
|
| 138 |
+
]
|
| 139 |
+
]
|
| 140 |
+
},
|
| 141 |
+
{
|
| 142 |
+
"title": "Algorithm 1 Certificate-Guided Pruning (CGP)",
|
| 143 |
+
"heading_level": null,
|
| 144 |
+
"page_id": 2,
|
| 145 |
+
"polygon": [
|
| 146 |
+
[
|
| 147 |
+
53.19140625,
|
| 148 |
+
68.25
|
| 149 |
+
],
|
| 150 |
+
[
|
| 151 |
+
246.0,
|
| 152 |
+
68.25
|
| 153 |
+
],
|
| 154 |
+
[
|
| 155 |
+
246.0,
|
| 156 |
+
78.1171875
|
| 157 |
+
],
|
| 158 |
+
[
|
| 159 |
+
53.19140625,
|
| 160 |
+
78.1171875
|
| 161 |
+
]
|
| 162 |
+
]
|
| 163 |
+
},
|
| 164 |
+
{
|
| 165 |
+
"title": "4. Theoretical Analysis",
|
| 166 |
+
"heading_level": null,
|
| 167 |
+
"page_id": 2,
|
| 168 |
+
"polygon": [
|
| 169 |
+
[
|
| 170 |
+
303.908203125,
|
| 171 |
+
145.01953125
|
| 172 |
+
],
|
| 173 |
+
[
|
| 174 |
+
424.5,
|
| 175 |
+
145.01953125
|
| 176 |
+
],
|
| 177 |
+
[
|
| 178 |
+
424.5,
|
| 179 |
+
155.07421875
|
| 180 |
+
],
|
| 181 |
+
[
|
| 182 |
+
303.908203125,
|
| 183 |
+
155.07421875
|
| 184 |
+
]
|
| 185 |
+
]
|
| 186 |
+
},
|
| 187 |
+
{
|
| 188 |
+
"title": "5. CGP-Adaptive: Learning L Online",
|
| 189 |
+
"heading_level": null,
|
| 190 |
+
"page_id": 3,
|
| 191 |
+
"polygon": [
|
| 192 |
+
[
|
| 193 |
+
53.19140625,
|
| 194 |
+
573.0
|
| 195 |
+
],
|
| 196 |
+
[
|
| 197 |
+
246.75,
|
| 198 |
+
573.0
|
| 199 |
+
],
|
| 200 |
+
[
|
| 201 |
+
246.75,
|
| 202 |
+
583.55859375
|
| 203 |
+
],
|
| 204 |
+
[
|
| 205 |
+
53.19140625,
|
| 206 |
+
583.55859375
|
| 207 |
+
]
|
| 208 |
+
]
|
| 209 |
+
},
|
| 210 |
+
{
|
| 211 |
+
"title": "Algorithm 2 CGP-Adaptive",
|
| 212 |
+
"heading_level": null,
|
| 213 |
+
"page_id": 3,
|
| 214 |
+
"polygon": [
|
| 215 |
+
[
|
| 216 |
+
305.40234375,
|
| 217 |
+
68.25
|
| 218 |
+
],
|
| 219 |
+
[
|
| 220 |
+
421.34765625,
|
| 221 |
+
68.25
|
| 222 |
+
],
|
| 223 |
+
[
|
| 224 |
+
421.34765625,
|
| 225 |
+
78.1171875
|
| 226 |
+
],
|
| 227 |
+
[
|
| 228 |
+
305.40234375,
|
| 229 |
+
78.1171875
|
| 230 |
+
]
|
| 231 |
+
]
|
| 232 |
+
},
|
| 233 |
+
{
|
| 234 |
+
"title": "6. CGP-TR: Trust Regions for High Dimensions",
|
| 235 |
+
"heading_level": null,
|
| 236 |
+
"page_id": 3,
|
| 237 |
+
"polygon": [
|
| 238 |
+
[
|
| 239 |
+
305.40234375,
|
| 240 |
+
647.75390625
|
| 241 |
+
],
|
| 242 |
+
[
|
| 243 |
+
490.5,
|
| 244 |
+
647.75390625
|
| 245 |
+
],
|
| 246 |
+
[
|
| 247 |
+
490.5,
|
| 248 |
+
672.0
|
| 249 |
+
],
|
| 250 |
+
[
|
| 251 |
+
305.40234375,
|
| 252 |
+
672.0
|
| 253 |
+
]
|
| 254 |
+
]
|
| 255 |
+
},
|
| 256 |
+
{
|
| 257 |
+
"title": "Algorithm 3 CGP-TR (Trust Region with Certified Restarts)",
|
| 258 |
+
"heading_level": null,
|
| 259 |
+
"page_id": 4,
|
| 260 |
+
"polygon": [
|
| 261 |
+
[
|
| 262 |
+
304.20703125,
|
| 263 |
+
237.75
|
| 264 |
+
],
|
| 265 |
+
[
|
| 266 |
+
542.25,
|
| 267 |
+
237.75
|
| 268 |
+
],
|
| 269 |
+
[
|
| 270 |
+
542.25,
|
| 271 |
+
247.5
|
| 272 |
+
],
|
| 273 |
+
[
|
| 274 |
+
304.20703125,
|
| 275 |
+
247.5
|
| 276 |
+
]
|
| 277 |
+
]
|
| 278 |
+
},
|
| 279 |
+
{
|
| 280 |
+
"title": "Algorithm 4 CGP-Hybrid",
|
| 281 |
+
"heading_level": null,
|
| 282 |
+
"page_id": 5,
|
| 283 |
+
"polygon": [
|
| 284 |
+
[
|
| 285 |
+
48.708984375,
|
| 286 |
+
68.44921875
|
| 287 |
+
],
|
| 288 |
+
[
|
| 289 |
+
160.2372283935547,
|
| 290 |
+
68.44921875
|
| 291 |
+
],
|
| 292 |
+
[
|
| 293 |
+
160.2372283935547,
|
| 294 |
+
80.05078125
|
| 295 |
+
],
|
| 296 |
+
[
|
| 297 |
+
48.708984375,
|
| 298 |
+
80.05078125
|
| 299 |
+
]
|
| 300 |
+
]
|
| 301 |
+
},
|
| 302 |
+
{
|
| 303 |
+
"title": "7. CGP-Hybrid: Best of Both Worlds",
|
| 304 |
+
"heading_level": null,
|
| 305 |
+
"page_id": 5,
|
| 306 |
+
"polygon": [
|
| 307 |
+
[
|
| 308 |
+
53.7890625,
|
| 309 |
+
427.91717529296875
|
| 310 |
+
],
|
| 311 |
+
[
|
| 312 |
+
243.88983154296875,
|
| 313 |
+
427.91717529296875
|
| 314 |
+
],
|
| 315 |
+
[
|
| 316 |
+
243.88983154296875,
|
| 317 |
+
439.87237548828125
|
| 318 |
+
],
|
| 319 |
+
[
|
| 320 |
+
53.7890625,
|
| 321 |
+
439.87237548828125
|
| 322 |
+
]
|
| 323 |
+
]
|
| 324 |
+
},
|
| 325 |
+
{
|
| 326 |
+
"title": "8. Related Work",
|
| 327 |
+
"heading_level": null,
|
| 328 |
+
"page_id": 5,
|
| 329 |
+
"polygon": [
|
| 330 |
+
[
|
| 331 |
+
307.4399719238281,
|
| 332 |
+
530.96484375
|
| 333 |
+
],
|
| 334 |
+
[
|
| 335 |
+
390.5525207519531,
|
| 336 |
+
530.96484375
|
| 337 |
+
],
|
| 338 |
+
[
|
| 339 |
+
390.5525207519531,
|
| 340 |
+
543.0493621826172
|
| 341 |
+
],
|
| 342 |
+
[
|
| 343 |
+
307.4399719238281,
|
| 344 |
+
543.0493621826172
|
| 345 |
+
]
|
| 346 |
+
]
|
| 347 |
+
},
|
| 348 |
+
{
|
| 349 |
+
"title": "9. Experiments",
|
| 350 |
+
"heading_level": null,
|
| 351 |
+
"page_id": 6,
|
| 352 |
+
"polygon": [
|
| 353 |
+
[
|
| 354 |
+
305.701171875,
|
| 355 |
+
407.21484375
|
| 356 |
+
],
|
| 357 |
+
[
|
| 358 |
+
384.47930908203125,
|
| 359 |
+
407.21484375
|
| 360 |
+
],
|
| 361 |
+
[
|
| 362 |
+
384.47930908203125,
|
| 363 |
+
419.5563659667969
|
| 364 |
+
],
|
| 365 |
+
[
|
| 366 |
+
305.701171875,
|
| 367 |
+
419.5563659667969
|
| 368 |
+
]
|
| 369 |
+
]
|
| 370 |
+
},
|
| 371 |
+
{
|
| 372 |
+
"title": "10. Conclusion",
|
| 373 |
+
"heading_level": null,
|
| 374 |
+
"page_id": 7,
|
| 375 |
+
"polygon": [
|
| 376 |
+
[
|
| 377 |
+
306.0,
|
| 378 |
+
315.0
|
| 379 |
+
],
|
| 380 |
+
[
|
| 381 |
+
383.25,
|
| 382 |
+
315.0
|
| 383 |
+
],
|
| 384 |
+
[
|
| 385 |
+
383.25,
|
| 386 |
+
325.5
|
| 387 |
+
],
|
| 388 |
+
[
|
| 389 |
+
306.0,
|
| 390 |
+
325.5
|
| 391 |
+
]
|
| 392 |
+
]
|
| 393 |
+
},
|
| 394 |
+
{
|
| 395 |
+
"title": "Impact Statement",
|
| 396 |
+
"heading_level": null,
|
| 397 |
+
"page_id": 7,
|
| 398 |
+
"polygon": [
|
| 399 |
+
[
|
| 400 |
+
305.40234375,
|
| 401 |
+
579.0
|
| 402 |
+
],
|
| 403 |
+
[
|
| 404 |
+
399.75,
|
| 405 |
+
579.0
|
| 406 |
+
],
|
| 407 |
+
[
|
| 408 |
+
399.75,
|
| 409 |
+
588.97265625
|
| 410 |
+
],
|
| 411 |
+
[
|
| 412 |
+
305.40234375,
|
| 413 |
+
588.97265625
|
| 414 |
+
]
|
| 415 |
+
]
|
| 416 |
+
},
|
| 417 |
+
{
|
| 418 |
+
"title": "References",
|
| 419 |
+
"heading_level": null,
|
| 420 |
+
"page_id": 8,
|
| 421 |
+
"polygon": [
|
| 422 |
+
[
|
| 423 |
+
52.892578125,
|
| 424 |
+
120.26953125
|
| 425 |
+
],
|
| 426 |
+
[
|
| 427 |
+
110.9838638305664,
|
| 428 |
+
120.26953125
|
| 429 |
+
],
|
| 430 |
+
[
|
| 431 |
+
110.9838638305664,
|
| 432 |
+
132.27838134765625
|
| 433 |
+
],
|
| 434 |
+
[
|
| 435 |
+
52.892578125,
|
| 436 |
+
132.27838134765625
|
| 437 |
+
]
|
| 438 |
+
]
|
| 439 |
+
},
|
| 440 |
+
{
|
| 441 |
+
"title": "A. Extended Problem Formulation",
|
| 442 |
+
"heading_level": null,
|
| 443 |
+
"page_id": 9,
|
| 444 |
+
"polygon": [
|
| 445 |
+
[
|
| 446 |
+
304.20703125,
|
| 447 |
+
67.67578125
|
| 448 |
+
],
|
| 449 |
+
[
|
| 450 |
+
486.0,
|
| 451 |
+
67.67578125
|
| 452 |
+
],
|
| 453 |
+
[
|
| 454 |
+
486.0,
|
| 455 |
+
77.25
|
| 456 |
+
],
|
| 457 |
+
[
|
| 458 |
+
304.20703125,
|
| 459 |
+
77.25
|
| 460 |
+
]
|
| 461 |
+
]
|
| 462 |
+
},
|
| 463 |
+
{
|
| 464 |
+
"title": "B. Algorithm Implementation Details",
|
| 465 |
+
"heading_level": null,
|
| 466 |
+
"page_id": 9,
|
| 467 |
+
"polygon": [
|
| 468 |
+
[
|
| 469 |
+
306.0,
|
| 470 |
+
480.75
|
| 471 |
+
],
|
| 472 |
+
[
|
| 473 |
+
498.0,
|
| 474 |
+
480.75
|
| 475 |
+
],
|
| 476 |
+
[
|
| 477 |
+
498.0,
|
| 478 |
+
490.5
|
| 479 |
+
],
|
| 480 |
+
[
|
| 481 |
+
306.0,
|
| 482 |
+
490.5
|
| 483 |
+
]
|
| 484 |
+
]
|
| 485 |
+
},
|
| 486 |
+
{
|
| 487 |
+
"title": "B.1. Score Maximization and Replication",
|
| 488 |
+
"heading_level": null,
|
| 489 |
+
"page_id": 9,
|
| 490 |
+
"polygon": [
|
| 491 |
+
[
|
| 492 |
+
306.0,
|
| 493 |
+
501.0
|
| 494 |
+
],
|
| 495 |
+
[
|
| 496 |
+
482.30859375,
|
| 497 |
+
501.0
|
| 498 |
+
],
|
| 499 |
+
[
|
| 500 |
+
482.30859375,
|
| 501 |
+
511.62890625
|
| 502 |
+
],
|
| 503 |
+
[
|
| 504 |
+
306.0,
|
| 505 |
+
511.62890625
|
| 506 |
+
]
|
| 507 |
+
]
|
| 508 |
+
},
|
| 509 |
+
{
|
| 510 |
+
"title": "B.2. Active Set Computation",
|
| 511 |
+
"heading_level": null,
|
| 512 |
+
"page_id": 10,
|
| 513 |
+
"polygon": [
|
| 514 |
+
[
|
| 515 |
+
52.59375,
|
| 516 |
+
93.0
|
| 517 |
+
],
|
| 518 |
+
[
|
| 519 |
+
177.0,
|
| 520 |
+
93.0
|
| 521 |
+
],
|
| 522 |
+
[
|
| 523 |
+
177.0,
|
| 524 |
+
104.02734375
|
| 525 |
+
],
|
| 526 |
+
[
|
| 527 |
+
52.59375,
|
| 528 |
+
104.02734375
|
| 529 |
+
]
|
| 530 |
+
]
|
| 531 |
+
},
|
| 532 |
+
{
|
| 533 |
+
"title": "C. Proofs",
|
| 534 |
+
"heading_level": null,
|
| 535 |
+
"page_id": 10,
|
| 536 |
+
"polygon": [
|
| 537 |
+
[
|
| 538 |
+
53.7890625,
|
| 539 |
+
599.02734375
|
| 540 |
+
],
|
| 541 |
+
[
|
| 542 |
+
104.25,
|
| 543 |
+
599.02734375
|
| 544 |
+
],
|
| 545 |
+
[
|
| 546 |
+
104.25,
|
| 547 |
+
610.5
|
| 548 |
+
],
|
| 549 |
+
[
|
| 550 |
+
53.7890625,
|
| 551 |
+
610.5
|
| 552 |
+
]
|
| 553 |
+
]
|
| 554 |
+
},
|
| 555 |
+
{
|
| 556 |
+
"title": "C.1. Proof to Lemma 4.1",
|
| 557 |
+
"heading_level": null,
|
| 558 |
+
"page_id": 10,
|
| 559 |
+
"polygon": [
|
| 560 |
+
[
|
| 561 |
+
53.490234375,
|
| 562 |
+
620.25
|
| 563 |
+
],
|
| 564 |
+
[
|
| 565 |
+
159.0,
|
| 566 |
+
620.25
|
| 567 |
+
],
|
| 568 |
+
[
|
| 569 |
+
159.0,
|
| 570 |
+
629.25
|
| 571 |
+
],
|
| 572 |
+
[
|
| 573 |
+
53.490234375,
|
| 574 |
+
629.25
|
| 575 |
+
]
|
| 576 |
+
]
|
| 577 |
+
},
|
| 578 |
+
{
|
| 579 |
+
"title": "C.2. Proof to Lemma 4.2 (UCB envelope is valid)",
|
| 580 |
+
"heading_level": null,
|
| 581 |
+
"page_id": 10,
|
| 582 |
+
"polygon": [
|
| 583 |
+
[
|
| 584 |
+
306.0,
|
| 585 |
+
273.41015625
|
| 586 |
+
],
|
| 587 |
+
[
|
| 588 |
+
513.75,
|
| 589 |
+
273.41015625
|
| 590 |
+
],
|
| 591 |
+
[
|
| 592 |
+
513.75,
|
| 593 |
+
283.46484375
|
| 594 |
+
],
|
| 595 |
+
[
|
| 596 |
+
306.0,
|
| 597 |
+
283.46484375
|
| 598 |
+
]
|
| 599 |
+
]
|
| 600 |
+
},
|
| 601 |
+
{
|
| 602 |
+
"title": "C.3. Proof to Lemma 4.3 (Envelope slack bound)",
|
| 603 |
+
"heading_level": null,
|
| 604 |
+
"page_id": 10,
|
| 605 |
+
"polygon": [
|
| 606 |
+
[
|
| 607 |
+
306.0,
|
| 608 |
+
465.99609375
|
| 609 |
+
],
|
| 610 |
+
[
|
| 611 |
+
514.5,
|
| 612 |
+
465.99609375
|
| 613 |
+
],
|
| 614 |
+
[
|
| 615 |
+
514.5,
|
| 616 |
+
475.5
|
| 617 |
+
],
|
| 618 |
+
[
|
| 619 |
+
306.0,
|
| 620 |
+
475.5
|
| 621 |
+
]
|
| 622 |
+
]
|
| 623 |
+
},
|
| 624 |
+
{
|
| 625 |
+
"title": "C.4. Proof to Theorem 4.5",
|
| 626 |
+
"heading_level": null,
|
| 627 |
+
"page_id": 11,
|
| 628 |
+
"polygon": [
|
| 629 |
+
[
|
| 630 |
+
52.59375,
|
| 631 |
+
197.2265625
|
| 632 |
+
],
|
| 633 |
+
[
|
| 634 |
+
165.47691345214844,
|
| 635 |
+
197.2265625
|
| 636 |
+
],
|
| 637 |
+
[
|
| 638 |
+
165.47691345214844,
|
| 639 |
+
209.09710693359375
|
| 640 |
+
],
|
| 641 |
+
[
|
| 642 |
+
52.59375,
|
| 643 |
+
209.09710693359375
|
| 644 |
+
]
|
| 645 |
+
]
|
| 646 |
+
},
|
| 647 |
+
{
|
| 648 |
+
"title": "C.5. Proof to Theorem 4.6",
|
| 649 |
+
"heading_level": null,
|
| 650 |
+
"page_id": 11,
|
| 651 |
+
"polygon": [
|
| 652 |
+
[
|
| 653 |
+
52.294921875,
|
| 654 |
+
617.58984375
|
| 655 |
+
],
|
| 656 |
+
[
|
| 657 |
+
165.47691345214844,
|
| 658 |
+
617.58984375
|
| 659 |
+
],
|
| 660 |
+
[
|
| 661 |
+
165.47691345214844,
|
| 662 |
+
628.0751190185547
|
| 663 |
+
],
|
| 664 |
+
[
|
| 665 |
+
52.294921875,
|
| 666 |
+
628.0751190185547
|
| 667 |
+
]
|
| 668 |
+
]
|
| 669 |
+
},
|
| 670 |
+
{
|
| 671 |
+
"title": "C.6. Proof to Theorem 4.8",
|
| 672 |
+
"heading_level": null,
|
| 673 |
+
"page_id": 11,
|
| 674 |
+
"polygon": [
|
| 675 |
+
[
|
| 676 |
+
306.59765625,
|
| 677 |
+
356.94140625
|
| 678 |
+
],
|
| 679 |
+
[
|
| 680 |
+
417.4769287109375,
|
| 681 |
+
356.94140625
|
| 682 |
+
],
|
| 683 |
+
[
|
| 684 |
+
417.4769287109375,
|
| 685 |
+
367.10406494140625
|
| 686 |
+
],
|
| 687 |
+
[
|
| 688 |
+
306.59765625,
|
| 689 |
+
367.10406494140625
|
| 690 |
+
]
|
| 691 |
+
]
|
| 692 |
+
},
|
| 693 |
+
{
|
| 694 |
+
"title": "C.7. Proof to Theorem 4.9",
|
| 695 |
+
"heading_level": null,
|
| 696 |
+
"page_id": 12,
|
| 697 |
+
"polygon": [
|
| 698 |
+
[
|
| 699 |
+
53.7890625,
|
| 700 |
+
175.5703125
|
| 701 |
+
],
|
| 702 |
+
[
|
| 703 |
+
165.75,
|
| 704 |
+
175.5703125
|
| 705 |
+
],
|
| 706 |
+
[
|
| 707 |
+
165.75,
|
| 708 |
+
185.25
|
| 709 |
+
],
|
| 710 |
+
[
|
| 711 |
+
53.7890625,
|
| 712 |
+
185.25
|
| 713 |
+
]
|
| 714 |
+
]
|
| 715 |
+
},
|
| 716 |
+
{
|
| 717 |
+
"title": "C.8. Proof to Theorem 5.1",
|
| 718 |
+
"heading_level": null,
|
| 719 |
+
"page_id": 12,
|
| 720 |
+
"polygon": [
|
| 721 |
+
[
|
| 722 |
+
306.0,
|
| 723 |
+
167.0625
|
| 724 |
+
],
|
| 725 |
+
[
|
| 726 |
+
417.0,
|
| 727 |
+
167.0625
|
| 728 |
+
],
|
| 729 |
+
[
|
| 730 |
+
417.0,
|
| 731 |
+
175.5
|
| 732 |
+
],
|
| 733 |
+
[
|
| 734 |
+
306.0,
|
| 735 |
+
175.5
|
| 736 |
+
]
|
| 737 |
+
]
|
| 738 |
+
},
|
| 739 |
+
{
|
| 740 |
+
"title": "C.9. Certificate Validity Under Adaptive Lipschitz Estimation",
|
| 741 |
+
"heading_level": null,
|
| 742 |
+
"page_id": 13,
|
| 743 |
+
"polygon": [
|
| 744 |
+
[
|
| 745 |
+
53.7890625,
|
| 746 |
+
69.0
|
| 747 |
+
],
|
| 748 |
+
[
|
| 749 |
+
269.25,
|
| 750 |
+
69.0
|
| 751 |
+
],
|
| 752 |
+
[
|
| 753 |
+
269.25,
|
| 754 |
+
90.0
|
| 755 |
+
],
|
| 756 |
+
[
|
| 757 |
+
53.7890625,
|
| 758 |
+
90.0
|
| 759 |
+
]
|
| 760 |
+
]
|
| 761 |
+
},
|
| 762 |
+
{
|
| 763 |
+
"title": "C.10. Global Safety of Certified Restarts (No False Elimination)",
|
| 764 |
+
"heading_level": null,
|
| 765 |
+
"page_id": 13,
|
| 766 |
+
"polygon": [
|
| 767 |
+
[
|
| 768 |
+
53.7890625,
|
| 769 |
+
441.24609375
|
| 770 |
+
],
|
| 771 |
+
[
|
| 772 |
+
269.25,
|
| 773 |
+
441.24609375
|
| 774 |
+
],
|
| 775 |
+
[
|
| 776 |
+
269.25,
|
| 777 |
+
462.12890625
|
| 778 |
+
],
|
| 779 |
+
[
|
| 780 |
+
53.7890625,
|
| 781 |
+
462.12890625
|
| 782 |
+
]
|
| 783 |
+
]
|
| 784 |
+
},
|
| 785 |
+
{
|
| 786 |
+
"title": "C.11. Proof to Theorem 6.1",
|
| 787 |
+
"heading_level": null,
|
| 788 |
+
"page_id": 13,
|
| 789 |
+
"polygon": [
|
| 790 |
+
[
|
| 791 |
+
304.505859375,
|
| 792 |
+
69.0
|
| 793 |
+
],
|
| 794 |
+
[
|
| 795 |
+
422.25,
|
| 796 |
+
69.0
|
| 797 |
+
],
|
| 798 |
+
[
|
| 799 |
+
422.25,
|
| 800 |
+
77.34375
|
| 801 |
+
],
|
| 802 |
+
[
|
| 803 |
+
304.505859375,
|
| 804 |
+
77.34375
|
| 805 |
+
]
|
| 806 |
+
]
|
| 807 |
+
},
|
| 808 |
+
{
|
| 809 |
+
"title": "C.12. Proof to Proposition 7.1",
|
| 810 |
+
"heading_level": null,
|
| 811 |
+
"page_id": 14,
|
| 812 |
+
"polygon": [
|
| 813 |
+
[
|
| 814 |
+
52.59375,
|
| 815 |
+
184.46484375
|
| 816 |
+
],
|
| 817 |
+
[
|
| 818 |
+
181.5,
|
| 819 |
+
184.46484375
|
| 820 |
+
],
|
| 821 |
+
[
|
| 822 |
+
181.5,
|
| 823 |
+
194.51953125
|
| 824 |
+
],
|
| 825 |
+
[
|
| 826 |
+
52.59375,
|
| 827 |
+
194.51953125
|
| 828 |
+
]
|
| 829 |
+
]
|
| 830 |
+
},
|
| 831 |
+
{
|
| 832 |
+
"title": "D. Practical Guidance",
|
| 833 |
+
"heading_level": null,
|
| 834 |
+
"page_id": 14,
|
| 835 |
+
"polygon": [
|
| 836 |
+
[
|
| 837 |
+
53.490234375,
|
| 838 |
+
655.5
|
| 839 |
+
],
|
| 840 |
+
[
|
| 841 |
+
168.75,
|
| 842 |
+
655.5
|
| 843 |
+
],
|
| 844 |
+
[
|
| 845 |
+
168.75,
|
| 846 |
+
666.31640625
|
| 847 |
+
],
|
| 848 |
+
[
|
| 849 |
+
53.490234375,
|
| 850 |
+
666.31640625
|
| 851 |
+
]
|
| 852 |
+
]
|
| 853 |
+
},
|
| 854 |
+
{
|
| 855 |
+
"title": "D.1. Adaptive Lipschitz Estimation",
|
| 856 |
+
"heading_level": null,
|
| 857 |
+
"page_id": 14,
|
| 858 |
+
"polygon": [
|
| 859 |
+
[
|
| 860 |
+
54.0,
|
| 861 |
+
676.37109375
|
| 862 |
+
],
|
| 863 |
+
[
|
| 864 |
+
204.99609375,
|
| 865 |
+
676.37109375
|
| 866 |
+
],
|
| 867 |
+
[
|
| 868 |
+
204.99609375,
|
| 869 |
+
686.25
|
| 870 |
+
],
|
| 871 |
+
[
|
| 872 |
+
54.0,
|
| 873 |
+
686.25
|
| 874 |
+
]
|
| 875 |
+
]
|
| 876 |
+
},
|
| 877 |
+
{
|
| 878 |
+
"title": "D.2. Trust Region Configuration",
|
| 879 |
+
"heading_level": null,
|
| 880 |
+
"page_id": 14,
|
| 881 |
+
"polygon": [
|
| 882 |
+
[
|
| 883 |
+
305.25,
|
| 884 |
+
221.25
|
| 885 |
+
],
|
| 886 |
+
[
|
| 887 |
+
444.75,
|
| 888 |
+
221.25
|
| 889 |
+
],
|
| 890 |
+
[
|
| 891 |
+
444.75,
|
| 892 |
+
232.03125
|
| 893 |
+
],
|
| 894 |
+
[
|
| 895 |
+
305.25,
|
| 896 |
+
232.03125
|
| 897 |
+
]
|
| 898 |
+
]
|
| 899 |
+
},
|
| 900 |
+
{
|
| 901 |
+
"title": "D.3. Smoothness Detection",
|
| 902 |
+
"heading_level": null,
|
| 903 |
+
"page_id": 14,
|
| 904 |
+
"polygon": [
|
| 905 |
+
[
|
| 906 |
+
306.0,
|
| 907 |
+
506.21484375
|
| 908 |
+
],
|
| 909 |
+
[
|
| 910 |
+
420.75,
|
| 911 |
+
506.21484375
|
| 912 |
+
],
|
| 913 |
+
[
|
| 914 |
+
420.75,
|
| 915 |
+
516.0
|
| 916 |
+
],
|
| 917 |
+
[
|
| 918 |
+
306.0,
|
| 919 |
+
516.0
|
| 920 |
+
]
|
| 921 |
+
]
|
| 922 |
+
},
|
| 923 |
+
{
|
| 924 |
+
"title": "D.4. When to Use CGP",
|
| 925 |
+
"heading_level": null,
|
| 926 |
+
"page_id": 14,
|
| 927 |
+
"polygon": [
|
| 928 |
+
[
|
| 929 |
+
304.8046875,
|
| 930 |
+
651.62109375
|
| 931 |
+
],
|
| 932 |
+
[
|
| 933 |
+
405.75,
|
| 934 |
+
651.62109375
|
| 935 |
+
],
|
| 936 |
+
[
|
| 937 |
+
405.75,
|
| 938 |
+
660.75
|
| 939 |
+
],
|
| 940 |
+
[
|
| 941 |
+
304.8046875,
|
| 942 |
+
660.75
|
| 943 |
+
]
|
| 944 |
+
]
|
| 945 |
+
},
|
| 946 |
+
{
|
| 947 |
+
"title": "E. Experimental Details",
|
| 948 |
+
"heading_level": null,
|
| 949 |
+
"page_id": 15,
|
| 950 |
+
"polygon": [
|
| 951 |
+
[
|
| 952 |
+
52.294921875,
|
| 953 |
+
257.16796875
|
| 954 |
+
],
|
| 955 |
+
[
|
| 956 |
+
177.75,
|
| 957 |
+
257.16796875
|
| 958 |
+
],
|
| 959 |
+
[
|
| 960 |
+
177.75,
|
| 961 |
+
268.76953125
|
| 962 |
+
],
|
| 963 |
+
[
|
| 964 |
+
52.294921875,
|
| 965 |
+
268.76953125
|
| 966 |
+
]
|
| 967 |
+
]
|
| 968 |
+
},
|
| 969 |
+
{
|
| 970 |
+
"title": "E.1. Baseline Configurations",
|
| 971 |
+
"heading_level": null,
|
| 972 |
+
"page_id": 15,
|
| 973 |
+
"polygon": [
|
| 974 |
+
[
|
| 975 |
+
54.0,
|
| 976 |
+
278.05078125
|
| 977 |
+
],
|
| 978 |
+
[
|
| 979 |
+
177.0,
|
| 980 |
+
278.05078125
|
| 981 |
+
],
|
| 982 |
+
[
|
| 983 |
+
177.0,
|
| 984 |
+
288.10546875
|
| 985 |
+
],
|
| 986 |
+
[
|
| 987 |
+
54.0,
|
| 988 |
+
288.10546875
|
| 989 |
+
]
|
| 990 |
+
]
|
| 991 |
+
},
|
| 992 |
+
{
|
| 993 |
+
"title": "E.2. Benchmark Details",
|
| 994 |
+
"heading_level": null,
|
| 995 |
+
"page_id": 15,
|
| 996 |
+
"polygon": [
|
| 997 |
+
[
|
| 998 |
+
53.490234375,
|
| 999 |
+
564.99609375
|
| 1000 |
+
],
|
| 1001 |
+
[
|
| 1002 |
+
157.5,
|
| 1003 |
+
564.99609375
|
| 1004 |
+
],
|
| 1005 |
+
[
|
| 1006 |
+
157.5,
|
| 1007 |
+
573.50390625
|
| 1008 |
+
],
|
| 1009 |
+
[
|
| 1010 |
+
53.490234375,
|
| 1011 |
+
573.50390625
|
| 1012 |
+
]
|
| 1013 |
+
]
|
| 1014 |
+
},
|
| 1015 |
+
{
|
| 1016 |
+
"title": "Low-dimensional.",
|
| 1017 |
+
"heading_level": null,
|
| 1018 |
+
"page_id": 15,
|
| 1019 |
+
"polygon": [
|
| 1020 |
+
[
|
| 1021 |
+
53.490234375,
|
| 1022 |
+
583.5
|
| 1023 |
+
],
|
| 1024 |
+
[
|
| 1025 |
+
132.0,
|
| 1026 |
+
583.5
|
| 1027 |
+
],
|
| 1028 |
+
[
|
| 1029 |
+
132.0,
|
| 1030 |
+
592.5
|
| 1031 |
+
],
|
| 1032 |
+
[
|
| 1033 |
+
53.490234375,
|
| 1034 |
+
592.5
|
| 1035 |
+
]
|
| 1036 |
+
]
|
| 1037 |
+
},
|
| 1038 |
+
{
|
| 1039 |
+
"title": "Medium-dimensional.",
|
| 1040 |
+
"heading_level": null,
|
| 1041 |
+
"page_id": 15,
|
| 1042 |
+
"polygon": [
|
| 1043 |
+
[
|
| 1044 |
+
306.59765625,
|
| 1045 |
+
69.0
|
| 1046 |
+
],
|
| 1047 |
+
[
|
| 1048 |
+
401.25,
|
| 1049 |
+
69.0
|
| 1050 |
+
],
|
| 1051 |
+
[
|
| 1052 |
+
401.25,
|
| 1053 |
+
77.34375
|
| 1054 |
+
],
|
| 1055 |
+
[
|
| 1056 |
+
306.59765625,
|
| 1057 |
+
77.34375
|
| 1058 |
+
]
|
| 1059 |
+
]
|
| 1060 |
+
},
|
| 1061 |
+
{
|
| 1062 |
+
"title": "High-dimensional.",
|
| 1063 |
+
"heading_level": null,
|
| 1064 |
+
"page_id": 15,
|
| 1065 |
+
"polygon": [
|
| 1066 |
+
[
|
| 1067 |
+
304.505859375,
|
| 1068 |
+
184.078125
|
| 1069 |
+
],
|
| 1070 |
+
[
|
| 1071 |
+
386.25,
|
| 1072 |
+
184.078125
|
| 1073 |
+
],
|
| 1074 |
+
[
|
| 1075 |
+
386.25,
|
| 1076 |
+
193.5
|
| 1077 |
+
],
|
| 1078 |
+
[
|
| 1079 |
+
304.505859375,
|
| 1080 |
+
193.5
|
| 1081 |
+
]
|
| 1082 |
+
]
|
| 1083 |
+
},
|
| 1084 |
+
{
|
| 1085 |
+
"title": "E.3. Computational Resources",
|
| 1086 |
+
"heading_level": null,
|
| 1087 |
+
"page_id": 15,
|
| 1088 |
+
"polygon": [
|
| 1089 |
+
[
|
| 1090 |
+
306.0,
|
| 1091 |
+
312.0
|
| 1092 |
+
],
|
| 1093 |
+
[
|
| 1094 |
+
436.5,
|
| 1095 |
+
312.0
|
| 1096 |
+
],
|
| 1097 |
+
[
|
| 1098 |
+
436.5,
|
| 1099 |
+
321.36328125
|
| 1100 |
+
],
|
| 1101 |
+
[
|
| 1102 |
+
306.0,
|
| 1103 |
+
321.36328125
|
| 1104 |
+
]
|
| 1105 |
+
]
|
| 1106 |
+
}
|
| 1107 |
+
],
|
| 1108 |
+
"page_stats": [
|
| 1109 |
+
{
|
| 1110 |
+
"page_id": 0,
|
| 1111 |
+
"text_extraction_method": "surya",
|
| 1112 |
+
"block_counts": [
|
| 1113 |
+
[
|
| 1114 |
+
"Line",
|
| 1115 |
+
155
|
| 1116 |
+
],
|
| 1117 |
+
[
|
| 1118 |
+
"Span",
|
| 1119 |
+
137
|
| 1120 |
+
],
|
| 1121 |
+
[
|
| 1122 |
+
"Text",
|
| 1123 |
+
24
|
| 1124 |
+
],
|
| 1125 |
+
[
|
| 1126 |
+
"SectionHeader",
|
| 1127 |
+
4
|
| 1128 |
+
],
|
| 1129 |
+
[
|
| 1130 |
+
"PageHeader",
|
| 1131 |
+
1
|
| 1132 |
+
],
|
| 1133 |
+
[
|
| 1134 |
+
"Footnote",
|
| 1135 |
+
1
|
| 1136 |
+
],
|
| 1137 |
+
[
|
| 1138 |
+
"PageFooter",
|
| 1139 |
+
1
|
| 1140 |
+
]
|
| 1141 |
+
],
|
| 1142 |
+
"block_metadata": {
|
| 1143 |
+
"llm_request_count": 0,
|
| 1144 |
+
"llm_error_count": 0,
|
| 1145 |
+
"llm_tokens_used": 0,
|
| 1146 |
+
"previous_text": "",
|
| 1147 |
+
"previous_type": "",
|
| 1148 |
+
"previous_order": 0
|
| 1149 |
+
}
|
| 1150 |
+
},
|
| 1151 |
+
{
|
| 1152 |
+
"page_id": 1,
|
| 1153 |
+
"text_extraction_method": "surya",
|
| 1154 |
+
"block_counts": [
|
| 1155 |
+
[
|
| 1156 |
+
"Span",
|
| 1157 |
+
177
|
| 1158 |
+
],
|
| 1159 |
+
[
|
| 1160 |
+
"Line",
|
| 1161 |
+
165
|
| 1162 |
+
],
|
| 1163 |
+
[
|
| 1164 |
+
"TableCell",
|
| 1165 |
+
62
|
| 1166 |
+
],
|
| 1167 |
+
[
|
| 1168 |
+
"Text",
|
| 1169 |
+
41
|
| 1170 |
+
],
|
| 1171 |
+
[
|
| 1172 |
+
"Reference",
|
| 1173 |
+
6
|
| 1174 |
+
],
|
| 1175 |
+
[
|
| 1176 |
+
"Caption",
|
| 1177 |
+
3
|
| 1178 |
+
],
|
| 1179 |
+
[
|
| 1180 |
+
"Table",
|
| 1181 |
+
2
|
| 1182 |
+
],
|
| 1183 |
+
[
|
| 1184 |
+
"SectionHeader",
|
| 1185 |
+
2
|
| 1186 |
+
],
|
| 1187 |
+
[
|
| 1188 |
+
"TableGroup",
|
| 1189 |
+
2
|
| 1190 |
+
],
|
| 1191 |
+
[
|
| 1192 |
+
"PageHeader",
|
| 1193 |
+
1
|
| 1194 |
+
],
|
| 1195 |
+
[
|
| 1196 |
+
"Figure",
|
| 1197 |
+
1
|
| 1198 |
+
],
|
| 1199 |
+
[
|
| 1200 |
+
"PageFooter",
|
| 1201 |
+
1
|
| 1202 |
+
]
|
| 1203 |
+
],
|
| 1204 |
+
"block_metadata": {
|
| 1205 |
+
"llm_request_count": 0,
|
| 1206 |
+
"llm_error_count": 0,
|
| 1207 |
+
"llm_tokens_used": 0,
|
| 1208 |
+
"previous_text": "",
|
| 1209 |
+
"previous_type": "",
|
| 1210 |
+
"previous_order": 0
|
| 1211 |
+
}
|
| 1212 |
+
},
|
| 1213 |
+
{
|
| 1214 |
+
"page_id": 2,
|
| 1215 |
+
"text_extraction_method": "surya",
|
| 1216 |
+
"block_counts": [
|
| 1217 |
+
[
|
| 1218 |
+
"Span",
|
| 1219 |
+
256
|
| 1220 |
+
],
|
| 1221 |
+
[
|
| 1222 |
+
"Line",
|
| 1223 |
+
230
|
| 1224 |
+
],
|
| 1225 |
+
[
|
| 1226 |
+
"Text",
|
| 1227 |
+
70
|
| 1228 |
+
],
|
| 1229 |
+
[
|
| 1230 |
+
"ListItem",
|
| 1231 |
+
8
|
| 1232 |
+
],
|
| 1233 |
+
[
|
| 1234 |
+
"Reference",
|
| 1235 |
+
8
|
| 1236 |
+
],
|
| 1237 |
+
[
|
| 1238 |
+
"Equation",
|
| 1239 |
+
3
|
| 1240 |
+
],
|
| 1241 |
+
[
|
| 1242 |
+
"SectionHeader",
|
| 1243 |
+
2
|
| 1244 |
+
],
|
| 1245 |
+
[
|
| 1246 |
+
"PageHeader",
|
| 1247 |
+
1
|
| 1248 |
+
],
|
| 1249 |
+
[
|
| 1250 |
+
"PageFooter",
|
| 1251 |
+
1
|
| 1252 |
+
],
|
| 1253 |
+
[
|
| 1254 |
+
"ListGroup",
|
| 1255 |
+
1
|
| 1256 |
+
]
|
| 1257 |
+
],
|
| 1258 |
+
"block_metadata": {
|
| 1259 |
+
"llm_request_count": 0,
|
| 1260 |
+
"llm_error_count": 0,
|
| 1261 |
+
"llm_tokens_used": 0,
|
| 1262 |
+
"previous_text": "",
|
| 1263 |
+
"previous_type": "",
|
| 1264 |
+
"previous_order": 0
|
| 1265 |
+
}
|
| 1266 |
+
},
|
| 1267 |
+
{
|
| 1268 |
+
"page_id": 3,
|
| 1269 |
+
"text_extraction_method": "surya",
|
| 1270 |
+
"block_counts": [
|
| 1271 |
+
[
|
| 1272 |
+
"Span",
|
| 1273 |
+
246
|
| 1274 |
+
],
|
| 1275 |
+
[
|
| 1276 |
+
"Line",
|
| 1277 |
+
224
|
| 1278 |
+
],
|
| 1279 |
+
[
|
| 1280 |
+
"Text",
|
| 1281 |
+
65
|
| 1282 |
+
],
|
| 1283 |
+
[
|
| 1284 |
+
"Reference",
|
| 1285 |
+
7
|
| 1286 |
+
],
|
| 1287 |
+
[
|
| 1288 |
+
"ListItem",
|
| 1289 |
+
4
|
| 1290 |
+
],
|
| 1291 |
+
[
|
| 1292 |
+
"SectionHeader",
|
| 1293 |
+
3
|
| 1294 |
+
],
|
| 1295 |
+
[
|
| 1296 |
+
"Code",
|
| 1297 |
+
2
|
| 1298 |
+
],
|
| 1299 |
+
[
|
| 1300 |
+
"PageHeader",
|
| 1301 |
+
1
|
| 1302 |
+
],
|
| 1303 |
+
[
|
| 1304 |
+
"PageFooter",
|
| 1305 |
+
1
|
| 1306 |
+
],
|
| 1307 |
+
[
|
| 1308 |
+
"ListGroup",
|
| 1309 |
+
1
|
| 1310 |
+
]
|
| 1311 |
+
],
|
| 1312 |
+
"block_metadata": {
|
| 1313 |
+
"llm_request_count": 0,
|
| 1314 |
+
"llm_error_count": 0,
|
| 1315 |
+
"llm_tokens_used": 0,
|
| 1316 |
+
"previous_text": "",
|
| 1317 |
+
"previous_type": "",
|
| 1318 |
+
"previous_order": 0
|
| 1319 |
+
}
|
| 1320 |
+
},
|
| 1321 |
+
{
|
| 1322 |
+
"page_id": 4,
|
| 1323 |
+
"text_extraction_method": "surya",
|
| 1324 |
+
"block_counts": [
|
| 1325 |
+
[
|
| 1326 |
+
"Span",
|
| 1327 |
+
208
|
| 1328 |
+
],
|
| 1329 |
+
[
|
| 1330 |
+
"Line",
|
| 1331 |
+
159
|
| 1332 |
+
],
|
| 1333 |
+
[
|
| 1334 |
+
"TableCell",
|
| 1335 |
+
80
|
| 1336 |
+
],
|
| 1337 |
+
[
|
| 1338 |
+
"Text",
|
| 1339 |
+
18
|
| 1340 |
+
],
|
| 1341 |
+
[
|
| 1342 |
+
"ListItem",
|
| 1343 |
+
18
|
| 1344 |
+
],
|
| 1345 |
+
[
|
| 1346 |
+
"Reference",
|
| 1347 |
+
3
|
| 1348 |
+
],
|
| 1349 |
+
[
|
| 1350 |
+
"PageHeader",
|
| 1351 |
+
2
|
| 1352 |
+
],
|
| 1353 |
+
[
|
| 1354 |
+
"ListGroup",
|
| 1355 |
+
2
|
| 1356 |
+
],
|
| 1357 |
+
[
|
| 1358 |
+
"Caption",
|
| 1359 |
+
1
|
| 1360 |
+
],
|
| 1361 |
+
[
|
| 1362 |
+
"Table",
|
| 1363 |
+
1
|
| 1364 |
+
],
|
| 1365 |
+
[
|
| 1366 |
+
"SectionHeader",
|
| 1367 |
+
1
|
| 1368 |
+
],
|
| 1369 |
+
[
|
| 1370 |
+
"Equation",
|
| 1371 |
+
1
|
| 1372 |
+
],
|
| 1373 |
+
[
|
| 1374 |
+
"PageFooter",
|
| 1375 |
+
1
|
| 1376 |
+
],
|
| 1377 |
+
[
|
| 1378 |
+
"TableGroup",
|
| 1379 |
+
1
|
| 1380 |
+
]
|
| 1381 |
+
],
|
| 1382 |
+
"block_metadata": {
|
| 1383 |
+
"llm_request_count": 0,
|
| 1384 |
+
"llm_error_count": 0,
|
| 1385 |
+
"llm_tokens_used": 0,
|
| 1386 |
+
"previous_text": "",
|
| 1387 |
+
"previous_type": "",
|
| 1388 |
+
"previous_order": 0
|
| 1389 |
+
}
|
| 1390 |
+
},
|
| 1391 |
+
{
|
| 1392 |
+
"page_id": 5,
|
| 1393 |
+
"text_extraction_method": "pdftext",
|
| 1394 |
+
"block_counts": [
|
| 1395 |
+
[
|
| 1396 |
+
"Span",
|
| 1397 |
+
847
|
| 1398 |
+
],
|
| 1399 |
+
[
|
| 1400 |
+
"Line",
|
| 1401 |
+
156
|
| 1402 |
+
],
|
| 1403 |
+
[
|
| 1404 |
+
"TableCell",
|
| 1405 |
+
32
|
| 1406 |
+
],
|
| 1407 |
+
[
|
| 1408 |
+
"Text",
|
| 1409 |
+
23
|
| 1410 |
+
],
|
| 1411 |
+
[
|
| 1412 |
+
"ListItem",
|
| 1413 |
+
11
|
| 1414 |
+
],
|
| 1415 |
+
[
|
| 1416 |
+
"Reference",
|
| 1417 |
+
4
|
| 1418 |
+
],
|
| 1419 |
+
[
|
| 1420 |
+
"SectionHeader",
|
| 1421 |
+
3
|
| 1422 |
+
],
|
| 1423 |
+
[
|
| 1424 |
+
"ListGroup",
|
| 1425 |
+
3
|
| 1426 |
+
],
|
| 1427 |
+
[
|
| 1428 |
+
"PageHeader",
|
| 1429 |
+
1
|
| 1430 |
+
],
|
| 1431 |
+
[
|
| 1432 |
+
"Caption",
|
| 1433 |
+
1
|
| 1434 |
+
],
|
| 1435 |
+
[
|
| 1436 |
+
"Table",
|
| 1437 |
+
1
|
| 1438 |
+
],
|
| 1439 |
+
[
|
| 1440 |
+
"Footnote",
|
| 1441 |
+
1
|
| 1442 |
+
],
|
| 1443 |
+
[
|
| 1444 |
+
"PageFooter",
|
| 1445 |
+
1
|
| 1446 |
+
],
|
| 1447 |
+
[
|
| 1448 |
+
"TableGroup",
|
| 1449 |
+
1
|
| 1450 |
+
]
|
| 1451 |
+
],
|
| 1452 |
+
"block_metadata": {
|
| 1453 |
+
"llm_request_count": 0,
|
| 1454 |
+
"llm_error_count": 0,
|
| 1455 |
+
"llm_tokens_used": 0,
|
| 1456 |
+
"previous_text": "",
|
| 1457 |
+
"previous_type": "",
|
| 1458 |
+
"previous_order": 0
|
| 1459 |
+
}
|
| 1460 |
+
},
|
| 1461 |
+
{
|
| 1462 |
+
"page_id": 6,
|
| 1463 |
+
"text_extraction_method": "pdftext",
|
| 1464 |
+
"block_counts": [
|
| 1465 |
+
[
|
| 1466 |
+
"Span",
|
| 1467 |
+
659
|
| 1468 |
+
],
|
| 1469 |
+
[
|
| 1470 |
+
"Line",
|
| 1471 |
+
125
|
| 1472 |
+
],
|
| 1473 |
+
[
|
| 1474 |
+
"TableCell",
|
| 1475 |
+
80
|
| 1476 |
+
],
|
| 1477 |
+
[
|
| 1478 |
+
"Text",
|
| 1479 |
+
12
|
| 1480 |
+
],
|
| 1481 |
+
[
|
| 1482 |
+
"Reference",
|
| 1483 |
+
3
|
| 1484 |
+
],
|
| 1485 |
+
[
|
| 1486 |
+
"PageHeader",
|
| 1487 |
+
2
|
| 1488 |
+
],
|
| 1489 |
+
[
|
| 1490 |
+
"Caption",
|
| 1491 |
+
2
|
| 1492 |
+
],
|
| 1493 |
+
[
|
| 1494 |
+
"Table",
|
| 1495 |
+
2
|
| 1496 |
+
],
|
| 1497 |
+
[
|
| 1498 |
+
"SectionHeader",
|
| 1499 |
+
1
|
| 1500 |
+
],
|
| 1501 |
+
[
|
| 1502 |
+
"PageFooter",
|
| 1503 |
+
1
|
| 1504 |
+
],
|
| 1505 |
+
[
|
| 1506 |
+
"TableGroup",
|
| 1507 |
+
1
|
| 1508 |
+
]
|
| 1509 |
+
],
|
| 1510 |
+
"block_metadata": {
|
| 1511 |
+
"llm_request_count": 0,
|
| 1512 |
+
"llm_error_count": 0,
|
| 1513 |
+
"llm_tokens_used": 0,
|
| 1514 |
+
"previous_text": "",
|
| 1515 |
+
"previous_type": "",
|
| 1516 |
+
"previous_order": 0
|
| 1517 |
+
}
|
| 1518 |
+
},
|
| 1519 |
+
{
|
| 1520 |
+
"page_id": 7,
|
| 1521 |
+
"text_extraction_method": "surya",
|
| 1522 |
+
"block_counts": [
|
| 1523 |
+
[
|
| 1524 |
+
"Line",
|
| 1525 |
+
187
|
| 1526 |
+
],
|
| 1527 |
+
[
|
| 1528 |
+
"Span",
|
| 1529 |
+
142
|
| 1530 |
+
],
|
| 1531 |
+
[
|
| 1532 |
+
"TableCell",
|
| 1533 |
+
105
|
| 1534 |
+
],
|
| 1535 |
+
[
|
| 1536 |
+
"Text",
|
| 1537 |
+
54
|
| 1538 |
+
],
|
| 1539 |
+
[
|
| 1540 |
+
"Caption",
|
| 1541 |
+
4
|
| 1542 |
+
],
|
| 1543 |
+
[
|
| 1544 |
+
"Table",
|
| 1545 |
+
4
|
| 1546 |
+
],
|
| 1547 |
+
[
|
| 1548 |
+
"Reference",
|
| 1549 |
+
4
|
| 1550 |
+
],
|
| 1551 |
+
[
|
| 1552 |
+
"TableGroup",
|
| 1553 |
+
3
|
| 1554 |
+
],
|
| 1555 |
+
[
|
| 1556 |
+
"SectionHeader",
|
| 1557 |
+
2
|
| 1558 |
+
],
|
| 1559 |
+
[
|
| 1560 |
+
"PageHeader",
|
| 1561 |
+
1
|
| 1562 |
+
],
|
| 1563 |
+
[
|
| 1564 |
+
"PageFooter",
|
| 1565 |
+
1
|
| 1566 |
+
]
|
| 1567 |
+
],
|
| 1568 |
+
"block_metadata": {
|
| 1569 |
+
"llm_request_count": 0,
|
| 1570 |
+
"llm_error_count": 0,
|
| 1571 |
+
"llm_tokens_used": 0,
|
| 1572 |
+
"previous_text": "",
|
| 1573 |
+
"previous_type": "",
|
| 1574 |
+
"previous_order": 0
|
| 1575 |
+
}
|
| 1576 |
+
},
|
| 1577 |
+
{
|
| 1578 |
+
"page_id": 8,
|
| 1579 |
+
"text_extraction_method": "pdftext",
|
| 1580 |
+
"block_counts": [
|
| 1581 |
+
[
|
| 1582 |
+
"Span",
|
| 1583 |
+
362
|
| 1584 |
+
],
|
| 1585 |
+
[
|
| 1586 |
+
"Line",
|
| 1587 |
+
149
|
| 1588 |
+
],
|
| 1589 |
+
[
|
| 1590 |
+
"ListItem",
|
| 1591 |
+
25
|
| 1592 |
+
],
|
| 1593 |
+
[
|
| 1594 |
+
"Reference",
|
| 1595 |
+
25
|
| 1596 |
+
],
|
| 1597 |
+
[
|
| 1598 |
+
"Text",
|
| 1599 |
+
2
|
| 1600 |
+
],
|
| 1601 |
+
[
|
| 1602 |
+
"ListGroup",
|
| 1603 |
+
2
|
| 1604 |
+
],
|
| 1605 |
+
[
|
| 1606 |
+
"PageHeader",
|
| 1607 |
+
1
|
| 1608 |
+
],
|
| 1609 |
+
[
|
| 1610 |
+
"SectionHeader",
|
| 1611 |
+
1
|
| 1612 |
+
],
|
| 1613 |
+
[
|
| 1614 |
+
"PageFooter",
|
| 1615 |
+
1
|
| 1616 |
+
]
|
| 1617 |
+
],
|
| 1618 |
+
"block_metadata": {
|
| 1619 |
+
"llm_request_count": 0,
|
| 1620 |
+
"llm_error_count": 0,
|
| 1621 |
+
"llm_tokens_used": 0,
|
| 1622 |
+
"previous_text": "",
|
| 1623 |
+
"previous_type": "",
|
| 1624 |
+
"previous_order": 0
|
| 1625 |
+
}
|
| 1626 |
+
},
|
| 1627 |
+
{
|
| 1628 |
+
"page_id": 9,
|
| 1629 |
+
"text_extraction_method": "surya",
|
| 1630 |
+
"block_counts": [
|
| 1631 |
+
[
|
| 1632 |
+
"Line",
|
| 1633 |
+
232
|
| 1634 |
+
],
|
| 1635 |
+
[
|
| 1636 |
+
"Span",
|
| 1637 |
+
193
|
| 1638 |
+
],
|
| 1639 |
+
[
|
| 1640 |
+
"Text",
|
| 1641 |
+
66
|
| 1642 |
+
],
|
| 1643 |
+
[
|
| 1644 |
+
"Reference",
|
| 1645 |
+
14
|
| 1646 |
+
],
|
| 1647 |
+
[
|
| 1648 |
+
"ListItem",
|
| 1649 |
+
12
|
| 1650 |
+
],
|
| 1651 |
+
[
|
| 1652 |
+
"SectionHeader",
|
| 1653 |
+
3
|
| 1654 |
+
],
|
| 1655 |
+
[
|
| 1656 |
+
"PageHeader",
|
| 1657 |
+
1
|
| 1658 |
+
],
|
| 1659 |
+
[
|
| 1660 |
+
"PageFooter",
|
| 1661 |
+
1
|
| 1662 |
+
],
|
| 1663 |
+
[
|
| 1664 |
+
"ListGroup",
|
| 1665 |
+
1
|
| 1666 |
+
]
|
| 1667 |
+
],
|
| 1668 |
+
"block_metadata": {
|
| 1669 |
+
"llm_request_count": 0,
|
| 1670 |
+
"llm_error_count": 0,
|
| 1671 |
+
"llm_tokens_used": 0,
|
| 1672 |
+
"previous_text": "",
|
| 1673 |
+
"previous_type": "",
|
| 1674 |
+
"previous_order": 0
|
| 1675 |
+
}
|
| 1676 |
+
},
|
| 1677 |
+
{
|
| 1678 |
+
"page_id": 10,
|
| 1679 |
+
"text_extraction_method": "surya",
|
| 1680 |
+
"block_counts": [
|
| 1681 |
+
[
|
| 1682 |
+
"Line",
|
| 1683 |
+
207
|
| 1684 |
+
],
|
| 1685 |
+
[
|
| 1686 |
+
"Span",
|
| 1687 |
+
157
|
| 1688 |
+
],
|
| 1689 |
+
[
|
| 1690 |
+
"Text",
|
| 1691 |
+
68
|
| 1692 |
+
],
|
| 1693 |
+
[
|
| 1694 |
+
"Equation",
|
| 1695 |
+
13
|
| 1696 |
+
],
|
| 1697 |
+
[
|
| 1698 |
+
"SectionHeader",
|
| 1699 |
+
5
|
| 1700 |
+
],
|
| 1701 |
+
[
|
| 1702 |
+
"Reference",
|
| 1703 |
+
2
|
| 1704 |
+
],
|
| 1705 |
+
[
|
| 1706 |
+
"PageHeader",
|
| 1707 |
+
1
|
| 1708 |
+
],
|
| 1709 |
+
[
|
| 1710 |
+
"PageFooter",
|
| 1711 |
+
1
|
| 1712 |
+
]
|
| 1713 |
+
],
|
| 1714 |
+
"block_metadata": {
|
| 1715 |
+
"llm_request_count": 0,
|
| 1716 |
+
"llm_error_count": 0,
|
| 1717 |
+
"llm_tokens_used": 0,
|
| 1718 |
+
"previous_text": "",
|
| 1719 |
+
"previous_type": "",
|
| 1720 |
+
"previous_order": 0
|
| 1721 |
+
}
|
| 1722 |
+
},
|
| 1723 |
+
{
|
| 1724 |
+
"page_id": 11,
|
| 1725 |
+
"text_extraction_method": "pdftext",
|
| 1726 |
+
"block_counts": [
|
| 1727 |
+
[
|
| 1728 |
+
"Span",
|
| 1729 |
+
1088
|
| 1730 |
+
],
|
| 1731 |
+
[
|
| 1732 |
+
"Line",
|
| 1733 |
+
213
|
| 1734 |
+
],
|
| 1735 |
+
[
|
| 1736 |
+
"Text",
|
| 1737 |
+
22
|
| 1738 |
+
],
|
| 1739 |
+
[
|
| 1740 |
+
"Equation",
|
| 1741 |
+
17
|
| 1742 |
+
],
|
| 1743 |
+
[
|
| 1744 |
+
"SectionHeader",
|
| 1745 |
+
3
|
| 1746 |
+
],
|
| 1747 |
+
[
|
| 1748 |
+
"PageHeader",
|
| 1749 |
+
1
|
| 1750 |
+
],
|
| 1751 |
+
[
|
| 1752 |
+
"PageFooter",
|
| 1753 |
+
1
|
| 1754 |
+
]
|
| 1755 |
+
],
|
| 1756 |
+
"block_metadata": {
|
| 1757 |
+
"llm_request_count": 0,
|
| 1758 |
+
"llm_error_count": 0,
|
| 1759 |
+
"llm_tokens_used": 0,
|
| 1760 |
+
"previous_text": "",
|
| 1761 |
+
"previous_type": "",
|
| 1762 |
+
"previous_order": 0
|
| 1763 |
+
}
|
| 1764 |
+
},
|
| 1765 |
+
{
|
| 1766 |
+
"page_id": 12,
|
| 1767 |
+
"text_extraction_method": "surya",
|
| 1768 |
+
"block_counts": [
|
| 1769 |
+
[
|
| 1770 |
+
"Line",
|
| 1771 |
+
212
|
| 1772 |
+
],
|
| 1773 |
+
[
|
| 1774 |
+
"Span",
|
| 1775 |
+
185
|
| 1776 |
+
],
|
| 1777 |
+
[
|
| 1778 |
+
"Text",
|
| 1779 |
+
69
|
| 1780 |
+
],
|
| 1781 |
+
[
|
| 1782 |
+
"Equation",
|
| 1783 |
+
10
|
| 1784 |
+
],
|
| 1785 |
+
[
|
| 1786 |
+
"SectionHeader",
|
| 1787 |
+
2
|
| 1788 |
+
],
|
| 1789 |
+
[
|
| 1790 |
+
"ListItem",
|
| 1791 |
+
2
|
| 1792 |
+
],
|
| 1793 |
+
[
|
| 1794 |
+
"PageHeader",
|
| 1795 |
+
1
|
| 1796 |
+
],
|
| 1797 |
+
[
|
| 1798 |
+
"PageFooter",
|
| 1799 |
+
1
|
| 1800 |
+
],
|
| 1801 |
+
[
|
| 1802 |
+
"ListGroup",
|
| 1803 |
+
1
|
| 1804 |
+
]
|
| 1805 |
+
],
|
| 1806 |
+
"block_metadata": {
|
| 1807 |
+
"llm_request_count": 0,
|
| 1808 |
+
"llm_error_count": 0,
|
| 1809 |
+
"llm_tokens_used": 0,
|
| 1810 |
+
"previous_text": "",
|
| 1811 |
+
"previous_type": "",
|
| 1812 |
+
"previous_order": 0
|
| 1813 |
+
}
|
| 1814 |
+
},
|
| 1815 |
+
{
|
| 1816 |
+
"page_id": 13,
|
| 1817 |
+
"text_extraction_method": "surya",
|
| 1818 |
+
"block_counts": [
|
| 1819 |
+
[
|
| 1820 |
+
"Span",
|
| 1821 |
+
262
|
| 1822 |
+
],
|
| 1823 |
+
[
|
| 1824 |
+
"Line",
|
| 1825 |
+
207
|
| 1826 |
+
],
|
| 1827 |
+
[
|
| 1828 |
+
"Text",
|
| 1829 |
+
70
|
| 1830 |
+
],
|
| 1831 |
+
[
|
| 1832 |
+
"Equation",
|
| 1833 |
+
10
|
| 1834 |
+
],
|
| 1835 |
+
[
|
| 1836 |
+
"SectionHeader",
|
| 1837 |
+
3
|
| 1838 |
+
],
|
| 1839 |
+
[
|
| 1840 |
+
"PageHeader",
|
| 1841 |
+
1
|
| 1842 |
+
],
|
| 1843 |
+
[
|
| 1844 |
+
"PageFooter",
|
| 1845 |
+
1
|
| 1846 |
+
]
|
| 1847 |
+
],
|
| 1848 |
+
"block_metadata": {
|
| 1849 |
+
"llm_request_count": 0,
|
| 1850 |
+
"llm_error_count": 0,
|
| 1851 |
+
"llm_tokens_used": 0,
|
| 1852 |
+
"previous_text": "",
|
| 1853 |
+
"previous_type": "",
|
| 1854 |
+
"previous_order": 0
|
| 1855 |
+
}
|
| 1856 |
+
},
|
| 1857 |
+
{
|
| 1858 |
+
"page_id": 14,
|
| 1859 |
+
"text_extraction_method": "surya",
|
| 1860 |
+
"block_counts": [
|
| 1861 |
+
[
|
| 1862 |
+
"Line",
|
| 1863 |
+
226
|
| 1864 |
+
],
|
| 1865 |
+
[
|
| 1866 |
+
"Span",
|
| 1867 |
+
165
|
| 1868 |
+
],
|
| 1869 |
+
[
|
| 1870 |
+
"Text",
|
| 1871 |
+
71
|
| 1872 |
+
],
|
| 1873 |
+
[
|
| 1874 |
+
"SectionHeader",
|
| 1875 |
+
6
|
| 1876 |
+
],
|
| 1877 |
+
[
|
| 1878 |
+
"ListItem",
|
| 1879 |
+
6
|
| 1880 |
+
],
|
| 1881 |
+
[
|
| 1882 |
+
"Equation",
|
| 1883 |
+
4
|
| 1884 |
+
],
|
| 1885 |
+
[
|
| 1886 |
+
"ListGroup",
|
| 1887 |
+
2
|
| 1888 |
+
],
|
| 1889 |
+
[
|
| 1890 |
+
"PageHeader",
|
| 1891 |
+
1
|
| 1892 |
+
],
|
| 1893 |
+
[
|
| 1894 |
+
"PageFooter",
|
| 1895 |
+
1
|
| 1896 |
+
],
|
| 1897 |
+
[
|
| 1898 |
+
"Reference",
|
| 1899 |
+
1
|
| 1900 |
+
]
|
| 1901 |
+
],
|
| 1902 |
+
"block_metadata": {
|
| 1903 |
+
"llm_request_count": 0,
|
| 1904 |
+
"llm_error_count": 0,
|
| 1905 |
+
"llm_tokens_used": 0,
|
| 1906 |
+
"previous_text": "",
|
| 1907 |
+
"previous_type": "",
|
| 1908 |
+
"previous_order": 0
|
| 1909 |
+
}
|
| 1910 |
+
},
|
| 1911 |
+
{
|
| 1912 |
+
"page_id": 15,
|
| 1913 |
+
"text_extraction_method": "surya",
|
| 1914 |
+
"block_counts": [
|
| 1915 |
+
[
|
| 1916 |
+
"Line",
|
| 1917 |
+
162
|
| 1918 |
+
],
|
| 1919 |
+
[
|
| 1920 |
+
"Span",
|
| 1921 |
+
110
|
| 1922 |
+
],
|
| 1923 |
+
[
|
| 1924 |
+
"ListItem",
|
| 1925 |
+
25
|
| 1926 |
+
],
|
| 1927 |
+
[
|
| 1928 |
+
"Text",
|
| 1929 |
+
21
|
| 1930 |
+
],
|
| 1931 |
+
[
|
| 1932 |
+
"SectionHeader",
|
| 1933 |
+
7
|
| 1934 |
+
],
|
| 1935 |
+
[
|
| 1936 |
+
"ListGroup",
|
| 1937 |
+
5
|
| 1938 |
+
],
|
| 1939 |
+
[
|
| 1940 |
+
"PageHeader",
|
| 1941 |
+
1
|
| 1942 |
+
],
|
| 1943 |
+
[
|
| 1944 |
+
"PageFooter",
|
| 1945 |
+
1
|
| 1946 |
+
],
|
| 1947 |
+
[
|
| 1948 |
+
"Reference",
|
| 1949 |
+
1
|
| 1950 |
+
]
|
| 1951 |
+
],
|
| 1952 |
+
"block_metadata": {
|
| 1953 |
+
"llm_request_count": 0,
|
| 1954 |
+
"llm_error_count": 0,
|
| 1955 |
+
"llm_tokens_used": 0,
|
| 1956 |
+
"previous_text": "",
|
| 1957 |
+
"previous_type": "",
|
| 1958 |
+
"previous_order": 0
|
| 1959 |
+
}
|
| 1960 |
+
}
|
| 1961 |
+
],
|
| 1962 |
+
"debug_data_path": "debug_data/5c3d6bff-e8ce-4d9f-840b-719084582491"
|
| 1963 |
+
}
|
icml26/5c3d6bff-e8ce-4d9f-840b-719084582491/model_text_v3.txt
ADDED
|
@@ -0,0 +1,320 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
[p. 1 | section: Abstract | type: Text]
|
| 2 |
+
We study black-box optimization of Lipschitz functions under noisy evaluations. Existing adaptive discretization methods implicitly avoid suboptimal regions but do not provide explicit certificates of optimality or measurable progress guarantees. We introduce Certificate-Guided Pruning (CGP), which maintains an explicit active set A_t of potentially optimal points via confidence-adjusted Lipschitz envelopes. Any point outside A_t is certifiably suboptimal with high probability, and under a margin condition with near-optimality dimension \alpha , we prove Vol(A_t) shrinks at a controlled rate yielding sample complexity \tilde{O}(\varepsilon^{-(2+\alpha)}) . We develop three extensions: CGP-Adaptive learns L online with O(\log T) overhead; CGP-TR scales to d > 50via trust regions with local certificates; and CGP-Hybrid switches to GP refinement when local smoothness is detected. Experiments on 12 benchmarks (d \in [2, 100]) show CGP variants match or exceed strong baselines while providing principled stopping criteria via certificate volume.
|
| 3 |
+
|
| 4 |
+
[p. 1 | section: 1. Introduction | type: Text]
|
| 5 |
+
Black-box optimization, the task of finding the maximum of a function f:\mathcal{X}\to\mathbb{R} accessible only through noisy point evaluations, is fundamental to machine learning, with applications spanning hyperparameter tuning (Snoek et al., 2012; Bergstra & Bengio, 2012), neural architecture search (Zoph & Le, 2017), and simulation-based optimization (Fu, 2015; Brochu et al., 2010). In many such settings, evaluations are expensive: training a neural network or running a physical simulation may cost hours or dollars per query. We call these "precious calls" where each evaluation must count, motivating the need for methods that provide explicit progress guarantees.
|
| 6 |
+
|
| 7 |
+
[p. 1 | section: 1. Introduction | type: Text]
|
| 8 |
+
The Lipschitz continuity assumption provides a natural framework for addressing this challenge. If |f(x)-f(y)| \le L \cdot d(x,y) for a known constant L, then observations at sampled points constrain f globally, enabling pruning of provably suboptimal regions. Classical methods exploiting this structure include DIRECT (Jones et al., 1993; 1998), Lipschitz bandits (Kleinberg et al., 2008; Auer et al., 2002), and adaptive discretization algorithms (Bubeck et al., 2011; Valko et al., 2013; Munos, 2014). However, existing methods implicitly avoid suboptimal regions via tree-based refinement without exposing two properties that matter for precious call optimization: (1) explicit certificates identifying which regions are provably suboptimal at any time t, and (2) measurable progress indicating how much of the domain remains plausibly optimal.
|
| 9 |
+
|
| 10 |
+
[p. 1 | section: 1. Introduction | type: Text]
|
| 11 |
+
To address these limitations, we introduce Certificate-Guided Pruning (CGP), which maintains an explicit active set A_t \subseteq \mathcal{X} of potentially optimal points. This set is defined via a Lipschitz UCB envelope U_t(x) that upper bounds f(x)with high probability, a global lower certificate \ell_t that lower bounds f(x^*) , and the active set A_t = \{x : U_t(x) \ge \ell_t\} . Points outside A_t are certifiably suboptimal, and as sampling proceeds, A_t shrinks, providing anytime valid progress certificates. Unlike prior work that uses similar mathematical tools implicitly, CGP exposes the pruning mechanism as a first-class algorithmic object: the certificate is computable in closed form, the shrinkage rate is provably controlled, and the certificate provides valid optimality bounds even when stopped early. Figure 1 illustrates this mechanism. To understand how CGP differs from existing approaches, consider zooming algorithms (Kleinberg et al., 2008; Bubeck et al., 2011). While zooming maintains a tree of "active arms" and expands nodes with high UCB, the implicit pruning is an analysis artifact not exposed to the user (Shihab et al., 2026). Table 1 makes this distinction precise: CGP provides explicit certificates, computable progress metrics, and principled stopping rules that zooming-based methods lack. Similarly, Thompson sampling (Thompson, 1933; Russo & Van Roy, 2014; Daniel et al., 2018) and information-directed methods (Hennig & Schuler, 2012; Hernández-Lobato et al., 2014) maintain implicit uncertainty without providing explicit geometric certificates.
|
| 12 |
+
|
| 13 |
+
[p. 1 | section: 1. Introduction | type: Text]
|
| 14 |
+
Building on this foundation, our contributions are fourfold. First, we present CGP with explicit active set maintenance
|
| 15 |
+
|
| 16 |
+
[p. 2 | section: 1. Introduction | type: Caption]
|
| 17 |
+
Figure 1. The active set A_t (shaded) consists of points where the Lipschitz envelope U_t(x) (red) exceeds the global lower bound \ell_t (green dashed). Regions where U_t(x) < \ell_t are certifiably suboptimal and pruned, causing A_t to shrink as sampling proceeds.
|
| 18 |
+
|
| 19 |
+
[p. 2 | section: 1. Introduction | type: TableGroup]
|
| 20 |
+
Table 1. Comparison with zooming-based bandits. CGP uniquely exports explicit certificates (active set A_t , gap bound \varepsilon_t , and volume \operatorname{Vol}(A_t) ), enabling principled stopping criteria that implicit zooming methods lack. Property CGP Zooming HOO/StoSOO Explicit active set A_t ✓ _ _ Computable Vol(A_t) ✓ _ - Anytime optimality bound ✓ _ _ Principled stopping rule ✓ _ _ Certificate export ✓ _ _ Sample complexity \tilde{O}(\varepsilon^{-(2+\alpha)}) \tilde{O}(\varepsilon^{-(2+\alpha)}) \tilde{O}(\varepsilon^{-(2+d)}) O(\log N_t) Per-iteration cost O(N_t) O(\log N_t) O(\log N_t) Adaptive L √(CGP-A) _ _ High-dim scaling √(CGP-TR) -
|
| 21 |
+
|
| 22 |
+
[p. 2 | section: 1. Introduction | type: Text]
|
| 23 |
+
and prove a shrinkage theorem: under a margin condition with near-optimality dimension \alpha (i.e., Vol( \{x: f(x) \geq f^* - \varepsilon ) \leq C\varepsilon^{d-\alpha} , we show Vol(A_t) \leq C \cdot (2(\beta_t + \beta_t)) (L\eta_t)^{d-\alpha} , yielding sample complexity T = \tilde{O}(\varepsilon^{-(2+\alpha)}) that improves on the worst case \tilde{O}(\varepsilon^{-(2+d)}) when \alpha < d (Section 4). Second, we develop CGP-Adaptive (Section 5), which learns L online via a doubling scheme, proving that unknown L adds only O(\log T) multiplicative overhead, the first such guarantee for Lipschitz optimization with certificates. Third, we introduce CGP-TR (Section 6), a trust region variant that scales to d > 50 by maintaining local certificates within adaptively sized regions, enabling high-dimensional applications previously intractable for Lipschitz methods. Fourth, we propose CGP-Hybrid (Section 7), which detects local smoothness via the ratio \rho = L_{\rm local}/L_{\rm global} and switches to GP refinement when \rho < 0.5 , achieving best of both worlds performance across diverse function classes.
|
| 24 |
+
|
| 25 |
+
[p. 2 | section: 1. Introduction | type: Text]
|
| 26 |
+
These theoretical contributions translate to strong empirical performance. Experiments (Section 9) demonstrate that CGP variants are competitive with strong baselines on 12 benchmarks spanning d \in [2, 100] , including Rover tra-
|
| 27 |
+
|
| 28 |
+
[p. 2 | section: 1. Introduction | type: TableGroup]
|
| 29 |
+
Table 2. Summary of Notation Symbol Description f^* Global maximum value of the objective function L Lipschitz constant (or global upper bound) A_t Active set at time t (contains potential optimizers) U_t(x) Lipschitz Upper Confidence Bound envelope \ell_t Global lower certificate (\max_i LCB_i) \alpha Near-optimality dimension (problem hardness) \beta_t Active confidence radius (uncertainty in A_t ) \eta_t Covering radius (resolution of A_t ) \gamma_t Gap to optimum proxy (f^* - \ell_t) ρ Local smoothness ratio ( L_{\rm local}/L_{\rm global} )
|
| 30 |
+
|
| 31 |
+
[p. 2 | section: 1. Introduction | type: Text]
|
| 32 |
+
jectory optimization (d=60), neural architecture search (d=36), and safe robotics where certificates enable stopping with guaranteed bounds (Shihab et al., 2025a). CGP-Hybrid performs best among tested methods on all 12 benchmarks under matched budgets, including Branin and Rosenbrock where vanilla CGP previously lost to GP-based methods.
|
| 33 |
+
|
| 34 |
+
[p. 2 | section: 2. Problem Formulation | type: Text]
|
| 35 |
+
Let (\mathcal{X}, d) be a compact metric space with diameter D = \sup_{x,y} d(x,y) . We consider \mathcal{X} = [0,1]^d with Euclidean metric. Let f: \mathcal{X} \to [0,1] satisfy:
|
| 36 |
+
|
| 37 |
+
[p. 2 | section: 2. Problem Formulation | type: Text]
|
| 38 |
+
Assumption 2.1 (Lipschitz continuity). There exists L > 0 such that for all x, y \in \mathcal{X} : |f(x) - f(y)| \le L \cdot d(x, y) .
|
| 39 |
+
|
| 40 |
+
[p. 2 | section: 2. Problem Formulation | type: Text]
|
| 41 |
+
We observe f through noisy queries: querying x returns y = f(x) + \epsilon , where:
|
| 42 |
+
|
| 43 |
+
[p. 2 | section: 2. Problem Formulation | type: Text]
|
| 44 |
+
Assumption 2.2 (Sub-Gaussian noise). The noise \epsilon is \sigma -sub-Gaussian: \mathbb{E}[e^{\lambda\epsilon}] \leq e^{\lambda^2 \sigma^2/2} for all \lambda \in \mathbb{R} .
|
| 45 |
+
|
| 46 |
+
[p. 2 | section: 2. Problem Formulation | type: Text]
|
| 47 |
+
After T samples, the algorithm outputs \hat{x}_T \in \mathcal{X} . The goal is to minimize simple regret r_T = f(x^*) - f(\hat{x}_T) . We seek PAC guarantees: r_T \leq \varepsilon with probability \geq 1 - \delta .
|
| 48 |
+
|
| 49 |
+
[p. 2 | section: 2. Problem Formulation | type: Text]
|
| 50 |
+
Assumption 2.3 (Margin / near-optimality dimension). There exist C>0 and \alpha\in[0,d] such that for all \varepsilon>0 : \operatorname{Vol}(\{x\in\mathcal{X}:f(x)\geq f^*-\varepsilon\})\leq C\varepsilon^{d-\alpha} .
|
| 51 |
+
|
| 52 |
+
[p. 2 | section: 2. Problem Formulation | type: Text]
|
| 53 |
+
The parameter \alpha is the near-optimality dimension: smaller \alpha means a sharper optimum (easier), \alpha=d is worst case. For isolated maxima with nondegenerate Hessian, \alpha=d/2 ; for f(x)\approx f^*-c\|x-x^*\|^p , we have \alpha=d/p . This assumption is standard in bandit theory (Audibert & Bubeck, 2010; Bubeck et al., 2011; Lattimore & Szepesvári, 2020); see Appendix A for extended discussion.
|
| 54 |
+
|
| 55 |
+
[p. 2 | section: 3. Algorithm: Certificate-Guided Pruning | type: Text]
|
| 56 |
+
With the problem formalized, we now describe the CGP algorithm (Algorithm 1). CGP maintains sampled points
|
| 57 |
+
|
| 58 |
+
[p. 3 | section: Algorithm 1 Certificate-Guided Pruning (CGP) | type: Text]
|
| 59 |
+
Require: Domain \mathcal{X} , Lipschitz constant L, noise \sigma , budget T, confidence \delta
|
| 60 |
+
|
| 61 |
+
[p. 3 | section: Algorithm 1 Certificate-Guided Pruning (CGP) | type: ListGroup]
|
| 62 |
+
1: Initialize: Sample x_1 uniformly, observe y_1 2: for t = 1, ..., T 1 do 3: Compute r_i(t) , \ell_t = \max_i LCB_i(t) , A_t = \{x : U_t(x) \ge \ell_t\} 4: x_{t+1} \leftarrow \arg\max_{x \in A_t} [U_t(x) L \cdot \min_i d(x, x_i)] 5: Query x_{t+1} , observe y_{t+1} , update statistics 6: Replicate active points with r_i(t) > \beta_{\text{target}}(t) 7: end for
|
| 63 |
+
|
| 64 |
+
[p. 3 | section: Algorithm 1 Certificate-Guided Pruning (CGP) | type: Text]
|
| 65 |
+
8: Output: \hat{x}_T = \arg \max_i \hat{\mu}_i(T) , certificate A_T
|
| 66 |
+
|
| 67 |
+
[p. 3 | section: Algorithm 1 Certificate-Guided Pruning (CGP) | type: Text]
|
| 68 |
+
with empirical estimates, confidence intervals, and the active set. At time t, let \{x_1,\ldots,x_{N_t}\} be distinct points sampled, with n_i observations at x_i . Define empirical mean \hat{\mu}_i(t) = \frac{1}{n_i} \sum_{j=1}^{n_i} y_{i,j} and confidence radius r_i(t) = \sigma \sqrt{2\log(2N_tT/\delta)}/n_i , ensuring |f(x_i) - \hat{\mu}_i(t)| \leq r_i(t) with high probability.
|
| 69 |
+
|
| 70 |
+
[p. 3 | section: Algorithm 1 Certificate-Guided Pruning (CGP) | type: Text]
|
| 71 |
+
The upper and lower confidence bounds are UCB_i(t) = \hat{\mu}_i(t) + r_i(t) and LCB_i(t) = \hat{\mu}_i(t) - r_i(t) . The global lower certificate \ell_t = \max_{i \leq N_t} LCB_i(t) satisfies \ell_t \leq f(x^*) under the good event. The Lipschitz UCB envelope propagates uncertainty:
|
| 72 |
+
|
| 73 |
+
[p. 3 | section: Algorithm 1 Certificate-Guided Pruning (CGP) | type: Equation]
|
| 74 |
+
U_t(x) = \min_{i \le N_t} \left\{ \text{UCB}_i(t) + L \cdot d(x, x_i) \right\}, \tag{1}
|
| 75 |
+
|
| 76 |
+
[p. 3 | section: Algorithm 1 Certificate-Guided Pruning (CGP) | type: Text]
|
| 77 |
+
which upper-bounds f(x) everywhere. The active set is
|
| 78 |
+
|
| 79 |
+
[p. 3 | section: Algorithm 1 Certificate-Guided Pruning (CGP) | type: Equation]
|
| 80 |
+
A_t = \left\{ x \in \mathcal{X} : U_t(x) \ge \ell_t \right\},\tag{2}
|
| 81 |
+
|
| 82 |
+
[p. 3 | section: Algorithm 1 Certificate-Guided Pruning (CGP) | type: Text]
|
| 83 |
+
and points outside A_t are certifiably suboptimal: their upper bound is below the lower bound on f^* .
|
| 84 |
+
|
| 85 |
+
[p. 3 | section: Algorithm 1 Certificate-Guided Pruning (CGP) | type: Text]
|
| 86 |
+
The algorithm selects queries via \mathrm{score}(x) = U_t(x) - \lambda \cdot \min_{i \leq N_t} d(x, x_i) where \lambda = L , selecting x_{t+1} = \arg\max_{x \in A_t} \mathrm{score}(x) . The first term favors high UCB regions while the second encourages coverage. CGP allocates additional samples to active points with r_i(t) > \beta_{\mathrm{target}}(t) to reduce confidence radii. The target confidence radius follows a schedule \beta_{\mathrm{target}}(t) = \sigma \sqrt{2\log(2T^2/\delta)/t} , ensuring that confidence radii decrease at rate O(1/\sqrt{t}) .
|
| 87 |
+
|
| 88 |
+
[p. 3 | section: Algorithm 1 Certificate-Guided Pruning (CGP) | type: Text]
|
| 89 |
+
We compute A_t via discretization for low dimensions and Monte Carlo sampling for d>5 (details in Appendix B.2). Theoretically, CGP assumes oracle access to \arg\max_{x\in A_t}\mathrm{score}(x) ; practically, we use CMA-ES with 10 random restarts within A_t (see Appendix B for details). For d\leq 3 , we additionally use Delaunay triangulation to identify candidate optima at Voronoi vertices. Membership in A_t is exact: checking U_t(x)\geq \ell_t requires O(N_t) time. Approximate maximization may slow convergence of \eta_t but does not invalidate certificates: any x\notin A_t remains certifiably suboptimal regardless of which x\in A_t is queried.
|
| 90 |
+
|
| 91 |
+
[p. 3 | section: Algorithm 1 Certificate-Guided Pruning (CGP) | type: Text]
|
| 92 |
+
Replication Strategy. When an active point x_i has r_i(t) > \beta_{\text{target}}(t) , we allocate \lceil (r_i(t)/\beta_{\text{target}}(t))^2 \rceil additional samples to reduce its confidence radius. This ensures all active points have comparable confidence, preventing any single point from dominating the envelope.
|
| 93 |
+
|
| 94 |
+
[p. 3 | section: 4. Theoretical Analysis | type: Text]
|
| 95 |
+
Having described the algorithm, we now establish its theoretical guarantees. We show that the active set is contained in the near-optimal set, its volume shrinks at a controlled rate, and this yields instance-dependent sample complexity. All results hold on the good event \mathcal E where |f(x_i) - \hat{\mu}_i(t)| \leq r_i(t) for all t,i. All proofs are deferred to Appendix C.
|
| 96 |
+
|
| 97 |
+
[p. 3 | section: 4. Theoretical Analysis | type: Text]
|
| 98 |
+
Lemma 4.1 (Good event). With r_i(t) = \sigma \sqrt{2 \log(2N_t T/\delta)/n_i} , we have \mathbb{P}[\mathcal{E}] \geq 1 - \delta .
|
| 99 |
+
|
| 100 |
+
[p. 3 | section: 4. Theoretical Analysis | type: Text]
|
| 101 |
+
Lemma 4.2 (UCB envelope is valid). On \mathcal{E} , for all x \in \mathcal{X} : f(x) \leq U_t(x) .
|
| 102 |
+
|
| 103 |
+
[p. 3 | section: 4. Theoretical Analysis | type: Text]
|
| 104 |
+
Proof sketch. For any sampled x_i , on \mathcal{E} : f(x_i) \leq \mathrm{UCB}_i(t) . By Lipschitz continuity: f(x) \leq f(x_i) + L \cdot d(x, x_i) \leq \mathrm{UCB}_i(t) + L \cdot d(x, x_i) . Taking min over i gives f(x) \leq U_t(x) .
|
| 105 |
+
|
| 106 |
+
[p. 3 | section: 4. Theoretical Analysis | type: Text]
|
| 107 |
+
Lemma 4.3 (Envelope slack bound). On \mathcal{E} , for all x \in \mathcal{X} : U_t(x) \leq f(x) + 2\rho_t(x) , where \rho_t(x) = \min_i \{r_i(t) + L \cdot d(x, x_i)\} .
|
| 108 |
+
|
| 109 |
+
[p. 3 | section: 4. Theoretical Analysis | type: Text]
|
| 110 |
+
Remark 4.4 (Slack in envelope bound). The factor of 2 arises from applying Lipschitz continuity (f(x_i) \leq f(x) + L \cdot d(x, x_i)) after bounding UCB_i \leq f(x_i) + 2r_i . A tighter bound separating the confidence and distance terms is possible but complicates notation without affecting rate dependencies. Constants throughout are not optimized.
|
| 111 |
+
|
| 112 |
+
[p. 3 | section: 4. Theoretical Analysis | type: Text]
|
| 113 |
+
Theorem 4.5 (Active set containment). On \mathcal{E} , A_t \subseteq \{x : f(x) \ge f^* - 2\Delta_t\} where \Delta_t = \sup_{x \in A_t} \rho_t(x) + (f^* - \ell_t) .
|
| 114 |
+
|
| 115 |
+
[p. 3 | section: 4. Theoretical Analysis | type: Text]
|
| 116 |
+
Proof sketch. On \mathcal{E} , \ell_t = \max_i \mathrm{LCB}_i(t) \leq f^* since each \mathrm{LCB}_i(t) \leq f(x_i) \leq f^* . For x \in A_t , by definition U_t(x) \geq \ell_t . Applying Lemma 4.3: f(x) + 2\rho_t(x) \geq U_t(x) \geq \ell_t . Rearranging: f(x) \geq \ell_t - 2\rho_t(x) = f^* - (f^* - \ell_t) - 2\rho_t(x) \geq f^* - 2\Delta_t .
|
| 117 |
+
|
| 118 |
+
[p. 3 | section: 4. Theoretical Analysis | type: Text]
|
| 119 |
+
The containment theorem bounds how far active points can be from optimal. To translate this into a volume bound, we introduce two key quantities: the covering radius \eta_t = \sup_{x \in A_t} \min_i d(x, x_i) measuring how well samples cover A_t , and the active confidence radius \beta_t = \max_{i:x_i \text{ active }} r_i(t) measuring confidence precision.
|
| 120 |
+
|
| 121 |
+
[p. 3 | section: 4. Theoretical Analysis | type: Text]
|
| 122 |
+
Theorem 4.6 (Shrinkage theorem). Under Assumptions 2.1–2.3, on \mathcal{E} :
|
| 123 |
+
|
| 124 |
+
[p. 3 | section: 4. Theoretical Analysis | type: Equation]
|
| 125 |
+
Vol(A_t) \le C \cdot \left(2(\beta_t + L\eta_t) + \gamma_t\right)^{d-\alpha},\tag{3}
|
| 126 |
+
|
| 127 |
+
[p. 4 | section: 4. Theoretical Analysis | type: Code]
|
| 128 |
+
where \gamma_t = f^* - \ell_t.
|
| 129 |
+
|
| 130 |
+
[p. 4 | section: 4. Theoretical Analysis | type: Text]
|
| 131 |
+
205206
|
| 132 |
+
|
| 133 |
+
[p. 4 | section: 4. Theoretical Analysis | type: Text]
|
| 134 |
+
Proof sketch. From Theorem 4.5, A_t \subseteq \{x: f(x) \ge f^* - 2\Delta_t\} . For x \in A_t , \rho_t(x) \le \beta_t + L\eta_t (the worst-case slack from active-point confidence plus covering distance), so \Delta_t \le \beta_t + L\eta_t + \gamma_t/2 . Applying Assumption 2.3 with \varepsilon = 2\Delta_t : \operatorname{Vol}(A_t) \le C(2\Delta_t)^{d-\alpha} \le C(2(\beta_t + L\eta_t) + \gamma_t)^{d-\alpha} .
|
| 135 |
+
|
| 136 |
+
[p. 4 | section: 4. Theoretical Analysis | type: Text]
|
| 137 |
+
This makes pruning measurable: \beta_t is controlled by replication, \eta_t by the query rule, \gamma_t by best-point improvement. All three quantities can be computed during the run, enabling practitioners to monitor progress.
|
| 138 |
+
|
| 139 |
+
[p. 4 | section: 4. Theoretical Analysis | type: Text]
|
| 140 |
+
Remark 4.7 (Certificate validity vs. progress estimation). The certificate itself is the set membership rule x \in A_t \Leftrightarrow U_t(x) \geq \ell_t , which is exact given (\hat{\mu}_i, r_i) and does not depend on any volume estimator. Approximations (grid/Monte Carlo) are used only to estimate \operatorname{Vol}(A_t) for monitoring and optional stopping heuristics; certificate validity is unaffected by volume estimation errors.
|
| 141 |
+
|
| 142 |
+
[p. 4 | section: 4. Theoretical Analysis | type: Text]
|
| 143 |
+
The shrinkage theorem directly yields sample complexity by bounding how many samples are needed to drive \beta_t , \eta_t , and \gamma_t below \varepsilon .
|
| 144 |
+
|
| 145 |
+
[p. 4 | section: 4. Theoretical Analysis | type: Text]
|
| 146 |
+
Theorem 4.8 (Sample complexity). Under Assumptions 2.1–2.3, CGP achieves r_T \le \varepsilon with probability \ge 1 - \delta using T = \tilde{O}(L^d \varepsilon^{-(2+\alpha)} \log(1/\delta)) samples. When \alpha < d , this improves upon the worst-case \tilde{O}(\varepsilon^{-(2+d)}) rate.
|
| 147 |
+
|
| 148 |
+
[p. 4 | section: 4. Theoretical Analysis | type: Text]
|
| 149 |
+
The following lower bound shows that our sample complexity is optimal up to logarithmic factors.
|
| 150 |
+
|
| 151 |
+
[p. 4 | section: 4. Theoretical Analysis | type: Text]
|
| 152 |
+
Theorem 4.9 (Lower bound). For any algorithm and \alpha \in (0, d] , there exists f satisfying Assumption 2.3 requiring T = \Omega(\varepsilon^{-(2+\alpha)}) samples for \varepsilon -optimality with probability \geq 2/3 .
|
| 153 |
+
|
| 154 |
+
[p. 4 | section: 4. Theoretical Analysis | type: Text]
|
| 155 |
+
This establishes CGP is minimax optimal up to logarithmic factors. A key property is anytime validity: at any t, any x \notin A_t satisfies f(x) < f^* - \varepsilon_t for computable \varepsilon_t > 0 . Full proofs are in Appendix C.
|
| 156 |
+
|
| 157 |
+
[p. 4 | section: 5. CGP-Adaptive: Learning L Online | type: Text]
|
| 158 |
+
The theoretical results above assume known L, which is often unavailable in practice. Underestimating L invalidates certificates, while overestimating is safe but conservative. We develop CGP-Adaptive (Algorithm 2), which learns L online via a doubling scheme with provable guarantees.
|
| 159 |
+
|
| 160 |
+
[p. 4 | section: 5. CGP-Adaptive: Learning L Online | type: Text]
|
| 161 |
+
The key insight is that Lipschitz violations are detectable. If |\hat{\mu}_i - \hat{\mu}_j| - 2(r_i + r_j) > \hat{L} \cdot d(x_i, x_j) , then \hat{L} underestimates L with high probability. CGP-Adaptive uses a doubling scheme: start with conservative \hat{L}_0 , and upon detecting a violation, double \hat{L} .
|
| 162 |
+
|
| 163 |
+
[p. 4 | section: Algorithm 2 CGP-Adaptive | type: Code]
|
| 164 |
+
Require: Domain \mathcal{X}, initial estimate \hat{L}_0, noise \sigma, budget T, confidence \delta 1: \hat{L} \leftarrow \hat{L}_0, k \leftarrow 0 (doubling counter) 2: for t = 1, ..., T do Run CGP iteration with current \hat{L} for all pairs (i, j) with n_i, n_j \ge \log(T/\delta) do 4: 5: if |\hat{\mu}_i - \hat{\mu}_j| - 2(r_i + r_j) > \hat{L} \cdot d(x_i, x_j) then \hat{L} \leftarrow 2\hat{L}, k \leftarrow k + 1 {Doubling event} 6: Recompute A_t with new \hat{L} 7: 8: end if end for 10: end for
|
| 165 |
+
|
| 166 |
+
[p. 4 | section: Algorithm 2 CGP-Adaptive | type: Text]
|
| 167 |
+
Theorem 5.1 (Adaptive L guarantee: learning regime). Let L^* = \sup_{x \neq y} |f(x) - f(y)|/d(x,y) be the true Lipschitz constant. CGP-Adaptive with initial \hat{L}_0 \leq L^* (learning from underestimation) satisfies:
|
| 168 |
+
|
| 169 |
+
[p. 4 | section: Algorithm 2 CGP-Adaptive | type: ListGroup]
|
| 170 |
+
1. The number of doubling events is at most K = \lceil \log_2(L^*/\hat{L}_0) \rceil . 2. After all doublings, \hat{L} \in [L^*, 2L^*] with probability \geq 1 \delta . 3. The total sample complexity is T = \tilde{O}(\varepsilon^{-(2+\alpha)} \cdot K) , i.e., O(\log(L^*/\hat{L}_0)) multiplicative overhead. 4. Certificate validity: Certificates are valid only after the final doubling (when \hat{L} \geq L^* ). Before this, certificates may falsely exclude near-optimal points.
|
| 171 |
+
|
| 172 |
+
[p. 4 | section: Algorithm 2 CGP-Adaptive | type: Text]
|
| 173 |
+
Remark 5.2 (Anytime-valid certificates). For applications requiring certificates valid at all times, use \hat{L}_0 \geq L^* (conservative overestimate). This ensures \hat{L} \geq L^* throughout, so all certificates are valid, but may be overly conservative. One can optionally decrease \hat{L} when evidence suggests overestimation, but this requires different analysis than the doubling scheme above.
|
| 174 |
+
|
| 175 |
+
[p. 4 | section: Algorithm 2 CGP-Adaptive | type: Text]
|
| 176 |
+
This is the first provably correct adaptive L estimation for Lipschitz optimization with certificates. Prior work (Malherbe & Vayatis, 2017) estimates L but without guarantees on certificate validity. Table 5 shows CGP-Adaptive matches oracle performance (known L) within 8% while being robust to 100\times underestimation of initial \hat{L}_0 .
|
| 177 |
+
|
| 178 |
+
[p. 4 | section: 6. CGP-TR: Trust Regions for High Dimensions | type: Text]
|
| 179 |
+
CGP-Adaptive addresses the unknown L problem, but another challenge remains: scalability. The covering number of A_t grows as O(\eta^{-d}) , making CGP intractable for d > 15.
|
| 180 |
+
|
| 181 |
+
[p. 5 | section: 6. CGP-TR: Trust Regions for High Dimensions | type: TableGroup]
|
| 182 |
+
Table 3. Simple regret ( \times 10^{-2} ) at T = 200. Bold: best; †: significant vs second-best. Method Needle Branin Hartmann Ackley Levy Rosen. SVM Random 8.2 12.1 15.3 22.4 18.7 14.2 11.2 GP-UCB 2.1 1.8 4.2 12.3 5.1 4.8 3.9 TuRBO 1.8 2.1 3.1 9.8 4.3 3.9 3.2 HEBO 1.9 1.6 3.3 9.4 4.1 3.7 3.1 BORE 2.0 1.9 3.5 10.1 4.5 4.1 3.4 HOO 3.4 5.2 8.7 14.2 9.8 8.1 7.1 CGP 1.2 2.0 2.9 8.1 3.8 3.8 2.8 CGP-A 1.3 2.1 3.0 8.3 3.9 3.9 2.9 CGP-H 1.1 ^{\dagger} 1.4 ^{\dagger} 2.7 ^{\dagger} 7.8 ^{\dagger} 3.5 ^{\dagger} 3.4 ^{\dagger} 2.6 ^{\dagger}
|
| 183 |
+
|
| 184 |
+
[p. 5 | section: 6. CGP-TR: Trust Regions for High Dimensions | type: Text]
|
| 185 |
+
To enable high-dimensional optimization, we develop CGP-TR (Algorithm 3), which maintains local certificates within trust regions that adapt based on observed progress.
|
| 186 |
+
|
| 187 |
+
[p. 5 | section: 6. CGP-TR: Trust Regions for High Dimensions | type: Text]
|
| 188 |
+
The key insight is that certificates need not be global. A local certificate A_t^{\mathcal{T}} within trust region \mathcal{T} \subset \mathcal{X} still provides valid bounds for \arg\max_{x \in \mathcal{T}} f(x) . CGP-TR maintains multiple trust regions \{\mathcal{T}_1, \dots, \mathcal{T}_m\} centered at promising points, with radii that expand on success and contract on failure (following TuRBO (Eriksson et al., 2019)).
|
| 189 |
+
|
| 190 |
+
[p. 5 | section: 6. CGP-TR: Trust Regions for High Dimensions | type: Text]
|
| 191 |
+
Certified restarts. We restart a trust region only when it is certifiably suboptimal: if u_t^{(j)} := \max_{x \in \mathcal{T}_j} U_t(x) satisfies u_t^{(j)} < \ell_t where \ell_t := \max_i \mathrm{LCB}_i(t) , then with high probability \sup_{x \in \mathcal{T}_j} f(x) < f(x^*) , so \mathcal{T}_j cannot contain x^* and can be safely restarted. This certified restart rule ensures that regions containing x^* are never falsely eliminated. In our implementation, contraction is lower-bounded by r_{\min} and centers are fixed, so a region that contains x^* cannot be contracted to exclude it; restarts occur only via the certified condition u_t^{(j)} < \ell_t .
|
| 192 |
+
|
| 193 |
+
[p. 5 | section: 6. CGP-TR: Trust Regions for High Dimensions | type: Text]
|
| 194 |
+
Theorem 6.1 (CGP-TR with certified restarts: correctness and allocation). Assume the good event \mathcal{E} holds for the confidence bounds used to construct U_t and \ell_t . CGP-TR uses certified restarts: restart \mathcal{T}_j only if u_t^{(j)} := \max_{x \in \mathcal{T}_j} U_t(x) < \ell_t .
|
| 195 |
+
|
| 196 |
+
[p. 5 | section: 6. CGP-TR: Trust Regions for High Dimensions | type: Text]
|
| 197 |
+
Let \mathcal{T}^* be a trust region that contains x^* at some time and is not contracted to exclude x^* (e.g., contraction is lower-bounded by r_{\min} and the center remains fixed). Then:
|
| 198 |
+
|
| 199 |
+
[p. 5 | section: 6. CGP-TR: Trust Regions for High Dimensions | type: ListGroup]
|
| 200 |
+
1. (No false restarts) On \mathcal{E} , \mathcal{T}^* is never restarted by the certified rule. 2. (Local certificate) Conditioned on receiving T^* evaluations inside T^* , the local active set A_t^{(T^*)} = \{x \in T^* : U_t(x) \geq \ell_t^{(T^*)}\} satisfies the same containment/shrinkage/sample-complexity bounds as CGP on the restricted domain T^* . 3. (Allocation bound) Define the region gap \Delta_i :=
|
| 201 |
+
|
| 202 |
+
[p. 5 | section: Algorithm 3 CGP-TR (Trust Region with Certified Restarts) | type: Text]
|
| 203 |
+
Require: Domain \mathcal{X}, L, \sigma , budget T, initial radius r_0, n_{\text{trust}} regions
|
| 204 |
+
|
| 205 |
+
[p. 5 | section: Algorithm 3 CGP-TR (Trust Region with Certified Restarts) | type: ListGroup]
|
| 206 |
+
1: Initialize n_{\text{trust}} trust regions at Sobol points with radius r_0 2: for t = 1, ..., T do 3: Compute \ell_t := \max_{i \leq N_t} LCB_i(t) (global lower certificate) 4: Select trust region \mathcal{T}_j with highest u_t^{(j)} := \max_{x \in \mathcal{T}_i} U_t(x) 5: Run CGP within \mathcal{T}_j : compute local A_t^{(j)} = \{x \in \mathcal{T}_j : U_t(x) \ge \ell_t^{(j)}\} 6: Query x_{t+1} \in A_t^{(j)} , observe y_{t+1} 7: if u_t^{(j)} < \ell_t then 8: Certified restart: restart \mathcal{T}_j at a new Sobol point with radius r_0 9: else if improvement in \mathcal{T}_i then 10: Expand: r_j \leftarrow \min(2r_j, D/2) 11: else if no improvement for \tau_{\text{fail}} iterations then 12: Contract: r_j \leftarrow \max(r_j/2, r_{\min}) 13: end if 14: end for 15: Output: Best point across all regions, local certificate A_T^{(j^*)}
|
| 207 |
+
|
| 208 |
+
[p. 5 | section: Algorithm 3 CGP-TR (Trust Region with Certified Restarts) | type: Text]
|
| 209 |
+
f^* - \sup_{x \in \mathcal{T}_j} f(x) (with \Delta_j > 0 for suboptimal regions). Assume each region runs CGP with replication ensuring its maximal active-point confidence radius after n within-region samples satisfies \beta_j(n) \le c_\sigma \sqrt{\log(c_T/\delta)/n} for constants c_\sigma, c_T matching the paper's confidence schedule. If the region radii are eventually bounded so that L \cdot \operatorname{diam}(\mathcal{T}_j) \le \Delta_j/8 , then any suboptimal region j is selected at most
|
| 210 |
+
|
| 211 |
+
[p. 5 | section: Algorithm 3 CGP-TR (Trust Region with Certified Restarts) | type: Equation]
|
| 212 |
+
N_j \le \frac{64c_\sigma^2}{\Delta_j^2} \log\left(\frac{c_T}{\delta}\right) + 1
|
| 213 |
+
|
| 214 |
+
[p. 5 | section: Algorithm 3 CGP-TR (Trust Region with Certified Restarts) | type: Text]
|
| 215 |
+
times before it is eliminated by the certified restart rule.
|
| 216 |
+
|
| 217 |
+
[p. 6 | section: Algorithm 4 CGP-Hybrid | type: Text]
|
| 218 |
+
Require: Domain X , L, σ, budget T, switch threshold ρthresh = 0.5
|
| 219 |
+
|
| 220 |
+
[p. 6 | section: Algorithm 4 CGP-Hybrid | type: ListGroup]
|
| 221 |
+
1: Phase 1: Run CGP until Vol(At) < 0.1 · Vol(X ) or t > T /3 2: Estimate ρ t = Lˆ local(t)/Lˆ global 3: if ρ t < ρthresh then 4: Phase 2: Switch to GP-UCB within A t (GP refinement) 5: Fit GP to points in At, continue with GP-UCB acquisition 6: else
|
| 222 |
+
|
| 223 |
+
[p. 6 | section: Algorithm 4 CGP-Hybrid | type: ListGroup]
|
| 224 |
+
7: Phase 2: Continue CGP within A t 8: end if 9: Output: Best point, certificate A T (from CGP phase)
|
| 225 |
+
|
| 226 |
+
[p. 6 | section: Algorithm 4 CGP-Hybrid | type: Text]
|
| 227 |
+
The key advantage is that covering T j requires O((rj/η) d ) points, and since r j ≪ D, this is tractable even for large d. With ntrust = O(log T) regions, CGP-TR explores globally while maintaining local certificates. The allocation bound (Theorem 6.1, item 3) ensures that suboptimal regions receive only O(log T /∆ 2 j ) evaluations before certified elimination, preventing wasted samples.
|
| 228 |
+
|
| 229 |
+
[p. 6 | section: Algorithm 4 CGP-Hybrid | type: Text]
|
| 230 |
+
CGP-TR provides local rather than global certificates, but the certified restart rule guarantees that the region containing x ∗ is never falsely eliminated. This enables scaling to d = 50 to 100 where global Lipschitz methods fail entirely.
|
| 231 |
+
|
| 232 |
+
[p. 6 | section: 7. CGP-Hybrid: Best of Both Worlds | type: Text]
|
| 233 |
+
While CGP-TR addresses scalability, some functions exhibit local smoothness that GPs can exploit more effectively than Lipschitz methods. CGP-Hybrid (Algorithm 4) preserves CGP's anytime certificates while allowing any optimizer to refine within the certified active set. The key point is modularity: Phase 1 constructs a certificate At; Phase 2 performs additional optimization restricted to A t without affecting certificate validity. We instantiate Phase 2 with GP-UCB when local smoothness is detected, but other optimizers can be used. This design captures the best of both worlds: CGP's explicit pruning guarantees and GP's ability to exploit local smoothness when present.
|
| 234 |
+
|
| 235 |
+
[p. 6 | section: 7. CGP-Hybrid: Best of Both Worlds | type: Text]
|
| 236 |
+
Define the effective smoothness ratio ρ t = Lˆ local(t)/Lˆ global, where Lˆ local(t) is estimated from points within At. When ρ t < 0.5, the function is significantly smoother near the optimum, and GP refinement is beneficial.
|
| 237 |
+
|
| 238 |
+
[p. 6 | section: 7. CGP-Hybrid: Best of Both Worlds | type: Text]
|
| 239 |
+
Proposition 7.1 (Hybrid guarantee). CGP-Hybrid achieves:
|
| 240 |
+
|
| 241 |
+
[p. 6 | section: 7. CGP-Hybrid: Best of Both Worlds | type: ListGroup]
|
| 242 |
+
1. If ρ ≥ 0.5 : same guarantee as CGP, T = O˜(ε −(2+α) ) . 2. If ρ < 0.5 : after CGP reduces A t to volume V , GP- UCB operates within a restricted domain of effective
|
| 243 |
+
|
| 244 |
+
[p. 6 | section: 7. CGP-Hybrid: Best of Both Worlds | type: Text]
|
| 245 |
+
diameter O(V 1 /d) . The additional sample complexity depends on the GP kernel's information gain γ T over At ; empirically, this yields faster convergence than continuing CGP when the function is locally smooth.
|
| 246 |
+
|
| 247 |
+
[p. 6 | section: 7. CGP-Hybrid: Best of Both Worlds | type: Text]
|
| 248 |
+
3. The certificate A T from Phase 1 remains valid regard less of Phase 2 method.
|
| 249 |
+
|
| 250 |
+
[p. 6 | section: 7. CGP-Hybrid: Best of Both Worlds | type: Text]
|
| 251 |
+
Proposition 7.2 (Certificate invariance under Phase 2). The certificate A t computed by CGP in Phase 1 remains valid regardless of the Phase-2 optimizer, since validity depends only on the confidence bounds and Lipschitz envelope used to define U t and ℓt . Specifically, any point x /∈ A t sat isfies f(x) < f ∗ − ε t with high probability, where ε t is computable from Phase 1 quantities alone.
|
| 252 |
+
|
| 253 |
+
[p. 6 | section: 7. CGP-Hybrid: Best of Both Worlds | type: Text]
|
| 254 |
+
The key insight is that CGP's certificate remains valid even when switching to GP: A t still contains x ∗ with high probability, so GP refinement within A t is safe. This provides the best of both worlds: CGP's certificates and pruning efficiency when ρ ≥ 0.5, and GP's smoothness exploitation when ρ < 0.5.
|
| 255 |
+
|
| 256 |
+
[p. 6 | section: 7. CGP-Hybrid: Best of Both Worlds | type: Text]
|
| 257 |
+
Table 6 shows CGP-Hybrid wins on all 12 benchmarks, including Branin (ρ = 0.31) and Rosenbrock (ρ = 0.28) where it detects low ρ and switches to GP refinement, and Needle (ρ = 0.98) where it stays with CGP.
|
| 258 |
+
|
| 259 |
+
[p. 6 | section: 7. CGP-Hybrid: Best of Both Worlds | type: TableGroup]
|
| 260 |
+
Table 4. High-dimensional benchmarks (d > 20) at T = 500. Method Rover-60 NAS-36 Ant-100 Random 42.1 ± 1.2 38.4 ± 0.9 51.2 ± 1.4 TuRBO 12.4 ± 0.4 11.2 ± 0.3 18.7 ± 0.6 HEBO 14.1 ± 0.5 12.8 ± 0.4 21.3 ± 0.7 CMA-ES 15.8 ± 0.6 14.1 ± 0.5 19.4 ± 0.6 CGP (intractable for d > 15) CGP-TR † 11.2 ± 0.3 † 10.4 ± 0.3 † 17.1 ± 0.5 † CGP-TR additionally provides local optimality certificates.
|
| 261 |
+
|
| 262 |
+
[p. 6 | section: 8. Related Work | type: Text]
|
| 263 |
+
Our work is primarily grounded in the literature on Lipschitz bandits and global optimization. Foundational approaches, such as the continuum-armed bandits of Kleinberg et al. (2008) and the X-armed bandit framework of Bubeck et al. (2011) , utilize zooming mechanisms to achieve regret bounds depending on the near-optimality dimension. These concepts were refined for deterministic and stochastic settings via tree-based algorithms like DOO/SOO (Munos, 2011) and StoSOO (Valko et al., 2013) . However, a key distinction is that zooming algorithms maintain pruning implicitly as an analysis artifact, whereas CGP exposes the active set A t as a computable object with measurable volume. This explicit geometric approach also relates to partitionbased global optimization methods like DIRECT (Jones
|
| 264 |
+
|
| 265 |
+
[p. 7 | section: 8. Related Work | type: TableGroup]
|
| 266 |
+
Table 5. CGP-Adaptive with varying initial Lˆ0. Robust to 100× underestimation. Initial Lˆ 0 Doublings Final L/L ˆ ∗ Regret (×10−2 ) Overhead ∗ L (oracle) 0 1.0 2.9 ± 0.1 1.0× ∗/2 L 1 1.0 3.0 ± 0.1 1.03× ∗/10 L 4 1.6 3.1 ± 0.1 1.07× ∗/100 L 7 1.3 3.2 ± 0.2 1.12× LIPO (adaptive) – – 6.2 ± 0.3 – Table 6. CGP-Hybrid smoothness detection. ρ < 0.5 triggers GP refinement.
|
| 267 |
+
|
| 268 |
+
[p. 7 | section: 8. Related Work | type: Table]
|
| 269 |
+
Benchmark ρˆ Phase 2 CGP-H Regret Best Baseline Needle-2D 0.98 CGP 1.1 ± 0.1 1.8 (TuRBO) Branin 0.31 GP 1.4 ± 0.1 1.6 (HEBO) Hartmann-6 0.72 CGP 2.7 ± 0.1 3.1 (TuRBO) Rosenbrock 0.28 GP 3.4 ± 0.1 3.7 (HEBO) Ackley-10 0.85 CGP 7.8 ± 0.3 9.4 (HEBO) Levy-5 0.67 CGP 3.5 ± 0.2 4.1 (HEBO) SVM-RBF 0.74 CGP 2.6 ± 0.1 3.1 (HEBO) LunarLander 0.81 CGP 6.1 ± 0.3 7.0 (HEBO)
|
| 270 |
+
|
| 271 |
+
[p. 7 | section: 8. Related Work | type: Text]
|
| 272 |
+
et al., 1993) and LIPO (Malherbe & Vayatis, 2017) . While LIPO addresses the unknown Lipschitz constant, it lacks the certificate guarantees provided by our CGP-Adaptive doubling scheme (Shihab et al., 2025c) . Our theoretical analysis further draws on sample complexity results under margin conditions from the finite-arm setting (Audibert & Bubeck, 2010; Jamieson & Nowak, 2014; Shihab et al., 2025b) , adapting confidence bounds from the UCB frame w ork (Auer et al., 2002) to continuous spaces with explicit uncertainty representation similar to safe optimization levelsets (Sui et al., 2015) .
|
| 273 |
+
|
| 274 |
+
[p. 7 | section: 8. Related Work | type: Text]
|
| 275 |
+
In the broader context of black-box optimization, Bayesian methods provide the standard alternative for uncertainty quantification. Classic approaches like GP-UCB (Srinivas et al., 2009) , Entropy Search (Hennig & Schuler, 2012; Hernandez-Lobato et al. ´ , 2014) , and Thompson Sampling (Thompson, 1933) offer strong performance but scale cubically with observations. To address high-dimensional scaling, recent work has introduced trust regions (TuRBO) (Eriksson et al., 2019) and nonstationary priors (HEBO) (Cowen-Rivers et al., 2022) . More recently, advanced heuristics such as Bounce (Papenmeier et al., 2024) have improved geometric adaptation for mixed spaces, while Prior-Fitted Networks (Hollmann et al., 2023) and generative diffusion models like Diff-BBO (Wu et al., 2024) exploit massive pretraining to minimize regret rapidly. While these emerging methods achieve impressive empirical results, they remain fundamentally heuristic, lacking the computable stopping criteria or active set containment guarantees that are central to CGP. We therefore focus our comparison on established
|
| 276 |
+
|
| 277 |
+
[p. 7 | section: 8. Related Work | type: Text]
|
| 278 |
+
baselines to isolate the specific utility of our certification mechanism, distinguishing our approach from heuristic hyperparameter tuners like Hyperband (Li et al., 2018) by prioritizing provable safety over raw speed.
|
| 279 |
+
|
| 280 |
+
[p. 7 | section: 9. Experiments | type: Text]
|
| 281 |
+
We evaluate CGP variants on 12 benchmarks spanning d ∈ [2, 100], measuring simple regret, certificate utility, and scalability. Code will be available upon acceptance.
|
| 282 |
+
|
| 283 |
+
[p. 7 | section: 9. Experiments | type: Text]
|
| 284 |
+
Setup. We compare against 9 baselines: Random Search, GP-UCB, TuRBO, HEBO, BORE, HOO, StoSOO, LIPO, and SAASBO (see Appendix E for configurations). We evaluate on 12 benchmarks spanning d ∈ [2, 100]: lowdimensional (Needle-2D, Branin, Hartmann-6, Levy-5, Rosenbrock-4), medium-dimensional (Ackley-10, SVM-RBF-6, LunarLander-12), and high-dimensional (Rover-60, NAS-36, MuJoCo-Ant-100). All experiments use 30 runs with σ = 0.1 noise; we report mean ± SE with Bonferronicorrected t-tests (p < 0.05).
|
| 285 |
+
|
| 286 |
+
[p. 7 | section: 9. Experiments | type: Text]
|
| 287 |
+
Table 3 shows CGP-Hybrid performs best among tested methods on all 7 low and medium-dimensional benchmarks. On Branin and Rosenbrock where vanilla CGP lost to HEBO, CGP-Hybrid detects ρ < 0.5 and switches to GP refinement, achieving 12% and 8% improvement over HEBO respectively. On Needle where ρ ≈ 1, CGP-Hybrid stays with CGP and matches vanilla CGP performance. For high-dimensional problems, Table 4 demonstrates CGP-TR scales to d = 100 while outperforming TuRBO by
|
| 288 |
+
|
| 289 |
+
[p. 8 | section: 9. Experiments | type: Text]
|
| 290 |
+
9 to 12%. Critically, CGP-TR provides local certificates within trust regions, enabling principled stopping—a capability TuRBO lacks. Regarding adaptive estimation, Table 5 shows CGP-Adaptive is robust to initial underestimation: even with \hat{L}_0=L^*/100 , performance degrades only 10% with 7 doublings, validating Theorem 5.1's O(\log(L^*/\hat{L}_0)) overhead. Finally, Table 6 confirms CGP-Hybrid correctly identifies when to switch: Branin ( \rho=0.31 ) and Rosenbrock ( \rho=0.28 ) trigger GP refinement, achieving 12% and 8% improvement over HEBO. Benchmarks with \rho>0.5 stay with CGP, maintaining certificate validity.
|
| 291 |
+
|
| 292 |
+
[p. 8 | section: 9. Experiments | type: Text]
|
| 293 |
+
Shrinkage validation. Across all benchmarks, we observe Vol(A_t) shrinks to < 5% by T = 100, with empirical decay rates closely matching the theoretical bound from Theorem 4.6. This confirms our analysis is tight and the margin condition captures the true problem difficulty.
|
| 294 |
+
|
| 295 |
+
[p. 8 | section: 9. Experiments | type: TableGroup]
|
| 296 |
+
Table 7. Certificate-enabled early stopping on Hartmann-6. CGP uniquely provides actionable stopping criteria. Stopping Rule Samples Regret ( \times 10^{-2} ) Savings Fixed T = 200 200 2.9 \pm 0.1 _ Vol(A_t) < 10\% 82 \pm 9 3.8 \pm 0.2 59% Vol(A_t) < 5\% 118 \pm 14 3.2 \pm 0.2 41% {\rm Gap\ bound} < 0.05 134 \pm 12 3.0 \pm 0.1 33%
|
| 297 |
+
|
| 298 |
+
[p. 8 | section: 9. Experiments | type: Text]
|
| 299 |
+
Table 7 demonstrates certificate utility: stopping at Vol(A_t) < 10\% saves 59% of samples with only 31% regret increase. No baseline provides such principled stopping rules. In d>20, we use the computable gap proxy \varepsilon_t:=2(\beta_t+L\eta_t)+\gamma_t as the primary criterion. Beyond sample efficiency, CGP also offers computational advantages. Table 8 shows CGP variants are 6 to 8 times faster than GP-based methods due to their O(n) per-iteration cost versus GP's O(n^3) . Finally, we ablate CGP's components to understand their individual contributions.
|
| 300 |
+
|
| 301 |
+
[p. 8 | section: 9. Experiments | type: TableGroup]
|
| 302 |
+
Table 8. Wall-clock time (seconds) for T=200 on Hartmann-6. Method Time (s) Regret ( \times 10^{-2} ) Speedup CGP 58 2.9 8× CGP-Adaptive 64 3.0 7.5 \times CGP-Hybrid 72 2.7 6.7 \times GP-UCB 480 4.2 1 \times TuRBO 620 3.1 0.8 \times HEBO 890 3.3 0.5 \times
|
| 303 |
+
|
| 304 |
+
[p. 8 | section: 9. Experiments | type: Text]
|
| 305 |
+
Table 9 shows all components contribute: removing pruning certificates increases regret 78%, coverage penalty 44%, replication 33%. GP refinement provides 7% improvement on Hartmann-6 where \rho=0.72 is borderline. Table 10 validates that \hat{\alpha} < d across all benchmarks, confirming the margin condition holds and our complexity bounds apply.
|
| 306 |
+
|
| 307 |
+
[p. 8 | section: 9. Experiments | type: TableGroup]
|
| 308 |
+
Table 9. Ablation study on Hartmann-6. All components contribute. Variant Regret ( \times 10^{-2} ) Vol(A_{200}) CGP-Hybrid (full) \textbf{2.7} \pm \textbf{0.1} 2.1% GP refinement 2.9 \pm 0.1 2.1% pruning certificate 4.8 \pm 0.2 _ coverage penalty 3.9 \pm 0.2 3.8% replication 3.6 \pm 0.2 2.9% CGP-TR (d=6) 2.8 \pm 0.1 2.4% (local) CGP-Adaptive 3.0 \pm 0.1 2.4% Table 10. Empirical \hat{\alpha} estimates from shrinkage trajectories.
|
| 309 |
+
|
| 310 |
+
[p. 8 | section: 9. Experiments | type: Table]
|
| 311 |
+
Benchmark d â (95% CI) True \alpha \hat{\alpha} < d ? Needle-2D 2 1.8 \pm 0.2 2.0 √ Branin 2 1.2 \pm 0.1 _ \checkmark Hartmann-6 6 2.4 \pm 0.3 _ \checkmark Ackley-10 10 3.2 \pm 0.4 _ \checkmark Rover-60 60 8.4 \pm 1.2 _ \checkmark
|
| 312 |
+
|
| 313 |
+
[p. 8 | section: 10. Conclusion | type: Text]
|
| 314 |
+
We introduced Certificate-Guided Pruning (CGP), an algorithm for stochastic Lipschitz optimization that maintains explicit active sets with provable shrinkage guarantees. Under a margin condition with near-optimality dimension \alpha , we prove Vol(A_t) \leq C \cdot (2(\beta_t + L\eta_t) + \gamma_t)^{d-\alpha} , yielding sample complexity \tilde{O}(\varepsilon^{-(2+\alpha)}) with anytime valid certificates. Three extensions broaden applicability: CGP-Adaptive learns L online with O(\log T) overhead, CGP-TR scales to d > 50 via trust regions, and CGP-Hybrid switches to GP refinement when local smoothness is detected. The margin condition holds broadly for isolated maxima with nondegenerate Hessian \alpha = d/2 , for polynomial decay \alpha = d/p and can be estimated online from shrinkage trajectories. Limitations include requiring Lipschitz continuity and dimension constraints ( d \le 15 for vanilla CGP, d \le 100 for CGP-TR); practical guidance is in Appendix D. Future directions include safe optimization using A_t for safety certificates and CGP-TR with random embeddings for global high-dimensional certificates.
|
| 315 |
+
|
| 316 |
+
[p. 8 | section: Impact Statement | type: Text]
|
| 317 |
+
This paper introduces Certificate-Guided Pruning (CGP), a method designed to improve the sample efficiency of black-box optimization in resource-constrained settings. By providing explicit optimality certificates and principled stopping criteria, our approach significantly reduces the computational budget required for expensive tasks such as neural architecture search and simulation-based engineering, directly contributing to lower energy consumption and carbon footprints. Furthermore, the ability to certify suboptimal regions enhances reliability in safety-critical applications like
|
| 318 |
+
|
| 319 |
+
[p. 9 | section: Impact Statement | type: Text]
|
| 320 |
+
robotics. However, practitioners must ensure the validity of the Lipschitz assumption, as violations could lead to the incorrect pruning of optimal solutions.
|
icml26/5c3d6bff-e8ce-4d9f-840b-719084582491/paper.blocks.json
ADDED
|
The diff for this file is too large to render.
See raw diff
|
|
|
icml26/5c3d6bff-e8ce-4d9f-840b-719084582491/paper.md
ADDED
|
@@ -0,0 +1,1244 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
|
| 2 |
+
|
| 3 |
+
{0}------------------------------------------------
|
| 4 |
+
|
| 5 |
+
032
|
| 6 |
+
|
| 7 |
+
034
|
| 8 |
+
|
| 9 |
+
035
|
| 10 |
+
|
| 11 |
+
038
|
| 12 |
+
|
| 13 |
+
039
|
| 14 |
+
|
| 15 |
+
041
|
| 16 |
+
|
| 17 |
+
043
|
| 18 |
+
|
| 19 |
+
044
|
| 20 |
+
|
| 21 |
+
045
|
| 22 |
+
|
| 23 |
+
046
|
| 24 |
+
|
| 25 |
+
047
|
| 26 |
+
|
| 27 |
+
048
|
| 28 |
+
|
| 29 |
+
049
|
| 30 |
+
|
| 31 |
+
051
|
| 32 |
+
|
| 33 |
+
052
|
| 34 |
+
|
| 35 |
+
053
|
| 36 |
+
|
| 37 |
+
054
|
| 38 |
+
|
| 39 |
+
## **Certificate-Guided Pruning for Stochastic Lipschitz Optimization**
|
| 40 |
+
|
| 41 |
+
#### Anonymous Authors1
|
| 42 |
+
|
| 43 |
+
#### **Abstract**
|
| 44 |
+
|
| 45 |
+
We study black-box optimization of Lipschitz functions under noisy evaluations. Existing adaptive discretization methods implicitly avoid suboptimal regions but do not provide explicit certificates of optimality or measurable progress guarantees. We introduce Certificate-Guided Pruning (CGP), which maintains an explicit active set $A_t$ of potentially optimal points via confidence-adjusted Lipschitz envelopes. Any point outside $A_t$ is certifiably suboptimal with high probability, and under a margin condition with near-optimality dimension $\alpha$ , we prove $Vol(A_t)$ shrinks at a controlled rate yielding sample complexity $\tilde{O}(\varepsilon^{-(2+\alpha)})$ . We develop three extensions: CGP-Adaptive learns L online with $O(\log T)$ overhead; CGP-TR scales to d > 50via trust regions with local certificates; and CGP-Hybrid switches to GP refinement when local smoothness is detected. Experiments on 12 benchmarks $(d \in [2, 100])$ show CGP variants match or exceed strong baselines while providing principled stopping criteria via certificate volume.
|
| 46 |
+
|
| 47 |
+
#### 1. Introduction
|
| 48 |
+
|
| 49 |
+
Black-box optimization, the task of finding the maximum of a function $f:\mathcal{X}\to\mathbb{R}$ accessible only through noisy point evaluations, is fundamental to machine learning, with applications spanning hyperparameter tuning (Snoek et al., 2012; Bergstra & Bengio, 2012), neural architecture search (Zoph & Le, 2017), and simulation-based optimization (Fu, 2015; Brochu et al., 2010). In many such settings, evaluations are expensive: training a neural network or running a physical simulation may cost hours or dollars per query. We call these "precious calls" where each evaluation must count, motivating the need for methods that provide explicit progress guarantees.
|
| 50 |
+
|
| 51 |
+
Preliminary work. Under review by the International Conference on Machine Learning (ICML). Do not distribute. The Lipschitz continuity assumption provides a natural framework for addressing this challenge. If $|f(x)-f(y)| \le L \cdot d(x,y)$ for a known constant L, then observations at sampled points constrain f globally, enabling pruning of provably suboptimal regions. Classical methods exploiting this structure include DIRECT (Jones et al., 1993; 1998), Lipschitz bandits (Kleinberg et al., 2008; Auer et al., 2002), and adaptive discretization algorithms (Bubeck et al., 2011; Valko et al., 2013; Munos, 2014). However, existing methods implicitly avoid suboptimal regions via tree-based refinement without exposing two properties that matter for precious call optimization: (1) explicit certificates identifying which regions are provably suboptimal at any time t, and (2) measurable progress indicating how much of the domain remains plausibly optimal.
|
| 52 |
+
|
| 53 |
+
To address these limitations, we introduce Certificate-Guided Pruning (CGP), which maintains an explicit active set $A_t \subseteq \mathcal{X}$ of potentially optimal points. This set is defined via a Lipschitz UCB envelope $U_t(x)$ that upper bounds f(x)with high probability, a global lower certificate $\ell_t$ that lower bounds $f(x^*)$ , and the active set $A_t = \{x : U_t(x) \ge \ell_t\}$ . Points outside $A_t$ are certifiably suboptimal, and as sampling proceeds, $A_t$ shrinks, providing anytime valid progress certificates. Unlike prior work that uses similar mathematical tools implicitly, CGP exposes the pruning mechanism as a first-class algorithmic object: the certificate is computable in closed form, the shrinkage rate is provably controlled, and the certificate provides valid optimality bounds even when stopped early. Figure 1 illustrates this mechanism. To understand how CGP differs from existing approaches, consider zooming algorithms (Kleinberg et al., 2008; Bubeck et al., 2011). While zooming maintains a tree of "active arms" and expands nodes with high UCB, the implicit pruning is an analysis artifact not exposed to the user (Shihab et al., 2026). Table 1 makes this distinction precise: CGP provides explicit certificates, computable progress metrics, and principled stopping rules that zooming-based methods lack. Similarly, Thompson sampling (Thompson, 1933; Russo & Van Roy, 2014; Daniel et al., 2018) and information-directed methods (Hennig & Schuler, 2012; Hernández-Lobato et al., 2014) maintain implicit uncertainty without providing explicit geometric certificates.
|
| 54 |
+
|
| 55 |
+
Building on this foundation, our contributions are fourfold. First, we present CGP with explicit active set maintenance
|
| 56 |
+
|
| 57 |
+
<sup>&</sup>lt;sup>1</sup>Anonymous Institution, Anonymous City, Anonymous Region, Anonymous Country. Correspondence to: Anonymous Author <anon.email@domain.com>.
|
| 58 |
+
|
| 59 |
+
{1}------------------------------------------------
|
| 60 |
+
|
| 61 |
+
<span id="page-1-0"></span>
|
| 62 |
+
|
| 63 |
+
067 068
|
| 64 |
+
|
| 65 |
+
069
|
| 66 |
+
|
| 67 |
+
071
|
| 68 |
+
|
| 69 |
+
072
|
| 70 |
+
|
| 71 |
+
074
|
| 72 |
+
|
| 73 |
+
075
|
| 74 |
+
|
| 75 |
+
076
|
| 76 |
+
|
| 77 |
+
078
|
| 78 |
+
|
| 79 |
+
081 082
|
| 80 |
+
|
| 81 |
+
083
|
| 82 |
+
|
| 83 |
+
084
|
| 84 |
+
|
| 85 |
+
087
|
| 86 |
+
|
| 87 |
+
088
|
| 88 |
+
|
| 89 |
+
089
|
| 90 |
+
|
| 91 |
+
090
|
| 92 |
+
|
| 93 |
+
091
|
| 94 |
+
|
| 95 |
+
092
|
| 96 |
+
|
| 97 |
+
093
|
| 98 |
+
|
| 99 |
+
094
|
| 100 |
+
|
| 101 |
+
095
|
| 102 |
+
|
| 103 |
+
096
|
| 104 |
+
|
| 105 |
+
097
|
| 106 |
+
|
| 107 |
+
098
|
| 108 |
+
|
| 109 |
+
099
|
| 110 |
+
|
| 111 |
+
100
|
| 112 |
+
|
| 113 |
+
104 105
|
| 114 |
+
|
| 115 |
+
106
|
| 116 |
+
|
| 117 |
+
109
|
| 118 |
+
|
| 119 |
+
Figure 1. The active set $A_t$ (shaded) consists of points where the Lipschitz envelope $U_t(x)$ (red) exceeds the global lower bound $\ell_t$ (green dashed). Regions where $U_t(x) < \ell_t$ are certifiably suboptimal and pruned, causing $A_t$ to shrink as sampling proceeds.
|
| 120 |
+
|
| 121 |
+
<span id="page-1-1"></span>Table 1. Comparison with zooming-based bandits. CGP uniquely exports explicit certificates (active set $A_t$ , gap bound $\varepsilon_t$ , and volume $\operatorname{Vol}(A_t)$ ), enabling principled stopping criteria that implicit zooming methods lack.
|
| 122 |
+
|
| 123 |
+
| Property | CGP | Zooming | HOO/StoSOO |
|
| 124 |
+
|---------------------------|----------------------------------------|----------------------------------------|----------------------------------------------------|
|
| 125 |
+
| Explicit active set $A_t$ | ✓ | _ | _ |
|
| 126 |
+
| Computable $Vol(A_t)$ | ✓ | _ | - |
|
| 127 |
+
| Anytime optimality bound | ✓ | _ | _ |
|
| 128 |
+
| Principled stopping rule | ✓ | _ | _ |
|
| 129 |
+
| Certificate export | ✓ | _ | _ |
|
| 130 |
+
| Sample complexity | $\tilde{O}(\varepsilon^{-(2+\alpha)})$ | $\tilde{O}(\varepsilon^{-(2+\alpha)})$ | $\tilde{O}(\varepsilon^{-(2+d)})$<br>$O(\log N_t)$ |
|
| 131 |
+
| Per-iteration cost | $O(N_t)$ | $O(\log N_t)$ | $O(\log N_t)$ |
|
| 132 |
+
| Adaptive L | √(CGP-A) | _ | _ |
|
| 133 |
+
| High-dim scaling | √(CGP-TR) | - | |
|
| 134 |
+
|
| 135 |
+
and prove a shrinkage theorem: under a margin condition with near-optimality dimension $\alpha$ (i.e., Vol( $\{x: f(x) \geq$ $f^* - \varepsilon$ ) $\leq C\varepsilon^{d-\alpha}$ , we show $Vol(A_t) \leq C \cdot (2(\beta_t + \beta_t))$ $(L\eta_t)^{d-\alpha}$ , yielding sample complexity $T = \tilde{O}(\varepsilon^{-(2+\alpha)})$ that improves on the worst case $\tilde{O}(\varepsilon^{-(2+d)})$ when $\alpha < d$ (Section 4). Second, we develop CGP-Adaptive (Section 5), which learns L online via a doubling scheme, proving that unknown L adds only $O(\log T)$ multiplicative overhead, the first such guarantee for Lipschitz optimization with certificates. Third, we introduce CGP-TR (Section 6), a trust region variant that scales to d > 50 by maintaining local certificates within adaptively sized regions, enabling high-dimensional applications previously intractable for Lipschitz methods. Fourth, we propose CGP-Hybrid (Section 7), which detects local smoothness via the ratio $\rho = L_{\rm local}/L_{\rm global}$ and switches to GP refinement when $\rho < 0.5$ , achieving best of both worlds performance across diverse function classes.
|
| 136 |
+
|
| 137 |
+
These theoretical contributions translate to strong empirical performance. Experiments (Section 9) demonstrate that CGP variants are competitive with strong baselines on 12 benchmarks spanning $d \in [2, 100]$ , including Rover tra-
|
| 138 |
+
|
| 139 |
+
*Table 2.* Summary of Notation
|
| 140 |
+
|
| 141 |
+
<span id="page-1-4"></span>
|
| 142 |
+
|
| 143 |
+
| Symbol | Description |
|
| 144 |
+
|------------|-----------------------------------------------------------|
|
| 145 |
+
| $f^*$ | Global maximum value of the objective function |
|
| 146 |
+
| L | Lipschitz constant (or global upper bound) |
|
| 147 |
+
| $A_t$ | Active set at time $t$ (contains potential optimizers) |
|
| 148 |
+
| $U_t(x)$ | Lipschitz Upper Confidence Bound envelope |
|
| 149 |
+
| $\ell_t$ | Global lower certificate $(\max_i LCB_i)$ |
|
| 150 |
+
| $\alpha$ | Near-optimality dimension (problem hardness) |
|
| 151 |
+
| $\beta_t$ | Active confidence radius (uncertainty in $A_t$ ) |
|
| 152 |
+
| $\eta_t$ | Covering radius (resolution of $A_t$ ) |
|
| 153 |
+
| $\gamma_t$ | Gap to optimum proxy $(f^* - \ell_t)$ |
|
| 154 |
+
| ρ | Local smoothness ratio ( $L_{\rm local}/L_{\rm global}$ ) |
|
| 155 |
+
|
| 156 |
+
jectory optimization (d=60), neural architecture search (d=36), and safe robotics where certificates enable stopping with guaranteed bounds (Shihab et al., 2025a). CGP-Hybrid performs best among tested methods on all 12 benchmarks under matched budgets, including Branin and Rosenbrock where vanilla CGP previously lost to GP-based methods.
|
| 157 |
+
|
| 158 |
+
#### 2. Problem Formulation
|
| 159 |
+
|
| 160 |
+
Let $(\mathcal{X}, d)$ be a compact metric space with diameter $D = \sup_{x,y} d(x,y)$ . We consider $\mathcal{X} = [0,1]^d$ with Euclidean metric. Let $f: \mathcal{X} \to [0,1]$ satisfy:
|
| 161 |
+
|
| 162 |
+
<span id="page-1-2"></span>**Assumption 2.1** (Lipschitz continuity). There exists L > 0 such that for all $x, y \in \mathcal{X}$ : $|f(x) - f(y)| \le L \cdot d(x, y)$ .
|
| 163 |
+
|
| 164 |
+
We observe f through noisy queries: querying x returns $y = f(x) + \epsilon$ , where:
|
| 165 |
+
|
| 166 |
+
<span id="page-1-5"></span>**Assumption 2.2** (Sub-Gaussian noise). The noise $\epsilon$ is $\sigma$ -sub-Gaussian: $\mathbb{E}[e^{\lambda\epsilon}] \leq e^{\lambda^2 \sigma^2/2}$ for all $\lambda \in \mathbb{R}$ .
|
| 167 |
+
|
| 168 |
+
After T samples, the algorithm outputs $\hat{x}_T \in \mathcal{X}$ . The goal is to minimize simple regret $r_T = f(x^*) - f(\hat{x}_T)$ . We seek PAC guarantees: $r_T \leq \varepsilon$ with probability $\geq 1 - \delta$ .
|
| 169 |
+
|
| 170 |
+
<span id="page-1-3"></span>**Assumption 2.3** (Margin / near-optimality dimension). There exist C>0 and $\alpha\in[0,d]$ such that for all $\varepsilon>0$ : $\operatorname{Vol}(\{x\in\mathcal{X}:f(x)\geq f^*-\varepsilon\})\leq C\varepsilon^{d-\alpha}$ .
|
| 171 |
+
|
| 172 |
+
The parameter $\alpha$ is the near-optimality dimension: smaller $\alpha$ means a sharper optimum (easier), $\alpha=d$ is worst case. For isolated maxima with nondegenerate Hessian, $\alpha=d/2$ ; for $f(x)\approx f^*-c\|x-x^*\|^p$ , we have $\alpha=d/p$ . This assumption is standard in bandit theory (Audibert & Bubeck, 2010; Bubeck et al., 2011; Lattimore & Szepesvári, 2020); see Appendix A for extended discussion.
|
| 173 |
+
|
| 174 |
+
#### 3. Algorithm: Certificate-Guided Pruning
|
| 175 |
+
|
| 176 |
+
With the problem formalized, we now describe the CGP algorithm (Algorithm 1). CGP maintains sampled points
|
| 177 |
+
|
| 178 |
+
{2}------------------------------------------------
|
| 179 |
+
|
| 180 |
+
#### <span id="page-2-1"></span>Algorithm 1 Certificate-Guided Pruning (CGP)
|
| 181 |
+
|
| 182 |
+
**Require:** Domain $\mathcal{X}$ , Lipschitz constant L, noise $\sigma$ , budget T, confidence $\delta$
|
| 183 |
+
|
| 184 |
+
- 1: Initialize: Sample $x_1$ uniformly, observe $y_1$
|
| 185 |
+
- 2: **for** t = 1, ..., T 1 **do**
|
| 186 |
+
- 3: Compute $r_i(t)$ , $\ell_t = \max_i LCB_i(t)$ , $A_t = \{x : U_t(x) \ge \ell_t\}$
|
| 187 |
+
- 4: $x_{t+1} \leftarrow \arg\max_{x \in A_t} [U_t(x) L \cdot \min_i d(x, x_i)]$
|
| 188 |
+
- 5: Query $x_{t+1}$ , observe $y_{t+1}$ , update statistics
|
| 189 |
+
- 6: Replicate active points with $r_i(t) > \beta_{\text{target}}(t)$
|
| 190 |
+
- 7: end for
|
| 191 |
+
|
| 192 |
+
8: Output: $\hat{x}_T = \arg \max_i \hat{\mu}_i(T)$ , certificate $A_T$
|
| 193 |
+
|
| 194 |
+
with empirical estimates, confidence intervals, and the active set. At time t, let $\{x_1,\ldots,x_{N_t}\}$ be distinct points sampled, with $n_i$ observations at $x_i$ . Define empirical mean $\hat{\mu}_i(t) = \frac{1}{n_i} \sum_{j=1}^{n_i} y_{i,j}$ and confidence radius $r_i(t) = \sigma \sqrt{2\log(2N_tT/\delta)}/n_i$ , ensuring $|f(x_i) - \hat{\mu}_i(t)| \leq r_i(t)$ with high probability.
|
| 195 |
+
|
| 196 |
+
The upper and lower confidence bounds are $UCB_i(t) = \hat{\mu}_i(t) + r_i(t)$ and $LCB_i(t) = \hat{\mu}_i(t) - r_i(t)$ . The global lower certificate $\ell_t = \max_{i \leq N_t} LCB_i(t)$ satisfies $\ell_t \leq f(x^*)$ under the good event. The Lipschitz UCB envelope propagates uncertainty:
|
| 197 |
+
|
| 198 |
+
$$U_t(x) = \min_{i \le N_t} \left\{ \text{UCB}_i(t) + L \cdot d(x, x_i) \right\}, \tag{1}$$
|
| 199 |
+
|
| 200 |
+
which upper-bounds f(x) everywhere. The active set is
|
| 201 |
+
|
| 202 |
+
<span id="page-2-7"></span>
|
| 203 |
+
$$A_t = \left\{ x \in \mathcal{X} : U_t(x) \ge \ell_t \right\},\tag{2}$$
|
| 204 |
+
|
| 205 |
+
and points outside $A_t$ are certifiably suboptimal: their upper bound is below the lower bound on $f^*$ .
|
| 206 |
+
|
| 207 |
+
The algorithm selects queries via $\mathrm{score}(x) = U_t(x) - \lambda \cdot \min_{i \leq N_t} d(x, x_i)$ where $\lambda = L$ , selecting $x_{t+1} = \arg\max_{x \in A_t} \mathrm{score}(x)$ . The first term favors high UCB regions while the second encourages coverage. CGP allocates additional samples to active points with $r_i(t) > \beta_{\mathrm{target}}(t)$ to reduce confidence radii. The target confidence radius follows a schedule $\beta_{\mathrm{target}}(t) = \sigma \sqrt{2\log(2T^2/\delta)/t}$ , ensuring that confidence radii decrease at rate $O(1/\sqrt{t})$ .
|
| 208 |
+
|
| 209 |
+
We compute $A_t$ via discretization for low dimensions and Monte Carlo sampling for d>5 (details in Appendix B.2). Theoretically, CGP assumes oracle access to $\arg\max_{x\in A_t}\mathrm{score}(x)$ ; practically, we use CMA-ES with 10 random restarts within $A_t$ (see Appendix B for details). For $d\leq 3$ , we additionally use Delaunay triangulation to identify candidate optima at Voronoi vertices. Membership in $A_t$ is exact: checking $U_t(x)\geq \ell_t$ requires $O(N_t)$ time. Approximate maximization may slow convergence of $\eta_t$ but does not invalidate certificates: any $x\notin A_t$ remains certifiably suboptimal regardless of which $x\in A_t$ is queried.
|
| 210 |
+
|
| 211 |
+
**Replication Strategy.** When an active point $x_i$ has $r_i(t) > \beta_{\text{target}}(t)$ , we allocate $\lceil (r_i(t)/\beta_{\text{target}}(t))^2 \rceil$ additional samples to reduce its confidence radius. This ensures all active points have comparable confidence, preventing any single point from dominating the envelope.
|
| 212 |
+
|
| 213 |
+
#### <span id="page-2-0"></span>4. Theoretical Analysis
|
| 214 |
+
|
| 215 |
+
Having described the algorithm, we now establish its theoretical guarantees. We show that the active set is contained in the near-optimal set, its volume shrinks at a controlled rate, and this yields instance-dependent sample complexity. All results hold on the good event $\mathcal E$ where $|f(x_i) - \hat{\mu}_i(t)| \leq r_i(t)$ for all t,i. All proofs are deferred to Appendix C.
|
| 216 |
+
|
| 217 |
+
<span id="page-2-5"></span>**Lemma 4.1** (Good event). With $r_i(t) = \sigma \sqrt{2 \log(2N_t T/\delta)/n_i}$ , we have $\mathbb{P}[\mathcal{E}] \geq 1 - \delta$ .
|
| 218 |
+
|
| 219 |
+
<span id="page-2-6"></span>**Lemma 4.2** (UCB envelope is valid). On $\mathcal{E}$ , for all $x \in \mathcal{X}$ : $f(x) \leq U_t(x)$ .
|
| 220 |
+
|
| 221 |
+
Proof sketch. For any sampled $x_i$ , on $\mathcal{E}$ : $f(x_i) \leq \mathrm{UCB}_i(t)$ . By Lipschitz continuity: $f(x) \leq f(x_i) + L \cdot d(x, x_i) \leq \mathrm{UCB}_i(t) + L \cdot d(x, x_i)$ . Taking min over i gives $f(x) \leq U_t(x)$ .
|
| 222 |
+
|
| 223 |
+
<span id="page-2-2"></span>**Lemma 4.3** (Envelope slack bound). On $\mathcal{E}$ , for all $x \in \mathcal{X}$ : $U_t(x) \leq f(x) + 2\rho_t(x)$ , where $\rho_t(x) = \min_i \{r_i(t) + L \cdot d(x, x_i)\}$ .
|
| 224 |
+
|
| 225 |
+
Remark 4.4 (Slack in envelope bound). The factor of 2 arises from applying Lipschitz continuity $(f(x_i) \leq f(x) + L \cdot d(x, x_i))$ after bounding $UCB_i \leq f(x_i) + 2r_i$ . A tighter bound separating the confidence and distance terms is possible but complicates notation without affecting rate dependencies. Constants throughout are not optimized.
|
| 226 |
+
|
| 227 |
+
<span id="page-2-3"></span>**Theorem 4.5** (Active set containment). On $\mathcal{E}$ , $A_t \subseteq \{x : f(x) \ge f^* - 2\Delta_t\}$ where $\Delta_t = \sup_{x \in A_t} \rho_t(x) + (f^* - \ell_t)$ .
|
| 228 |
+
|
| 229 |
+
Proof sketch. On $\mathcal{E}$ , $\ell_t = \max_i \mathrm{LCB}_i(t) \leq f^*$ since each $\mathrm{LCB}_i(t) \leq f(x_i) \leq f^*$ . For $x \in A_t$ , by definition $U_t(x) \geq \ell_t$ . Applying Lemma 4.3: $f(x) + 2\rho_t(x) \geq U_t(x) \geq \ell_t$ . Rearranging: $f(x) \geq \ell_t - 2\rho_t(x) = f^* - (f^* - \ell_t) - 2\rho_t(x) \geq f^* - 2\Delta_t$ .
|
| 230 |
+
|
| 231 |
+
The containment theorem bounds how far active points can be from optimal. To translate this into a volume bound, we introduce two key quantities: the covering radius $\eta_t = \sup_{x \in A_t} \min_i d(x, x_i)$ measuring how well samples cover $A_t$ , and the active confidence radius $\beta_t = \max_{i:x_i \text{ active }} r_i(t)$ measuring confidence precision.
|
| 232 |
+
|
| 233 |
+
<span id="page-2-4"></span>**Theorem 4.6** (Shrinkage theorem). *Under Assumptions 2.1–2.3, on \mathcal{E}:*
|
| 234 |
+
|
| 235 |
+
$$Vol(A_t) \le C \cdot \left(2(\beta_t + L\eta_t) + \gamma_t\right)^{d-\alpha},\tag{3}$$
|
| 236 |
+
|
| 237 |
+
{3}------------------------------------------------
|
| 238 |
+
|
| 239 |
+
```
|
| 240 |
+
where \gamma_t = f^* - \ell_t.
|
| 241 |
+
```
|
| 242 |
+
|
| 243 |
+
166
|
| 244 |
+
|
| 245 |
+
167
|
| 246 |
+
|
| 247 |
+
168
|
| 248 |
+
|
| 249 |
+
169
|
| 250 |
+
|
| 251 |
+
170
|
| 252 |
+
|
| 253 |
+
171
|
| 254 |
+
|
| 255 |
+
172
|
| 256 |
+
|
| 257 |
+
173
|
| 258 |
+
|
| 259 |
+
174
|
| 260 |
+
|
| 261 |
+
175
|
| 262 |
+
|
| 263 |
+
176
|
| 264 |
+
|
| 265 |
+
177
|
| 266 |
+
|
| 267 |
+
178
|
| 268 |
+
|
| 269 |
+
179
|
| 270 |
+
|
| 271 |
+
180
|
| 272 |
+
|
| 273 |
+
181
|
| 274 |
+
|
| 275 |
+
182
|
| 276 |
+
|
| 277 |
+
183
|
| 278 |
+
|
| 279 |
+
184
|
| 280 |
+
|
| 281 |
+
185 186
|
| 282 |
+
|
| 283 |
+
187
|
| 284 |
+
|
| 285 |
+
188
|
| 286 |
+
|
| 287 |
+
189
|
| 288 |
+
|
| 289 |
+
190
|
| 290 |
+
|
| 291 |
+
191
|
| 292 |
+
|
| 293 |
+
192
|
| 294 |
+
|
| 295 |
+
193 194
|
| 296 |
+
|
| 297 |
+
195
|
| 298 |
+
|
| 299 |
+
196
|
| 300 |
+
|
| 301 |
+
197
|
| 302 |
+
|
| 303 |
+
198
|
| 304 |
+
|
| 305 |
+
199
|
| 306 |
+
|
| 307 |
+
200
|
| 308 |
+
|
| 309 |
+
201
|
| 310 |
+
|
| 311 |
+
202
|
| 312 |
+
|
| 313 |
+
204
|
| 314 |
+
|
| 315 |
+
205206
|
| 316 |
+
|
| 317 |
+
207
|
| 318 |
+
|
| 319 |
+
208
|
| 320 |
+
|
| 321 |
+
209
|
| 322 |
+
|
| 323 |
+
210
|
| 324 |
+
|
| 325 |
+
211
|
| 326 |
+
|
| 327 |
+
212
|
| 328 |
+
|
| 329 |
+
213
|
| 330 |
+
|
| 331 |
+
214
|
| 332 |
+
|
| 333 |
+
215
|
| 334 |
+
|
| 335 |
+
216
|
| 336 |
+
|
| 337 |
+
217
|
| 338 |
+
|
| 339 |
+
218
|
| 340 |
+
|
| 341 |
+
<span id="page-3-3"></span>219
|
| 342 |
+
|
| 343 |
+
Proof sketch. From Theorem 4.5, $A_t \subseteq \{x: f(x) \ge f^* - 2\Delta_t\}$ . For $x \in A_t$ , $\rho_t(x) \le \beta_t + L\eta_t$ (the worst-case slack from active-point confidence plus covering distance), so $\Delta_t \le \beta_t + L\eta_t + \gamma_t/2$ . Applying Assumption 2.3 with $\varepsilon = 2\Delta_t$ : $\operatorname{Vol}(A_t) \le C(2\Delta_t)^{d-\alpha} \le C(2(\beta_t + L\eta_t) + \gamma_t)^{d-\alpha}$ .
|
| 344 |
+
|
| 345 |
+
This makes pruning measurable: $\beta_t$ is controlled by replication, $\eta_t$ by the query rule, $\gamma_t$ by best-point improvement. All three quantities can be computed during the run, enabling practitioners to monitor progress.
|
| 346 |
+
|
| 347 |
+
<span id="page-3-4"></span>Remark 4.7 (Certificate validity vs. progress estimation). The certificate itself is the set membership rule $x \in A_t \Leftrightarrow U_t(x) \geq \ell_t$ , which is exact given $(\hat{\mu}_i, r_i)$ and does not depend on any volume estimator. Approximations (grid/Monte Carlo) are used only to estimate $\operatorname{Vol}(A_t)$ for monitoring and optional stopping heuristics; certificate validity is unaffected by volume estimation errors.
|
| 348 |
+
|
| 349 |
+
The shrinkage theorem directly yields sample complexity by bounding how many samples are needed to drive $\beta_t$ , $\eta_t$ , and $\gamma_t$ below $\varepsilon$ .
|
| 350 |
+
|
| 351 |
+
<span id="page-3-5"></span>**Theorem 4.8** (Sample complexity). Under Assumptions 2.1–2.3, CGP achieves $r_T \le \varepsilon$ with probability $\ge 1 - \delta$ using $T = \tilde{O}(L^d \varepsilon^{-(2+\alpha)} \log(1/\delta))$ samples. When $\alpha < d$ , this improves upon the worst-case $\tilde{O}(\varepsilon^{-(2+d)})$ rate.
|
| 352 |
+
|
| 353 |
+
The following lower bound shows that our sample complexity is optimal up to logarithmic factors.
|
| 354 |
+
|
| 355 |
+
<span id="page-3-6"></span>**Theorem 4.9** (Lower bound). For any algorithm and $\alpha \in (0, d]$ , there exists f satisfying Assumption 2.3 requiring $T = \Omega(\varepsilon^{-(2+\alpha)})$ samples for $\varepsilon$ -optimality with probability $\geq 2/3$ .
|
| 356 |
+
|
| 357 |
+
This establishes CGP is minimax optimal up to logarithmic factors. A key property is anytime validity: at any t, any $x \notin A_t$ satisfies $f(x) < f^* - \varepsilon_t$ for computable $\varepsilon_t > 0$ . Full proofs are in Appendix C.
|
| 358 |
+
|
| 359 |
+
#### <span id="page-3-0"></span>**5.** CGP-Adaptive: Learning *L* Online
|
| 360 |
+
|
| 361 |
+
The theoretical results above assume known L, which is often unavailable in practice. Underestimating L invalidates certificates, while overestimating is safe but conservative. We develop CGP-Adaptive (Algorithm 2), which learns L online via a doubling scheme with provable guarantees.
|
| 362 |
+
|
| 363 |
+
The key insight is that Lipschitz violations are detectable. If $|\hat{\mu}_i - \hat{\mu}_j| - 2(r_i + r_j) > \hat{L} \cdot d(x_i, x_j)$ , then $\hat{L}$ underestimates L with high probability. CGP-Adaptive uses a doubling scheme: start with conservative $\hat{L}_0$ , and upon detecting a violation, double $\hat{L}$ .
|
| 364 |
+
|
| 365 |
+
#### <span id="page-3-2"></span>Algorithm 2 CGP-Adaptive
|
| 366 |
+
|
| 367 |
+
```
|
| 368 |
+
Require: Domain \mathcal{X}, initial estimate \hat{L}_0, noise \sigma, budget
|
| 369 |
+
T, confidence \delta
|
| 370 |
+
1: \hat{L} \leftarrow \hat{L}_0, k \leftarrow 0 (doubling counter)
|
| 371 |
+
2: for t = 1, ..., T do
|
| 372 |
+
Run CGP iteration with current \hat{L}
|
| 373 |
+
for all pairs (i, j) with n_i, n_j \ge \log(T/\delta) do
|
| 374 |
+
4:
|
| 375 |
+
5:
|
| 376 |
+
if |\hat{\mu}_i - \hat{\mu}_j| - 2(r_i + r_j) > \hat{L} \cdot d(x_i, x_j) then
|
| 377 |
+
\hat{L} \leftarrow 2\hat{L}, k \leftarrow k + 1 {Doubling event}
|
| 378 |
+
6:
|
| 379 |
+
Recompute A_t with new \hat{L}
|
| 380 |
+
7:
|
| 381 |
+
8:
|
| 382 |
+
end if
|
| 383 |
+
end for
|
| 384 |
+
10: end for
|
| 385 |
+
```
|
| 386 |
+
|
| 387 |
+
**Theorem 5.1** (Adaptive L guarantee: learning regime). Let $L^* = \sup_{x \neq y} |f(x) - f(y)|/d(x,y)$ be the true Lipschitz constant. CGP-Adaptive with initial $\hat{L}_0 \leq L^*$ (learning from underestimation) satisfies:
|
| 388 |
+
|
| 389 |
+
- 1. The number of doubling events is at most $K = \lceil \log_2(L^*/\hat{L}_0) \rceil$ .
|
| 390 |
+
- 2. After all doublings, $\hat{L} \in [L^*, 2L^*]$ with probability $\geq 1 \delta$ .
|
| 391 |
+
- 3. The total sample complexity is $T = \tilde{O}(\varepsilon^{-(2+\alpha)} \cdot K)$ , i.e., $O(\log(L^*/\hat{L}_0))$ multiplicative overhead.
|
| 392 |
+
- 4. Certificate validity: Certificates are valid only after the final doubling (when $\hat{L} \geq L^*$ ). Before this, certificates may falsely exclude near-optimal points.
|
| 393 |
+
|
| 394 |
+
Remark 5.2 (Anytime-valid certificates). For applications requiring certificates valid at all times, use $\hat{L}_0 \geq L^*$ (conservative overestimate). This ensures $\hat{L} \geq L^*$ throughout, so all certificates are valid, but may be overly conservative. One can optionally decrease $\hat{L}$ when evidence suggests overestimation, but this requires different analysis than the doubling scheme above.
|
| 395 |
+
|
| 396 |
+
This is the first provably correct adaptive L estimation for Lipschitz optimization with certificates. Prior work (Malherbe & Vayatis, 2017) estimates L but without guarantees on certificate validity. Table 5 shows CGP-Adaptive matches oracle performance (known L) within 8% while being robust to $100\times$ underestimation of initial $\hat{L}_0$ .
|
| 397 |
+
|
| 398 |
+
# <span id="page-3-1"></span>6. CGP-TR: Trust Regions for High Dimensions
|
| 399 |
+
|
| 400 |
+
CGP-Adaptive addresses the unknown L problem, but another challenge remains: scalability. The covering number of $A_t$ grows as $O(\eta^{-d})$ , making CGP intractable for d > 15.
|
| 401 |
+
|
| 402 |
+
{4}------------------------------------------------
|
| 403 |
+
|
| 404 |
+
Table 3. Simple regret ( $\times 10^{-2}$ ) at T = 200. Bold: best; †: significant vs second-best.
|
| 405 |
+
|
| 406 |
+
<span id="page-4-2"></span>
|
| 407 |
+
|
| 408 |
+
| Method | Needle | Branin | Hartmann | Ackley | Levy | Rosen. | SVM |
|
| 409 |
+
|--------|-----------------|-----------------|-----------------|-----------------|-----------------|-----------------|-----------------|
|
| 410 |
+
| Random | 8.2 | 12.1 | 15.3 | 22.4 | 18.7 | 14.2 | 11.2 |
|
| 411 |
+
| GP-UCB | 2.1 | 1.8 | 4.2 | 12.3 | 5.1 | 4.8 | 3.9 |
|
| 412 |
+
| TuRBO | 1.8 | 2.1 | 3.1 | 9.8 | 4.3 | 3.9 | 3.2 |
|
| 413 |
+
| HEBO | 1.9 | 1.6 | 3.3 | 9.4 | 4.1 | 3.7 | 3.1 |
|
| 414 |
+
| BORE | 2.0 | 1.9 | 3.5 | 10.1 | 4.5 | 4.1 | 3.4 |
|
| 415 |
+
| HOO | 3.4 | 5.2 | 8.7 | 14.2 | 9.8 | 8.1 | 7.1 |
|
| 416 |
+
| CGP | 1.2 | 2.0 | 2.9 | 8.1 | 3.8 | 3.8 | 2.8 |
|
| 417 |
+
| CGP-A | 1.3 | 2.1 | 3.0 | 8.3 | 3.9 | 3.9 | 2.9 |
|
| 418 |
+
| CGP-H | $1.1^{\dagger}$ | $1.4^{\dagger}$ | $2.7^{\dagger}$ | $7.8^{\dagger}$ | $3.5^{\dagger}$ | $3.4^{\dagger}$ | $2.6^{\dagger}$ |
|
| 419 |
+
|
| 420 |
+
To enable high-dimensional optimization, we develop CGP-TR (Algorithm 3), which maintains local certificates within trust regions that adapt based on observed progress.
|
| 421 |
+
|
| 422 |
+
The key insight is that certificates need not be global. A local certificate $A_t^{\mathcal{T}}$ within trust region $\mathcal{T} \subset \mathcal{X}$ still provides valid bounds for $\arg\max_{x \in \mathcal{T}} f(x)$ . CGP-TR maintains multiple trust regions $\{\mathcal{T}_1, \dots, \mathcal{T}_m\}$ centered at promising points, with radii that expand on success and contract on failure (following TuRBO (Eriksson et al., 2019)).
|
| 423 |
+
|
| 424 |
+
**Certified restarts.** We restart a trust region only when it is certifiably suboptimal: if $u_t^{(j)} := \max_{x \in \mathcal{T}_j} U_t(x)$ satisfies $u_t^{(j)} < \ell_t$ where $\ell_t := \max_i \mathrm{LCB}_i(t)$ , then with high probability $\sup_{x \in \mathcal{T}_j} f(x) < f(x^*)$ , so $\mathcal{T}_j$ cannot contain $x^*$ and can be safely restarted. This certified restart rule ensures that regions containing $x^*$ are never falsely eliminated. In our implementation, contraction is lower-bounded by $r_{\min}$ and centers are fixed, so a region that contains $x^*$ cannot be contracted to exclude it; restarts occur only via the certified condition $u_t^{(j)} < \ell_t$ .
|
| 425 |
+
|
| 426 |
+
<span id="page-4-1"></span>**Theorem 6.1** (CGP-TR with certified restarts: correctness and allocation). Assume the good event $\mathcal{E}$ holds for the confidence bounds used to construct $U_t$ and $\ell_t$ . CGP-TR uses certified restarts: restart $\mathcal{T}_j$ only if $u_t^{(j)} := \max_{x \in \mathcal{T}_j} U_t(x) < \ell_t$ .
|
| 427 |
+
|
| 428 |
+
Let $\mathcal{T}^*$ be a trust region that contains $x^*$ at some time and is not contracted to exclude $x^*$ (e.g., contraction is lower-bounded by $r_{\min}$ and the center remains fixed). Then:
|
| 429 |
+
|
| 430 |
+
- 1. (No false restarts) On $\mathcal{E}$ , $\mathcal{T}^*$ is never restarted by the certified rule.
|
| 431 |
+
- 2. (Local certificate) Conditioned on receiving $T^*$ evaluations inside $T^*$ , the local active set $A_t^{(T^*)} = \{x \in T^* : U_t(x) \geq \ell_t^{(T^*)}\}$ satisfies the same containment/shrinkage/sample-complexity bounds as CGP on the restricted domain $T^*$ .
|
| 432 |
+
- 3. (Allocation bound) Define the region gap $\Delta_i :=$
|
| 433 |
+
|
| 434 |
+
#### <span id="page-4-0"></span>**Algorithm 3** CGP-TR (Trust Region with Certified Restarts)
|
| 435 |
+
|
| 436 |
+
**Require:** Domain $\mathcal{X}, L, \sigma$ , budget T, initial radius $r_0, n_{\text{trust}}$ regions
|
| 437 |
+
|
| 438 |
+
- 1: Initialize $n_{\text{trust}}$ trust regions at Sobol points with radius $r_0$
|
| 439 |
+
- 2: **for** t = 1, ..., T **do**
|
| 440 |
+
- 3: Compute $\ell_t := \max_{i \leq N_t} LCB_i(t)$ (global lower certificate)
|
| 441 |
+
- 4: Select trust region $\mathcal{T}_j$ with highest $u_t^{(j)} := \max_{x \in \mathcal{T}_i} U_t(x)$
|
| 442 |
+
- 5: Run CGP within $\mathcal{T}_j$ : compute local $A_t^{(j)} = \{x \in \mathcal{T}_j : U_t(x) \ge \ell_t^{(j)}\}$
|
| 443 |
+
- 6: Query $x_{t+1} \in A_t^{(j)}$ , observe $y_{t+1}$
|
| 444 |
+
- 7: **if** $u_t^{(j)} < \ell_t$ **then**
|
| 445 |
+
- 8: **Certified restart:** restart $\mathcal{T}_j$ at a new Sobol point with radius $r_0$
|
| 446 |
+
- 9: **else if** improvement in $\mathcal{T}_i$ **then**
|
| 447 |
+
- 10: Expand: $r_j \leftarrow \min(2r_j, D/2)$
|
| 448 |
+
- 11: **else if** no improvement for $\tau_{\text{fail}}$ iterations **then**
|
| 449 |
+
- 12: Contract: $r_j \leftarrow \max(r_j/2, r_{\min})$
|
| 450 |
+
- 13: **end if**
|
| 451 |
+
- 14: **end for**
|
| 452 |
+
- 15: **Output:** Best point across all regions, local certificate $A_T^{(j^*)}$
|
| 453 |
+
|
| 454 |
+
$f^* - \sup_{x \in \mathcal{T}_j} f(x)$ (with $\Delta_j > 0$ for suboptimal regions). Assume each region runs CGP with replication ensuring its maximal active-point confidence radius after n within-region samples satisfies $\beta_j(n) \le c_\sigma \sqrt{\log(c_T/\delta)/n}$ for constants $c_\sigma, c_T$ matching the paper's confidence schedule. If the region radii are eventually bounded so that $L \cdot \operatorname{diam}(\mathcal{T}_j) \le \Delta_j/8$ , then any suboptimal region j is selected at most
|
| 455 |
+
|
| 456 |
+
$$N_j \le \frac{64c_\sigma^2}{\Delta_j^2} \log\left(\frac{c_T}{\delta}\right) + 1$$
|
| 457 |
+
|
| 458 |
+
times before it is eliminated by the certified restart rule.
|
| 459 |
+
|
| 460 |
+
{5}------------------------------------------------
|
| 461 |
+
|
| 462 |
+
## <span id="page-5-1"></span>Algorithm 4 CGP-Hybrid
|
| 463 |
+
|
| 464 |
+
Require: Domain X , L, σ, budget T, switch threshold ρthresh = 0.5
|
| 465 |
+
|
| 466 |
+
- 1: Phase 1: Run CGP until Vol(At) < 0.1 · Vol(X ) or t > T /3
|
| 467 |
+
- 2: Estimate ρ<sup>t</sup> = Lˆ local(t)/Lˆ global
|
| 468 |
+
- 3: if ρ<sup>t</sup> < ρthresh then
|
| 469 |
+
- 4: Phase 2: Switch to GP-UCB within A<sup>t</sup> (GP refinement)
|
| 470 |
+
- 5: Fit GP to points in At, continue with GP-UCB acquisition
|
| 471 |
+
- 6: else
|
| 472 |
+
|
| 473 |
+
276
|
| 474 |
+
|
| 475 |
+
294
|
| 476 |
+
|
| 477 |
+
296
|
| 478 |
+
|
| 479 |
+
313 314
|
| 480 |
+
|
| 481 |
+
316
|
| 482 |
+
|
| 483 |
+
324
|
| 484 |
+
|
| 485 |
+
326
|
| 486 |
+
|
| 487 |
+
328 329
|
| 488 |
+
|
| 489 |
+
- 7: Phase 2: Continue CGP within A<sup>t</sup>
|
| 490 |
+
- 8: end if
|
| 491 |
+
- 9: Output: Best point, certificate A<sup>T</sup> (from CGP phase)
|
| 492 |
+
|
| 493 |
+
The key advantage is that covering T<sup>j</sup> requires O((rj/η) d ) points, and since r<sup>j</sup> ≪ D, this is tractable even for large d. With ntrust = O(log T) regions, CGP-TR explores globally while maintaining local certificates. The allocation bound (Theorem [6.1,](#page-4-1) item 3) ensures that suboptimal regions receive only O(log T /∆<sup>2</sup> j ) evaluations before certified elimination, preventing wasted samples.
|
| 494 |
+
|
| 495 |
+
CGP-TR provides local rather than global certificates, but the certified restart rule guarantees that the region containing x ∗ is never falsely eliminated. This enables scaling to d = 50 to 100 where global Lipschitz methods fail entirely.
|
| 496 |
+
|
| 497 |
+
## <span id="page-5-0"></span>7. CGP-Hybrid: Best of Both Worlds
|
| 498 |
+
|
| 499 |
+
While CGP-TR addresses scalability, some functions exhibit local smoothness that GPs can exploit more effectively than Lipschitz methods. CGP-Hybrid (Algorithm [4\)](#page-5-1) preserves CGP's anytime certificates while allowing any optimizer to refine within the certified active set. The key point is modularity: Phase 1 constructs a certificate At; Phase 2 performs additional optimization restricted to A<sup>t</sup> without affecting certificate validity. We instantiate Phase 2 with GP-UCB when local smoothness is detected, but other optimizers can be used. This design captures the best of both worlds: CGP's explicit pruning guarantees and GP's ability to exploit local smoothness when present.
|
| 500 |
+
|
| 501 |
+
Define the *effective smoothness ratio* ρ<sup>t</sup> = Lˆ local(t)/Lˆ global, where Lˆ local(t) is estimated from points within At. When ρ<sup>t</sup> < 0.5, the function is significantly smoother near the optimum, and GP refinement is beneficial.
|
| 502 |
+
|
| 503 |
+
<span id="page-5-3"></span>Proposition 7.1 (Hybrid guarantee). *CGP-Hybrid achieves:*
|
| 504 |
+
|
| 505 |
+
- *1. If* ρ ≥ 0.5*: same guarantee as CGP,* T = O˜(ε −(2+α) )*.*
|
| 506 |
+
- *2. If* ρ < 0.5*: after CGP reduces* A<sup>t</sup> *to volume* V *, GP-UCB operates within a restricted domain of effective*
|
| 507 |
+
|
| 508 |
+
*diameter* O(V <sup>1</sup>/d)*. The additional sample complexity depends on the GP kernel's information gain* γ<sup>T</sup> *over* At*; empirically, this yields faster convergence than continuing CGP when the function is locally smooth.*
|
| 509 |
+
|
| 510 |
+
*3. The certificate* A<sup>T</sup> *from Phase 1 remains valid regardless of Phase 2 method.*
|
| 511 |
+
|
| 512 |
+
Proposition 7.2 (Certificate invariance under Phase 2). *The certificate* A<sup>t</sup> *computed by CGP in Phase 1 remains valid regardless of the Phase-2 optimizer, since validity depends only on the confidence bounds and Lipschitz envelope used to define* U<sup>t</sup> *and* ℓt*. Specifically, any point* x /∈ A<sup>t</sup> *satisfies* f(x) < f <sup>∗</sup> − ε<sup>t</sup> *with high probability, where* ε<sup>t</sup> *is computable from Phase 1 quantities alone.*
|
| 513 |
+
|
| 514 |
+
The key insight is that CGP's certificate remains valid even when switching to GP: A<sup>t</sup> still contains x <sup>∗</sup> with high probability, so GP refinement within A<sup>t</sup> is safe. This provides the best of both worlds: CGP's certificates and pruning efficiency when ρ ≥ 0.5, and GP's smoothness exploitation when ρ < 0.5.
|
| 515 |
+
|
| 516 |
+
Table [6](#page-6-2) shows CGP-Hybrid wins on all 12 benchmarks, including Branin (ρ = 0.31) and Rosenbrock (ρ = 0.28) where it detects low ρ and switches to GP refinement, and Needle (ρ = 0.98) where it stays with CGP.
|
| 517 |
+
|
| 518 |
+
<span id="page-5-2"></span>*Table 4.* High-dimensional benchmarks (d > 20) at T = 500.
|
| 519 |
+
|
| 520 |
+
| Method | Rover-60 | NAS-36 | Ant-100 |
|
| 521 |
+
|--------|-----------------|--------------------------|-----------------|
|
| 522 |
+
| Random | 42.1 ± 1.2 | 38.4 ± 0.9 | 51.2 ± 1.4 |
|
| 523 |
+
| TuRBO | 12.4 ± 0.4 | 11.2 ± 0.3 | 18.7 ± 0.6 |
|
| 524 |
+
| HEBO | 14.1 ± 0.5 | 12.8 ± 0.4 | 21.3 ± 0.7 |
|
| 525 |
+
| CMA-ES | 15.8 ± 0.6 | 14.1 ± 0.5 | 19.4 ± 0.6 |
|
| 526 |
+
| CGP | | (intractable for d > 15) | |
|
| 527 |
+
| CGP-TR | †<br>11.2 ± 0.3 | †<br>10.4 ± 0.3 | †<br>17.1 ± 0.5 |
|
| 528 |
+
|
| 529 |
+
<sup>†</sup> CGP-TR additionally provides local optimality certificates.
|
| 530 |
+
|
| 531 |
+
## 8. Related Work
|
| 532 |
+
|
| 533 |
+
Our work is primarily grounded in the literature on Lipschitz bandits and global optimization. Foundational approaches, such as the continuum-armed bandits of [Kleinberg](#page-8-5) [et al.](#page-8-5) [\(2008\)](#page-8-5) and the X-armed bandit framework of [Bubeck](#page-8-7) [et al.](#page-8-7) [\(2011\)](#page-8-7), utilize zooming mechanisms to achieve regret bounds depending on the near-optimality dimension. These concepts were refined for deterministic and stochastic settings via tree-based algorithms like DOO/SOO [\(Munos,](#page-8-17) [2011\)](#page-8-17) and StoSOO [\(Valko et al.,](#page-9-2) [2013\)](#page-9-2). However, a key distinction is that zooming algorithms maintain pruning implicitly as an analysis artifact, whereas CGP exposes the active set A<sup>t</sup> as a computable object with measurable volume. This explicit geometric approach also relates to partitionbased global optimization methods like DIRECT [\(Jones](#page-8-3)
|
| 534 |
+
|
| 535 |
+
{6}------------------------------------------------
|
| 536 |
+
|
| 537 |
+
356
|
| 538 |
+
|
| 539 |
+
374
|
| 540 |
+
|
| 541 |
+
376
|
| 542 |
+
|
| 543 |
+
*Table 5.* CGP-Adaptive with varying initial Lˆ0. Robust to 100× underestimation.
|
| 544 |
+
|
| 545 |
+
<span id="page-6-1"></span>
|
| 546 |
+
|
| 547 |
+
| Initial Lˆ<br>0 | Doublings | Final L/L ˆ ∗ | Regret (×10−2<br>) | Overhead |
|
| 548 |
+
|--------------------|-----------|---------------|--------------------|----------|
|
| 549 |
+
| ∗<br>L<br>(oracle) | 0 | 1.0 | 2.9 ± 0.1 | 1.0× |
|
| 550 |
+
| ∗/2<br>L | 1 | 1.0 | 3.0 ± 0.1 | 1.03× |
|
| 551 |
+
| ∗/10<br>L | 4 | 1.6 | 3.1 ± 0.1 | 1.07× |
|
| 552 |
+
| ∗/100<br>L | 7 | 1.3 | 3.2 ± 0.2 | 1.12× |
|
| 553 |
+
| LIPO (adaptive) | – | – | 6.2 ± 0.3 | – |
|
| 554 |
+
|
| 555 |
+
<span id="page-6-2"></span>*Table 6.* CGP-Hybrid smoothness detection. ρ < 0.5 triggers GP refinement.
|
| 556 |
+
|
| 557 |
+
| Benchmark | ρˆ | Phase 2 | CGP-H Regret | Best Baseline |
|
| 558 |
+
|-------------|------|---------|--------------|---------------|
|
| 559 |
+
| Needle-2D | 0.98 | CGP | 1.1 ± 0.1 | 1.8 (TuRBO) |
|
| 560 |
+
| Branin | 0.31 | GP | 1.4 ± 0.1 | 1.6 (HEBO) |
|
| 561 |
+
| Hartmann-6 | 0.72 | CGP | 2.7 ± 0.1 | 3.1 (TuRBO) |
|
| 562 |
+
| Rosenbrock | 0.28 | GP | 3.4 ± 0.1 | 3.7 (HEBO) |
|
| 563 |
+
| Ackley-10 | 0.85 | CGP | 7.8 ± 0.3 | 9.4 (HEBO) |
|
| 564 |
+
| Levy-5 | 0.67 | CGP | 3.5 ± 0.2 | 4.1 (HEBO) |
|
| 565 |
+
| SVM-RBF | 0.74 | CGP | 2.6 ± 0.1 | 3.1 (HEBO) |
|
| 566 |
+
| LunarLander | 0.81 | CGP | 6.1 ± 0.3 | 7.0 (HEBO) |
|
| 567 |
+
|
| 568 |
+
[et al.,](#page-8-3) [1993\)](#page-8-3) and LIPO [\(Malherbe & Vayatis,](#page-8-15) [2017\)](#page-8-15). While LIPO addresses the unknown Lipschitz constant, it lacks the certificate guarantees provided by our CGP-Adaptive doubling scheme [\(Shihab et al.,](#page-9-8) [2025c\)](#page-9-8). Our theoretical analysis further draws on sample complexity results under margin conditions from the finite-arm setting [\(Audibert &](#page-8-13) [Bubeck,](#page-8-13) [2010;](#page-8-13) [Jamieson & Nowak,](#page-8-18) [2014;](#page-8-18) [Shihab et al.,](#page-9-9) [2025b\)](#page-9-9), adapting confidence bounds from the UCB frame[w](#page-8-3)ork [\(Auer et al.,](#page-8-6) [2002\)](#page-8-6) to continuous spaces with explicit uncertainty representation similar to safe optimization levelsets [\(Sui et al.,](#page-9-10) [2015\)](#page-9-10).
|
| 569 |
+
|
| 570 |
+
In the broader context of black-box optimization, Bayesian methods provide the standard alternative for uncertainty quantification. Classic approaches like GP-UCB [\(Srinivas](#page-9-11) [et al.,](#page-9-11) [2009\)](#page-9-11), Entropy Search [\(Hennig & Schuler,](#page-8-11) [2012;](#page-8-11) [Hernandez-Lobato et al.](#page-8-12) ´ , [2014\)](#page-8-12), and Thompson Sampling [\(Thompson,](#page-9-4) [1933\)](#page-9-4) offer strong performance but scale cubically with observations. To address high-dimensional scaling, recent work has introduced trust regions (TuRBO) [\(Eriksson et al.,](#page-8-16) [2019\)](#page-8-16) and nonstationary priors (HEBO) [\(Cowen-Rivers et al.,](#page-8-19) [2022\)](#page-8-19). More recently, advanced heuristics such as Bounce [\(Papenmeier et al.,](#page-8-20) [2024\)](#page-8-20) have improved geometric adaptation for mixed spaces, while Prior-Fitted Networks [\(Hollmann et al.,](#page-8-21) [2023\)](#page-8-21) and generative diffusion models like Diff-BBO [\(Wu et al.,](#page-9-12) [2024\)](#page-9-12) exploit massive pretraining to minimize regret rapidly. While these emerging methods achieve impressive empirical results, they remain fundamentally heuristic, lacking the computable stopping criteria or active set containment guarantees that are central to CGP. We therefore focus our comparison on established
|
| 571 |
+
|
| 572 |
+
baselines to isolate the specific utility of our certification mechanism, distinguishing our approach from heuristic hyperparameter tuners like Hyperband [\(Li et al.,](#page-8-22) [2018\)](#page-8-22) by prioritizing provable safety over raw speed.
|
| 573 |
+
|
| 574 |
+
## <span id="page-6-0"></span>9. Experiments
|
| 575 |
+
|
| 576 |
+
We evaluate CGP variants on 12 benchmarks spanning d ∈ [2, 100], measuring simple regret, certificate utility, and scalability. Code will be available upon acceptance.
|
| 577 |
+
|
| 578 |
+
Setup. We compare against 9 baselines: Random Search, GP-UCB, TuRBO, HEBO, BORE, HOO, StoSOO, LIPO, and SAASBO (see Appendix [E](#page-15-0) for configurations). We evaluate on 12 benchmarks spanning d ∈ [2, 100]: lowdimensional (Needle-2D, Branin, Hartmann-6, Levy-5, Rosenbrock-4), medium-dimensional (Ackley-10, SVM-RBF-6, LunarLander-12), and high-dimensional (Rover-60, NAS-36, MuJoCo-Ant-100). All experiments use 30 runs with σ = 0.1 noise; we report mean ± SE with Bonferronicorrected t-tests (p < 0.05).
|
| 579 |
+
|
| 580 |
+
Table [3](#page-4-2) shows CGP-Hybrid performs best among tested methods on all 7 low and medium-dimensional benchmarks. On Branin and Rosenbrock where vanilla CGP lost to HEBO, CGP-Hybrid detects ρ < 0.5 and switches to GP refinement, achieving 12% and 8% improvement over HEBO respectively. On Needle where ρ ≈ 1, CGP-Hybrid stays with CGP and matches vanilla CGP performance. For high-dimensional problems, Table [4](#page-5-2) demonstrates CGP-TR scales to d = 100 while outperforming TuRBO by
|
| 581 |
+
|
| 582 |
+
{7}------------------------------------------------
|
| 583 |
+
|
| 584 |
+
9 to 12%. Critically, CGP-TR provides local certificates within trust regions, enabling principled stopping—a capability TuRBO lacks. Regarding adaptive estimation, Table 5 shows CGP-Adaptive is robust to initial underestimation: even with $\hat{L}_0=L^*/100$ , performance degrades only 10% with 7 doublings, validating Theorem 5.1's $O(\log(L^*/\hat{L}_0))$ overhead. Finally, Table 6 confirms CGP-Hybrid correctly identifies when to switch: Branin ( $\rho=0.31$ ) and Rosenbrock ( $\rho=0.28$ ) trigger GP refinement, achieving 12% and 8% improvement over HEBO. Benchmarks with $\rho>0.5$ stay with CGP, maintaining certificate validity.
|
| 585 |
+
|
| 586 |
+
**Shrinkage validation.** Across all benchmarks, we observe $Vol(A_t)$ shrinks to < 5% by T = 100, with empirical decay rates closely matching the theoretical bound from Theorem 4.6. This confirms our analysis is tight and the margin condition captures the true problem difficulty.
|
| 587 |
+
|
| 588 |
+
<span id="page-7-0"></span>*Table 7.* Certificate-enabled early stopping on Hartmann-6. CGP uniquely provides actionable stopping criteria.
|
| 589 |
+
|
| 590 |
+
| Stopping Rule | Samples | Regret ( $\times 10^{-2}$ ) | Savings |
|
| 591 |
+
|---------------------------|--------------|-----------------------------|---------|
|
| 592 |
+
| Fixed $T = 200$ | 200 | $2.9 \pm 0.1$ | _ |
|
| 593 |
+
| $Vol(A_t) < 10\%$ | $82 \pm 9$ | $3.8 \pm 0.2$ | 59% |
|
| 594 |
+
| $Vol(A_t) < 5\%$ | $118 \pm 14$ | $3.2 \pm 0.2$ | 41% |
|
| 595 |
+
| ${\rm Gap\ bound} < 0.05$ | $134 \pm 12$ | $3.0 \pm 0.1$ | 33% |
|
| 596 |
+
|
| 597 |
+
Table 7 demonstrates certificate utility: stopping at $Vol(A_t) < 10\%$ saves 59% of samples with only 31% regret increase. No baseline provides such principled stopping rules. In d>20, we use the computable gap proxy $\varepsilon_t:=2(\beta_t+L\eta_t)+\gamma_t$ as the primary criterion. Beyond sample efficiency, CGP also offers computational advantages. Table 8 shows CGP variants are 6 to 8 times faster than GP-based methods due to their O(n) per-iteration cost versus GP's $O(n^3)$ . Finally, we ablate CGP's components to understand their individual contributions.
|
| 598 |
+
|
| 599 |
+
<span id="page-7-1"></span>Table 8. Wall-clock time (seconds) for T=200 on Hartmann-6.
|
| 600 |
+
|
| 601 |
+
| Method | Time (s) | Regret ( $\times 10^{-2}$ ) | Speedup |
|
| 602 |
+
|--------------|----------|-----------------------------|--------------|
|
| 603 |
+
| CGP | 58 | 2.9 | 8× |
|
| 604 |
+
| CGP-Adaptive | 64 | 3.0 | $7.5 \times$ |
|
| 605 |
+
| CGP-Hybrid | 72 | 2.7 | $6.7 \times$ |
|
| 606 |
+
| GP-UCB | 480 | 4.2 | $1 \times$ |
|
| 607 |
+
| TuRBO | 620 | 3.1 | $0.8 \times$ |
|
| 608 |
+
| HEBO | 890 | 3.3 | $0.5 \times$ |
|
| 609 |
+
|
| 610 |
+
Table 9 shows all components contribute: removing pruning certificates increases regret 78%, coverage penalty 44%, replication 33%. GP refinement provides 7% improvement on Hartmann-6 where $\rho=0.72$ is borderline. Table 10 validates that $\hat{\alpha} < d$ across all benchmarks, confirming the margin condition holds and our complexity bounds apply.
|
| 611 |
+
|
| 612 |
+
<span id="page-7-2"></span>*Table 9.* Ablation study on Hartmann-6. All components contribute.
|
| 613 |
+
|
| 614 |
+
| Variant | Regret ( $\times 10^{-2}$ ) | $Vol(A_{200})$ |
|
| 615 |
+
|-----------------------------------------|---------------------------------|----------------|
|
| 616 |
+
| CGP-Hybrid (full) | $\textbf{2.7} \pm \textbf{0.1}$ | 2.1% |
|
| 617 |
+
| <ul> <li>GP refinement</li> </ul> | $2.9 \pm 0.1$ | 2.1% |
|
| 618 |
+
| <ul> <li>pruning certificate</li> </ul> | $4.8 \pm 0.2$ | _ |
|
| 619 |
+
| <ul> <li>coverage penalty</li> </ul> | $3.9 \pm 0.2$ | 3.8% |
|
| 620 |
+
| <ul><li>replication</li></ul> | $3.6 \pm 0.2$ | 2.9% |
|
| 621 |
+
| CGP-TR $(d=6)$ | $2.8 \pm 0.1$ | 2.4% (local) |
|
| 622 |
+
| CGP-Adaptive | $3.0 \pm 0.1$ | 2.4% |
|
| 623 |
+
|
| 624 |
+
<span id="page-7-3"></span>*Table 10.* Empirical $\hat{\alpha}$ estimates from shrinkage trajectories.
|
| 625 |
+
|
| 626 |
+
| Benchmark | d | â (95% CI) | True $\alpha$ | $\hat{\alpha} < d$ ? |
|
| 627 |
+
|------------|----|---------------|---------------|----------------------|
|
| 628 |
+
| Needle-2D | 2 | $1.8 \pm 0.2$ | 2.0 | <u>√</u> |
|
| 629 |
+
| Branin | 2 | $1.2 \pm 0.1$ | _ | $\checkmark$ |
|
| 630 |
+
| Hartmann-6 | 6 | $2.4 \pm 0.3$ | _ | $\checkmark$ |
|
| 631 |
+
| Ackley-10 | 10 | $3.2 \pm 0.4$ | _ | $\checkmark$ |
|
| 632 |
+
| Rover-60 | 60 | $8.4 \pm 1.2$ | _ | $\checkmark$ |
|
| 633 |
+
|
| 634 |
+
#### 10. Conclusion
|
| 635 |
+
|
| 636 |
+
We introduced Certificate-Guided Pruning (CGP), an algorithm for stochastic Lipschitz optimization that maintains explicit active sets with provable shrinkage guarantees. Under a margin condition with near-optimality dimension $\alpha$ , we prove $Vol(A_t) \leq C \cdot (2(\beta_t + L\eta_t) + \gamma_t)^{d-\alpha}$ , yielding sample complexity $\tilde{O}(\varepsilon^{-(2+\alpha)})$ with anytime valid certificates. Three extensions broaden applicability: CGP-Adaptive learns L online with $O(\log T)$ overhead, CGP-TR scales to d > 50 via trust regions, and CGP-Hybrid switches to GP refinement when local smoothness is detected. The margin condition holds broadly for isolated maxima with nondegenerate Hessian $\alpha = d/2$ , for polynomial decay $\alpha = d/p$ and can be estimated online from shrinkage trajectories. Limitations include requiring Lipschitz continuity and dimension constraints ( $d \le 15$ for vanilla CGP, $d \le 100$ for CGP-TR); practical guidance is in Appendix D. Future directions include safe optimization using $A_t$ for safety certificates and CGP-TR with random embeddings for global high-dimensional certificates.
|
| 637 |
+
|
| 638 |
+
#### **Impact Statement**
|
| 639 |
+
|
| 640 |
+
This paper introduces Certificate-Guided Pruning (CGP), a method designed to improve the sample efficiency of black-box optimization in resource-constrained settings. By providing explicit optimality certificates and principled stopping criteria, our approach significantly reduces the computational budget required for expensive tasks such as neural architecture search and simulation-based engineering, directly contributing to lower energy consumption and carbon footprints. Furthermore, the ability to certify suboptimal regions enhances reliability in safety-critical applications like
|
| 641 |
+
|
| 642 |
+
{8}------------------------------------------------
|
| 643 |
+
|
| 644 |
+
robotics. However, practitioners must ensure the validity of the Lipschitz assumption, as violations could lead to the incorrect pruning of optimal solutions.
|
| 645 |
+
|
| 646 |
+
## References
|
| 647 |
+
|
| 648 |
+
- <span id="page-8-13"></span>Audibert, J.-Y. and Bubeck, S. Best arm identification in multi-armed bandits. In *Conference on Learning Theory*, pp. 41–53, 2010.
|
| 649 |
+
- <span id="page-8-6"></span>Auer, P., Cesa-Bianchi, N., and Fischer, P. Finite-time analysis of the multiarmed bandit problem. *Machine Learning*, 47(2-3):235–256, 2002.
|
| 650 |
+
- <span id="page-8-0"></span>Bergstra, J. and Bengio, Y. Random search for hyperparameter optimization. *Journal of Machine Learning Research*, 13(10):281–305, 2012.
|
| 651 |
+
- <span id="page-8-2"></span>Brochu, E., Cora, V. M., and De Freitas, N. A tutorial on Bayesian optimization of expensive cost functions. *arXiv preprint arXiv:1012.2599*, 2010.
|
| 652 |
+
- <span id="page-8-7"></span>Bubeck, S., Munos, R., Stoltz, G., and Szepesvari, C. X- ´ armed bandits. In *Journal of Machine Learning Research*, volume 12, pp. 1655–1695, 2011.
|
| 653 |
+
- <span id="page-8-19"></span>Cowen-Rivers, A. I., Lyu, W., Tutunov, R., Wang, Z., Grosnit, A., Griffiths, R. R., Maraval, A. M., Jianye, H., Wang, J., Peters, J., et al. HEBO: Pushing the limits of sampleefficient hyper-parameter optimisation. In *Journal of Artificial Intelligence Research*, volume 74, pp. 1269–1349, 2022.
|
| 654 |
+
- <span id="page-8-10"></span>Daniel, J. R., Benjamin, V. R., Abbas, K., Ian, O., and Zheng, W. A tutorial on thompson sampling. *Foundations and trends® in machine learning*, 11(1):1–99, 2018.
|
| 655 |
+
- <span id="page-8-16"></span>Eriksson, D., Pearce, M., Gardner, J., Turner, R. D., and Poloczek, M. Scalable global optimization via local bayesian optimization. *Advances in neural information processing systems*, 32, 2019.
|
| 656 |
+
- <span id="page-8-1"></span>Fu, M. C. *Handbook of simulation optimization*. Springer, 2015.
|
| 657 |
+
- <span id="page-8-24"></span>Garivier, A. and Kaufmann, E. Optimal best arm identification with fixed confidence. *Conference on Learning Theory*, pp. 998–1027, 2016.
|
| 658 |
+
- <span id="page-8-11"></span>Hennig, P. and Schuler, C. J. Entropy search for informationefficient global optimization. *Journal of Machine Learning Research*, 13(57):1809–1837, 2012.
|
| 659 |
+
- <span id="page-8-12"></span>Hernandez-Lobato, J. M., Hoffman, M. W., and Ghahra- ´ mani, Z. Predictive entropy search for efficient global optimization of black-box functions. In *Advances in Neural Information Processing Systems*, volume 27, pp. 918–926, 2014.
|
| 660 |
+
|
| 661 |
+
- <span id="page-8-21"></span>Hollmann, N., Muller, S., Eggensperger, K., and Hutter, F. ¨ Large-scale transfer learning for bayesian optimization with prior-data fitted networks. In *Advances in Neural Information Processing Systems (NeurIPS)*, 2023.
|
| 662 |
+
- <span id="page-8-18"></span>Jamieson, K. and Nowak, R. Best-arm identification algorithms for multi-armed bandits in the fixed confidence setting. In *Conference on Information Sciences and Systems*, pp. 1–6. IEEE, 2014.
|
| 663 |
+
- <span id="page-8-3"></span>Jones, D. R., Perttunen, C. D., and Stuckman, B. E. Lipschitzian optimization without the Lipschitz constant. *Journal of Optimization Theory and Applications*, 79(1): 157–181, 1993.
|
| 664 |
+
- <span id="page-8-4"></span>Jones, D. R., Schonlau, M., and Welch, W. J. Efficient global optimization of expensive black-box functions. *Journal of Global Optimization*, 13(4):455–492, 1998.
|
| 665 |
+
- <span id="page-8-23"></span>Kaufmann, E., Cappe, O., and Garivier, A. On the com- ´ plexity of best-arm identification in multi-armed bandit models. *Journal of Machine Learning Research*, 17:1–42, 2016.
|
| 666 |
+
- <span id="page-8-5"></span>Kleinberg, R., Slivkins, A., and Upfal, E. Multi-armed bandits in metric spaces. In *Proceedings of the 40th Annual ACM Symposium on Theory of Computing*, pp. 681–690, 2008.
|
| 667 |
+
- <span id="page-8-14"></span>Lattimore, T. and Szepesvari, C. Bandit algorithms. ´ *Cambridge University Press*, 2020.
|
| 668 |
+
- <span id="page-8-22"></span>Li, L., Jamieson, K., DeSalvo, G., Rostamizadeh, A., and Talwalkar, A. Hyperband: A novel bandit-based approach to hyperparameter optimization. *Journal of Machine Learning Research*, 18:1–52, 2018.
|
| 669 |
+
- <span id="page-8-15"></span>Malherbe, C. and Vayatis, N. Global optimization of lipschitz functions. In *International conference on machine learning*, pp. 2314–2323. PMLR, 2017.
|
| 670 |
+
- <span id="page-8-17"></span>Munos, R. Optimistic optimization of a deterministic function without the knowledge of its smoothness. *Advances in Neural Information Processing Systems*, 24:783–791, 2011.
|
| 671 |
+
- <span id="page-8-8"></span>Munos, R. From bandits to Monte-Carlo tree search: The optimistic principle applied to optimization and planning. *Foundations and Trends in Machine Learning*, 7:1–129, 2014.
|
| 672 |
+
- <span id="page-8-20"></span>Papenmeier, L., Nardi, L., and Poloczek, M. Bounce: Reliable high-dimensional bayesian optimization for combinatorial and mixed spaces. In *International Conference on Learning Representations (ICLR)*, 2024.
|
| 673 |
+
- <span id="page-8-9"></span>Russo, D. and Van Roy, B. Learning to optimize via posterior sampling. *Mathematics of Operations Research*, 39: 1221–1243, 2014.
|
| 674 |
+
|
| 675 |
+
{9}------------------------------------------------
|
| 676 |
+
|
| 677 |
+
Shihab, I. F., Akter, S., and Sharma, A. Detecting proxy gaming in rl and llm alignment via evaluator stress tests. *arXiv preprint arXiv:2507.05619*, 2025a.
|
| 678 |
+
|
| 679 |
+
<span id="page-9-5"></span>
|
| 680 |
+
|
| 681 |
+
<span id="page-9-9"></span>
|
| 682 |
+
|
| 683 |
+
<span id="page-9-8"></span>
|
| 684 |
+
|
| 685 |
+
<span id="page-9-3"></span>
|
| 686 |
+
|
| 687 |
+
<span id="page-9-0"></span>
|
| 688 |
+
|
| 689 |
+
<span id="page-9-11"></span>
|
| 690 |
+
|
| 691 |
+
<span id="page-9-10"></span>
|
| 692 |
+
|
| 693 |
+
<span id="page-9-4"></span>
|
| 694 |
+
|
| 695 |
+
<span id="page-9-2"></span>
|
| 696 |
+
|
| 697 |
+
<span id="page-9-13"></span>
|
| 698 |
+
|
| 699 |
+
<span id="page-9-12"></span>
|
| 700 |
+
|
| 701 |
+
<span id="page-9-1"></span>
|
| 702 |
+
|
| 703 |
+
- Shihab, I. F., Akter, S., and Sharma, A. What fundamental structure in reward functions enables efficient sparsereward learning? *arXiv preprint arXiv:2509.03790*, 2025b.
|
| 704 |
+
- Shihab, I. F., Akter, S., and Sharma, A. Universal adaptive constraint propagation: Scaling structured inference for large language models via meta-reinforcement learning. arXiv preprint arXiv:2601.00095, 2025c.
|
| 705 |
+
- Shihab, I. F., Akter, S., and Sharma, A. Beyond variance: Knowledge-aware llm compression via fisher-aligned subspace diagnostics. *arXiv preprint arXiv:2601.07197*, 2026.
|
| 706 |
+
- Snoek, J., Larochelle, H., and Adams, R. P. Practical Bayesian optimization of machine learning algorithms. In *Advances in Neural Information Processing Systems*, volume 25, 2012.
|
| 707 |
+
- Srinivas, N., Krause, A., Kakade, S. M., and Seeger, M. Gaussian process optimization in the bandit setting: No regret and experimental design. arXiv preprint arXiv:0912.3995, 2009.
|
| 708 |
+
- Sui, Y., Gotovos, A., Burdick, J., and Krause, A. Safe exploration for optimization with gaussian processes. In *International conference on machine learning*, pp. 997–1005. PMLR, 2015.
|
| 709 |
+
- Thompson, W. R. On the likelihood that one unknown probability exceeds another in view of the evidence of two samples. *Biometrika*, 25(3/4):285–294, 1933.
|
| 710 |
+
- Valko, M., Carpentier, A., and Munos, R. Stochastic simultaneous optimistic optimization. In *Proceedings of the 30th International Conference on Machine Learning*, pp. 19–27, 2013.
|
| 711 |
+
- Wang, Z., Hutter, F., Zoghi, M., Matheson, D., and de Freitas, N. Bayesian optimization in a billion dimensions via random embeddings. *Journal of Artificial Intelligence Research*, 55:361–387, 2016.
|
| 712 |
+
- Wu, D., Kuang, N. L., Niu, R., Ma, Y., and Yu, R. Diffbbo: Diffusion-based inverse modeling for black-box optimization. 2024.
|
| 713 |
+
- Zoph, B. and Le, Q. V. Neural architecture search with reinforcement learning. In *International Conference on Learning Representations*, 2017.
|
| 714 |
+
|
| 715 |
+
#### <span id="page-9-6"></span>A. Extended Problem Formulation
|
| 716 |
+
|
| 717 |
+
We restate the assumptions from Section 2 for completeness. Let $(\mathcal{X}, d)$ be a compact metric space with diameter $D = \sup_{x,y \in \mathcal{X}} d(x,y)$ . We consider $\mathcal{X} = [0,1]^d$ with Euclidean metric, though our results extend to general metric spaces. Let $f: \mathcal{X} \to [0,1]$ be an unknown function satisfying:
|
| 718 |
+
|
| 719 |
+
**Assumption 2.1 (Lipschitz continuity).** There exists L > 0 such that for all $x, y \in \mathcal{X}$ : $|f(x) - f(y)| \le L \cdot d(x, y)$ .
|
| 720 |
+
|
| 721 |
+
We observe f through noisy queries: querying x returns $y = f(x) + \epsilon$ , where:
|
| 722 |
+
|
| 723 |
+
**Assumption 2.2 (Sub-Gaussian noise).** The noise $\epsilon$ is $\sigma$ -sub-Gaussian: $\mathbb{E}[e^{\lambda\epsilon}] \leq e^{\lambda^2\sigma^2/2}$ for all $\lambda \in \mathbb{R}$ .
|
| 724 |
+
|
| 725 |
+
After T samples, the algorithm outputs $\hat{x}_T \in \mathcal{X}$ . The goal is to minimize simple regret $r_T = f(x^*) - f(\hat{x}_T)$ , where $x^* \in \arg\max_{x \in \mathcal{X}} f(x)$ . We seek PAC-style guarantees: with probability at least $1 - \delta$ , achieve $r_T \leq \varepsilon$ .
|
| 726 |
+
|
| 727 |
+
To obtain instance-dependent rates that improve upon worst case bounds, we assume margin structure:
|
| 728 |
+
|
| 729 |
+
Assumption 2.3 (Margin / near-optimality dimension). There exist C>0 and $\alpha\in[0,d]$ such that for all $\varepsilon>0$ : $\operatorname{Vol}(\{x\in\mathcal{X}:f(x)\geq f^*-\varepsilon\})\leq C\,\varepsilon^{d-\alpha}$ .
|
| 730 |
+
|
| 731 |
+
The parameter $\alpha$ is the near-optimality dimension: smaller $\alpha$ corresponds to a sharper optimum (easier), while larger $\alpha$ corresponds to a broader near-optimal region (harder). The worst case is $\alpha=d$ , which recovers the standard d-dimensional Lipschitz difficulty. This assumption is standard in bandit theory (Audibert & Bubeck, 2010; Bubeck et al., 2011; Lattimore & Szepesvári, 2020; Kaufmann et al., 2016; Garivier & Kaufmann, 2016).
|
| 732 |
+
|
| 733 |
+
#### <span id="page-9-7"></span>**B.** Algorithm Implementation Details
|
| 734 |
+
|
| 735 |
+
#### **B.1. Score Maximization and Replication**
|
| 736 |
+
|
| 737 |
+
CGP optimizes $\operatorname{score}(x) = U_t(x) - \lambda \cdot \min_i d(x, x_i)$ , which is piecewise linear. Since this is non-smooth, we use CMA-ES (Covariance Matrix Adaptation Evolution Strategy) with bounded domain and 10 random restarts within $A_t$ . For $d \leq 3$ , we additionally use Delaunay triangulation to identify candidate optima at Voronoi vertices.
|
| 738 |
+
|
| 739 |
+
Membership in $A_t$ is exact: given $\{x_i, \hat{\mu}_i, r_i\}$ , checking $U_t(x) \geq \ell_t$ requires $O(N_t)$ time. Approximate maximization may slow convergence of $\eta_t$ but does not invalidate certificates: any $x \notin A_t$ remains certifiably suboptimal regardless of which $x \in A_t$ is queried.
|
| 740 |
+
|
| 741 |
+
**Replication Strategy.** When an active point $x_i$ has $r_i(t) > \beta_{\text{target}}(t)$ , we allocate $\lceil (r_i(t)/\beta_{\text{target}}(t))^2 \rceil$ additional samples to reduce its confidence radius. This ensures all active points have comparable confidence, preventing
|
| 742 |
+
|
| 743 |
+
{10}------------------------------------------------
|
| 744 |
+
|
| 745 |
+
any single point from dominating the envelope.
|
| 746 |
+
|
| 747 |
+
## <span id="page-10-0"></span>**B.2.** Active Set Computation
|
| 748 |
+
|
| 749 |
+
For low dimensions $(d \le 5)$ , we compute $A_t$ exactly using grid discretization with resolution $\eta = D/\sqrt[d]{N_{\rm grid}}$ where $N_{\rm grid} = 10^4$ . For each grid point x, we evaluate $U_t(x)$ in $O(N_t)$ time and check if $U_t(x) \ge \ell_t$ . The volume $\operatorname{Vol}(A_t)$ is estimated as the fraction of grid points in $A_t$ .
|
| 750 |
+
|
| 751 |
+
For higher dimensions (d>5), uniform Monte Carlo becomes ineffective once $\operatorname{Vol}(A_t)$ is small. We therefore estimate $\operatorname{Vol}(A_t)$ via a nested-set ratio estimator (subset simulation): define thresholds $\ell_t - \tau_0 < \ell_t - \tau_1 < \cdots < \ell_t$ inducing nested sets
|
| 752 |
+
|
| 753 |
+
$$A_t^{(k)} = \{x : U_t(x) \ge \ell_t - \tau_k\}, \quad A_t^{(K)} = A_t.$$
|
| 754 |
+
|
| 755 |
+
We estimate
|
| 756 |
+
|
| 757 |
+
$$Vol(A_t) = Vol(A_t^{(0)}) \prod_{k=1}^{K} \mathbb{P}_{x \sim Unif(A_t^{(k-1)})} [x \in A_t^{(k)}],$$
|
| 758 |
+
|
| 759 |
+
sampling approximately uniformly from $A_t^{(k-1)}$ using a hit-and-run Markov chain with the membership oracle $U_t(x) \geq \ell_t - \tau_{k-1}$ . This yields stable estimates even when $\operatorname{Vol}(A_t)$ is very small. In all high-dimensional experiments, we report confidence intervals of $\log \operatorname{Vol}(A_t)$ from repeated estimator runs.
|
| 760 |
+
|
| 761 |
+
Crucially, certificate validity (Remark 4.7) is independent of volume estimation accuracy: the set membership rule $x \in A_t \Leftrightarrow U_t(x) \ge \ell_t$ is exact.
|
| 762 |
+
|
| 763 |
+
On volume-based stopping. The shrinkage bound (Theorem 4.6) provides an upper bound on $\operatorname{Vol}(A_t)$ as a function of the algorithmic gap proxy
|
| 764 |
+
|
| 765 |
+
$$\varepsilon_t := 2(\beta_t + L\eta_t) + \gamma_t, \quad \gamma_t = f^* - \ell_t,$$
|
| 766 |
+
|
| 767 |
+
and therefore supports using $\operatorname{Vol}(A_t)$ as a practical progress diagnostic. However, $\operatorname{Vol}(A_t)$ alone does not yield an anytime upper bound on regret without additional lower-regularity assumptions linking volume back to function values. In our experiments we therefore use $\varepsilon_t$ as the primary certificate-based stopping criterion, and treat $\operatorname{Vol}(A_t)$ as a secondary monitoring signal.
|
| 768 |
+
|
| 769 |
+
## <span id="page-10-1"></span>C. Proofs
|
| 770 |
+
|
| 771 |
+
#### C.1. Proof to Lemma 4.1
|
| 772 |
+
|
| 773 |
+
*Proof.* Define the good event $\mathcal{E} = \bigcap_{t=1}^T \bigcap_{i=1}^{N_t} \{|\hat{\mu}_i(t) - f(x_i)| \leq r_i(t)\}.$
|
| 774 |
+
|
| 775 |
+
Step 1: Single-point concentration. By Hoeffding's inequality for $\sigma$ -sub-Gaussian random variables:
|
| 776 |
+
|
| 777 |
+
$$\mathbb{P}\big[|\hat{\mu}_i(t) - f(x_i)| > r\big] \le 2\exp\left(-\frac{n_i r^2}{2\sigma^2}\right). \tag{4}$$
|
| 778 |
+
|
| 779 |
+
Step 2: Calibration of confidence radius. Substituting $r = r_i(t) = \sigma \sqrt{2 \log(2N_t T/\delta)/n_i}$ :
|
| 780 |
+
|
| 781 |
+
$$\mathbb{P}[|\hat{\mu}_{i}(t) - f(x_{i})| > r_{i}(t)] \leq 2 \exp\left(-\frac{n_{i} \cdot 2\sigma^{2} \log(2N_{t}T/\delta)}{2\sigma^{2} \cdot n_{i}}\right)$$
|
| 782 |
+
|
| 783 |
+
$$= 2 \exp\left(-\log(2N_{t}T/\delta)\right) = \frac{\delta}{N_{t}T}.$$
|
| 784 |
+
(5)
|
| 785 |
+
|
| 786 |
+
Step 3: Union bound. Applying the union bound over all $i \in \{1, ..., N_t\}$ and $t \in \{1, ..., T\}$ :
|
| 787 |
+
|
| 788 |
+
$$\mathbb{P}[\mathcal{E}^c] \le \sum_{t=1}^T \sum_{i=1}^{N_t} \frac{\delta}{N_t T} \le \sum_{t=1}^T \frac{\delta}{T} = \delta. \tag{6}$$
|
| 789 |
+
|
| 790 |
+
Hence
|
| 791 |
+
$$\mathbb{P}[\mathcal{E}] \geq 1 - \delta$$
|
| 792 |
+
.
|
| 793 |
+
|
| 794 |
+
#### C.2. Proof to Lemma 4.2 (UCB envelope is valid)
|
| 795 |
+
|
| 796 |
+
*Proof.* Fix any $x \in \mathcal{X}$ . For any sampled point $x_i$ , on the good event $\mathcal{E}$ :
|
| 797 |
+
|
| 798 |
+
$$f(x_i) \le \hat{\mu}_i(t) + r_i(t) = UCB_i(t). \tag{7}$$
|
| 799 |
+
|
| 800 |
+
By Lipschitz continuity of f:
|
| 801 |
+
|
| 802 |
+
$$f(x) \le f(x_i) + L \cdot d(x, x_i) \le UCB_i(t) + L \cdot d(x, x_i)$$
|
| 803 |
+
. (8)
|
| 804 |
+
|
| 805 |
+
Since this holds for all sampled *i*, taking the minimum over *i*:
|
| 806 |
+
|
| 807 |
+
$$f(x) \le \min_{i \le N_t} \{ \text{UCB}_i(t) + L \cdot d(x, x_i) \} = U_t(x).$$
|
| 808 |
+
(9)
|
| 809 |
+
|
| 810 |
+
#### C.3. Proof to Lemma 4.3 (Envelope slack bound)
|
| 811 |
+
|
| 812 |
+
*Proof.* Fix any $x \in \mathcal{X}$ and any sampled point $x_i$ .
|
| 813 |
+
|
| 814 |
+
Step 1: Upper confidence bound. On the good event $\mathcal{E}$ , we have $\hat{\mu}_i(t) \leq f(x_i) + r_i(t)$ . Therefore:
|
| 815 |
+
|
| 816 |
+
$$UCB_{i}(t) = \hat{\mu}_{i}(t) + r_{i}(t) \le f(x_{i}) + 2r_{i}(t).$$
|
| 817 |
+
(10)
|
| 818 |
+
|
| 819 |
+
Step 2: Lipschitz propagation. By Assumption 2.1 (Lipschitz continuity):
|
| 820 |
+
|
| 821 |
+
$$f(x_i) \le f(x) + L \cdot d(x, x_i). \tag{11}$$
|
| 822 |
+
|
| 823 |
+
Step 3: Combining bounds. Substituting the Lipschitz bound into Step 1:
|
| 824 |
+
|
| 825 |
+
$$UCB_{i}(t) + L d(x, x_{i}) \leq f(x_{i}) + 2r_{i}(t) + L d(x, x_{i})$$
|
| 826 |
+
|
| 827 |
+
|
| 828 |
+
$$\leq f(x) + L d(x, x_{i}) + 2r_{i}(t) + L d(x, x_{i})$$
|
| 829 |
+
|
| 830 |
+
|
| 831 |
+
$$= f(x) + 2(r_{i}(t) + L d(x, x_{i})). \quad (12)$$
|
| 832 |
+
|
| 833 |
+
{11}------------------------------------------------
|
| 834 |
+
|
| 835 |
+
*Step 4: Taking the minimum.* Since the above holds for all i, taking the minimum over i on both sides:
|
| 836 |
+
|
| 837 |
+
$$U_{t}(x) = \min_{i \leq N_{t}} \left\{ \text{UCB}_{i}(t) + L d(x, x_{i}) \right\}$$
|
| 838 |
+
|
| 839 |
+
$$\leq f(x) + 2 \min_{i} \left\{ r_{i}(t) + L d(x, x_{i}) \right\}$$
|
| 840 |
+
|
| 841 |
+
$$= f(x) + 2\rho_{t}(x).$$
|
| 842 |
+
(13)
|
| 843 |
+
|
| 844 |
+
## C.4. Proof to Theorem [4.5](#page-2-3)
|
| 845 |
+
|
| 846 |
+
*Proof.* We show that any point in the active set must have function value close to optimal.
|
| 847 |
+
|
| 848 |
+
*Step 1: Lower certificate validity.* On E, for any sampled point x<sup>i</sup> :
|
| 849 |
+
|
| 850 |
+
$$LCB_i(t) = \hat{\mu}_i(t) - r_i(t) \le f(x_i) \le f^*.$$
|
| 851 |
+
(14)
|
| 852 |
+
|
| 853 |
+
Taking the maximum over all i:
|
| 854 |
+
|
| 855 |
+
$$\ell_t = \max_{i \le N_t} LCB_i(t) \le f^*. \tag{15}$$
|
| 856 |
+
|
| 857 |
+
*Step 2: Active set membership implies high UCB.* Let x ∈ At. By definition of the active set [\(2\)](#page-2-7):
|
| 858 |
+
|
| 859 |
+
$$U_t(x) \ge \ell_t. \tag{16}$$
|
| 860 |
+
|
| 861 |
+
*Step 3: Applying the envelope bound.* By Lemma [4.3:](#page-2-2)
|
| 862 |
+
|
| 863 |
+
$$f(x) + 2\rho_t(x) \ge U_t(x) \ge \ell_t. \tag{17}$$
|
| 864 |
+
|
| 865 |
+
*Step 4: Rearranging to obtain the containment.* Solving for f(x):
|
| 866 |
+
|
| 867 |
+
$$f(x) \ge \ell_t - 2\rho_t(x)$$
|
| 868 |
+
|
| 869 |
+
$$= f^* - (f^* - \ell_t) - 2\rho_t(x)$$
|
| 870 |
+
|
| 871 |
+
$$\ge f^* - (f^* - \ell_t) - 2 \sup_{x' \in A_t} \rho_t(x')$$
|
| 872 |
+
|
| 873 |
+
$$= f^* - 2\Delta_t.$$
|
| 874 |
+
(18)
|
| 875 |
+
|
| 876 |
+
Hence
|
| 877 |
+
$$A_t \subseteq \{x : f(x) \ge f^* - 2\Delta_t\}.$$
|
| 878 |
+
|
| 879 |
+
#### C.5. Proof to Theorem [4.6](#page-2-4)
|
| 880 |
+
|
| 881 |
+
*Proof.* We connect the active set volume to the margin condition via the containment theorem.
|
| 882 |
+
|
| 883 |
+
*Step 1: Bounding* ∆t*.* From Theorem [4.5,](#page-2-3) A<sup>t</sup> ⊆ {x : f(x) ≥ f <sup>∗</sup> − 2∆t} where:
|
| 884 |
+
|
| 885 |
+
$$\Delta_t = \sup_{x \in A_t} \rho_t(x) + (f^* - \ell_t).$$
|
| 886 |
+
(19)
|
| 887 |
+
|
| 888 |
+
For any x ∈ At:
|
| 889 |
+
|
| 890 |
+
$$\rho_t(x) = \min_{i} \left\{ r_i(t) + L \cdot d(x, x_i) \right\}$$
|
| 891 |
+
|
| 892 |
+
$$\leq \max_{i:x_i \text{ active}} r_i(t) + L \cdot \sup_{x \in A_t} \min_{i} d(x, x_i)$$
|
| 893 |
+
|
| 894 |
+
$$= \beta_t + L\eta_t. \tag{20}$$
|
| 895 |
+
|
| 896 |
+
Therefore:
|
| 897 |
+
|
| 898 |
+
$$\Delta_t \le \beta_t + L\eta_t + \frac{\gamma_t}{2}.\tag{21}$$
|
| 899 |
+
|
| 900 |
+
*Step 2: Applying the margin condition.* By Assumption [2.3,](#page-1-3) for any ε > 0:
|
| 901 |
+
|
| 902 |
+
$$\operatorname{Vol}(\{x: f(x) \ge f^* - \varepsilon\}) \le C \cdot \varepsilon^{d-\alpha}. \tag{22}$$
|
| 903 |
+
|
| 904 |
+
*Step 3: Combining the bounds.* Setting ε = 2∆<sup>t</sup> ≤ 2(β<sup>t</sup> + Lηt) + γt:
|
| 905 |
+
|
| 906 |
+
$$\operatorname{Vol}(A_t) \leq \operatorname{Vol}(\{x : f(x) \geq f^* - 2\Delta_t\})$$
|
| 907 |
+
|
| 908 |
+
$$\leq C \cdot (2\Delta_t)^{d-\alpha}$$
|
| 909 |
+
|
| 910 |
+
$$\leq C \cdot (2(\beta_t + L\eta_t) + \gamma_t)^{d-\alpha}.$$
|
| 911 |
+
(23)
|
| 912 |
+
|
| 913 |
+
#### C.6. Proof to Theorem [4.8](#page-3-5)
|
| 914 |
+
|
| 915 |
+
*Proof.* We derive the sample complexity by analyzing the requirements for ε-optimality.
|
| 916 |
+
|
| 917 |
+
*Step 1: Optimality condition.* To achieve simple regret r<sup>T</sup> ≤ ε, it suffices to ensure:
|
| 918 |
+
|
| 919 |
+
$$2(\beta_t + L\eta_t) + \gamma_t \le \varepsilon. \tag{24}$$
|
| 920 |
+
|
| 921 |
+
This requires β<sup>t</sup> ≤ ε/6, η<sup>t</sup> �� ε/(6L), and γ<sup>t</sup> ≤ ε/3.
|
| 922 |
+
|
| 923 |
+
*Step 2: Covering the active set.* By the shrinkage theorem (Theorem [4.6\)](#page-2-4), once 2(β<sup>t</sup> + Lηt) + γ<sup>t</sup> ≤ ε we have:
|
| 924 |
+
|
| 925 |
+
$$Vol(A_t) \le C \cdot \varepsilon^{d-\alpha}.$$
|
| 926 |
+
(25)
|
| 927 |
+
|
| 928 |
+
To achieve covering radius η = ε/(6L) over a region of volume Cε<sup>d</sup>−<sup>α</sup>, we need:
|
| 929 |
+
|
| 930 |
+
$$N_{\text{cover}} = O\left(\frac{\text{Vol}(A_t)}{\eta^d}\right) = O\left(\frac{C\varepsilon^{d-\alpha}}{(\varepsilon/(6L))^d}\right) = O\left(L^d\varepsilon^{-\alpha}\right)$$
|
| 931 |
+
(26)
|
| 932 |
+
|
| 933 |
+
distinct sample locations.
|
| 934 |
+
|
| 935 |
+
*Step 3: Samples per location.* To achieve confidence radius β<sup>t</sup> ≤ ε/6 at each location, we need:
|
| 936 |
+
|
| 937 |
+
$$\sigma\sqrt{\frac{2\log(2N_tT/\delta)}{n_i}} \le \frac{\varepsilon}{6}.$$
|
| 938 |
+
(27)
|
| 939 |
+
|
| 940 |
+
Solving for n<sup>i</sup> :
|
| 941 |
+
|
| 942 |
+
$$n_i = O\left(\frac{\sigma^2 \log(T/\delta)}{\varepsilon^2}\right).$$
|
| 943 |
+
(28)
|
| 944 |
+
|
| 945 |
+
{12}------------------------------------------------
|
| 946 |
+
|
| 947 |
+
Step 4: Total sample complexity. Combining Steps 2 and 3:
|
| 948 |
+
|
| 949 |
+
$$T = N_{\text{cover}} \cdot n_{i}$$
|
| 950 |
+
|
| 951 |
+
$$= O(L^{d} \varepsilon^{-\alpha}) \cdot O\left(\frac{\sigma^{2} \log(T/\delta)}{\varepsilon^{2}}\right)$$
|
| 952 |
+
|
| 953 |
+
$$= \tilde{O}(L^{d} \varepsilon^{-(2+\alpha)}). \tag{29}$$
|
| 954 |
+
|
| 955 |
+
#### C.7. Proof to Theorem 4.9
|
| 956 |
+
|
| 957 |
+
*Proof.* We construct a hard instance via a randomized reduction.
|
| 958 |
+
|
| 959 |
+
Step 1: Hard instance construction (continuous bump). We use a standard "one bump among M locations" construction that ensures global Lipschitz continuity. Partition $\mathcal{X} = [0,1]^d$ into $M = \varepsilon^{-\alpha}$ disjoint cells with centers $\{c_1,\ldots,c_M\}$ , each cell having diameter $\Theta(\varepsilon^{\alpha/d})$ . Select one cell $i^*$ uniformly at random to contain the optimum, and place $x^* = c_{i^*}$ . Define:
|
| 960 |
+
|
| 961 |
+
$$f(x) = (1 - \varepsilon) + \varepsilon \cdot \max \left\{ 0, 1 - \frac{L \|x - x^*\|}{\varepsilon} \right\}. \quad (30)$$
|
| 962 |
+
|
| 963 |
+
This is a cone/bump centered at $x^*$ that peaks at $f(x^*)=1$ and decreases linearly with slope L until it reaches the baseline value $1-\varepsilon$ at radius $\varepsilon/L$ from $x^*$ . Outside this radius, $f(x)\equiv 1-\varepsilon$ . The function is globally L-Lipschitz: within the bump, the gradient has magnitude L; outside, the function is constant; and at the boundary $\|x-x^*\|=\varepsilon/L$ , both pieces match at value $1-\varepsilon$ .
|
| 964 |
+
|
| 965 |
+
Verification of Assumption 2.3. The $\varepsilon$ -near-optimal set $\{x:f(x)\geq 1-\varepsilon\}$ is exactly the ball of radius $\varepsilon/L$ around $x^*$ , which has volume $\operatorname{Vol}(B_{\varepsilon/L})=O((\varepsilon/L)^d)=O(\varepsilon^d)$ . With $M=\varepsilon^{-\alpha}$ candidate locations, only one contains the bump, so the near-optimal fraction of the domain is $O(\varepsilon^d)$ . Since the domain has unit volume, $\operatorname{Vol}(\{f\geq f^*-\varepsilon\})=O(\varepsilon^d)\leq C\varepsilon^{d-\alpha}$ for $\alpha\geq 0$ , satisfying Assumption 2.3.
|
| 966 |
+
|
| 967 |
+
Step 2: Information-theoretic lower bound. To identify the correct cell with probability $\geq 2/3$ , the algorithm must distinguish between M hypotheses. By Fano's inequality, this requires:
|
| 968 |
+
|
| 969 |
+
$$\sum_{i=1}^{M} n_i \cdot \text{KL}(P_i || P_0) \ge \log(M/3), \tag{31}$$
|
| 970 |
+
|
| 971 |
+
where $n_i$ is the number of samples in cell i, and $\mathrm{KL}(P_i || P_0)$ is the KL divergence between observations under hypothesis i versus the null.
|
| 972 |
+
|
| 973 |
+
Step 3: Per-cell sample requirement. For $\sigma$ -sub-Gaussian noise, distinguishing a cell with optimum from one without requires:
|
| 974 |
+
|
| 975 |
+
$$n_i = \Omega\left(\frac{\sigma^2}{\varepsilon^2}\right) \tag{32}$$
|
| 976 |
+
|
| 977 |
+
samples per cell to detect the $\varepsilon$ gap with constant probability. Step 4: Total sample complexity. Summing over all M cells:
|
| 978 |
+
|
| 979 |
+
$$T = \Omega \left( M \cdot \frac{\sigma^2}{\varepsilon^2} \right) = \Omega \left( \varepsilon^{-\alpha} \cdot \varepsilon^{-2} \right) = \Omega \left( \varepsilon^{-(2+\alpha)} \right). \tag{33}$$
|
| 980 |
+
|
| 981 |
+
#### C.8. Proof to Theorem 5.1
|
| 982 |
+
|
| 983 |
+
*Proof.* We prove each claim separately.
|
| 984 |
+
|
| 985 |
+
Proof of (1): Bounded doubling events. Each doubling event multiplies $\hat{L}$ by 2. Starting from $\hat{L}_0 \leq L^*$ :
|
| 986 |
+
|
| 987 |
+
$$\hat{L}_k = 2^k \hat{L}_0$$
|
| 988 |
+
after $k$ doublings. (34)
|
| 989 |
+
|
| 990 |
+
The algorithm stops doubling when $\hat{L} \geq L^*$ , which requires:
|
| 991 |
+
|
| 992 |
+
$$2^K \hat{L}_0 \ge L^* \implies K \ge \log_2(L^*/\hat{L}_0).$$
|
| 993 |
+
(35)
|
| 994 |
+
|
| 995 |
+
Hence $K \leq \lceil \log_2(L^*/\hat{L}_0) \rceil$ .
|
| 996 |
+
|
| 997 |
+
*Proof of (2): Final estimate accuracy.* A violation is detected when:
|
| 998 |
+
|
| 999 |
+
$$|\hat{\mu}_i - \hat{\mu}_j| - 2(r_i + r_j) > \hat{L} \cdot d(x_i, x_j).$$
|
| 1000 |
+
(36)
|
| 1001 |
+
|
| 1002 |
+
On the good event $\mathcal{E}$ :
|
| 1003 |
+
|
| 1004 |
+
$$|f(x_i) - f(x_j)| \le |\hat{\mu}_i - \hat{\mu}_j| + 2(r_i + r_j).$$
|
| 1005 |
+
(37)
|
| 1006 |
+
|
| 1007 |
+
If $\hat{L} \geq L^*$ , then by Lipschitz continuity:
|
| 1008 |
+
|
| 1009 |
+
$$|f(x_i) - f(x_i)| \le L^* \cdot d(x_i, x_i) \le \hat{L} \cdot d(x_i, x_i), \quad (38)$$
|
| 1010 |
+
|
| 1011 |
+
so no violation can occur. Thus violations only occur when $\hat{L} < L^*$ , and after all doublings complete, $\hat{L} \geq L^*$ . Since we double (rather than increase by smaller factors), $\hat{L} \leq 2L^*$ .
|
| 1012 |
+
|
| 1013 |
+
*Proof of (3): Sample complexity overhead.* Between doublings, CGP runs with either:
|
| 1014 |
+
|
| 1015 |
+
- Invalid $\hat{L} < L^*$ (before sufficient doublings): certificates may be incorrect, but each such phase has at most O(T/K) samples before a violation triggers doubling.
|
| 1016 |
+
- Valid $\hat{L} \geq L^*$ (after final doubling): CGP achieves $\tilde{O}(\varepsilon^{-(2+\alpha)})$ complexity by Theorem 4.8.
|
| 1017 |
+
|
| 1018 |
+
There are at most K invalid phases, each contributing O(T/K) samples. The final valid phase dominates, giving total complexity $T = \tilde{O}(\varepsilon^{-(2+\alpha)} \cdot K) = \tilde{O}(\varepsilon^{-(2+\alpha)} \cdot \log(L^*/\hat{L}_0))$ .
|
| 1019 |
+
|
| 1020 |
+
{13}------------------------------------------------
|
| 1021 |
+
|
| 1022 |
+
## C.9. Certificate Validity Under Adaptive Lipschitz Estimation
|
| 1023 |
+
|
| 1024 |
+
**Lemma C.1** (Certificate validity once $\hat{L} \geq L^*$ ). Assume the good event $\mathcal{E}$ holds. Fix any time t at which the current estimate satisfies $\hat{L} \geq L^*$ . Then the envelope constructed with $\hat{L}$ satisfies $f(x) \leq U_t(x)$ for all $x \in \mathcal{X}$ , and consequently the active set
|
| 1025 |
+
|
| 1026 |
+
$$A_t = \{ x \in \mathcal{X} : U_t(x) \ge \ell_t \}$$
|
| 1027 |
+
|
| 1028 |
+
contains $x^*$ and certifies that any $x \notin A_t$ is suboptimal (in the sense $U_t(x) < \ell_t \le f^*$ ).
|
| 1029 |
+
|
| 1030 |
+
*Proof.* If $\hat{L} \geq L^*$ , then for all $x, y \in \mathcal{X}$ we have $|f(x) - f(y)| \leq L^* d(x, y) \leq \hat{L} d(x, y)$ , i.e., f is $\hat{L}$ -Lipschitz. On $\mathcal{E}$ , for each sampled point $x_i$ , $f(x_i) \leq \mathrm{UCB}_i(t)$ . Therefore for all x,
|
| 1031 |
+
|
| 1032 |
+
$$f(x) \le f(x_i) + \hat{L} d(x, x_i) \le UCB_i(t) + \hat{L} d(x, x_i).$$
|
| 1033 |
+
|
| 1034 |
+
Taking the minimum over i yields $f(x) \leq U_t(x)$ for all x. In particular, $U_t(x^*) \geq f^*$ . Also $\ell_t = \max_i \mathrm{LCB}_i(t) \leq f^*$ on $\mathcal{E}$ , hence $U_t(x^*) \geq \ell_t$ and so $x^* \in A_t$ . Finally, if $x \notin A_t$ , then $U_t(x) < \ell_t \leq f^*$ , certifying x cannot be optimal under $\mathcal{E}$ .
|
| 1035 |
+
|
| 1036 |
+
Remark C.2 (Why pre-final certificates need not be valid). If $\hat{L} < L^*$ , then the envelope may fail to upper-bound f globally, and the rule $U_t(x) \geq \ell_t$ can (in principle) exclude near-optimal points. CGP-Adaptive therefore guarantees certificate validity only after the final doubling event ensures $\hat{L} \geq L^*$ .
|
| 1037 |
+
|
| 1038 |
+
## C.10. Global Safety of Certified Restarts (No False Elimination)
|
| 1039 |
+
|
| 1040 |
+
We restate and prove the key safety property underlying certified restarts in CGP-TR.
|
| 1041 |
+
|
| 1042 |
+
**Lemma C.3** (No false certified restart for the region containing $x^*$ ). Fix any trust region $\mathcal{T}^*$ such that $x^* \in \mathcal{T}^*$ . On the good event $\mathcal{E}$ , the certified restart condition
|
| 1043 |
+
|
| 1044 |
+
$$u_t^{(\mathcal{T}^*)} := \max_{x \in \mathcal{T}^*} U_t(x) < \ell_t$$
|
| 1045 |
+
|
| 1046 |
+
never holds. Hence, $\mathcal{T}^*$ is never restarted by the certified rule.
|
| 1047 |
+
|
| 1048 |
+
*Proof.* On the good event $\mathcal{E}$ , Lemma 4.2 implies $f(x) \leq U_t(x)$ for all $x \in \mathcal{X}$ . Since $x^* \in \mathcal{T}^*$ ,
|
| 1049 |
+
|
| 1050 |
+
$$u_t^{(\mathcal{T}^*)} = \max_{x \in \mathcal{T}^*} U_t(x) \ge U_t(x^*) \ge f(x^*) = f^*.$$
|
| 1051 |
+
|
| 1052 |
+
Also, by definition $\ell_t = \max_i \mathrm{LCB}_i(t)$ . On $\mathcal{E}$ , $\mathrm{LCB}_i(t) \leq f(x_i) \leq f^*$ for every sampled point $x_i$ , hence $\ell_t \leq f^*$ . Therefore,
|
| 1053 |
+
|
| 1054 |
+
$$u_t^{(\mathcal{T}^*)} \geq f^* \geq \ell_t,$$
|
| 1055 |
+
|
| 1056 |
+
so the strict inequality $u_t^{(\mathcal{T}^*)} < \ell_t$ cannot occur on $\mathcal{E}$ .
|
| 1057 |
+
|
| 1058 |
+
#### C.11. Proof to Theorem 6.1
|
| 1059 |
+
|
| 1060 |
+
We first establish a key lemma showing that certified elimination is safe.
|
| 1061 |
+
|
| 1062 |
+
**Lemma C.4** (Certified elimination). On the good event $\mathcal{E}$ , for any trust region $\mathcal{T}_j$ ,
|
| 1063 |
+
|
| 1064 |
+
$$u_t^{(j)} := \max_{x \in \mathcal{T}_j} U_t(x) < \ell_t \quad \Longrightarrow \quad \sup_{x \in \mathcal{T}_i} f(x) < f(x^*).$$
|
| 1065 |
+
|
| 1066 |
+
*Proof.* On $\mathcal{E}$ , by Lemma 4.2, $f(x) \leq U_t(x)$ for all $x \in \mathcal{X}$ . Also $\ell_t = \max_i \mathrm{LCB}_i(t) \leq f(x^*)$ on $\mathcal{E}$ since each $\mathrm{LCB}_i(t) \leq f(x_i) \leq f(x^*)$ . Therefore
|
| 1067 |
+
|
| 1068 |
+
$$\sup_{x \in \mathcal{T}_j} f(x) \le \sup_{x \in \mathcal{T}_j} U_t(x) = u_t^{(j)} < \ell_t \le f(x^*),$$
|
| 1069 |
+
|
| 1070 |
+
which proves the claim.
|
| 1071 |
+
|
| 1072 |
+
**Proof of Theorem 6.1(1): No false restarts.** If $x^* \in \mathcal{T}^*$ , then $u_t^{(\mathcal{T}^*)} = \max_{x \in \mathcal{T}^*} U_t(x) \geq U_t(x^*) \geq f(x^*)$ on $\mathcal{E}$ . Also $\ell_t \leq f(x^*)$ on $\mathcal{E}$ . Hence $u_t^{(\mathcal{T}^*)} \geq \ell_t$ and the restart condition $u_t^{(\mathcal{T}^*)} < \ell_t$ never triggers.
|
| 1073 |
+
|
| 1074 |
+
**Proof of Theorem 6.1(2): Local certificate.** Conditioned on $T^*$ evaluations within $\mathcal{T}^*$ , the CGP analysis applies verbatim on the restricted domain $\mathcal{T}^*$ : the good event $\mathcal{E}$ implies all within-region confidence bounds hold; Lipschitz continuity holds on $\mathcal{T}^*$ ; and Assumption 2.3 holds restricted to $\mathcal{T}^*$ . Therefore the containment and shrinkage results follow with $\mathcal{X}$ replaced by $\mathcal{T}^*$ , yielding $\operatorname{Vol}(A_T^{(\mathcal{T}^*)}) \leq C\varepsilon^{d-\alpha}$ after $T^* = \tilde{O}(\varepsilon^{-(2+\alpha)})$ within-region samples.
|
| 1075 |
+
|
| 1076 |
+
**Proof of Theorem 6.1(3): Allocation bound.** Fix a sub-optimal region j with gap $\Delta_j > 0$ . On $\mathcal{E}$ , using the envelope bound from Lemma 4.3, we have for all x:
|
| 1077 |
+
|
| 1078 |
+
$$U_t(x) < f(x) + 2\rho_t(x),$$
|
| 1079 |
+
|
| 1080 |
+
where $\rho_t(x) = \min_i \{r_i(t) + Ld(x, x_i)\}$ . Restricting to points sampled in $\mathcal{T}_j$ and using the within-region replication schedule, we bound $\rho_t(x) \leq \beta_j(n) + L \cdot \operatorname{diam}(\mathcal{T}_j)$ , yielding
|
| 1081 |
+
|
| 1082 |
+
$$u_t^{(j)} = \max_{x \in \mathcal{T}_j} U_t(x) \le \sup_{x \in \mathcal{T}_j} f(x) + 2\beta_j(n) + 2L \cdot \operatorname{diam}(\mathcal{T}_j),$$
|
| 1083 |
+
|
| 1084 |
+
after n within-region samples. By the diameter condition $L \operatorname{diam}(\mathcal{T}_j) \leq \Delta_j/8$ and by requiring $\beta_j(n) \leq \Delta_j/8$ , we obtain
|
| 1085 |
+
|
| 1086 |
+
$$u_t^{(j)} \le \sup_{\mathcal{T}_j} f + \Delta_j/4 + \Delta_j/4 = f^* - \Delta_j/2.$$
|
| 1087 |
+
|
| 1088 |
+
Meanwhile, on $\mathcal{E}$ we have $\ell_t \leq f^*$ always, and once the region containing $x^*$ has been sampled sufficiently (which occurs because it is never restarted and is favored by UCB
|
| 1089 |
+
|
| 1090 |
+
{14}------------------------------------------------
|
| 1091 |
+
|
| 1092 |
+
selection), $\ell_t$ becomes at least $f^* - \Delta_j/4$ . Hence eventually $u_t^{(j)} < \ell_t$ , triggering certified restart/elimination.
|
| 1093 |
+
|
| 1094 |
+
Solving $\beta_j(n) \leq \Delta_j/8$ under $\beta_j(n) \leq c_\sigma \sqrt{\log(c_T/\delta)/n}$ gives
|
| 1095 |
+
|
| 1096 |
+
$$n \ge \frac{64c_{\sigma}^2}{\Delta_i^2} \log \left(\frac{c_T}{\delta}\right),\,$$
|
| 1097 |
+
|
| 1098 |
+
yielding the stated bound $N_j \leq \frac{64c_\sigma^2}{\Delta_j^2}\log(c_T/\delta) + 1$ .
|
| 1099 |
+
|
| 1100 |
+
#### C.12. Proof to Proposition 7.1
|
| 1101 |
+
|
| 1102 |
+
*Proof.* We analyze each case and the certificate validity separately.
|
| 1103 |
+
|
| 1104 |
+
*Proof of (1): High smoothness ratio case.* When $\rho \geq 0.5$ , CGP-Hybrid continues with CGP in Phase 2. The algorithm is identical to vanilla CGP, so by Theorem 4.8:
|
| 1105 |
+
|
| 1106 |
+
$$T = \tilde{O}(\varepsilon^{-(2+\alpha)}). \tag{39}$$
|
| 1107 |
+
|
| 1108 |
+
Proof of (2): Low smoothness ratio case. When $\rho < 0.5$ , the function is significantly smoother near the optimum. After Phase 1, CGP has reduced the active set to volume $V < 0.1 \cdot \mathrm{Vol}(\mathcal{X})$ . In Phase 2, GP-UCB operates within $A_t$ , which has:
|
| 1109 |
+
|
| 1110 |
+
- Effective diameter diam $(A_t) = O(V^{1/d})$ .
|
| 1111 |
+
- Local Lipschitz constant $L_{local} = \rho \cdot L < 0.5L$ .
|
| 1112 |
+
|
| 1113 |
+
The sample complexity of GP-UCB on this restricted domain depends on the kernel's maximum information gain $\gamma_T$ over $A_t$ (Srinivas et al., 2009). For commonly used kernels (Matérn, SE), $\gamma_T$ scales polylogarithmically with T when the domain is bounded. The reduced diameter $O(V^{1/d})$ and local smoothness $\rho < 0.5$ empirically yield faster convergence than continuing CGP; we validate this empirically in Section 9 rather than claiming a specific rate.
|
| 1114 |
+
|
| 1115 |
+
*Proof of (3): Certificate validity.* The certificate $A_T$ is computed in Phase 1 using CGP's Lipschitz envelope construction. By Theorem 4.5, on the good event $\mathcal{E}$ :
|
| 1116 |
+
|
| 1117 |
+
$$x^* \in A_T$$
|
| 1118 |
+
with probability $\geq 1 - \delta$ . (40)
|
| 1119 |
+
|
| 1120 |
+
This guarantee depends only on the Lipschitz assumption and confidence bounds, not on the Phase 2 optimization method. Therefore, switching to GP-UCB in Phase 2 does not invalidate the certificate: $A_T$ still contains $x^*$ with high probability, and any point outside $A_T$ remains certifiably suboptimal.
|
| 1121 |
+
|
| 1122 |
+
### <span id="page-14-0"></span>**D. Practical Guidance**
|
| 1123 |
+
|
| 1124 |
+
#### **D.1. Adaptive Lipschitz Estimation**
|
| 1125 |
+
|
| 1126 |
+
CGP-Adaptive removes the requirement for known L with $O(\log T)$ overhead. The doubling scheme is conservative
|
| 1127 |
+
|
| 1128 |
+
but provably correct; more aggressive schemes (e.g., multiplicative updates with factor 1.5) may reduce overhead but risk certificate invalidation.
|
| 1129 |
+
|
| 1130 |
+
We recommend initializing $\hat{L}_0$ from finite differences on initial Sobol samples:
|
| 1131 |
+
|
| 1132 |
+
$$\hat{L}_0 = \max_{i \neq j} \frac{|y_i - y_j|}{d(x_i, x_j)}$$
|
| 1133 |
+
|
| 1134 |
+
over the first 10 samples. This typically underestimates L by a factor of 2 to 10, requiring 1 to 4 doublings to reach $\hat{L} \geq L^*$ .
|
| 1135 |
+
|
| 1136 |
+
### **D.2. Trust Region Configuration**
|
| 1137 |
+
|
| 1138 |
+
CGP-TR trades global certificates for scalability. The local certificates within trust regions still enable principled stopping and progress assessment, but do not guarantee global optimality.
|
| 1139 |
+
|
| 1140 |
+
Recommended settings:
|
| 1141 |
+
|
| 1142 |
+
- Number of trust regions: $n_{\text{trust}} = 5$ (balances exploration vs. overhead)
|
| 1143 |
+
- Initial radius: $r_0 = 0.2$ (covers 20% of domain diameter per region)
|
| 1144 |
+
- Minimum radius: $r_{\min} = 0.01$ (prevents over-contraction)
|
| 1145 |
+
- Failure threshold: $\tau_{\text{fail}} = 10$ (triggers contraction after 10 non-improving samples)
|
| 1146 |
+
|
| 1147 |
+
For applications requiring global certificates in high dimensions, combining CGP-TR with random embeddings (Wang et al., 2016) is promising: project to a low-dimensional subspace, run CGP with global certificates, then lift back.
|
| 1148 |
+
|
| 1149 |
+
#### **D.3. Smoothness Detection**
|
| 1150 |
+
|
| 1151 |
+
CGP-Hybrid's smoothness detection via $\rho = L_{\rm local}/L_{\rm global}$ is heuristic but effective. We estimate $L_{\rm local}$ from points within $A_t$ using the same finite difference approach as $L_{\rm global}$ .
|
| 1152 |
+
|
| 1153 |
+
The threshold $\rho_{\rm thresh}=0.5$ was selected via cross-validation on held-out benchmarks. More sophisticated detection could use local GP posterior variance or curvature estimates. The key insight is that CGP's certificate remains valid regardless of Phase 2 method, so switching is always safe.
|
| 1154 |
+
|
| 1155 |
+
#### D.4. When to Use CGP
|
| 1156 |
+
|
| 1157 |
+
CGP is well suited when:
|
| 1158 |
+
|
| 1159 |
+
Evaluations are expensive and interpretable progress is valued
|
| 1160 |
+
|
| 1161 |
+
{15}------------------------------------------------
|
| 1162 |
+
|
| 1163 |
+
851852
|
| 1164 |
+
|
| 1165 |
+
853
|
| 1166 |
+
|
| 1167 |
+
854855
|
| 1168 |
+
|
| 1169 |
+
856
|
| 1170 |
+
|
| 1171 |
+
857
|
| 1172 |
+
|
| 1173 |
+
858859
|
| 1174 |
+
|
| 1175 |
+
860861
|
| 1176 |
+
|
| 1177 |
+
862863
|
| 1178 |
+
|
| 1179 |
+
864
|
| 1180 |
+
|
| 1181 |
+
865866
|
| 1182 |
+
|
| 1183 |
+
867868
|
| 1184 |
+
|
| 1185 |
+
869870
|
| 1186 |
+
|
| 1187 |
+
871
|
| 1188 |
+
|
| 1189 |
+
872873
|
| 1190 |
+
|
| 1191 |
+
874
|
| 1192 |
+
|
| 1193 |
+
875876
|
| 1194 |
+
|
| 1195 |
+
877878
|
| 1196 |
+
|
| 1197 |
+
879
|
| 1198 |
+
|
| 1199 |
+
- 2. The objective has margin structure (sharp peak rather than wide plateau)
|
| 1200 |
+
- 3. Lipschitz continuity is a reasonable assumption
|
| 1201 |
+
- 4. Dimension is moderate ( $d \le 15$ for vanilla CGP, $d \le 100$ for CGP-TR)
|
| 1202 |
+
- 5. Anytime stopping decisions are needed
|
| 1203 |
+
|
| 1204 |
+
For very high-dimensional problems (d>100), trust region methods like TuRBO may scale better. For smooth problems with cheap evaluations, GP-based methods may be more sample efficient due to their ability to exploit higher-order smoothness.
|
| 1205 |
+
|
| 1206 |
+
## <span id="page-15-0"></span>E. Experimental Details
|
| 1207 |
+
|
| 1208 |
+
#### E.1. Baseline Configurations
|
| 1209 |
+
|
| 1210 |
+
- Random Search: Sobol sequences for quasi-random sampling
|
| 1211 |
+
- **GP-UCB**: Matérn-5/2 kernel via BoTorch, $\beta_t = 2\log(t^2\pi^2/6\delta)$
|
| 1212 |
+
- **TuRBO**: Default settings from Eriksson et al. (2019), 1 trust region
|
| 1213 |
+
- HEBO: Heteroscedastic GP with input warping, default settings
|
| 1214 |
+
- **BORE**: Tree-Parzen estimator with density ratio, default settings
|
| 1215 |
+
- **HOO**: Binary tree with $\nu_1 = 1$ , $\rho = 0.5$
|
| 1216 |
+
- StoSOO: k = 3 children per node, $h_{\text{max}} = 20$
|
| 1217 |
+
- LIPO: Pure Lipschitz optimization, L estimated online
|
| 1218 |
+
- SAASBO: Sparse axis-aligned GP, 10 active dimensions
|
| 1219 |
+
|
| 1220 |
+
#### E.2. Benchmark Details
|
| 1221 |
+
|
| 1222 |
+
#### Low-dimensional.
|
| 1223 |
+
|
| 1224 |
+
- Needle-2D: $f(x) = 1 \|x x^*\|^{1/\alpha}$ with $\alpha = 2$ , sharp peak
|
| 1225 |
+
- Branin: Standard 2D benchmark with 3 global optima
|
| 1226 |
+
- Hartmann-6: 6D benchmark with narrow global basin
|
| 1227 |
+
- Levy-5: 5D benchmark with global structure
|
| 1228 |
+
- Rosenbrock-4: 4D benchmark with curved valley
|
| 1229 |
+
|
| 1230 |
+
#### Medium-dimensional.
|
| 1231 |
+
|
| 1232 |
+
- Ackley-10: 10D benchmark with many local optima
|
| 1233 |
+
- **SVM-RBF-6**: Real hyperparameter tuning $(C, \gamma, 4)$ preprocessing) on MNIST
|
| 1234 |
+
- LunarLander-12: RL reward optimization with 12 policy parameters
|
| 1235 |
+
|
| 1236 |
+
#### High-dimensional.
|
| 1237 |
+
|
| 1238 |
+
- **Rover-60**: Mars rover trajectory with 60 waypoint parameters (Wang et al., 2016)
|
| 1239 |
+
- NAS-36: Neural architecture search on CIFAR-10, 36 continuous encodings
|
| 1240 |
+
- Ant-100: MuJoCo Ant locomotion, 100 morphology and control parameters
|
| 1241 |
+
|
| 1242 |
+
#### E.3. Computational Resources
|
| 1243 |
+
|
| 1244 |
+
All experiments run on AMD EPYC 7763 with 256GB RAM. CGP variants use NumPy/SciPy; GP baselines use BoTorch/GPyTorch with GPU acceleration (NVIDIA A100) where available.
|
icml26/5c3d6bff-e8ce-4d9f-840b-719084582491/parse_report.json
ADDED
|
@@ -0,0 +1,76 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
{
|
| 2 |
+
"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491",
|
| 3 |
+
"pipeline": "marker_non_llm_v3",
|
| 4 |
+
"parser": "marker_single",
|
| 5 |
+
"formats": [
|
| 6 |
+
"markdown",
|
| 7 |
+
"chunks"
|
| 8 |
+
],
|
| 9 |
+
"llm_enabled": false,
|
| 10 |
+
"pdf_path": "/network/scratch/j/jianan.zhao/ReviewAgent/data/processed_papers/icml26_20260429_1952_duequeue/raw/5c3d6bff-e8ce-4d9f-840b-719084582491.pdf",
|
| 11 |
+
"pdf_sha256": "e9cc905fb13f976490024de8d9451345fa5e1aada57f53d21e11bf40683a247a",
|
| 12 |
+
"bytes": 408679,
|
| 13 |
+
"source": "https://koala.science/storage/pdfs/5c3d6bff-e8ce-4d9f-840b-719084582491.pdf",
|
| 14 |
+
"page_count": 16,
|
| 15 |
+
"ok": true,
|
| 16 |
+
"elapsed_seconds": 223.15,
|
| 17 |
+
"paper2markdown_v3": {
|
| 18 |
+
"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491",
|
| 19 |
+
"pipeline": "Paper2Markdown-V3",
|
| 20 |
+
"ok": true,
|
| 21 |
+
"page_count": 16,
|
| 22 |
+
"chunk_count": 297,
|
| 23 |
+
"main_body_chunk_count": 107,
|
| 24 |
+
"appendix_chunk_count": 186,
|
| 25 |
+
"reference_chunk_count": 4,
|
| 26 |
+
"model_text_chars": 43413,
|
| 27 |
+
"raw_markdown_chars": 83365,
|
| 28 |
+
"sanitized_chars": 76947,
|
| 29 |
+
"page_provenance": {
|
| 30 |
+
"min_page": 1,
|
| 31 |
+
"max_page": 16,
|
| 32 |
+
"invalid_count": 0
|
| 33 |
+
},
|
| 34 |
+
"marker_block_type_counts": {
|
| 35 |
+
"Caption": 1,
|
| 36 |
+
"Code": 2,
|
| 37 |
+
"Equation": 58,
|
| 38 |
+
"Figure": 1,
|
| 39 |
+
"Footnote": 1,
|
| 40 |
+
"ListGroup": 18,
|
| 41 |
+
"PageFooter": 16,
|
| 42 |
+
"PageHeader": 18,
|
| 43 |
+
"SectionHeader": 48,
|
| 44 |
+
"Table": 2,
|
| 45 |
+
"TableGroup": 8,
|
| 46 |
+
"Text": 696
|
| 47 |
+
},
|
| 48 |
+
"asset_count_raw": 1,
|
| 49 |
+
"asset_count_model_kept": 1,
|
| 50 |
+
"asset_count_rejected": 0,
|
| 51 |
+
"asset_reject_reasons": {
|
| 52 |
+
"kept": 1
|
| 53 |
+
},
|
| 54 |
+
"artifact_leak_audit": {
|
| 55 |
+
"ok": true,
|
| 56 |
+
"hits": {
|
| 57 |
+
"Anonymous Authors": [],
|
| 58 |
+
"ACKNOWLEDGMENT": [],
|
| 59 |
+
"OpenReview": [],
|
| 60 |
+
"\"accept_label\"": [],
|
| 61 |
+
"\"decision\"": [],
|
| 62 |
+
"\"decision_tier\"": [],
|
| 63 |
+
"\"source_status\"": [],
|
| 64 |
+
"Meta-review": [],
|
| 65 |
+
"Official Review": [],
|
| 66 |
+
"official_reviews": [],
|
| 67 |
+
"meta_reviews": [],
|
| 68 |
+
"suggested_verdict_score": []
|
| 69 |
+
},
|
| 70 |
+
"artifact_count": 2
|
| 71 |
+
},
|
| 72 |
+
"default_model_input": "model_text_v3.txt",
|
| 73 |
+
"appendix_input": "appendix_text_v3.txt",
|
| 74 |
+
"reference_input": "reference_text_v3.txt"
|
| 75 |
+
}
|
| 76 |
+
}
|
icml26/5c3d6bff-e8ce-4d9f-840b-719084582491/reference_chunks.jsonl
ADDED
|
@@ -0,0 +1,4 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0107", "section": "References", "page_start": 9, "page_end": 9, "type": "ListGroup", "text": "Audibert, J.-Y. and Bubeck, S. Best arm identification in multi-armed bandits. In Conference on Learning Theory , pp. 41–53, 2010. Auer, P., Cesa-Bianchi, N., and Fischer, P. Finite-time analysis of the multiarmed bandit problem. Machine Learning , 47(2-3):235–256, 2002. Bergstra, J. and Bengio, Y. Random search for hyperparameter optimization. Journal of Machine Learning Research , 13(10):281–305, 2012. Brochu, E., Cora, V. M., and De Freitas, N. A tutorial on Bayesian optimization of expensive cost functions. arXiv preprint arXiv:1012.2599 , 2010. Bubeck, S., Munos, R., Stoltz, G., and Szepesvari, C. X- ´ armed bandits. In Journal of Machine Learning Research , volume 12, pp. 1655–1695, 2011. Cowen-Rivers, A. I., Lyu, W., Tutunov, R., Wang, Z., Grosnit, A., Griffiths, R. R., Maraval, A. M., Jianye, H., Wang, J., Peters, J., et al. HEBO: Pushing the limits of sampleefficient hyper-parameter optimisation. In Journal of Ar tificial Intelligence Research , volume 74, pp. 1269–1349, 2022. Daniel, J. R., Benjamin, V. R., Abbas, K., Ian, O., and Zheng, W. A tutorial on thompson sampling. Foundations and trends® in machine learning , 11(1):1–99, 2018. Eriksson, D., Pearce, M., Gardner, J., Turner, R. D., and Poloczek, M. Scalable global optimization via local bayesian optimization. Advances in neural information processing systems , 32, 2019. Fu, M. C. Handbook of simulation optimization . Springer, 2015. Garivier, A. and Kaufmann, E. Optimal best arm identification with fixed confidence. Conference on Learning Theory , pp. 998–1027, 2016. Hennig, P. and Schuler, C. J. Entropy search for informationefficient global optimization. Journal of Machine Learn ing Research , 13(57):1809–1837, 2012. Hernandez-Lobato, J. M., Hoffman, M. W., and Ghahra- ´ mani, Z. Predictive entropy search for efficient global optimization of black-box functions. In Advances in Neural Information Processing Systems , volume 27, pp. 918–926, 2014.", "source": "marker_v2", "marker_block_id": "/page/8/ListGroup/541"}
|
| 2 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0108", "section": "References", "page_start": 9, "page_end": 9, "type": "ListGroup", "text": "Hollmann, N., Muller, S., Eggensperger, K., and Hutter, F. ¨ Large-scale transfer learning for bayesian optimization with prior-data fitted networks. In Advances in Neural Information Processing Systems (NeurIPS) , 2023. Jamieson, K. and Nowak, R. Best-arm identification algorithms for multi-armed bandits in the fixed confidence setting. In Conference on Information Sciences and Sys tems , pp. 1–6. IEEE, 2014. Jones, D. R., Perttunen, C. D., and Stuckman, B. E. Lipschitzian optimization without the Lipschitz constant. Journal of Optimization Theory and Applications , 79(1): 157–181, 1993. Jones, D. R., Schonlau, M., and Welch, W. J. Efficient global optimization of expensive black-box functions. Journal of Global Optimization , 13(4):455–492, 1998. Kaufmann, E., Cappe, O., and Garivier, A. On the com- ´ plexity of best-arm identification in multi-armed bandit models. Journal of Machine Learning Research , 17:1–42, 2016. Kleinberg, R., Slivkins, A., and Upfal, E. Multi-armed bandits in metric spaces. In Proceedings of the 40th Annual ACM Symposium on Theory of Computing , pp. 681–690, 2008. Lattimore, T. and Szepesvari, C. Bandit algorithms. ´ Cam bridge University Press , 2020. Li, L., Jamieson, K., DeSalvo, G., Rostamizadeh, A., and Talwalkar, A. Hyperband: A novel bandit-based approach to hyperparameter optimization. Journal of Machine Learning Research , 18:1–52, 2018. Malherbe, C. and Vayatis, N. Global optimization of lipschitz functions. In International conference on machine learning , pp. 2314–2323. PMLR, 2017. Munos, R. Optimistic optimization of a deterministic function without the knowledge of its smoothness. Advances in Neural Information Processing Systems , 24:783–791, 2011. Munos, R. From bandits to Monte-Carlo tree search: The optimistic principle applied to optimization and planning. Foundations and Trends in Machine Learning , 7:1–129, 2014. Papenmeier, L., Nardi, L., and Poloczek, M. Bounce: Reliable high-dimensional bayesian optimization for combinatorial and mixed spaces. In International Conference on Learning Representations (ICLR) , 2024. Russo, D. and Van Roy, B. Learning to optimize via posterior sampling. Mathematics of Operations Research , 39: 1221–1243, 2014.", "source": "marker_v2", "marker_block_id": "/page/8/ListGroup/542"}
|
| 3 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0109", "section": "References", "page_start": 10, "page_end": 10, "type": "Text", "text": "Shihab, I. F., Akter, S., and Sharma, A. Detecting proxy gaming in rl and llm alignment via evaluator stress tests. arXiv preprint arXiv:2507.05619 , 2025a.", "source": "marker_v2", "marker_block_id": "/page/9/Text/508"}
|
| 4 |
+
{"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491", "chunk_id": "5c3d6bff-e8ce-4d9f-840b-719084582491:0110", "section": "References", "page_start": 10, "page_end": 10, "type": "ListGroup", "text": "Shihab, I. F., Akter, S., and Sharma, A. What fundamental structure in reward functions enables efficient sparsereward learning? arXiv preprint arXiv:2509.03790 , 2025b. Shihab, I. F., Akter, S., and Sharma, A. Universal adaptive constraint propagation: Scaling structured inference for large language models via meta-reinforcement learning. arXiv preprint arXiv:2601.00095, 2025c. Shihab, I. F., Akter, S., and Sharma, A. Beyond variance: Knowledge-aware llm compression via fisher-aligned subspace diagnostics. arXiv preprint arXiv:2601.07197 , 2026. Snoek, J., Larochelle, H., and Adams, R. P. Practical Bayesian optimization of machine learning algorithms. In Advances in Neural Information Processing Systems , volume 25, 2012. Srinivas, N., Krause, A., Kakade, S. M., and Seeger, M. Gaussian process optimization in the bandit setting: No regret and experimental design. arXiv preprint arXiv:0912.3995, 2009. Sui, Y., Gotovos, A., Burdick, J., and Krause, A. Safe exploration for optimization with gaussian processes. In International conference on machine learning , pp. 997–1005. PMLR, 2015. Thompson, W. R. On the likelihood that one unknown probability exceeds another in view of the evidence of two samples. Biometrika , 25(3/4):285–294, 1933. Valko, M., Carpentier, A., and Munos, R. Stochastic simultaneous optimistic optimization. In Proceedings of the 30th International Conference on Machine Learning , pp. 19–27, 2013. Wang, Z., Hutter, F., Zoghi, M., Matheson, D., and de Freitas, N. Bayesian optimization in a billion dimensions via random embeddings. Journal of Artificial Intelligence Research , 55:361–387, 2016. Wu, D., Kuang, N. L., Niu, R., Ma, Y., and Yu, R. Diffbbo: Diffusion-based inverse modeling for black-box optimization. 2024. Zoph, B. and Le, Q. V. Neural architecture search with reinforcement learning. In International Conference on Learning Representations , 2017.", "source": "marker_v2", "marker_block_id": "/page/9/ListGroup/507"}
|
icml26/5c3d6bff-e8ce-4d9f-840b-719084582491/reference_text_v3.txt
ADDED
|
@@ -0,0 +1,11 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
[p. 9 | section: References | type: ListGroup]
|
| 2 |
+
Audibert, J.-Y. and Bubeck, S. Best arm identification in multi-armed bandits. In Conference on Learning Theory , pp. 41–53, 2010. Auer, P., Cesa-Bianchi, N., and Fischer, P. Finite-time analysis of the multiarmed bandit problem. Machine Learning , 47(2-3):235–256, 2002. Bergstra, J. and Bengio, Y. Random search for hyperparameter optimization. Journal of Machine Learning Research , 13(10):281–305, 2012. Brochu, E., Cora, V. M., and De Freitas, N. A tutorial on Bayesian optimization of expensive cost functions. arXiv preprint arXiv:1012.2599 , 2010. Bubeck, S., Munos, R., Stoltz, G., and Szepesvari, C. X- ´ armed bandits. In Journal of Machine Learning Research , volume 12, pp. 1655–1695, 2011. Cowen-Rivers, A. I., Lyu, W., Tutunov, R., Wang, Z., Grosnit, A., Griffiths, R. R., Maraval, A. M., Jianye, H., Wang, J., Peters, J., et al. HEBO: Pushing the limits of sampleefficient hyper-parameter optimisation. In Journal of Ar tificial Intelligence Research , volume 74, pp. 1269–1349, 2022. Daniel, J. R., Benjamin, V. R., Abbas, K., Ian, O., and Zheng, W. A tutorial on thompson sampling. Foundations and trends® in machine learning , 11(1):1–99, 2018. Eriksson, D., Pearce, M., Gardner, J., Turner, R. D., and Poloczek, M. Scalable global optimization via local bayesian optimization. Advances in neural information processing systems , 32, 2019. Fu, M. C. Handbook of simulation optimization . Springer, 2015. Garivier, A. and Kaufmann, E. Optimal best arm identification with fixed confidence. Conference on Learning Theory , pp. 998–1027, 2016. Hennig, P. and Schuler, C. J. Entropy search for informationefficient global optimization. Journal of Machine Learn ing Research , 13(57):1809–1837, 2012. Hernandez-Lobato, J. M., Hoffman, M. W., and Ghahra- ´ mani, Z. Predictive entropy search for efficient global optimization of black-box functions. In Advances in Neural Information Processing Systems , volume 27, pp. 918–926, 2014.
|
| 3 |
+
|
| 4 |
+
[p. 9 | section: References | type: ListGroup]
|
| 5 |
+
Hollmann, N., Muller, S., Eggensperger, K., and Hutter, F. ¨ Large-scale transfer learning for bayesian optimization with prior-data fitted networks. In Advances in Neural Information Processing Systems (NeurIPS) , 2023. Jamieson, K. and Nowak, R. Best-arm identification algorithms for multi-armed bandits in the fixed confidence setting. In Conference on Information Sciences and Sys tems , pp. 1–6. IEEE, 2014. Jones, D. R., Perttunen, C. D., and Stuckman, B. E. Lipschitzian optimization without the Lipschitz constant. Journal of Optimization Theory and Applications , 79(1): 157–181, 1993. Jones, D. R., Schonlau, M., and Welch, W. J. Efficient global optimization of expensive black-box functions. Journal of Global Optimization , 13(4):455–492, 1998. Kaufmann, E., Cappe, O., and Garivier, A. On the com- ´ plexity of best-arm identification in multi-armed bandit models. Journal of Machine Learning Research , 17:1–42, 2016. Kleinberg, R., Slivkins, A., and Upfal, E. Multi-armed bandits in metric spaces. In Proceedings of the 40th Annual ACM Symposium on Theory of Computing , pp. 681–690, 2008. Lattimore, T. and Szepesvari, C. Bandit algorithms. ´ Cam bridge University Press , 2020. Li, L., Jamieson, K., DeSalvo, G., Rostamizadeh, A., and Talwalkar, A. Hyperband: A novel bandit-based approach to hyperparameter optimization. Journal of Machine Learning Research , 18:1–52, 2018. Malherbe, C. and Vayatis, N. Global optimization of lipschitz functions. In International conference on machine learning , pp. 2314–2323. PMLR, 2017. Munos, R. Optimistic optimization of a deterministic function without the knowledge of its smoothness. Advances in Neural Information Processing Systems , 24:783–791, 2011. Munos, R. From bandits to Monte-Carlo tree search: The optimistic principle applied to optimization and planning. Foundations and Trends in Machine Learning , 7:1–129, 2014. Papenmeier, L., Nardi, L., and Poloczek, M. Bounce: Reliable high-dimensional bayesian optimization for combinatorial and mixed spaces. In International Conference on Learning Representations (ICLR) , 2024. Russo, D. and Van Roy, B. Learning to optimize via posterior sampling. Mathematics of Operations Research , 39: 1221–1243, 2014.
|
| 6 |
+
|
| 7 |
+
[p. 10 | section: References | type: Text]
|
| 8 |
+
Shihab, I. F., Akter, S., and Sharma, A. Detecting proxy gaming in rl and llm alignment via evaluator stress tests. arXiv preprint arXiv:2507.05619 , 2025a.
|
| 9 |
+
|
| 10 |
+
[p. 10 | section: References | type: ListGroup]
|
| 11 |
+
Shihab, I. F., Akter, S., and Sharma, A. What fundamental structure in reward functions enables efficient sparsereward learning? arXiv preprint arXiv:2509.03790 , 2025b. Shihab, I. F., Akter, S., and Sharma, A. Universal adaptive constraint propagation: Scaling structured inference for large language models via meta-reinforcement learning. arXiv preprint arXiv:2601.00095, 2025c. Shihab, I. F., Akter, S., and Sharma, A. Beyond variance: Knowledge-aware llm compression via fisher-aligned subspace diagnostics. arXiv preprint arXiv:2601.07197 , 2026. Snoek, J., Larochelle, H., and Adams, R. P. Practical Bayesian optimization of machine learning algorithms. In Advances in Neural Information Processing Systems , volume 25, 2012. Srinivas, N., Krause, A., Kakade, S. M., and Seeger, M. Gaussian process optimization in the bandit setting: No regret and experimental design. arXiv preprint arXiv:0912.3995, 2009. Sui, Y., Gotovos, A., Burdick, J., and Krause, A. Safe exploration for optimization with gaussian processes. In International conference on machine learning , pp. 997–1005. PMLR, 2015. Thompson, W. R. On the likelihood that one unknown probability exceeds another in view of the evidence of two samples. Biometrika , 25(3/4):285–294, 1933. Valko, M., Carpentier, A., and Munos, R. Stochastic simultaneous optimistic optimization. In Proceedings of the 30th International Conference on Machine Learning , pp. 19–27, 2013. Wang, Z., Hutter, F., Zoghi, M., Matheson, D., and de Freitas, N. Bayesian optimization in a billion dimensions via random embeddings. Journal of Artificial Intelligence Research , 55:361–387, 2016. Wu, D., Kuang, N. L., Niu, R., Ma, Y., and Yu, R. Diffbbo: Diffusion-based inverse modeling for black-box optimization. 2024. Zoph, B. and Le, Q. V. Neural architecture search with reinforcement learning. In International Conference on Learning Representations , 2017.
|
icml26/5c3d6bff-e8ce-4d9f-840b-719084582491/sanitization_report.json
ADDED
|
@@ -0,0 +1,59 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
{
|
| 2 |
+
"paper_id": "5c3d6bff-e8ce-4d9f-840b-719084582491",
|
| 3 |
+
"pipeline": "Paper2Markdown-V3",
|
| 4 |
+
"ok": true,
|
| 5 |
+
"page_count": 16,
|
| 6 |
+
"chunk_count": 297,
|
| 7 |
+
"main_body_chunk_count": 107,
|
| 8 |
+
"appendix_chunk_count": 186,
|
| 9 |
+
"reference_chunk_count": 4,
|
| 10 |
+
"model_text_chars": 43413,
|
| 11 |
+
"raw_markdown_chars": 83365,
|
| 12 |
+
"sanitized_chars": 76947,
|
| 13 |
+
"page_provenance": {
|
| 14 |
+
"min_page": 1,
|
| 15 |
+
"max_page": 16,
|
| 16 |
+
"invalid_count": 0
|
| 17 |
+
},
|
| 18 |
+
"marker_block_type_counts": {
|
| 19 |
+
"Caption": 1,
|
| 20 |
+
"Code": 2,
|
| 21 |
+
"Equation": 58,
|
| 22 |
+
"Figure": 1,
|
| 23 |
+
"Footnote": 1,
|
| 24 |
+
"ListGroup": 18,
|
| 25 |
+
"PageFooter": 16,
|
| 26 |
+
"PageHeader": 18,
|
| 27 |
+
"SectionHeader": 48,
|
| 28 |
+
"Table": 2,
|
| 29 |
+
"TableGroup": 8,
|
| 30 |
+
"Text": 696
|
| 31 |
+
},
|
| 32 |
+
"asset_count_raw": 1,
|
| 33 |
+
"asset_count_model_kept": 1,
|
| 34 |
+
"asset_count_rejected": 0,
|
| 35 |
+
"asset_reject_reasons": {
|
| 36 |
+
"kept": 1
|
| 37 |
+
},
|
| 38 |
+
"artifact_leak_audit": {
|
| 39 |
+
"ok": true,
|
| 40 |
+
"hits": {
|
| 41 |
+
"Anonymous Authors": [],
|
| 42 |
+
"ACKNOWLEDGMENT": [],
|
| 43 |
+
"OpenReview": [],
|
| 44 |
+
"\"accept_label\"": [],
|
| 45 |
+
"\"decision\"": [],
|
| 46 |
+
"\"decision_tier\"": [],
|
| 47 |
+
"\"source_status\"": [],
|
| 48 |
+
"Meta-review": [],
|
| 49 |
+
"Official Review": [],
|
| 50 |
+
"official_reviews": [],
|
| 51 |
+
"meta_reviews": [],
|
| 52 |
+
"suggested_verdict_score": []
|
| 53 |
+
},
|
| 54 |
+
"artifact_count": 2
|
| 55 |
+
},
|
| 56 |
+
"default_model_input": "model_text_v3.txt",
|
| 57 |
+
"appendix_input": "appendix_text_v3.txt",
|
| 58 |
+
"reference_input": "reference_text_v3.txt"
|
| 59 |
+
}
|
icml26/5c3d6bff-e8ce-4d9f-840b-719084582491/sanitized_v3.txt
ADDED
|
@@ -0,0 +1,611 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
{0}
|
| 2 |
+
#### **Abstract**
|
| 3 |
+
We study black-box optimization of Lipschitz functions under noisy evaluations. Existing adaptive discretization methods implicitly avoid suboptimal regions but do not provide explicit certificates of optimality or measurable progress guarantees. We introduce Certificate-Guided Pruning (CGP), which maintains an explicit active set $A_t$ of potentially optimal points via confidence-adjusted Lipschitz envelopes. Any point outside $A_t$ is certifiably suboptimal with high probability, and under a margin condition with near-optimality dimension $\alpha$ , we prove $Vol(A_t)$ shrinks at a controlled rate yielding sample complexity $\tilde{O}(\varepsilon^{-(2+\alpha)})$ . We develop three extensions: CGP-Adaptive learns L online with $O(\log T)$ overhead; CGP-TR scales to d > 50via trust regions with local certificates; and CGP-Hybrid switches to GP refinement when local smoothness is detected. Experiments on 12 benchmarks $(d \in [2, 100])$ show CGP variants match or exceed strong baselines while providing principled stopping criteria via certificate volume.
|
| 4 |
+
#### 1. Introduction
|
| 5 |
+
Black-box optimization, the task of finding the maximum of a function $f:\mathcal{X}\to\mathbb{R}$ accessible only through noisy point evaluations, is fundamental to machine learning, with applications spanning hyperparameter tuning (Snoek et al., 2012; Bergstra & Bengio, 2012), neural architecture search (Zoph & Le, 2017), and simulation-based optimization (Fu, 2015; Brochu et al., 2010). In many such settings, evaluations are expensive: training a neural network or running a physical simulation may cost hours or dollars per query. We call these "precious calls" where each evaluation must count, motivating the need for methods that provide explicit progress guarantees.
|
| 6 |
+
To address these limitations, we introduce Certificate-Guided Pruning (CGP), which maintains an explicit active set $A_t \subseteq \mathcal{X}$ of potentially optimal points. This set is defined via a Lipschitz UCB envelope $U_t(x)$ that upper bounds f(x)with high probability, a global lower certificate $\ell_t$ that lower bounds $f(x^*)$ , and the active set $A_t = \{x : U_t(x) \ge \ell_t\}$ . Points outside $A_t$ are certifiably suboptimal, and as sampling proceeds, $A_t$ shrinks, providing anytime valid progress certificates. Unlike prior work that uses similar mathematical tools implicitly, CGP exposes the pruning mechanism as a first-class algorithmic object: the certificate is computable in closed form, the shrinkage rate is provably controlled, and the certificate provides valid optimality bounds even when stopped early. Figure 1 illustrates this mechanism. To understand how CGP differs from existing approaches, consider zooming algorithms (Kleinberg et al., 2008; Bubeck et al., 2011). While zooming maintains a tree of "active arms" and expands nodes with high UCB, the implicit pruning is an analysis artifact not exposed to the user (Shihab et al., 2026). Table 1 makes this distinction precise: CGP provides explicit certificates, computable progress metrics, and principled stopping rules that zooming-based methods lack. Similarly, Thompson sampling (Thompson, 1933; Russo & Van Roy, 2014; Daniel et al., 2018) and information-directed methods (Hennig & Schuler, 2012; Hernández-Lobato et al., 2014) maintain implicit uncertainty without providing explicit geometric certificates.
|
| 7 |
+
Building on this foundation, our contributions are fourfold. First, we present CGP with explicit active set maintenance
|
| 8 |
+
{1}------------------------------------------------
|
| 9 |
+
<span id="page-1-0"></span>
|
| 10 |
+
Figure 1. The active set $A_t$ (shaded) consists of points where the Lipschitz envelope $U_t(x)$ (red) exceeds the global lower bound $\ell_t$ (green dashed). Regions where $U_t(x) < \ell_t$ are certifiably suboptimal and pruned, causing $A_t$ to shrink as sampling proceeds.
|
| 11 |
+
<span id="page-1-1"></span>Table 1. Comparison with zooming-based bandits. CGP uniquely exports explicit certificates (active set $A_t$ , gap bound $\varepsilon_t$ , and volume $\operatorname{Vol}(A_t)$ ), enabling principled stopping criteria that implicit zooming methods lack.
|
| 12 |
+
| Property | CGP | Zooming | HOO/StoSOO |
|
| 13 |
+
|---------------------------|----------------------------------------|----------------------------------------|----------------------------------------------------|
|
| 14 |
+
| Explicit active set $A_t$ | ✓ | _ | _ |
|
| 15 |
+
| Computable $Vol(A_t)$ | ✓ | _ | - |
|
| 16 |
+
| Anytime optimality bound | ✓ | _ | _ |
|
| 17 |
+
| Principled stopping rule | ✓ | _ | _ |
|
| 18 |
+
| Certificate export | ✓ | _ | _ |
|
| 19 |
+
| Sample complexity | $\tilde{O}(\varepsilon^{-(2+\alpha)})$ | $\tilde{O}(\varepsilon^{-(2+\alpha)})$ | $\tilde{O}(\varepsilon^{-(2+d)})$<br>$O(\log N_t)$ |
|
| 20 |
+
| Per-iteration cost | $O(N_t)$ | $O(\log N_t)$ | $O(\log N_t)$ |
|
| 21 |
+
| Adaptive L | √(CGP-A) | _ | _ |
|
| 22 |
+
| High-dim scaling | √(CGP-TR) | - | |
|
| 23 |
+
and prove a shrinkage theorem: under a margin condition with near-optimality dimension $\alpha$ (i.e., Vol( $\{x: f(x) \geq$ $f^* - \varepsilon$ ) $\leq C\varepsilon^{d-\alpha}$ , we show $Vol(A_t) \leq C \cdot (2(\beta_t + \beta_t))$ $(L\eta_t)^{d-\alpha}$ , yielding sample complexity $T = \tilde{O}(\varepsilon^{-(2+\alpha)})$ that improves on the worst case $\tilde{O}(\varepsilon^{-(2+d)})$ when $\alpha < d$ (Section 4). Second, we develop CGP-Adaptive (Section 5), which learns L online via a doubling scheme, proving that unknown L adds only $O(\log T)$ multiplicative overhead, the first such guarantee for Lipschitz optimization with certificates. Third, we introduce CGP-TR (Section 6), a trust region variant that scales to d > 50 by maintaining local certificates within adaptively sized regions, enabling high-dimensional applications previously intractable for Lipschitz methods. Fourth, we propose CGP-Hybrid (Section 7), which detects local smoothness via the ratio $\rho = L_{\rm local}/L_{\rm global}$ and switches to GP refinement when $\rho < 0.5$ , achieving best of both worlds performance across diverse function classes.
|
| 24 |
+
These theoretical contributions translate to strong empirical performance. Experiments (Section 9) demonstrate that CGP variants are competitive with strong baselines on 12 benchmarks spanning $d \in [2, 100]$ , including Rover tra-
|
| 25 |
+
*Table 2.* Summary of Notation
|
| 26 |
+
<span id="page-1-4"></span>
|
| 27 |
+
| Symbol | Description |
|
| 28 |
+
|------------|-----------------------------------------------------------|
|
| 29 |
+
| $f^*$ | Global maximum value of the objective function |
|
| 30 |
+
| L | Lipschitz constant (or global upper bound) |
|
| 31 |
+
| $A_t$ | Active set at time $t$ (contains potential optimizers) |
|
| 32 |
+
| $U_t(x)$ | Lipschitz Upper Confidence Bound envelope |
|
| 33 |
+
| $\ell_t$ | Global lower certificate $(\max_i LCB_i)$ |
|
| 34 |
+
| $\alpha$ | Near-optimality dimension (problem hardness) |
|
| 35 |
+
| $\beta_t$ | Active confidence radius (uncertainty in $A_t$ ) |
|
| 36 |
+
| $\eta_t$ | Covering radius (resolution of $A_t$ ) |
|
| 37 |
+
| $\gamma_t$ | Gap to optimum proxy $(f^* - \ell_t)$ |
|
| 38 |
+
| ρ | Local smoothness ratio ( $L_{\rm local}/L_{\rm global}$ ) |
|
| 39 |
+
jectory optimization (d=60), neural architecture search (d=36), and safe robotics where certificates enable stopping with guaranteed bounds (Shihab et al., 2025a). CGP-Hybrid performs best among tested methods on all 12 benchmarks under matched budgets, including Branin and Rosenbrock where vanilla CGP previously lost to GP-based methods.
|
| 40 |
+
#### 2. Problem Formulation
|
| 41 |
+
Let $(\mathcal{X}, d)$ be a compact metric space with diameter $D = \sup_{x,y} d(x,y)$ . We consider $\mathcal{X} = [0,1]^d$ with Euclidean metric. Let $f: \mathcal{X} \to [0,1]$ satisfy:
|
| 42 |
+
<span id="page-1-2"></span>**Assumption 2.1** (Lipschitz continuity). There exists L > 0 such that for all $x, y \in \mathcal{X}$ : $|f(x) - f(y)| \le L \cdot d(x, y)$ .
|
| 43 |
+
We observe f through noisy queries: querying x returns $y = f(x) + \epsilon$ , where:
|
| 44 |
+
<span id="page-1-5"></span>**Assumption 2.2** (Sub-Gaussian noise). The noise $\epsilon$ is $\sigma$ -sub-Gaussian: $\mathbb{E}[e^{\lambda\epsilon}] \leq e^{\lambda^2 \sigma^2/2}$ for all $\lambda \in \mathbb{R}$ .
|
| 45 |
+
After T samples, the algorithm outputs $\hat{x}_T \in \mathcal{X}$ . The goal is to minimize simple regret $r_T = f(x^*) - f(\hat{x}_T)$ . We seek PAC guarantees: $r_T \leq \varepsilon$ with probability $\geq 1 - \delta$ .
|
| 46 |
+
<span id="page-1-3"></span>**Assumption 2.3** (Margin / near-optimality dimension). There exist C>0 and $\alpha\in[0,d]$ such that for all $\varepsilon>0$ : $\operatorname{Vol}(\{x\in\mathcal{X}:f(x)\geq f^*-\varepsilon\})\leq C\varepsilon^{d-\alpha}$ .
|
| 47 |
+
The parameter $\alpha$ is the near-optimality dimension: smaller $\alpha$ means a sharper optimum (easier), $\alpha=d$ is worst case. For isolated maxima with nondegenerate Hessian, $\alpha=d/2$ ; for $f(x)\approx f^*-c\|x-x^*\|^p$ , we have $\alpha=d/p$ . This assumption is standard in bandit theory (Audibert & Bubeck, 2010; Bubeck et al., 2011; Lattimore & Szepesvári, 2020); see Appendix A for extended discussion.
|
| 48 |
+
#### 3. Algorithm: Certificate-Guided Pruning
|
| 49 |
+
With the problem formalized, we now describe the CGP algorithm (Algorithm 1). CGP maintains sampled points
|
| 50 |
+
{2}------------------------------------------------
|
| 51 |
+
#### <span id="page-2-1"></span>Algorithm 1 Certificate-Guided Pruning (CGP)
|
| 52 |
+
**Require:** Domain $\mathcal{X}$ , Lipschitz constant L, noise $\sigma$ , budget T, confidence $\delta$
|
| 53 |
+
- 1: Initialize: Sample $x_1$ uniformly, observe $y_1$
|
| 54 |
+
- 2: **for** t = 1, ..., T 1 **do**
|
| 55 |
+
- 3: Compute $r_i(t)$ , $\ell_t = \max_i LCB_i(t)$ , $A_t = \{x : U_t(x) \ge \ell_t\}$
|
| 56 |
+
- 4: $x_{t+1} \leftarrow \arg\max_{x \in A_t} [U_t(x) L \cdot \min_i d(x, x_i)]$
|
| 57 |
+
- 5: Query $x_{t+1}$ , observe $y_{t+1}$ , update statistics
|
| 58 |
+
- 6: Replicate active points with $r_i(t) > \beta_{\text{target}}(t)$
|
| 59 |
+
- 7: end for
|
| 60 |
+
8: Output: $\hat{x}_T = \arg \max_i \hat{\mu}_i(T)$ , certificate $A_T$
|
| 61 |
+
with empirical estimates, confidence intervals, and the active set. At time t, let $\{x_1,\ldots,x_{N_t}\}$ be distinct points sampled, with $n_i$ observations at $x_i$ . Define empirical mean $\hat{\mu}_i(t) = \frac{1}{n_i} \sum_{j=1}^{n_i} y_{i,j}$ and confidence radius $r_i(t) = \sigma \sqrt{2\log(2N_tT/\delta)}/n_i$ , ensuring $|f(x_i) - \hat{\mu}_i(t)| \leq r_i(t)$ with high probability.
|
| 62 |
+
The upper and lower confidence bounds are $UCB_i(t) = \hat{\mu}_i(t) + r_i(t)$ and $LCB_i(t) = \hat{\mu}_i(t) - r_i(t)$ . The global lower certificate $\ell_t = \max_{i \leq N_t} LCB_i(t)$ satisfies $\ell_t \leq f(x^*)$ under the good event. The Lipschitz UCB envelope propagates uncertainty:
|
| 63 |
+
$$U_t(x) = \min_{i \le N_t} \left\{ \text{UCB}_i(t) + L \cdot d(x, x_i) \right\}, \tag{1}$$
|
| 64 |
+
which upper-bounds f(x) everywhere. The active set is
|
| 65 |
+
<span id="page-2-7"></span>
|
| 66 |
+
$$A_t = \left\{ x \in \mathcal{X} : U_t(x) \ge \ell_t \right\},\tag{2}$$
|
| 67 |
+
and points outside $A_t$ are certifiably suboptimal: their upper bound is below the lower bound on $f^*$ .
|
| 68 |
+
The algorithm selects queries via $\mathrm{score}(x) = U_t(x) - \lambda \cdot \min_{i \leq N_t} d(x, x_i)$ where $\lambda = L$ , selecting $x_{t+1} = \arg\max_{x \in A_t} \mathrm{score}(x)$ . The first term favors high UCB regions while the second encourages coverage. CGP allocates additional samples to active points with $r_i(t) > \beta_{\mathrm{target}}(t)$ to reduce confidence radii. The target confidence radius follows a schedule $\beta_{\mathrm{target}}(t) = \sigma \sqrt{2\log(2T^2/\delta)/t}$ , ensuring that confidence radii decrease at rate $O(1/\sqrt{t})$ .
|
| 69 |
+
We compute $A_t$ via discretization for low dimensions and Monte Carlo sampling for d>5 (details in Appendix B.2). Theoretically, CGP assumes oracle access to $\arg\max_{x\in A_t}\mathrm{score}(x)$ ; practically, we use CMA-ES with 10 random restarts within $A_t$ (see Appendix B for details). For $d\leq 3$ , we additionally use Delaunay triangulation to identify candidate optima at Voronoi vertices. Membership in $A_t$ is exact: checking $U_t(x)\geq \ell_t$ requires $O(N_t)$ time. Approximate maximization may slow convergence of $\eta_t$ but does not invalidate certificates: any $x\notin A_t$ remains certifiably suboptimal regardless of which $x\in A_t$ is queried.
|
| 70 |
+
**Replication Strategy.** When an active point $x_i$ has $r_i(t) > \beta_{\text{target}}(t)$ , we allocate $\lceil (r_i(t)/\beta_{\text{target}}(t))^2 \rceil$ additional samples to reduce its confidence radius. This ensures all active points have comparable confidence, preventing any single point from dominating the envelope.
|
| 71 |
+
#### <span id="page-2-0"></span>4. Theoretical Analysis
|
| 72 |
+
Having described the algorithm, we now establish its theoretical guarantees. We show that the active set is contained in the near-optimal set, its volume shrinks at a controlled rate, and this yields instance-dependent sample complexity. All results hold on the good event $\mathcal E$ where $|f(x_i) - \hat{\mu}_i(t)| \leq r_i(t)$ for all t,i. All proofs are deferred to Appendix C.
|
| 73 |
+
<span id="page-2-5"></span>**Lemma 4.1** (Good event). With $r_i(t) = \sigma \sqrt{2 \log(2N_t T/\delta)/n_i}$ , we have $\mathbb{P}[\mathcal{E}] \geq 1 - \delta$ .
|
| 74 |
+
<span id="page-2-6"></span>**Lemma 4.2** (UCB envelope is valid). On $\mathcal{E}$ , for all $x \in \mathcal{X}$ : $f(x) \leq U_t(x)$ .
|
| 75 |
+
Proof sketch. For any sampled $x_i$ , on $\mathcal{E}$ : $f(x_i) \leq \mathrm{UCB}_i(t)$ . By Lipschitz continuity: $f(x) \leq f(x_i) + L \cdot d(x, x_i) \leq \mathrm{UCB}_i(t) + L \cdot d(x, x_i)$ . Taking min over i gives $f(x) \leq U_t(x)$ .
|
| 76 |
+
<span id="page-2-2"></span>**Lemma 4.3** (Envelope slack bound). On $\mathcal{E}$ , for all $x \in \mathcal{X}$ : $U_t(x) \leq f(x) + 2\rho_t(x)$ , where $\rho_t(x) = \min_i \{r_i(t) + L \cdot d(x, x_i)\}$ .
|
| 77 |
+
Remark 4.4 (Slack in envelope bound). The factor of 2 arises from applying Lipschitz continuity $(f(x_i) \leq f(x) + L \cdot d(x, x_i))$ after bounding $UCB_i \leq f(x_i) + 2r_i$ . A tighter bound separating the confidence and distance terms is possible but complicates notation without affecting rate dependencies. Constants throughout are not optimized.
|
| 78 |
+
<span id="page-2-3"></span>**Theorem 4.5** (Active set containment). On $\mathcal{E}$ , $A_t \subseteq \{x : f(x) \ge f^* - 2\Delta_t\}$ where $\Delta_t = \sup_{x \in A_t} \rho_t(x) + (f^* - \ell_t)$ .
|
| 79 |
+
Proof sketch. On $\mathcal{E}$ , $\ell_t = \max_i \mathrm{LCB}_i(t) \leq f^*$ since each $\mathrm{LCB}_i(t) \leq f(x_i) \leq f^*$ . For $x \in A_t$ , by definition $U_t(x) \geq \ell_t$ . Applying Lemma 4.3: $f(x) + 2\rho_t(x) \geq U_t(x) \geq \ell_t$ . Rearranging: $f(x) \geq \ell_t - 2\rho_t(x) = f^* - (f^* - \ell_t) - 2\rho_t(x) \geq f^* - 2\Delta_t$ .
|
| 80 |
+
The containment theorem bounds how far active points can be from optimal. To translate this into a volume bound, we introduce two key quantities: the covering radius $\eta_t = \sup_{x \in A_t} \min_i d(x, x_i)$ measuring how well samples cover $A_t$ , and the active confidence radius $\beta_t = \max_{i:x_i \text{ active }} r_i(t)$ measuring confidence precision.
|
| 81 |
+
<span id="page-2-4"></span>**Theorem 4.6** (Shrinkage theorem). *Under Assumptions 2.1–2.3, on \mathcal{E}:*
|
| 82 |
+
$$Vol(A_t) \le C \cdot \left(2(\beta_t + L\eta_t) + \gamma_t\right)^{d-\alpha},\tag{3}$$
|
| 83 |
+
{3}------------------------------------------------
|
| 84 |
+
```
|
| 85 |
+
where \gamma_t = f^* - \ell_t.
|
| 86 |
+
```
|
| 87 |
+
205206
|
| 88 |
+
<span id="page-3-3"></span>219
|
| 89 |
+
Proof sketch. From Theorem 4.5, $A_t \subseteq \{x: f(x) \ge f^* - 2\Delta_t\}$ . For $x \in A_t$ , $\rho_t(x) \le \beta_t + L\eta_t$ (the worst-case slack from active-point confidence plus covering distance), so $\Delta_t \le \beta_t + L\eta_t + \gamma_t/2$ . Applying Assumption 2.3 with $\varepsilon = 2\Delta_t$ : $\operatorname{Vol}(A_t) \le C(2\Delta_t)^{d-\alpha} \le C(2(\beta_t + L\eta_t) + \gamma_t)^{d-\alpha}$ .
|
| 90 |
+
This makes pruning measurable: $\beta_t$ is controlled by replication, $\eta_t$ by the query rule, $\gamma_t$ by best-point improvement. All three quantities can be computed during the run, enabling practitioners to monitor progress.
|
| 91 |
+
<span id="page-3-4"></span>Remark 4.7 (Certificate validity vs. progress estimation). The certificate itself is the set membership rule $x \in A_t \Leftrightarrow U_t(x) \geq \ell_t$ , which is exact given $(\hat{\mu}_i, r_i)$ and does not depend on any volume estimator. Approximations (grid/Monte Carlo) are used only to estimate $\operatorname{Vol}(A_t)$ for monitoring and optional stopping heuristics; certificate validity is unaffected by volume estimation errors.
|
| 92 |
+
The shrinkage theorem directly yields sample complexity by bounding how many samples are needed to drive $\beta_t$ , $\eta_t$ , and $\gamma_t$ below $\varepsilon$ .
|
| 93 |
+
<span id="page-3-5"></span>**Theorem 4.8** (Sample complexity). Under Assumptions 2.1–2.3, CGP achieves $r_T \le \varepsilon$ with probability $\ge 1 - \delta$ using $T = \tilde{O}(L^d \varepsilon^{-(2+\alpha)} \log(1/\delta))$ samples. When $\alpha < d$ , this improves upon the worst-case $\tilde{O}(\varepsilon^{-(2+d)})$ rate.
|
| 94 |
+
The following lower bound shows that our sample complexity is optimal up to logarithmic factors.
|
| 95 |
+
<span id="page-3-6"></span>**Theorem 4.9** (Lower bound). For any algorithm and $\alpha \in (0, d]$ , there exists f satisfying Assumption 2.3 requiring $T = \Omega(\varepsilon^{-(2+\alpha)})$ samples for $\varepsilon$ -optimality with probability $\geq 2/3$ .
|
| 96 |
+
This establishes CGP is minimax optimal up to logarithmic factors. A key property is anytime validity: at any t, any $x \notin A_t$ satisfies $f(x) < f^* - \varepsilon_t$ for computable $\varepsilon_t > 0$ . Full proofs are in Appendix C.
|
| 97 |
+
#### <span id="page-3-0"></span>**5.** CGP-Adaptive: Learning *L* Online
|
| 98 |
+
The theoretical results above assume known L, which is often unavailable in practice. Underestimating L invalidates certificates, while overestimating is safe but conservative. We develop CGP-Adaptive (Algorithm 2), which learns L online via a doubling scheme with provable guarantees.
|
| 99 |
+
The key insight is that Lipschitz violations are detectable. If $|\hat{\mu}_i - \hat{\mu}_j| - 2(r_i + r_j) > \hat{L} \cdot d(x_i, x_j)$ , then $\hat{L}$ underestimates L with high probability. CGP-Adaptive uses a doubling scheme: start with conservative $\hat{L}_0$ , and upon detecting a violation, double $\hat{L}$ .
|
| 100 |
+
#### <span id="page-3-2"></span>Algorithm 2 CGP-Adaptive
|
| 101 |
+
```
|
| 102 |
+
Require: Domain \mathcal{X}, initial estimate \hat{L}_0, noise \sigma, budget
|
| 103 |
+
T, confidence \delta
|
| 104 |
+
1: \hat{L} \leftarrow \hat{L}_0, k \leftarrow 0 (doubling counter)
|
| 105 |
+
2: for t = 1, ..., T do
|
| 106 |
+
Run CGP iteration with current \hat{L}
|
| 107 |
+
for all pairs (i, j) with n_i, n_j \ge \log(T/\delta) do
|
| 108 |
+
4:
|
| 109 |
+
5:
|
| 110 |
+
if |\hat{\mu}_i - \hat{\mu}_j| - 2(r_i + r_j) > \hat{L} \cdot d(x_i, x_j) then
|
| 111 |
+
\hat{L} \leftarrow 2\hat{L}, k \leftarrow k + 1 {Doubling event}
|
| 112 |
+
6:
|
| 113 |
+
Recompute A_t with new \hat{L}
|
| 114 |
+
7:
|
| 115 |
+
8:
|
| 116 |
+
end if
|
| 117 |
+
end for
|
| 118 |
+
10: end for
|
| 119 |
+
```
|
| 120 |
+
**Theorem 5.1** (Adaptive L guarantee: learning regime). Let $L^* = \sup_{x \neq y} |f(x) - f(y)|/d(x,y)$ be the true Lipschitz constant. CGP-Adaptive with initial $\hat{L}_0 \leq L^*$ (learning from underestimation) satisfies:
|
| 121 |
+
- 1. The number of doubling events is at most $K = \lceil \log_2(L^*/\hat{L}_0) \rceil$ .
|
| 122 |
+
- 2. After all doublings, $\hat{L} \in [L^*, 2L^*]$ with probability $\geq 1 \delta$ .
|
| 123 |
+
- 3. The total sample complexity is $T = \tilde{O}(\varepsilon^{-(2+\alpha)} \cdot K)$ , i.e., $O(\log(L^*/\hat{L}_0))$ multiplicative overhead.
|
| 124 |
+
- 4. Certificate validity: Certificates are valid only after the final doubling (when $\hat{L} \geq L^*$ ). Before this, certificates may falsely exclude near-optimal points.
|
| 125 |
+
Remark 5.2 (Anytime-valid certificates). For applications requiring certificates valid at all times, use $\hat{L}_0 \geq L^*$ (conservative overestimate). This ensures $\hat{L} \geq L^*$ throughout, so all certificates are valid, but may be overly conservative. One can optionally decrease $\hat{L}$ when evidence suggests overestimation, but this requires different analysis than the doubling scheme above.
|
| 126 |
+
This is the first provably correct adaptive L estimation for Lipschitz optimization with certificates. Prior work (Malherbe & Vayatis, 2017) estimates L but without guarantees on certificate validity. Table 5 shows CGP-Adaptive matches oracle performance (known L) within 8% while being robust to $100\times$ underestimation of initial $\hat{L}_0$ .
|
| 127 |
+
# <span id="page-3-1"></span>6. CGP-TR: Trust Regions for High Dimensions
|
| 128 |
+
CGP-Adaptive addresses the unknown L problem, but another challenge remains: scalability. The covering number of $A_t$ grows as $O(\eta^{-d})$ , making CGP intractable for d > 15.
|
| 129 |
+
{4}------------------------------------------------
|
| 130 |
+
Table 3. Simple regret ( $\times 10^{-2}$ ) at T = 200. Bold: best; †: significant vs second-best.
|
| 131 |
+
<span id="page-4-2"></span>
|
| 132 |
+
| Method | Needle | Branin | Hartmann | Ackley | Levy | Rosen. | SVM |
|
| 133 |
+
|--------|-----------------|-----------------|-----------------|-----------------|-----------------|-----------------|-----------------|
|
| 134 |
+
| Random | 8.2 | 12.1 | 15.3 | 22.4 | 18.7 | 14.2 | 11.2 |
|
| 135 |
+
| GP-UCB | 2.1 | 1.8 | 4.2 | 12.3 | 5.1 | 4.8 | 3.9 |
|
| 136 |
+
| TuRBO | 1.8 | 2.1 | 3.1 | 9.8 | 4.3 | 3.9 | 3.2 |
|
| 137 |
+
| HEBO | 1.9 | 1.6 | 3.3 | 9.4 | 4.1 | 3.7 | 3.1 |
|
| 138 |
+
| BORE | 2.0 | 1.9 | 3.5 | 10.1 | 4.5 | 4.1 | 3.4 |
|
| 139 |
+
| HOO | 3.4 | 5.2 | 8.7 | 14.2 | 9.8 | 8.1 | 7.1 |
|
| 140 |
+
| CGP | 1.2 | 2.0 | 2.9 | 8.1 | 3.8 | 3.8 | 2.8 |
|
| 141 |
+
| CGP-A | 1.3 | 2.1 | 3.0 | 8.3 | 3.9 | 3.9 | 2.9 |
|
| 142 |
+
| CGP-H | $1.1^{\dagger}$ | $1.4^{\dagger}$ | $2.7^{\dagger}$ | $7.8^{\dagger}$ | $3.5^{\dagger}$ | $3.4^{\dagger}$ | $2.6^{\dagger}$ |
|
| 143 |
+
To enable high-dimensional optimization, we develop CGP-TR (Algorithm 3), which maintains local certificates within trust regions that adapt based on observed progress.
|
| 144 |
+
The key insight is that certificates need not be global. A local certificate $A_t^{\mathcal{T}}$ within trust region $\mathcal{T} \subset \mathcal{X}$ still provides valid bounds for $\arg\max_{x \in \mathcal{T}} f(x)$ . CGP-TR maintains multiple trust regions $\{\mathcal{T}_1, \dots, \mathcal{T}_m\}$ centered at promising points, with radii that expand on success and contract on failure (following TuRBO (Eriksson et al., 2019)).
|
| 145 |
+
**Certified restarts.** We restart a trust region only when it is certifiably suboptimal: if $u_t^{(j)} := \max_{x \in \mathcal{T}_j} U_t(x)$ satisfies $u_t^{(j)} < \ell_t$ where $\ell_t := \max_i \mathrm{LCB}_i(t)$ , then with high probability $\sup_{x \in \mathcal{T}_j} f(x) < f(x^*)$ , so $\mathcal{T}_j$ cannot contain $x^*$ and can be safely restarted. This certified restart rule ensures that regions containing $x^*$ are never falsely eliminated. In our implementation, contraction is lower-bounded by $r_{\min}$ and centers are fixed, so a region that contains $x^*$ cannot be contracted to exclude it; restarts occur only via the certified condition $u_t^{(j)} < \ell_t$ .
|
| 146 |
+
<span id="page-4-1"></span>**Theorem 6.1** (CGP-TR with certified restarts: correctness and allocation). Assume the good event $\mathcal{E}$ holds for the confidence bounds used to construct $U_t$ and $\ell_t$ . CGP-TR uses certified restarts: restart $\mathcal{T}_j$ only if $u_t^{(j)} := \max_{x \in \mathcal{T}_j} U_t(x) < \ell_t$ .
|
| 147 |
+
Let $\mathcal{T}^*$ be a trust region that contains $x^*$ at some time and is not contracted to exclude $x^*$ (e.g., contraction is lower-bounded by $r_{\min}$ and the center remains fixed). Then:
|
| 148 |
+
- 1. (No false restarts) On $\mathcal{E}$ , $\mathcal{T}^*$ is never restarted by the certified rule.
|
| 149 |
+
- 2. (Local certificate) Conditioned on receiving $T^*$ evaluations inside $T^*$ , the local active set $A_t^{(T^*)} = \{x \in T^* : U_t(x) \geq \ell_t^{(T^*)}\}$ satisfies the same containment/shrinkage/sample-complexity bounds as CGP on the restricted domain $T^*$ .
|
| 150 |
+
- 3. (Allocation bound) Define the region gap $\Delta_i :=$
|
| 151 |
+
#### <span id="page-4-0"></span>**Algorithm 3** CGP-TR (Trust Region with Certified Restarts)
|
| 152 |
+
**Require:** Domain $\mathcal{X}, L, \sigma$ , budget T, initial radius $r_0, n_{\text{trust}}$ regions
|
| 153 |
+
- 1: Initialize $n_{\text{trust}}$ trust regions at Sobol points with radius $r_0$
|
| 154 |
+
- 2: **for** t = 1, ..., T **do**
|
| 155 |
+
- 3: Compute $\ell_t := \max_{i \leq N_t} LCB_i(t)$ (global lower certificate)
|
| 156 |
+
- 4: Select trust region $\mathcal{T}_j$ with highest $u_t^{(j)} := \max_{x \in \mathcal{T}_i} U_t(x)$
|
| 157 |
+
- 5: Run CGP within $\mathcal{T}_j$ : compute local $A_t^{(j)} = \{x \in \mathcal{T}_j : U_t(x) \ge \ell_t^{(j)}\}$
|
| 158 |
+
- 6: Query $x_{t+1} \in A_t^{(j)}$ , observe $y_{t+1}$
|
| 159 |
+
- 7: **if** $u_t^{(j)} < \ell_t$ **then**
|
| 160 |
+
- 8: **Certified restart:** restart $\mathcal{T}_j$ at a new Sobol point with radius $r_0$
|
| 161 |
+
- 9: **else if** improvement in $\mathcal{T}_i$ **then**
|
| 162 |
+
- 10: Expand: $r_j \leftarrow \min(2r_j, D/2)$
|
| 163 |
+
- 11: **else if** no improvement for $\tau_{\text{fail}}$ iterations **then**
|
| 164 |
+
- 12: Contract: $r_j \leftarrow \max(r_j/2, r_{\min})$
|
| 165 |
+
- 13: **end if**
|
| 166 |
+
- 14: **end for**
|
| 167 |
+
- 15: **Output:** Best point across all regions, local certificate $A_T^{(j^*)}$
|
| 168 |
+
$f^* - \sup_{x \in \mathcal{T}_j} f(x)$ (with $\Delta_j > 0$ for suboptimal regions). Assume each region runs CGP with replication ensuring its maximal active-point confidence radius after n within-region samples satisfies $\beta_j(n) \le c_\sigma \sqrt{\log(c_T/\delta)/n}$ for constants $c_\sigma, c_T$ matching the paper's confidence schedule. If the region radii are eventually bounded so that $L \cdot \operatorname{diam}(\mathcal{T}_j) \le \Delta_j/8$ , then any suboptimal region j is selected at most
|
| 169 |
+
$$N_j \le \frac{64c_\sigma^2}{\Delta_j^2} \log\left(\frac{c_T}{\delta}\right) + 1$$
|
| 170 |
+
times before it is eliminated by the certified restart rule.
|
| 171 |
+
{5}------------------------------------------------
|
| 172 |
+
## <span id="page-5-1"></span>Algorithm 4 CGP-Hybrid
|
| 173 |
+
Require: Domain X , L, σ, budget T, switch threshold ρthresh = 0.5
|
| 174 |
+
- 1: Phase 1: Run CGP until Vol(At) < 0.1 · Vol(X ) or t > T /3
|
| 175 |
+
- 2: Estimate ρ<sup>t</sup> = Lˆ local(t)/Lˆ global
|
| 176 |
+
- 3: if ρ<sup>t</sup> < ρthresh then
|
| 177 |
+
- 4: Phase 2: Switch to GP-UCB within A<sup>t</sup> (GP refinement)
|
| 178 |
+
- 5: Fit GP to points in At, continue with GP-UCB acquisition
|
| 179 |
+
- 6: else
|
| 180 |
+
- 7: Phase 2: Continue CGP within A<sup>t</sup>
|
| 181 |
+
- 8: end if
|
| 182 |
+
- 9: Output: Best point, certificate A<sup>T</sup> (from CGP phase)
|
| 183 |
+
The key advantage is that covering T<sup>j</sup> requires O((rj/η) d ) points, and since r<sup>j</sup> ≪ D, this is tractable even for large d. With ntrust = O(log T) regions, CGP-TR explores globally while maintaining local certificates. The allocation bound (Theorem [6.1,](#page-4-1) item 3) ensures that suboptimal regions receive only O(log T /∆<sup>2</sup> j ) evaluations before certified elimination, preventing wasted samples.
|
| 184 |
+
CGP-TR provides local rather than global certificates, but the certified restart rule guarantees that the region containing x ∗ is never falsely eliminated. This enables scaling to d = 50 to 100 where global Lipschitz methods fail entirely.
|
| 185 |
+
## <span id="page-5-0"></span>7. CGP-Hybrid: Best of Both Worlds
|
| 186 |
+
While CGP-TR addresses scalability, some functions exhibit local smoothness that GPs can exploit more effectively than Lipschitz methods. CGP-Hybrid (Algorithm [4\)](#page-5-1) preserves CGP's anytime certificates while allowing any optimizer to refine within the certified active set. The key point is modularity: Phase 1 constructs a certificate At; Phase 2 performs additional optimization restricted to A<sup>t</sup> without affecting certificate validity. We instantiate Phase 2 with GP-UCB when local smoothness is detected, but other optimizers can be used. This design captures the best of both worlds: CGP's explicit pruning guarantees and GP's ability to exploit local smoothness when present.
|
| 187 |
+
Define the *effective smoothness ratio* ρ<sup>t</sup> = Lˆ local(t)/Lˆ global, where Lˆ local(t) is estimated from points within At. When ρ<sup>t</sup> < 0.5, the function is significantly smoother near the optimum, and GP refinement is beneficial.
|
| 188 |
+
<span id="page-5-3"></span>Proposition 7.1 (Hybrid guarantee). *CGP-Hybrid achieves:*
|
| 189 |
+
- *1. If* ρ ≥ 0.5*: same guarantee as CGP,* T = O˜(ε −(2+α) )*.*
|
| 190 |
+
- *2. If* ρ < 0.5*: after CGP reduces* A<sup>t</sup> *to volume* V *, GP-UCB operates within a restricted domain of effective*
|
| 191 |
+
*diameter* O(V <sup>1</sup>/d)*. The additional sample complexity depends on the GP kernel's information gain* γ<sup>T</sup> *over* At*; empirically, this yields faster convergence than continuing CGP when the function is locally smooth.*
|
| 192 |
+
*3. The certificate* A<sup>T</sup> *from Phase 1 remains valid regardless of Phase 2 method.*
|
| 193 |
+
Proposition 7.2 (Certificate invariance under Phase 2). *The certificate* A<sup>t</sup> *computed by CGP in Phase 1 remains valid regardless of the Phase-2 optimizer, since validity depends only on the confidence bounds and Lipschitz envelope used to define* U<sup>t</sup> *and* ℓt*. Specifically, any point* x /∈ A<sup>t</sup> *satisfies* f(x) < f <sup>∗</sup> − ε<sup>t</sup> *with high probability, where* ε<sup>t</sup> *is computable from Phase 1 quantities alone.*
|
| 194 |
+
The key insight is that CGP's certificate remains valid even when switching to GP: A<sup>t</sup> still contains x <sup>∗</sup> with high probability, so GP refinement within A<sup>t</sup> is safe. This provides the best of both worlds: CGP's certificates and pruning efficiency when ρ ≥ 0.5, and GP's smoothness exploitation when ρ < 0.5.
|
| 195 |
+
Table [6](#page-6-2) shows CGP-Hybrid wins on all 12 benchmarks, including Branin (ρ = 0.31) and Rosenbrock (ρ = 0.28) where it detects low ρ and switches to GP refinement, and Needle (ρ = 0.98) where it stays with CGP.
|
| 196 |
+
<span id="page-5-2"></span>*Table 4.* High-dimensional benchmarks (d > 20) at T = 500.
|
| 197 |
+
| Method | Rover-60 | NAS-36 | Ant-100 |
|
| 198 |
+
|--------|-----------------|--------------------------|-----------------|
|
| 199 |
+
| Random | 42.1 ± 1.2 | 38.4 ± 0.9 | 51.2 ± 1.4 |
|
| 200 |
+
| TuRBO | 12.4 ± 0.4 | 11.2 ± 0.3 | 18.7 ± 0.6 |
|
| 201 |
+
| HEBO | 14.1 ± 0.5 | 12.8 ± 0.4 | 21.3 ± 0.7 |
|
| 202 |
+
| CMA-ES | 15.8 ± 0.6 | 14.1 ± 0.5 | 19.4 ± 0.6 |
|
| 203 |
+
| CGP | | (intractable for d > 15) | |
|
| 204 |
+
| CGP-TR | †<br>11.2 ± 0.3 | †<br>10.4 ± 0.3 | †<br>17.1 ± 0.5 |
|
| 205 |
+
<sup>†</sup> CGP-TR additionally provides local optimality certificates.
|
| 206 |
+
## 8. Related Work
|
| 207 |
+
Our work is primarily grounded in the literature on Lipschitz bandits and global optimization. Foundational approaches, such as the continuum-armed bandits of [Kleinberg](#page-8-5) [et al.](#page-8-5) [\(2008\)](#page-8-5) and the X-armed bandit framework of [Bubeck](#page-8-7) [et al.](#page-8-7) [\(2011\)](#page-8-7), utilize zooming mechanisms to achieve regret bounds depending on the near-optimality dimension. These concepts were refined for deterministic and stochastic settings via tree-based algorithms like DOO/SOO [\(Munos,](#page-8-17) [2011\)](#page-8-17) and StoSOO [\(Valko et al.,](#page-9-2) [2013\)](#page-9-2). However, a key distinction is that zooming algorithms maintain pruning implicitly as an analysis artifact, whereas CGP exposes the active set A<sup>t</sup> as a computable object with measurable volume. This explicit geometric approach also relates to partitionbased global optimization methods like DIRECT [\(Jones](#page-8-3)
|
| 208 |
+
{6}------------------------------------------------
|
| 209 |
+
*Table 5.* CGP-Adaptive with varying initial Lˆ0. Robust to 100× underestimation.
|
| 210 |
+
<span id="page-6-1"></span>
|
| 211 |
+
| Initial Lˆ<br>0 | Doublings | Final L/L ˆ ∗ | Regret (×10−2<br>) | Overhead |
|
| 212 |
+
|--------------------|-----------|---------------|--------------------|----------|
|
| 213 |
+
| ∗<br>L<br>(oracle) | 0 | 1.0 | 2.9 ± 0.1 | 1.0× |
|
| 214 |
+
| ∗/2<br>L | 1 | 1.0 | 3.0 ± 0.1 | 1.03× |
|
| 215 |
+
| ∗/10<br>L | 4 | 1.6 | 3.1 ± 0.1 | 1.07× |
|
| 216 |
+
| ∗/100<br>L | 7 | 1.3 | 3.2 ± 0.2 | 1.12× |
|
| 217 |
+
| LIPO (adaptive) | – | – | 6.2 ± 0.3 | – |
|
| 218 |
+
<span id="page-6-2"></span>*Table 6.* CGP-Hybrid smoothness detection. ρ < 0.5 triggers GP refinement.
|
| 219 |
+
| Benchmark | ρˆ | Phase 2 | CGP-H Regret | Best Baseline |
|
| 220 |
+
|-------------|------|---------|--------------|---------------|
|
| 221 |
+
| Needle-2D | 0.98 | CGP | 1.1 ± 0.1 | 1.8 (TuRBO) |
|
| 222 |
+
| Branin | 0.31 | GP | 1.4 ± 0.1 | 1.6 (HEBO) |
|
| 223 |
+
| Hartmann-6 | 0.72 | CGP | 2.7 ± 0.1 | 3.1 (TuRBO) |
|
| 224 |
+
| Rosenbrock | 0.28 | GP | 3.4 ± 0.1 | 3.7 (HEBO) |
|
| 225 |
+
| Ackley-10 | 0.85 | CGP | 7.8 ± 0.3 | 9.4 (HEBO) |
|
| 226 |
+
| Levy-5 | 0.67 | CGP | 3.5 ± 0.2 | 4.1 (HEBO) |
|
| 227 |
+
| SVM-RBF | 0.74 | CGP | 2.6 ± 0.1 | 3.1 (HEBO) |
|
| 228 |
+
| LunarLander | 0.81 | CGP | 6.1 ± 0.3 | 7.0 (HEBO) |
|
| 229 |
+
[et al.,](#page-8-3) [1993\)](#page-8-3) and LIPO [\(Malherbe & Vayatis,](#page-8-15) [2017\)](#page-8-15). While LIPO addresses the unknown Lipschitz constant, it lacks the certificate guarantees provided by our CGP-Adaptive doubling scheme [\(Shihab et al.,](#page-9-8) [2025c\)](#page-9-8). Our theoretical analysis further draws on sample complexity results under margin conditions from the finite-arm setting [\(Audibert &](#page-8-13) [Bubeck,](#page-8-13) [2010;](#page-8-13) [Jamieson & Nowak,](#page-8-18) [2014;](#page-8-18) [Shihab et al.,](#page-9-9) [2025b\)](#page-9-9), adapting confidence bounds from the UCB frame[w](#page-8-3)ork [\(Auer et al.,](#page-8-6) [2002\)](#page-8-6) to continuous spaces with explicit uncertainty representation similar to safe optimization levelsets [\(Sui et al.,](#page-9-10) [2015\)](#page-9-10).
|
| 230 |
+
In the broader context of black-box optimization, Bayesian methods provide the standard alternative for uncertainty quantification. Classic approaches like GP-UCB [\(Srinivas](#page-9-11) [et al.,](#page-9-11) [2009\)](#page-9-11), Entropy Search [\(Hennig & Schuler,](#page-8-11) [2012;](#page-8-11) [Hernandez-Lobato et al.](#page-8-12) ´ , [2014\)](#page-8-12), and Thompson Sampling [\(Thompson,](#page-9-4) [1933\)](#page-9-4) offer strong performance but scale cubically with observations. To address high-dimensional scaling, recent work has introduced trust regions (TuRBO) [\(Eriksson et al.,](#page-8-16) [2019\)](#page-8-16) and nonstationary priors (HEBO) [\(Cowen-Rivers et al.,](#page-8-19) [2022\)](#page-8-19). More recently, advanced heuristics such as Bounce [\(Papenmeier et al.,](#page-8-20) [2024\)](#page-8-20) have improved geometric adaptation for mixed spaces, while Prior-Fitted Networks [\(Hollmann et al.,](#page-8-21) [2023\)](#page-8-21) and generative diffusion models like Diff-BBO [\(Wu et al.,](#page-9-12) [2024\)](#page-9-12) exploit massive pretraining to minimize regret rapidly. While these emerging methods achieve impressive empirical results, they remain fundamentally heuristic, lacking the computable stopping criteria or active set containment guarantees that are central to CGP. We therefore focus our comparison on established
|
| 231 |
+
baselines to isolate the specific utility of our certification mechanism, distinguishing our approach from heuristic hyperparameter tuners like Hyperband [\(Li et al.,](#page-8-22) [2018\)](#page-8-22) by prioritizing provable safety over raw speed.
|
| 232 |
+
## <span id="page-6-0"></span>9. Experiments
|
| 233 |
+
We evaluate CGP variants on 12 benchmarks spanning d ∈ [2, 100], measuring simple regret, certificate utility, and scalability. Code will be available upon acceptance.
|
| 234 |
+
Setup. We compare against 9 baselines: Random Search, GP-UCB, TuRBO, HEBO, BORE, HOO, StoSOO, LIPO, and SAASBO (see Appendix [E](#page-15-0) for configurations). We evaluate on 12 benchmarks spanning d ∈ [2, 100]: lowdimensional (Needle-2D, Branin, Hartmann-6, Levy-5, Rosenbrock-4), medium-dimensional (Ackley-10, SVM-RBF-6, LunarLander-12), and high-dimensional (Rover-60, NAS-36, MuJoCo-Ant-100). All experiments use 30 runs with σ = 0.1 noise; we report mean ± SE with Bonferronicorrected t-tests (p < 0.05).
|
| 235 |
+
Table [3](#page-4-2) shows CGP-Hybrid performs best among tested methods on all 7 low and medium-dimensional benchmarks. On Branin and Rosenbrock where vanilla CGP lost to HEBO, CGP-Hybrid detects ρ < 0.5 and switches to GP refinement, achieving 12% and 8% improvement over HEBO respectively. On Needle where ρ ≈ 1, CGP-Hybrid stays with CGP and matches vanilla CGP performance. For high-dimensional problems, Table [4](#page-5-2) demonstrates CGP-TR scales to d = 100 while outperforming TuRBO by
|
| 236 |
+
{7}------------------------------------------------
|
| 237 |
+
9 to 12%. Critically, CGP-TR provides local certificates within trust regions, enabling principled stopping—a capability TuRBO lacks. Regarding adaptive estimation, Table 5 shows CGP-Adaptive is robust to initial underestimation: even with $\hat{L}_0=L^*/100$ , performance degrades only 10% with 7 doublings, validating Theorem 5.1's $O(\log(L^*/\hat{L}_0))$ overhead. Finally, Table 6 confirms CGP-Hybrid correctly identifies when to switch: Branin ( $\rho=0.31$ ) and Rosenbrock ( $\rho=0.28$ ) trigger GP refinement, achieving 12% and 8% improvement over HEBO. Benchmarks with $\rho>0.5$ stay with CGP, maintaining certificate validity.
|
| 238 |
+
**Shrinkage validation.** Across all benchmarks, we observe $Vol(A_t)$ shrinks to < 5% by T = 100, with empirical decay rates closely matching the theoretical bound from Theorem 4.6. This confirms our analysis is tight and the margin condition captures the true problem difficulty.
|
| 239 |
+
<span id="page-7-0"></span>*Table 7.* Certificate-enabled early stopping on Hartmann-6. CGP uniquely provides actionable stopping criteria.
|
| 240 |
+
| Stopping Rule | Samples | Regret ( $\times 10^{-2}$ ) | Savings |
|
| 241 |
+
|---------------------------|--------------|-----------------------------|---------|
|
| 242 |
+
| Fixed $T = 200$ | 200 | $2.9 \pm 0.1$ | _ |
|
| 243 |
+
| $Vol(A_t) < 10\%$ | $82 \pm 9$ | $3.8 \pm 0.2$ | 59% |
|
| 244 |
+
| $Vol(A_t) < 5\%$ | $118 \pm 14$ | $3.2 \pm 0.2$ | 41% |
|
| 245 |
+
| ${\rm Gap\ bound} < 0.05$ | $134 \pm 12$ | $3.0 \pm 0.1$ | 33% |
|
| 246 |
+
Table 7 demonstrates certificate utility: stopping at $Vol(A_t) < 10\%$ saves 59% of samples with only 31% regret increase. No baseline provides such principled stopping rules. In d>20, we use the computable gap proxy $\varepsilon_t:=2(\beta_t+L\eta_t)+\gamma_t$ as the primary criterion. Beyond sample efficiency, CGP also offers computational advantages. Table 8 shows CGP variants are 6 to 8 times faster than GP-based methods due to their O(n) per-iteration cost versus GP's $O(n^3)$ . Finally, we ablate CGP's components to understand their individual contributions.
|
| 247 |
+
<span id="page-7-1"></span>Table 8. Wall-clock time (seconds) for T=200 on Hartmann-6.
|
| 248 |
+
| Method | Time (s) | Regret ( $\times 10^{-2}$ ) | Speedup |
|
| 249 |
+
|--------------|----------|-----------------------------|--------------|
|
| 250 |
+
| CGP | 58 | 2.9 | 8× |
|
| 251 |
+
| CGP-Adaptive | 64 | 3.0 | $7.5 \times$ |
|
| 252 |
+
| CGP-Hybrid | 72 | 2.7 | $6.7 \times$ |
|
| 253 |
+
| GP-UCB | 480 | 4.2 | $1 \times$ |
|
| 254 |
+
| TuRBO | 620 | 3.1 | $0.8 \times$ |
|
| 255 |
+
| HEBO | 890 | 3.3 | $0.5 \times$ |
|
| 256 |
+
Table 9 shows all components contribute: removing pruning certificates increases regret 78%, coverage penalty 44%, replication 33%. GP refinement provides 7% improvement on Hartmann-6 where $\rho=0.72$ is borderline. Table 10 validates that $\hat{\alpha} < d$ across all benchmarks, confirming the margin condition holds and our complexity bounds apply.
|
| 257 |
+
<span id="page-7-2"></span>*Table 9.* Ablation study on Hartmann-6. All components contribute.
|
| 258 |
+
| Variant | Regret ( $\times 10^{-2}$ ) | $Vol(A_{200})$ |
|
| 259 |
+
|-----------------------------------------|---------------------------------|----------------|
|
| 260 |
+
| CGP-Hybrid (full) | $\textbf{2.7} \pm \textbf{0.1}$ | 2.1% |
|
| 261 |
+
| <ul> <li>GP refinement</li> </ul> | $2.9 \pm 0.1$ | 2.1% |
|
| 262 |
+
| <ul> <li>pruning certificate</li> </ul> | $4.8 \pm 0.2$ | _ |
|
| 263 |
+
| <ul> <li>coverage penalty</li> </ul> | $3.9 \pm 0.2$ | 3.8% |
|
| 264 |
+
| <ul><li>replication</li></ul> | $3.6 \pm 0.2$ | 2.9% |
|
| 265 |
+
| CGP-TR $(d=6)$ | $2.8 \pm 0.1$ | 2.4% (local) |
|
| 266 |
+
| CGP-Adaptive | $3.0 \pm 0.1$ | 2.4% |
|
| 267 |
+
<span id="page-7-3"></span>*Table 10.* Empirical $\hat{\alpha}$ estimates from shrinkage trajectories.
|
| 268 |
+
| Benchmark | d | â (95% CI) | True $\alpha$ | $\hat{\alpha} < d$ ? |
|
| 269 |
+
|------------|----|---------------|---------------|----------------------|
|
| 270 |
+
| Needle-2D | 2 | $1.8 \pm 0.2$ | 2.0 | <u>√</u> |
|
| 271 |
+
| Branin | 2 | $1.2 \pm 0.1$ | _ | $\checkmark$ |
|
| 272 |
+
| Hartmann-6 | 6 | $2.4 \pm 0.3$ | _ | $\checkmark$ |
|
| 273 |
+
| Ackley-10 | 10 | $3.2 \pm 0.4$ | _ | $\checkmark$ |
|
| 274 |
+
| Rover-60 | 60 | $8.4 \pm 1.2$ | _ | $\checkmark$ |
|
| 275 |
+
#### 10. Conclusion
|
| 276 |
+
We introduced Certificate-Guided Pruning (CGP), an algorithm for stochastic Lipschitz optimization that maintains explicit active sets with provable shrinkage guarantees. Under a margin condition with near-optimality dimension $\alpha$ , we prove $Vol(A_t) \leq C \cdot (2(\beta_t + L\eta_t) + \gamma_t)^{d-\alpha}$ , yielding sample complexity $\tilde{O}(\varepsilon^{-(2+\alpha)})$ with anytime valid certificates. Three extensions broaden applicability: CGP-Adaptive learns L online with $O(\log T)$ overhead, CGP-TR scales to d > 50 via trust regions, and CGP-Hybrid switches to GP refinement when local smoothness is detected. The margin condition holds broadly for isolated maxima with nondegenerate Hessian $\alpha = d/2$ , for polynomial decay $\alpha = d/p$ and can be estimated online from shrinkage trajectories. Limitations include requiring Lipschitz continuity and dimension constraints ( $d \le 15$ for vanilla CGP, $d \le 100$ for CGP-TR); practical guidance is in Appendix D. Future directions include safe optimization using $A_t$ for safety certificates and CGP-TR with random embeddings for global high-dimensional certificates.
|
| 277 |
+
#### **Impact Statement**
|
| 278 |
+
This paper introduces Certificate-Guided Pruning (CGP), a method designed to improve the sample efficiency of black-box optimization in resource-constrained settings. By providing explicit optimality certificates and principled stopping criteria, our approach significantly reduces the computational budget required for expensive tasks such as neural architecture search and simulation-based engineering, directly contributing to lower energy consumption and carbon footprints. Furthermore, the ability to certify suboptimal regions enhances reliability in safety-critical applications like
|
| 279 |
+
{8}------------------------------------------------
|
| 280 |
+
robotics. However, practitioners must ensure the validity of the Lipschitz assumption, as violations could lead to the incorrect pruning of optimal solutions.
|
| 281 |
+
## References
|
| 282 |
+
- <span id="page-8-13"></span>Audibert, J.-Y. and Bubeck, S. Best arm identification in multi-armed bandits. In *Conference on Learning Theory*, pp. 41–53, 2010.
|
| 283 |
+
- <span id="page-8-6"></span>Auer, P., Cesa-Bianchi, N., and Fischer, P. Finite-time analysis of the multiarmed bandit problem. *Machine Learning*, 47(2-3):235–256, 2002.
|
| 284 |
+
- <span id="page-8-0"></span>Bergstra, J. and Bengio, Y. Random search for hyperparameter optimization. *Journal of Machine Learning Research*, 13(10):281–305, 2012.
|
| 285 |
+
- <span id="page-8-2"></span>Brochu, E., Cora, V. M., and De Freitas, N. A tutorial on Bayesian optimization of expensive cost functions. *arXiv preprint arXiv:1012.2599*, 2010.
|
| 286 |
+
- <span id="page-8-7"></span>Bubeck, S., Munos, R., Stoltz, G., and Szepesvari, C. X- ´ armed bandits. In *Journal of Machine Learning Research*, volume 12, pp. 1655–1695, 2011.
|
| 287 |
+
- <span id="page-8-19"></span>Cowen-Rivers, A. I., Lyu, W., Tutunov, R., Wang, Z., Grosnit, A., Griffiths, R. R., Maraval, A. M., Jianye, H., Wang, J., Peters, J., et al. HEBO: Pushing the limits of sampleefficient hyper-parameter optimisation. In *Journal of Artificial Intelligence Research*, volume 74, pp. 1269–1349, 2022.
|
| 288 |
+
- <span id="page-8-10"></span>Daniel, J. R., Benjamin, V. R., Abbas, K., Ian, O., and Zheng, W. A tutorial on thompson sampling. *Foundations and trends® in machine learning*, 11(1):1–99, 2018.
|
| 289 |
+
- <span id="page-8-16"></span>Eriksson, D., Pearce, M., Gardner, J., Turner, R. D., and Poloczek, M. Scalable global optimization via local bayesian optimization. *Advances in neural information processing systems*, 32, 2019.
|
| 290 |
+
- <span id="page-8-1"></span>Fu, M. C. *Handbook of simulation optimization*. Springer, 2015.
|
| 291 |
+
- <span id="page-8-24"></span>Garivier, A. and Kaufmann, E. Optimal best arm identification with fixed confidence. *Conference on Learning Theory*, pp. 998–1027, 2016.
|
| 292 |
+
- <span id="page-8-11"></span>Hennig, P. and Schuler, C. J. Entropy search for informationefficient global optimization. *Journal of Machine Learning Research*, 13(57):1809–1837, 2012.
|
| 293 |
+
- <span id="page-8-12"></span>Hernandez-Lobato, J. M., Hoffman, M. W., and Ghahra- ´ mani, Z. Predictive entropy search for efficient global optimization of black-box functions. In *Advances in Neural Information Processing Systems*, volume 27, pp. 918–926, 2014.
|
| 294 |
+
- <span id="page-8-21"></span>Hollmann, N., Muller, S., Eggensperger, K., and Hutter, F. ¨ Large-scale transfer learning for bayesian optimization with prior-data fitted networks. In *Advances in Neural Information Processing Systems (NeurIPS)*, 2023.
|
| 295 |
+
- <span id="page-8-18"></span>Jamieson, K. and Nowak, R. Best-arm identification algorithms for multi-armed bandits in the fixed confidence setting. In *Conference on Information Sciences and Systems*, pp. 1–6. IEEE, 2014.
|
| 296 |
+
- <span id="page-8-3"></span>Jones, D. R., Perttunen, C. D., and Stuckman, B. E. Lipschitzian optimization without the Lipschitz constant. *Journal of Optimization Theory and Applications*, 79(1): 157–181, 1993.
|
| 297 |
+
- <span id="page-8-4"></span>Jones, D. R., Schonlau, M., and Welch, W. J. Efficient global optimization of expensive black-box functions. *Journal of Global Optimization*, 13(4):455–492, 1998.
|
| 298 |
+
- <span id="page-8-23"></span>Kaufmann, E., Cappe, O., and Garivier, A. On the com- ´ plexity of best-arm identification in multi-armed bandit models. *Journal of Machine Learning Research*, 17:1–42, 2016.
|
| 299 |
+
- <span id="page-8-5"></span>Kleinberg, R., Slivkins, A., and Upfal, E. Multi-armed bandits in metric spaces. In *Proceedings of the 40th Annual ACM Symposium on Theory of Computing*, pp. 681–690, 2008.
|
| 300 |
+
- <span id="page-8-14"></span>Lattimore, T. and Szepesvari, C. Bandit algorithms. ´ *Cambridge University Press*, 2020.
|
| 301 |
+
- <span id="page-8-22"></span>Li, L., Jamieson, K., DeSalvo, G., Rostamizadeh, A., and Talwalkar, A. Hyperband: A novel bandit-based approach to hyperparameter optimization. *Journal of Machine Learning Research*, 18:1–52, 2018.
|
| 302 |
+
- <span id="page-8-15"></span>Malherbe, C. and Vayatis, N. Global optimization of lipschitz functions. In *International conference on machine learning*, pp. 2314–2323. PMLR, 2017.
|
| 303 |
+
- <span id="page-8-17"></span>Munos, R. Optimistic optimization of a deterministic function without the knowledge of its smoothness. *Advances in Neural Information Processing Systems*, 24:783–791, 2011.
|
| 304 |
+
- <span id="page-8-8"></span>Munos, R. From bandits to Monte-Carlo tree search: The optimistic principle applied to optimization and planning. *Foundations and Trends in Machine Learning*, 7:1–129, 2014.
|
| 305 |
+
- <span id="page-8-20"></span>Papenmeier, L., Nardi, L., and Poloczek, M. Bounce: Reliable high-dimensional bayesian optimization for combinatorial and mixed spaces. In *International Conference on Learning Representations (ICLR)*, 2024.
|
| 306 |
+
- <span id="page-8-9"></span>Russo, D. and Van Roy, B. Learning to optimize via posterior sampling. *Mathematics of Operations Research*, 39: 1221–1243, 2014.
|
| 307 |
+
{9}------------------------------------------------
|
| 308 |
+
Shihab, I. F., Akter, S., and Sharma, A. Detecting proxy gaming in rl and llm alignment via evaluator stress tests. *arXiv preprint arXiv:2507.05619*, 2025a.
|
| 309 |
+
<span id="page-9-5"></span>
|
| 310 |
+
<span id="page-9-9"></span>
|
| 311 |
+
<span id="page-9-8"></span>
|
| 312 |
+
<span id="page-9-3"></span>
|
| 313 |
+
<span id="page-9-0"></span>
|
| 314 |
+
<span id="page-9-11"></span>
|
| 315 |
+
<span id="page-9-10"></span>
|
| 316 |
+
<span id="page-9-4"></span>
|
| 317 |
+
<span id="page-9-2"></span>
|
| 318 |
+
<span id="page-9-13"></span>
|
| 319 |
+
<span id="page-9-12"></span>
|
| 320 |
+
<span id="page-9-1"></span>
|
| 321 |
+
- Shihab, I. F., Akter, S., and Sharma, A. What fundamental structure in reward functions enables efficient sparsereward learning? *arXiv preprint arXiv:2509.03790*, 2025b.
|
| 322 |
+
- Shihab, I. F., Akter, S., and Sharma, A. Universal adaptive constraint propagation: Scaling structured inference for large language models via meta-reinforcement learning. arXiv preprint arXiv:2601.00095, 2025c.
|
| 323 |
+
- Shihab, I. F., Akter, S., and Sharma, A. Beyond variance: Knowledge-aware llm compression via fisher-aligned subspace diagnostics. *arXiv preprint arXiv:2601.07197*, 2026.
|
| 324 |
+
- Snoek, J., Larochelle, H., and Adams, R. P. Practical Bayesian optimization of machine learning algorithms. In *Advances in Neural Information Processing Systems*, volume 25, 2012.
|
| 325 |
+
- Srinivas, N., Krause, A., Kakade, S. M., and Seeger, M. Gaussian process optimization in the bandit setting: No regret and experimental design. arXiv preprint arXiv:0912.3995, 2009.
|
| 326 |
+
- Sui, Y., Gotovos, A., Burdick, J., and Krause, A. Safe exploration for optimization with gaussian processes. In *International conference on machine learning*, pp. 997–1005. PMLR, 2015.
|
| 327 |
+
- Thompson, W. R. On the likelihood that one unknown probability exceeds another in view of the evidence of two samples. *Biometrika*, 25(3/4):285–294, 1933.
|
| 328 |
+
- Valko, M., Carpentier, A., and Munos, R. Stochastic simultaneous optimistic optimization. In *Proceedings of the 30th International Conference on Machine Learning*, pp. 19–27, 2013.
|
| 329 |
+
- Wang, Z., Hutter, F., Zoghi, M., Matheson, D., and de Freitas, N. Bayesian optimization in a billion dimensions via random embeddings. *Journal of Artificial Intelligence Research*, 55:361–387, 2016.
|
| 330 |
+
- Wu, D., Kuang, N. L., Niu, R., Ma, Y., and Yu, R. Diffbbo: Diffusion-based inverse modeling for black-box optimization. 2024.
|
| 331 |
+
- Zoph, B. and Le, Q. V. Neural architecture search with reinforcement learning. In *International Conference on Learning Representations*, 2017.
|
| 332 |
+
#### <span id="page-9-6"></span>A. Extended Problem Formulation
|
| 333 |
+
We restate the assumptions from Section 2 for completeness. Let $(\mathcal{X}, d)$ be a compact metric space with diameter $D = \sup_{x,y \in \mathcal{X}} d(x,y)$ . We consider $\mathcal{X} = [0,1]^d$ with Euclidean metric, though our results extend to general metric spaces. Let $f: \mathcal{X} \to [0,1]$ be an unknown function satisfying:
|
| 334 |
+
**Assumption 2.1 (Lipschitz continuity).** There exists L > 0 such that for all $x, y \in \mathcal{X}$ : $|f(x) - f(y)| \le L \cdot d(x, y)$ .
|
| 335 |
+
We observe f through noisy queries: querying x returns $y = f(x) + \epsilon$ , where:
|
| 336 |
+
**Assumption 2.2 (Sub-Gaussian noise).** The noise $\epsilon$ is $\sigma$ -sub-Gaussian: $\mathbb{E}[e^{\lambda\epsilon}] \leq e^{\lambda^2\sigma^2/2}$ for all $\lambda \in \mathbb{R}$ .
|
| 337 |
+
After T samples, the algorithm outputs $\hat{x}_T \in \mathcal{X}$ . The goal is to minimize simple regret $r_T = f(x^*) - f(\hat{x}_T)$ , where $x^* \in \arg\max_{x \in \mathcal{X}} f(x)$ . We seek PAC-style guarantees: with probability at least $1 - \delta$ , achieve $r_T \leq \varepsilon$ .
|
| 338 |
+
To obtain instance-dependent rates that improve upon worst case bounds, we assume margin structure:
|
| 339 |
+
Assumption 2.3 (Margin / near-optimality dimension). There exist C>0 and $\alpha\in[0,d]$ such that for all $\varepsilon>0$ : $\operatorname{Vol}(\{x\in\mathcal{X}:f(x)\geq f^*-\varepsilon\})\leq C\,\varepsilon^{d-\alpha}$ .
|
| 340 |
+
The parameter $\alpha$ is the near-optimality dimension: smaller $\alpha$ corresponds to a sharper optimum (easier), while larger $\alpha$ corresponds to a broader near-optimal region (harder). The worst case is $\alpha=d$ , which recovers the standard d-dimensional Lipschitz difficulty. This assumption is standard in bandit theory (Audibert & Bubeck, 2010; Bubeck et al., 2011; Lattimore & Szepesvári, 2020; Kaufmann et al., 2016; Garivier & Kaufmann, 2016).
|
| 341 |
+
#### <span id="page-9-7"></span>**B.** Algorithm Implementation Details
|
| 342 |
+
#### **B.1. Score Maximization and Replication**
|
| 343 |
+
CGP optimizes $\operatorname{score}(x) = U_t(x) - \lambda \cdot \min_i d(x, x_i)$ , which is piecewise linear. Since this is non-smooth, we use CMA-ES (Covariance Matrix Adaptation Evolution Strategy) with bounded domain and 10 random restarts within $A_t$ . For $d \leq 3$ , we additionally use Delaunay triangulation to identify candidate optima at Voronoi vertices.
|
| 344 |
+
Membership in $A_t$ is exact: given $\{x_i, \hat{\mu}_i, r_i\}$ , checking $U_t(x) \geq \ell_t$ requires $O(N_t)$ time. Approximate maximization may slow convergence of $\eta_t$ but does not invalidate certificates: any $x \notin A_t$ remains certifiably suboptimal regardless of which $x \in A_t$ is queried.
|
| 345 |
+
**Replication Strategy.** When an active point $x_i$ has $r_i(t) > \beta_{\text{target}}(t)$ , we allocate $\lceil (r_i(t)/\beta_{\text{target}}(t))^2 \rceil$ additional samples to reduce its confidence radius. This ensures all active points have comparable confidence, preventing
|
| 346 |
+
{10}------------------------------------------------
|
| 347 |
+
any single point from dominating the envelope.
|
| 348 |
+
## <span id="page-10-0"></span>**B.2.** Active Set Computation
|
| 349 |
+
For low dimensions $(d \le 5)$ , we compute $A_t$ exactly using grid discretization with resolution $\eta = D/\sqrt[d]{N_{\rm grid}}$ where $N_{\rm grid} = 10^4$ . For each grid point x, we evaluate $U_t(x)$ in $O(N_t)$ time and check if $U_t(x) \ge \ell_t$ . The volume $\operatorname{Vol}(A_t)$ is estimated as the fraction of grid points in $A_t$ .
|
| 350 |
+
For higher dimensions (d>5), uniform Monte Carlo becomes ineffective once $\operatorname{Vol}(A_t)$ is small. We therefore estimate $\operatorname{Vol}(A_t)$ via a nested-set ratio estimator (subset simulation): define thresholds $\ell_t - \tau_0 < \ell_t - \tau_1 < \cdots < \ell_t$ inducing nested sets
|
| 351 |
+
$$A_t^{(k)} = \{x : U_t(x) \ge \ell_t - \tau_k\}, \quad A_t^{(K)} = A_t.$$
|
| 352 |
+
We estimate
|
| 353 |
+
$$Vol(A_t) = Vol(A_t^{(0)}) \prod_{k=1}^{K} \mathbb{P}_{x \sim Unif(A_t^{(k-1)})} [x \in A_t^{(k)}],$$
|
| 354 |
+
sampling approximately uniformly from $A_t^{(k-1)}$ using a hit-and-run Markov chain with the membership oracle $U_t(x) \geq \ell_t - \tau_{k-1}$ . This yields stable estimates even when $\operatorname{Vol}(A_t)$ is very small. In all high-dimensional experiments, we report confidence intervals of $\log \operatorname{Vol}(A_t)$ from repeated estimator runs.
|
| 355 |
+
Crucially, certificate validity (Remark 4.7) is independent of volume estimation accuracy: the set membership rule $x \in A_t \Leftrightarrow U_t(x) \ge \ell_t$ is exact.
|
| 356 |
+
On volume-based stopping. The shrinkage bound (Theorem 4.6) provides an upper bound on $\operatorname{Vol}(A_t)$ as a function of the algorithmic gap proxy
|
| 357 |
+
$$\varepsilon_t := 2(\beta_t + L\eta_t) + \gamma_t, \quad \gamma_t = f^* - \ell_t,$$
|
| 358 |
+
and therefore supports using $\operatorname{Vol}(A_t)$ as a practical progress diagnostic. However, $\operatorname{Vol}(A_t)$ alone does not yield an anytime upper bound on regret without additional lower-regularity assumptions linking volume back to function values. In our experiments we therefore use $\varepsilon_t$ as the primary certificate-based stopping criterion, and treat $\operatorname{Vol}(A_t)$ as a secondary monitoring signal.
|
| 359 |
+
## <span id="page-10-1"></span>C. Proofs
|
| 360 |
+
#### C.1. Proof to Lemma 4.1
|
| 361 |
+
*Proof.* Define the good event $\mathcal{E} = \bigcap_{t=1}^T \bigcap_{i=1}^{N_t} \{|\hat{\mu}_i(t) - f(x_i)| \leq r_i(t)\}.$
|
| 362 |
+
Step 1: Single-point concentration. By Hoeffding's inequality for $\sigma$ -sub-Gaussian random variables:
|
| 363 |
+
$$\mathbb{P}\big[|\hat{\mu}_i(t) - f(x_i)| > r\big] \le 2\exp\left(-\frac{n_i r^2}{2\sigma^2}\right). \tag{4}$$
|
| 364 |
+
Step 2: Calibration of confidence radius. Substituting $r = r_i(t) = \sigma \sqrt{2 \log(2N_t T/\delta)/n_i}$ :
|
| 365 |
+
$$\mathbb{P}[|\hat{\mu}_{i}(t) - f(x_{i})| > r_{i}(t)] \leq 2 \exp\left(-\frac{n_{i} \cdot 2\sigma^{2} \log(2N_{t}T/\delta)}{2\sigma^{2} \cdot n_{i}}\right)$$
|
| 366 |
+
$$= 2 \exp\left(-\log(2N_{t}T/\delta)\right) = \frac{\delta}{N_{t}T}.$$
|
| 367 |
+
(5)
|
| 368 |
+
Step 3: Union bound. Applying the union bound over all $i \in \{1, ..., N_t\}$ and $t \in \{1, ..., T\}$ :
|
| 369 |
+
$$\mathbb{P}[\mathcal{E}^c] \le \sum_{t=1}^T \sum_{i=1}^{N_t} \frac{\delta}{N_t T} \le \sum_{t=1}^T \frac{\delta}{T} = \delta. \tag{6}$$
|
| 370 |
+
Hence
|
| 371 |
+
$$\mathbb{P}[\mathcal{E}] \geq 1 - \delta$$
|
| 372 |
+
.
|
| 373 |
+
#### C.2. Proof to Lemma 4.2 (UCB envelope is valid)
|
| 374 |
+
*Proof.* Fix any $x \in \mathcal{X}$ . For any sampled point $x_i$ , on the good event $\mathcal{E}$ :
|
| 375 |
+
$$f(x_i) \le \hat{\mu}_i(t) + r_i(t) = UCB_i(t). \tag{7}$$
|
| 376 |
+
By Lipschitz continuity of f:
|
| 377 |
+
$$f(x) \le f(x_i) + L \cdot d(x, x_i) \le UCB_i(t) + L \cdot d(x, x_i)$$
|
| 378 |
+
. (8)
|
| 379 |
+
Since this holds for all sampled *i*, taking the minimum over *i*:
|
| 380 |
+
$$f(x) \le \min_{i \le N_t} \{ \text{UCB}_i(t) + L \cdot d(x, x_i) \} = U_t(x).$$
|
| 381 |
+
(9)
|
| 382 |
+
#### C.3. Proof to Lemma 4.3 (Envelope slack bound)
|
| 383 |
+
*Proof.* Fix any $x \in \mathcal{X}$ and any sampled point $x_i$ .
|
| 384 |
+
Step 1: Upper confidence bound. On the good event $\mathcal{E}$ , we have $\hat{\mu}_i(t) \leq f(x_i) + r_i(t)$ . Therefore:
|
| 385 |
+
$$UCB_{i}(t) = \hat{\mu}_{i}(t) + r_{i}(t) \le f(x_{i}) + 2r_{i}(t).$$
|
| 386 |
+
(10)
|
| 387 |
+
Step 2: Lipschitz propagation. By Assumption 2.1 (Lipschitz continuity):
|
| 388 |
+
$$f(x_i) \le f(x) + L \cdot d(x, x_i). \tag{11}$$
|
| 389 |
+
Step 3: Combining bounds. Substituting the Lipschitz bound into Step 1:
|
| 390 |
+
$$UCB_{i}(t) + L d(x, x_{i}) \leq f(x_{i}) + 2r_{i}(t) + L d(x, x_{i})$$
|
| 391 |
+
$$\leq f(x) + L d(x, x_{i}) + 2r_{i}(t) + L d(x, x_{i})$$
|
| 392 |
+
$$= f(x) + 2(r_{i}(t) + L d(x, x_{i})). \quad (12)$$
|
| 393 |
+
{11}------------------------------------------------
|
| 394 |
+
*Step 4: Taking the minimum.* Since the above holds for all i, taking the minimum over i on both sides:
|
| 395 |
+
$$U_{t}(x) = \min_{i \leq N_{t}} \left\{ \text{UCB}_{i}(t) + L d(x, x_{i}) \right\}$$
|
| 396 |
+
$$\leq f(x) + 2 \min_{i} \left\{ r_{i}(t) + L d(x, x_{i}) \right\}$$
|
| 397 |
+
$$= f(x) + 2\rho_{t}(x).$$
|
| 398 |
+
(13)
|
| 399 |
+
## C.4. Proof to Theorem [4.5](#page-2-3)
|
| 400 |
+
*Proof.* We show that any point in the active set must have function value close to optimal.
|
| 401 |
+
*Step 1: Lower certificate validity.* On E, for any sampled point x<sup>i</sup> :
|
| 402 |
+
$$LCB_i(t) = \hat{\mu}_i(t) - r_i(t) \le f(x_i) \le f^*.$$
|
| 403 |
+
(14)
|
| 404 |
+
Taking the maximum over all i:
|
| 405 |
+
$$\ell_t = \max_{i \le N_t} LCB_i(t) \le f^*. \tag{15}$$
|
| 406 |
+
*Step 2: Active set membership implies high UCB.* Let x ∈ At. By definition of the active set [\(2\)](#page-2-7):
|
| 407 |
+
$$U_t(x) \ge \ell_t. \tag{16}$$
|
| 408 |
+
*Step 3: Applying the envelope bound.* By Lemma [4.3:](#page-2-2)
|
| 409 |
+
$$f(x) + 2\rho_t(x) \ge U_t(x) \ge \ell_t. \tag{17}$$
|
| 410 |
+
*Step 4: Rearranging to obtain the containment.* Solving for f(x):
|
| 411 |
+
$$f(x) \ge \ell_t - 2\rho_t(x)$$
|
| 412 |
+
$$= f^* - (f^* - \ell_t) - 2\rho_t(x)$$
|
| 413 |
+
$$\ge f^* - (f^* - \ell_t) - 2 \sup_{x' \in A_t} \rho_t(x')$$
|
| 414 |
+
$$= f^* - 2\Delta_t.$$
|
| 415 |
+
(18)
|
| 416 |
+
Hence
|
| 417 |
+
$$A_t \subseteq \{x : f(x) \ge f^* - 2\Delta_t\}.$$
|
| 418 |
+
#### C.5. Proof to Theorem [4.6](#page-2-4)
|
| 419 |
+
*Proof.* We connect the active set volume to the margin condition via the containment theorem.
|
| 420 |
+
*Step 1: Bounding* ∆t*.* From Theorem [4.5,](#page-2-3) A<sup>t</sup> ⊆ {x : f(x) ≥ f <sup>∗</sup> − 2∆t} where:
|
| 421 |
+
$$\Delta_t = \sup_{x \in A_t} \rho_t(x) + (f^* - \ell_t).$$
|
| 422 |
+
(19)
|
| 423 |
+
For any x ∈ At:
|
| 424 |
+
$$\rho_t(x) = \min_{i} \left\{ r_i(t) + L \cdot d(x, x_i) \right\}$$
|
| 425 |
+
$$\leq \max_{i:x_i \text{ active}} r_i(t) + L \cdot \sup_{x \in A_t} \min_{i} d(x, x_i)$$
|
| 426 |
+
$$= \beta_t + L\eta_t. \tag{20}$$
|
| 427 |
+
Therefore:
|
| 428 |
+
$$\Delta_t \le \beta_t + L\eta_t + \frac{\gamma_t}{2}.\tag{21}$$
|
| 429 |
+
*Step 2: Applying the margin condition.* By Assumption [2.3,](#page-1-3) for any ε > 0:
|
| 430 |
+
$$\operatorname{Vol}(\{x: f(x) \ge f^* - \varepsilon\}) \le C \cdot \varepsilon^{d-\alpha}. \tag{22}$$
|
| 431 |
+
*Step 3: Combining the bounds.* Setting ε = 2∆<sup>t</sup> ≤ 2(β<sup>t</sup> + Lηt) + γt:
|
| 432 |
+
$$\operatorname{Vol}(A_t) \leq \operatorname{Vol}(\{x : f(x) \geq f^* - 2\Delta_t\})$$
|
| 433 |
+
$$\leq C \cdot (2\Delta_t)^{d-\alpha}$$
|
| 434 |
+
$$\leq C \cdot (2(\beta_t + L\eta_t) + \gamma_t)^{d-\alpha}.$$
|
| 435 |
+
(23)
|
| 436 |
+
#### C.6. Proof to Theorem [4.8](#page-3-5)
|
| 437 |
+
*Proof.* We derive the sample complexity by analyzing the requirements for ε-optimality.
|
| 438 |
+
*Step 1: Optimality condition.* To achieve simple regret r<sup>T</sup> ≤ ε, it suffices to ensure:
|
| 439 |
+
$$2(\beta_t + L\eta_t) + \gamma_t \le \varepsilon. \tag{24}$$
|
| 440 |
+
This requires β<sup>t</sup> ≤ ε/6, η<sup>t</sup> ≤ ε/(6L), and γ<sup>t</sup> ≤ ε/3.
|
| 441 |
+
*Step 2: Covering the active set.* By the shrinkage theorem (Theorem [4.6\)](#page-2-4), once 2(β<sup>t</sup> + Lηt) + γ<sup>t</sup> ≤ ε we have:
|
| 442 |
+
$$Vol(A_t) \le C \cdot \varepsilon^{d-\alpha}.$$
|
| 443 |
+
(25)
|
| 444 |
+
To achieve covering radius η = ε/(6L) over a region of volume Cε<sup>d</sup>−<sup>α</sup>, we need:
|
| 445 |
+
$$N_{\text{cover}} = O\left(\frac{\text{Vol}(A_t)}{\eta^d}\right) = O\left(\frac{C\varepsilon^{d-\alpha}}{(\varepsilon/(6L))^d}\right) = O\left(L^d\varepsilon^{-\alpha}\right)$$
|
| 446 |
+
(26)
|
| 447 |
+
distinct sample locations.
|
| 448 |
+
*Step 3: Samples per location.* To achieve confidence radius β<sup>t</sup> ≤ ε/6 at each location, we need:
|
| 449 |
+
$$\sigma\sqrt{\frac{2\log(2N_tT/\delta)}{n_i}} \le \frac{\varepsilon}{6}.$$
|
| 450 |
+
(27)
|
| 451 |
+
Solving for n<sup>i</sup> :
|
| 452 |
+
$$n_i = O\left(\frac{\sigma^2 \log(T/\delta)}{\varepsilon^2}\right).$$
|
| 453 |
+
(28)
|
| 454 |
+
{12}------------------------------------------------
|
| 455 |
+
Step 4: Total sample complexity. Combining Steps 2 and 3:
|
| 456 |
+
$$T = N_{\text{cover}} \cdot n_{i}$$
|
| 457 |
+
$$= O(L^{d} \varepsilon^{-\alpha}) \cdot O\left(\frac{\sigma^{2} \log(T/\delta)}{\varepsilon^{2}}\right)$$
|
| 458 |
+
$$= \tilde{O}(L^{d} \varepsilon^{-(2+\alpha)}). \tag{29}$$
|
| 459 |
+
#### C.7. Proof to Theorem 4.9
|
| 460 |
+
*Proof.* We construct a hard instance via a randomized reduction.
|
| 461 |
+
Step 1: Hard instance construction (continuous bump). We use a standard "one bump among M locations" construction that ensures global Lipschitz continuity. Partition $\mathcal{X} = [0,1]^d$ into $M = \varepsilon^{-\alpha}$ disjoint cells with centers $\{c_1,\ldots,c_M\}$ , each cell having diameter $\Theta(\varepsilon^{\alpha/d})$ . Select one cell $i^*$ uniformly at random to contain the optimum, and place $x^* = c_{i^*}$ . Define:
|
| 462 |
+
$$f(x) = (1 - \varepsilon) + \varepsilon \cdot \max \left\{ 0, 1 - \frac{L \|x - x^*\|}{\varepsilon} \right\}. \quad (30)$$
|
| 463 |
+
This is a cone/bump centered at $x^*$ that peaks at $f(x^*)=1$ and decreases linearly with slope L until it reaches the baseline value $1-\varepsilon$ at radius $\varepsilon/L$ from $x^*$ . Outside this radius, $f(x)\equiv 1-\varepsilon$ . The function is globally L-Lipschitz: within the bump, the gradient has magnitude L; outside, the function is constant; and at the boundary $\|x-x^*\|=\varepsilon/L$ , both pieces match at value $1-\varepsilon$ .
|
| 464 |
+
Verification of Assumption 2.3. The $\varepsilon$ -near-optimal set $\{x:f(x)\geq 1-\varepsilon\}$ is exactly the ball of radius $\varepsilon/L$ around $x^*$ , which has volume $\operatorname{Vol}(B_{\varepsilon/L})=O((\varepsilon/L)^d)=O(\varepsilon^d)$ . With $M=\varepsilon^{-\alpha}$ candidate locations, only one contains the bump, so the near-optimal fraction of the domain is $O(\varepsilon^d)$ . Since the domain has unit volume, $\operatorname{Vol}(\{f\geq f^*-\varepsilon\})=O(\varepsilon^d)\leq C\varepsilon^{d-\alpha}$ for $\alpha\geq 0$ , satisfying Assumption 2.3.
|
| 465 |
+
Step 2: Information-theoretic lower bound. To identify the correct cell with probability $\geq 2/3$ , the algorithm must distinguish between M hypotheses. By Fano's inequality, this requires:
|
| 466 |
+
$$\sum_{i=1}^{M} n_i \cdot \text{KL}(P_i || P_0) \ge \log(M/3), \tag{31}$$
|
| 467 |
+
where $n_i$ is the number of samples in cell i, and $\mathrm{KL}(P_i || P_0)$ is the KL divergence between observations under hypothesis i versus the null.
|
| 468 |
+
Step 3: Per-cell sample requirement. For $\sigma$ -sub-Gaussian noise, distinguishing a cell with optimum from one without requires:
|
| 469 |
+
$$n_i = \Omega\left(\frac{\sigma^2}{\varepsilon^2}\right) \tag{32}$$
|
| 470 |
+
samples per cell to detect the $\varepsilon$ gap with constant probability. Step 4: Total sample complexity. Summing over all M cells:
|
| 471 |
+
$$T = \Omega \left( M \cdot \frac{\sigma^2}{\varepsilon^2} \right) = \Omega \left( \varepsilon^{-\alpha} \cdot \varepsilon^{-2} \right) = \Omega \left( \varepsilon^{-(2+\alpha)} \right). \tag{33}$$
|
| 472 |
+
#### C.8. Proof to Theorem 5.1
|
| 473 |
+
*Proof.* We prove each claim separately.
|
| 474 |
+
Proof of (1): Bounded doubling events. Each doubling event multiplies $\hat{L}$ by 2. Starting from $\hat{L}_0 \leq L^*$ :
|
| 475 |
+
$$\hat{L}_k = 2^k \hat{L}_0$$
|
| 476 |
+
after $k$ doublings. (34)
|
| 477 |
+
The algorithm stops doubling when $\hat{L} \geq L^*$ , which requires:
|
| 478 |
+
$$2^K \hat{L}_0 \ge L^* \implies K \ge \log_2(L^*/\hat{L}_0).$$
|
| 479 |
+
(35)
|
| 480 |
+
Hence $K \leq \lceil \log_2(L^*/\hat{L}_0) \rceil$ .
|
| 481 |
+
*Proof of (2): Final estimate accuracy.* A violation is detected when:
|
| 482 |
+
$$|\hat{\mu}_i - \hat{\mu}_j| - 2(r_i + r_j) > \hat{L} \cdot d(x_i, x_j).$$
|
| 483 |
+
(36)
|
| 484 |
+
On the good event $\mathcal{E}$ :
|
| 485 |
+
$$|f(x_i) - f(x_j)| \le |\hat{\mu}_i - \hat{\mu}_j| + 2(r_i + r_j).$$
|
| 486 |
+
(37)
|
| 487 |
+
If $\hat{L} \geq L^*$ , then by Lipschitz continuity:
|
| 488 |
+
$$|f(x_i) - f(x_i)| \le L^* \cdot d(x_i, x_i) \le \hat{L} \cdot d(x_i, x_i), \quad (38)$$
|
| 489 |
+
so no violation can occur. Thus violations only occur when $\hat{L} < L^*$ , and after all doublings complete, $\hat{L} \geq L^*$ . Since we double (rather than increase by smaller factors), $\hat{L} \leq 2L^*$ .
|
| 490 |
+
*Proof of (3): Sample complexity overhead.* Between doublings, CGP runs with either:
|
| 491 |
+
- Invalid $\hat{L} < L^*$ (before sufficient doublings): certificates may be incorrect, but each such phase has at most O(T/K) samples before a violation triggers doubling.
|
| 492 |
+
- Valid $\hat{L} \geq L^*$ (after final doubling): CGP achieves $\tilde{O}(\varepsilon^{-(2+\alpha)})$ complexity by Theorem 4.8.
|
| 493 |
+
There are at most K invalid phases, each contributing O(T/K) samples. The final valid phase dominates, giving total complexity $T = \tilde{O}(\varepsilon^{-(2+\alpha)} \cdot K) = \tilde{O}(\varepsilon^{-(2+\alpha)} \cdot \log(L^*/\hat{L}_0))$ .
|
| 494 |
+
{13}------------------------------------------------
|
| 495 |
+
## C.9. Certificate Validity Under Adaptive Lipschitz Estimation
|
| 496 |
+
**Lemma C.1** (Certificate validity once $\hat{L} \geq L^*$ ). Assume the good event $\mathcal{E}$ holds. Fix any time t at which the current estimate satisfies $\hat{L} \geq L^*$ . Then the envelope constructed with $\hat{L}$ satisfies $f(x) \leq U_t(x)$ for all $x \in \mathcal{X}$ , and consequently the active set
|
| 497 |
+
$$A_t = \{ x \in \mathcal{X} : U_t(x) \ge \ell_t \}$$
|
| 498 |
+
contains $x^*$ and certifies that any $x \notin A_t$ is suboptimal (in the sense $U_t(x) < \ell_t \le f^*$ ).
|
| 499 |
+
*Proof.* If $\hat{L} \geq L^*$ , then for all $x, y \in \mathcal{X}$ we have $|f(x) - f(y)| \leq L^* d(x, y) \leq \hat{L} d(x, y)$ , i.e., f is $\hat{L}$ -Lipschitz. On $\mathcal{E}$ , for each sampled point $x_i$ , $f(x_i) \leq \mathrm{UCB}_i(t)$ . Therefore for all x,
|
| 500 |
+
$$f(x) \le f(x_i) + \hat{L} d(x, x_i) \le UCB_i(t) + \hat{L} d(x, x_i).$$
|
| 501 |
+
Taking the minimum over i yields $f(x) \leq U_t(x)$ for all x. In particular, $U_t(x^*) \geq f^*$ . Also $\ell_t = \max_i \mathrm{LCB}_i(t) \leq f^*$ on $\mathcal{E}$ , hence $U_t(x^*) \geq \ell_t$ and so $x^* \in A_t$ . Finally, if $x \notin A_t$ , then $U_t(x) < \ell_t \leq f^*$ , certifying x cannot be optimal under $\mathcal{E}$ .
|
| 502 |
+
Remark C.2 (Why pre-final certificates need not be valid). If $\hat{L} < L^*$ , then the envelope may fail to upper-bound f globally, and the rule $U_t(x) \geq \ell_t$ can (in principle) exclude near-optimal points. CGP-Adaptive therefore guarantees certificate validity only after the final doubling event ensures $\hat{L} \geq L^*$ .
|
| 503 |
+
## C.10. Global Safety of Certified Restarts (No False Elimination)
|
| 504 |
+
We restate and prove the key safety property underlying certified restarts in CGP-TR.
|
| 505 |
+
**Lemma C.3** (No false certified restart for the region containing $x^*$ ). Fix any trust region $\mathcal{T}^*$ such that $x^* \in \mathcal{T}^*$ . On the good event $\mathcal{E}$ , the certified restart condition
|
| 506 |
+
$$u_t^{(\mathcal{T}^*)} := \max_{x \in \mathcal{T}^*} U_t(x) < \ell_t$$
|
| 507 |
+
never holds. Hence, $\mathcal{T}^*$ is never restarted by the certified rule.
|
| 508 |
+
*Proof.* On the good event $\mathcal{E}$ , Lemma 4.2 implies $f(x) \leq U_t(x)$ for all $x \in \mathcal{X}$ . Since $x^* \in \mathcal{T}^*$ ,
|
| 509 |
+
$$u_t^{(\mathcal{T}^*)} = \max_{x \in \mathcal{T}^*} U_t(x) \ge U_t(x^*) \ge f(x^*) = f^*.$$
|
| 510 |
+
Also, by definition $\ell_t = \max_i \mathrm{LCB}_i(t)$ . On $\mathcal{E}$ , $\mathrm{LCB}_i(t) \leq f(x_i) \leq f^*$ for every sampled point $x_i$ , hence $\ell_t \leq f^*$ . Therefore,
|
| 511 |
+
$$u_t^{(\mathcal{T}^*)} \geq f^* \geq \ell_t,$$
|
| 512 |
+
so the strict inequality $u_t^{(\mathcal{T}^*)} < \ell_t$ cannot occur on $\mathcal{E}$ .
|
| 513 |
+
#### C.11. Proof to Theorem 6.1
|
| 514 |
+
We first establish a key lemma showing that certified elimination is safe.
|
| 515 |
+
**Lemma C.4** (Certified elimination). On the good event $\mathcal{E}$ , for any trust region $\mathcal{T}_j$ ,
|
| 516 |
+
$$u_t^{(j)} := \max_{x \in \mathcal{T}_j} U_t(x) < \ell_t \quad \Longrightarrow \quad \sup_{x \in \mathcal{T}_i} f(x) < f(x^*).$$
|
| 517 |
+
*Proof.* On $\mathcal{E}$ , by Lemma 4.2, $f(x) \leq U_t(x)$ for all $x \in \mathcal{X}$ . Also $\ell_t = \max_i \mathrm{LCB}_i(t) \leq f(x^*)$ on $\mathcal{E}$ since each $\mathrm{LCB}_i(t) \leq f(x_i) \leq f(x^*)$ . Therefore
|
| 518 |
+
$$\sup_{x \in \mathcal{T}_j} f(x) \le \sup_{x \in \mathcal{T}_j} U_t(x) = u_t^{(j)} < \ell_t \le f(x^*),$$
|
| 519 |
+
which proves the claim.
|
| 520 |
+
**Proof of Theorem 6.1(1): No false restarts.** If $x^* \in \mathcal{T}^*$ , then $u_t^{(\mathcal{T}^*)} = \max_{x \in \mathcal{T}^*} U_t(x) \geq U_t(x^*) \geq f(x^*)$ on $\mathcal{E}$ . Also $\ell_t \leq f(x^*)$ on $\mathcal{E}$ . Hence $u_t^{(\mathcal{T}^*)} \geq \ell_t$ and the restart condition $u_t^{(\mathcal{T}^*)} < \ell_t$ never triggers.
|
| 521 |
+
**Proof of Theorem 6.1(2): Local certificate.** Conditioned on $T^*$ evaluations within $\mathcal{T}^*$ , the CGP analysis applies verbatim on the restricted domain $\mathcal{T}^*$ : the good event $\mathcal{E}$ implies all within-region confidence bounds hold; Lipschitz continuity holds on $\mathcal{T}^*$ ; and Assumption 2.3 holds restricted to $\mathcal{T}^*$ . Therefore the containment and shrinkage results follow with $\mathcal{X}$ replaced by $\mathcal{T}^*$ , yielding $\operatorname{Vol}(A_T^{(\mathcal{T}^*)}) \leq C\varepsilon^{d-\alpha}$ after $T^* = \tilde{O}(\varepsilon^{-(2+\alpha)})$ within-region samples.
|
| 522 |
+
**Proof of Theorem 6.1(3): Allocation bound.** Fix a sub-optimal region j with gap $\Delta_j > 0$ . On $\mathcal{E}$ , using the envelope bound from Lemma 4.3, we have for all x:
|
| 523 |
+
$$U_t(x) < f(x) + 2\rho_t(x),$$
|
| 524 |
+
where $\rho_t(x) = \min_i \{r_i(t) + Ld(x, x_i)\}$ . Restricting to points sampled in $\mathcal{T}_j$ and using the within-region replication schedule, we bound $\rho_t(x) \leq \beta_j(n) + L \cdot \operatorname{diam}(\mathcal{T}_j)$ , yielding
|
| 525 |
+
$$u_t^{(j)} = \max_{x \in \mathcal{T}_j} U_t(x) \le \sup_{x \in \mathcal{T}_j} f(x) + 2\beta_j(n) + 2L \cdot \operatorname{diam}(\mathcal{T}_j),$$
|
| 526 |
+
after n within-region samples. By the diameter condition $L \operatorname{diam}(\mathcal{T}_j) \leq \Delta_j/8$ and by requiring $\beta_j(n) \leq \Delta_j/8$ , we obtain
|
| 527 |
+
$$u_t^{(j)} \le \sup_{\mathcal{T}_j} f + \Delta_j/4 + \Delta_j/4 = f^* - \Delta_j/2.$$
|
| 528 |
+
Meanwhile, on $\mathcal{E}$ we have $\ell_t \leq f^*$ always, and once the region containing $x^*$ has been sampled sufficiently (which occurs because it is never restarted and is favored by UCB
|
| 529 |
+
{14}------------------------------------------------
|
| 530 |
+
selection), $\ell_t$ becomes at least $f^* - \Delta_j/4$ . Hence eventually $u_t^{(j)} < \ell_t$ , triggering certified restart/elimination.
|
| 531 |
+
Solving $\beta_j(n) \leq \Delta_j/8$ under $\beta_j(n) \leq c_\sigma \sqrt{\log(c_T/\delta)/n}$ gives
|
| 532 |
+
$$n \ge \frac{64c_{\sigma}^2}{\Delta_i^2} \log \left(\frac{c_T}{\delta}\right),\,$$
|
| 533 |
+
yielding the stated bound $N_j \leq \frac{64c_\sigma^2}{\Delta_j^2}\log(c_T/\delta) + 1$ .
|
| 534 |
+
#### C.12. Proof to Proposition 7.1
|
| 535 |
+
*Proof.* We analyze each case and the certificate validity separately.
|
| 536 |
+
*Proof of (1): High smoothness ratio case.* When $\rho \geq 0.5$ , CGP-Hybrid continues with CGP in Phase 2. The algorithm is identical to vanilla CGP, so by Theorem 4.8:
|
| 537 |
+
$$T = \tilde{O}(\varepsilon^{-(2+\alpha)}). \tag{39}$$
|
| 538 |
+
Proof of (2): Low smoothness ratio case. When $\rho < 0.5$ , the function is significantly smoother near the optimum. After Phase 1, CGP has reduced the active set to volume $V < 0.1 \cdot \mathrm{Vol}(\mathcal{X})$ . In Phase 2, GP-UCB operates within $A_t$ , which has:
|
| 539 |
+
- Effective diameter diam $(A_t) = O(V^{1/d})$ .
|
| 540 |
+
- Local Lipschitz constant $L_{local} = \rho \cdot L < 0.5L$ .
|
| 541 |
+
The sample complexity of GP-UCB on this restricted domain depends on the kernel's maximum information gain $\gamma_T$ over $A_t$ (Srinivas et al., 2009). For commonly used kernels (Matérn, SE), $\gamma_T$ scales polylogarithmically with T when the domain is bounded. The reduced diameter $O(V^{1/d})$ and local smoothness $\rho < 0.5$ empirically yield faster convergence than continuing CGP; we validate this empirically in Section 9 rather than claiming a specific rate.
|
| 542 |
+
*Proof of (3): Certificate validity.* The certificate $A_T$ is computed in Phase 1 using CGP's Lipschitz envelope construction. By Theorem 4.5, on the good event $\mathcal{E}$ :
|
| 543 |
+
$$x^* \in A_T$$
|
| 544 |
+
with probability $\geq 1 - \delta$ . (40)
|
| 545 |
+
This guarantee depends only on the Lipschitz assumption and confidence bounds, not on the Phase 2 optimization method. Therefore, switching to GP-UCB in Phase 2 does not invalidate the certificate: $A_T$ still contains $x^*$ with high probability, and any point outside $A_T$ remains certifiably suboptimal.
|
| 546 |
+
### <span id="page-14-0"></span>**D. Practical Guidance**
|
| 547 |
+
#### **D.1. Adaptive Lipschitz Estimation**
|
| 548 |
+
CGP-Adaptive removes the requirement for known L with $O(\log T)$ overhead. The doubling scheme is conservative
|
| 549 |
+
but provably correct; more aggressive schemes (e.g., multiplicative updates with factor 1.5) may reduce overhead but risk certificate invalidation.
|
| 550 |
+
We recommend initializing $\hat{L}_0$ from finite differences on initial Sobol samples:
|
| 551 |
+
$$\hat{L}_0 = \max_{i \neq j} \frac{|y_i - y_j|}{d(x_i, x_j)}$$
|
| 552 |
+
over the first 10 samples. This typically underestimates L by a factor of 2 to 10, requiring 1 to 4 doublings to reach $\hat{L} \geq L^*$ .
|
| 553 |
+
### **D.2. Trust Region Configuration**
|
| 554 |
+
CGP-TR trades global certificates for scalability. The local certificates within trust regions still enable principled stopping and progress assessment, but do not guarantee global optimality.
|
| 555 |
+
Recommended settings:
|
| 556 |
+
- Number of trust regions: $n_{\text{trust}} = 5$ (balances exploration vs. overhead)
|
| 557 |
+
- Initial radius: $r_0 = 0.2$ (covers 20% of domain diameter per region)
|
| 558 |
+
- Minimum radius: $r_{\min} = 0.01$ (prevents over-contraction)
|
| 559 |
+
- Failure threshold: $\tau_{\text{fail}} = 10$ (triggers contraction after 10 non-improving samples)
|
| 560 |
+
For applications requiring global certificates in high dimensions, combining CGP-TR with random embeddings (Wang et al., 2016) is promising: project to a low-dimensional subspace, run CGP with global certificates, then lift back.
|
| 561 |
+
#### **D.3. Smoothness Detection**
|
| 562 |
+
CGP-Hybrid's smoothness detection via $\rho = L_{\rm local}/L_{\rm global}$ is heuristic but effective. We estimate $L_{\rm local}$ from points within $A_t$ using the same finite difference approach as $L_{\rm global}$ .
|
| 563 |
+
The threshold $\rho_{\rm thresh}=0.5$ was selected via cross-validation on held-out benchmarks. More sophisticated detection could use local GP posterior variance or curvature estimates. The key insight is that CGP's certificate remains valid regardless of Phase 2 method, so switching is always safe.
|
| 564 |
+
#### D.4. When to Use CGP
|
| 565 |
+
CGP is well suited when:
|
| 566 |
+
Evaluations are expensive and interpretable progress is valued
|
| 567 |
+
{15}------------------------------------------------
|
| 568 |
+
851852
|
| 569 |
+
854855
|
| 570 |
+
858859
|
| 571 |
+
860861
|
| 572 |
+
862863
|
| 573 |
+
865866
|
| 574 |
+
867868
|
| 575 |
+
869870
|
| 576 |
+
872873
|
| 577 |
+
875876
|
| 578 |
+
877878
|
| 579 |
+
- 2. The objective has margin structure (sharp peak rather than wide plateau)
|
| 580 |
+
- 3. Lipschitz continuity is a reasonable assumption
|
| 581 |
+
- 4. Dimension is moderate ( $d \le 15$ for vanilla CGP, $d \le 100$ for CGP-TR)
|
| 582 |
+
- 5. Anytime stopping decisions are needed
|
| 583 |
+
For very high-dimensional problems (d>100), trust region methods like TuRBO may scale better. For smooth problems with cheap evaluations, GP-based methods may be more sample efficient due to their ability to exploit higher-order smoothness.
|
| 584 |
+
## <span id="page-15-0"></span>E. Experimental Details
|
| 585 |
+
#### E.1. Baseline Configurations
|
| 586 |
+
- Random Search: Sobol sequences for quasi-random sampling
|
| 587 |
+
- **GP-UCB**: Matérn-5/2 kernel via BoTorch, $\beta_t = 2\log(t^2\pi^2/6\delta)$
|
| 588 |
+
- **TuRBO**: Default settings from Eriksson et al. (2019), 1 trust region
|
| 589 |
+
- HEBO: Heteroscedastic GP with input warping, default settings
|
| 590 |
+
- **BORE**: Tree-Parzen estimator with density ratio, default settings
|
| 591 |
+
- **HOO**: Binary tree with $\nu_1 = 1$ , $\rho = 0.5$
|
| 592 |
+
- StoSOO: k = 3 children per node, $h_{\text{max}} = 20$
|
| 593 |
+
- LIPO: Pure Lipschitz optimization, L estimated online
|
| 594 |
+
- SAASBO: Sparse axis-aligned GP, 10 active dimensions
|
| 595 |
+
#### E.2. Benchmark Details
|
| 596 |
+
#### Low-dimensional.
|
| 597 |
+
- Needle-2D: $f(x) = 1 \|x x^*\|^{1/\alpha}$ with $\alpha = 2$ , sharp peak
|
| 598 |
+
- Branin: Standard 2D benchmark with 3 global optima
|
| 599 |
+
- Hartmann-6: 6D benchmark with narrow global basin
|
| 600 |
+
- Levy-5: 5D benchmark with global structure
|
| 601 |
+
- Rosenbrock-4: 4D benchmark with curved valley
|
| 602 |
+
#### Medium-dimensional.
|
| 603 |
+
- Ackley-10: 10D benchmark with many local optima
|
| 604 |
+
- **SVM-RBF-6**: Real hyperparameter tuning $(C, \gamma, 4)$ preprocessing) on MNIST
|
| 605 |
+
- LunarLander-12: RL reward optimization with 12 policy parameters
|
| 606 |
+
#### High-dimensional.
|
| 607 |
+
- **Rover-60**: Mars rover trajectory with 60 waypoint parameters (Wang et al., 2016)
|
| 608 |
+
- NAS-36: Neural architecture search on CIFAR-10, 36 continuous encodings
|
| 609 |
+
- Ant-100: MuJoCo Ant locomotion, 100 morphology and control parameters
|
| 610 |
+
#### E.3. Computational Resources
|
| 611 |
+
All experiments run on AMD EPYC 7763 with 256GB RAM. CGP variants use NumPy/SciPy; GP baselines use BoTorch/GPyTorch with GPU acceleration (NVIDIA A100) where available.
|