{"id":"c0a6dbb4-141a-444d-a87a-bcee3071a30c","arxiv_id":"2412.11485","paper_version":4,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Inexact proximal point methods, with Gibbs-sampling or tensor-train estimates of the proximal operator, converge to the global minimum of nonconvex black-box functions under a gap assumption.","lead":"This paper develops inexact proximal point algorithms that can provably converge to the global minimum of nonconvex functions using only function evaluations, provided the proximal estimates are accurate and the landscape satisfies a gap assumption. It also introduces a tensor-train-based estimator that makes the iteration practical, and validates both approaches on benchmarks and engineering problems.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The implemented TT- and MC-IPP variants do not verify the summability condition (4)/(13) on which the global-convergence theorems hinge; the tested parameters are admitted to be outside the corollaries' sufficient regimes, so the practical algorithms' global convergence is unproven.","rationale":"The reader's CONDITIONAL verdict is appropriate. My stress test did not find a flaw in Theorem 1's main argument: the summation in (6) plus condition (4) does give boundedness and f(x_k)→f_min under the stated assumptions, and the T>M/µ device works. The concern is the bridge from theorem to algorithms. The paper is honest about the unrealistic mesh bound and calls its parameter choices heuristic, but the abstract and contribution section claim practical global-convergence guarantees for TT-IPP and MC-IPP; that claim is not supported unless (4)/(13) is verified or replaced. Corollary 4 is the weakest formal step: it asserts, rather than proves, that the MC schedule satisfies (13), and the rejection loop at Line 7 of Algorithm 4 is not part of the standard MC estimator analyzed. The proposed experiment would settle whether the tested regime actually meets the hypotheses. Since this is an addressable gap rather than an internal inconsistency, the verdict should remain CONDITIONAL; no change from the reader's assessment.","tokens_in":33086,"tokens_out":18541,"duration_ms":163987,"concrete_test":"Run TT-IPP and MC-IPP on Griewank R100 (TT) and R20 (MC) with the paper's Table 1 parameters, but at each iteration replace the estimator by a high-accuracy reference proximal point (e.g., multi-start local minimization or a fine-mesh TT with h=δ^2) and compute e_k = ||raw_estimate_k - prox_{t_k} f(x_k)||. Record the partial sums S_K = Σ_{k≤K} e_k^2 (for MC use the undamped prox_MC estimate) and the MC tail counts P(e_k^2 > ε_k) over repeated runs. If S_K diverges or the tail probabilities do not form a summable sequence, then conditions (4)/(13) fail for the implemented parameter schedule. As a control, run TT-IPP with γ large enough to satisfy Corollary 3 (e.g., γ=5 for d=10, s=2) and compare convergence; this distinguishes a merely conservative bound from a genuine violation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1 itself is internally coherent, but the paper's practical claim depends on the hypotheses (4) and (13) being satisfied by the estimators in Algorithms 3 and 4. That link is the weakest point. For TT-IPP, Corollary 3 requires h_k = O(δ_k^{(d+2)/(2s)+1}) and ǫ_TT ≤ exp(-φ_min/δ_k) at every iteration. Algorithm 3 fixes h_0=0.1, γ=1.1, C=10 and refines only when h_k > C δ_k^γ. For d=100 and s=2, the corollary's exponent is about 26, while the implemented refinement policy keeps h_k ~ δ_k^{1.1} up to the constant C; this is many orders of magnitude too coarse for the theoretical error bound to vanish. The paper itself states the mesh requirement is unrealistic, so condition (4) is not established for the implemented schedule. For MC-IPP, Corollary 4 asserts that C=1.1, c=0.9 and α_min=0.2 make (13) hold without verifying the implied constants, and the Line 7 rejection/resampling step is an adaptive stopping rule that the standard MC variance computation in Section 5.2 does not cover. If (4)/(13) fail, Theorems 1–2 do not apply, so the global-convergence guarantee for the tested algorithms rests on an unverified condition.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript develops an inexact proximal point (IPP) framework for zeroth-order global minimization of nonconvex functions with a unique global minimizer. Under Assumptions 1–3, Theorem 1 proves convergence of the deterministic IPP iterate values to the global minimum when the proximal-estimation errors satisfy the summability condition (4), and Theorem 2 gives an almost-sure analogue under the probabilistic condition (13). Section 3 relates the proximal operator to the mean of a Gibbs measure and proves convergence of that mean to the proximal point as δ→0+, with an O(δ) rate under a nondegeneracy condition. Section 4 introduces tensor-train (TT) approximations of the Gibbs integrand, using the randomized TT cross algorithm, and provides error bounds for the resulting proximal estimates in Proposition 2 and Corollary 2. Section 5 defines two practical algorithms, TT-IPP and MC-IPP, with adaptive refinement and sampling schedules, and claims their convergence as corollaries of Theorems 1 and 2. Section 6 reports experiments on benchmark functions and two applications, showing competitive accuracy and function-evaluation counts for the proposed methods.","tokens_in":33474,"tokens_out":6643,"duration_ms":64304,"significance":"If the theoretical claims are fully established, the paper would provide the first convergence guarantee of an inexact proximal point method to a global minimizer under only summability of estimation errors, and it offers a coherent Gibbs-measure justification for sampling-based proximal estimates. The TT-based estimator is a genuine algorithmic contribution: it exploits Sobolev smoothness and low-rank structure to reduce the cost of the discretized integrals, and the code is publicly available. The experimental section is broad, covering many benchmarks and dimensions up to 100, with TT-IPP matching or beating several standard global solvers. However, the central link between the implemented algorithms and the theorem hypotheses is the weakest point: the sufficient conditions (4) and (13) are not verified for the parameter schedules actually used in the experiments, and the paper explicitly admits that the mesh requirement behind Corollary 3 is unrealistic in high dimensions. The proof of the deterministic theorem is essentially correct but contains a circularity in the choice of T, and the stochastic theorem inherits the same issue in a sample-path-dependent form.","major_comments":[{"comment":"The choice of T via (12) and (37) is circular. The bound M in (9) is derived after T is fixed, yet (12) requires T > M/μ, so no argument shows that a single T can be chosen in advance. In Theorem 2, M is sample-path-dependent, making (37) a condition on a random constant rather than a verifiable parameter choice. The proof needs either a uniform-in-T bound on M, an explicit construction of T, or a reformulation as an existence statement. This is load-bearing because the entire convergence proof relies on the stationarity of the limit point via (11)-(12).","section":"Section 2, Theorem 1 and Appendix A.1"},{"comment":"The implemented TT-IPP schedule does not satisfy the sufficient conditions used to invoke Theorem 1. Corollary 3 requires γ > (d+2)/(2s)+1 and, via (31), h_k = O(δ_k^{(d+2)/(2s)+1}) and ε_TT ≤ exp(-φ_min/δ_k) at every iteration. Algorithm 3 uses h0=0.1, γ=1.1, C=10, and refines only when h_k > C δ_k^γ, so h_k tracks C δ_k^{1.1} rather than the required power; for d=100 and s=2 the required exponent is about 26.5. The paper itself states after (29) that such a mesh bound is unrealistic, and no numerical or analytic evidence is provided that (4) holds for the tested parameter values. Consequently Theorem 1 does not currently prove global convergence of TT-IPP as implemented.","section":"Section 5.1, Corollary 3 and Algorithm 3"},{"comment":"The proof of Corollary 4 asserts that choosing C > 1 sufficiently large makes condition (13) hold, but the implemented values C=1.1, c=0.9, α_min=0.2, p=0.8 are not accompanied by any verification of the implied constants. The MC variance analysis preceding (35) is derived for a fixed δ, fixed sample size N, and no rejection step; Algorithm 4 adaptively changes δ, N, and α, and Line 7 introduces probability-p rejection with resampling, which the standard MC covariance computation at the top of Section 5.2 does not cover. Thus condition (13) of Theorem 2 is not established for MC-IPP as implemented, and the almost-sure convergence claim for the practical algorithm remains unsupported.","section":"Section 5.2, Corollary 4 and Algorithm 4"}],"minor_comments":[{"comment":"After Eq. (5), 'inequity' should be 'inequality', and later in the proof 'arbitray κ' should be 'arbitrary κ'.","section":"Section 2, Theorem 1 proof"},{"comment":"The subdifferential ∂f(x) is stated as a subset of R^n but should be R^d to match the domain of f.","section":"Definition 1"},{"comment":"The summation limits in the definition of z̄* are written as 'j−1' instead of 'j=1' in both places; this typo makes the formula unintelligible.","section":"Corollary 1 and Appendix A.5"},{"comment":"The while-loop condition uses '<' for the error test; presumably the loop should continue while the error exceeds τ_stop, so the inequality direction should be '>' or the text should clarify the stopping convention.","section":"Algorithm 2"},{"comment":"In the proof of Theorem 2, the sentence 'Combining with (4)' after Eq. (36) should refer to the almost-sure consequence of (13) together with the Borel–Cantelli argument, since (4) is not an assumption of Theorem 2.","section":"Appendix A.1"},{"comment":"There are several typographical errors in the table entries, including 'Brwon R10' (should be 'Brown R10') and 'Alphine 1' (should be 'Alpine 1').","section":"Tables 2, 3, and 6"}],"recommendation":"major_revision","confidential_remarks":"The paper is within scope for a mathematical optimization journal and the central theoretical framework is plausible, but the practical convergence claims in Corollaries 3 and 4 rest on unverified hypotheses. The referee report focuses on the verification gap and on the circular choice of T. Additionally, the handling editor may wish to scrutinize the relationship to [26], [52], and [71]: the IPP framework and MC estimator are close to the HJ-MAD method, and the novelty statement should be carefully bounded. These are scientific rather than ethical concerns."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nYou should know two things about this paper. First, the theory is genuinely new: Theorem 1 shows that an inexact proximal point method converges to the global minimum when the estimation errors are square-summable, and Theorem 2 gives the almost-sure stochastic version. The claim that this is the first IPP global convergence result in the literature seems justified – prior IPP analyses either used exact proximal operators or converged to critical points. Second, the implemented algorithms, TT-IPP and MC-IPP, never verify the error conditions those theorems require. The paper is honest about this, but it means the headline numerical results are not covered by the paper's own convergence guarantees.\n\nWhat I think is solid: Theorem 1's telescoping argument is careful, and the Gibbs-expectation O(δ) rate in Section 3 is standard Laplace asymptotics, cleanly stated. Proposition 2 and Corollary 2 give explicit worst-case bounds for TT-based proximal estimation; reusing a TT approximation via Hadamard products when δ halves is an elegant trick. The experiments are broad, the baselines are handled fairly, and the code is public.\n\nNow the soft spots, in proportion. The largest is the gap between Corollaries 3–4 and the experiments. Corollary 3 requires h_k = O(δ_k^{(d+2)/(2s)+1}) and ǫ_TT ≤ exp(-φ_min/δ_k). For d=100, s=2, that exponent is about 26; the code uses h_0=0.1, γ=1.1, and refines only when h_k > C δ_k^γ, many orders of magnitude coarser. The authors call this mesh condition unrealistic and say they choose parameters heuristically – that is honest, but it means condition (4) is unverified. Corollary 4 asserts condition (13) holds for sufficiently large C without checking constants, and the rejection resampling on line 7 of Algorithm 4 is absent from the variance analysis. If (4)/(13) fail, Theorems 1–2 do not apply, so the practical algorithms' global convergence is unproven.\n\nMinor items: the choice of T in Theorem 1 is existential (M/μ) rather than explicit; Theorem 2's proof is a sketch; a few table entries have typo-level problems (e.g., 'Brown', 'Alphine').\n\nWho gets value: researchers in derivative-free global optimization, especially those working with tensor trains, will find the IPP framework and the TT error analysis worth having. Practitioners in moderate dimensions should treat the empirical success as promising but not as certified convergence.\n\nRecommendation: send it to peer review; it deserves a serious referee. The revision should align claims: either verify the error conditions under realistic assumptions, or clearly label the practical algorithms as heuristics, with the corollaries stated as sufficient conditions that the experiments do not meet.","headline":"A genuinely new theoretical result on global convergence of inexact proximal point methods, but the implemented TT- and MC-IPP algorithms do not meet the conditions the theorems require – worth peer review, needs a revision aligning claims with hypotheses.","tokens_in":34041,"tokens_out":6886,"would_cite":true,"duration_ms":54166,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["49M37","65K05","90C26","90C56"],"pacs":[],"model":"deepseek-v4-flash","headline":"Sample-based proximal estimates can provably drive iterates to the global minimum","keywords":["global optimization","zeroth-order optimization","inexact proximal point method","Gibbs measure","tensor train approximation","Monte Carlo integration","nonconvex optimization","derivative-free optimization"],"falsifier":"Implement TT-IPP or MC-IPP on a coercive continuous nonconvex function with multiple local minima using the Section 6 parameter values ($h=0.1$, $\\gamma=1.1$ for TT-IPP; $C=1.1$, $c=0.9$ for MC-IPP) and record $\\sum_{k=0}^\\infty \\|y_k-\\hat{x}_k\\|^2$; if this sum diverges while $f(x_k)$ still approaches $f_{\\min}$, the observed convergence is not explained by Theorem 1's hypotheses, and if the iterates stall at a local minimizer whenever the sum diverges, the practical algorithms fail exactly where the theory's condition is violated.","tokens_in":32824,"feed_emoji":"🎯","tokens_out":10227,"duration_ms":86167,"temperature":0.7,"pith_summary":"This paper tries to establish that a proximal point iteration can find the global minimum of a continuous nonconvex function even when the proximal operator is only estimated approximately, provided the squared estimation errors are summable over iterations. Its central theorem (Theorem 1) states that under mild coercivity and non-oscillation assumptions, the values $f(x_k)$ converge to $f_{\\min}$ for deterministic inexact estimates satisfying $\\sum_{k=0}^\\infty \\|y_k-\\hat{x}_k\\|^2<\\infty$; Theorem 2 gives the same almost-sure conclusion for stochastic estimates satisfying a Borel-Cantelli-type tail condition. The paper also proves that the Gibbs-measure expectation converges to the proximal point at rate $O(\\delta)$ as $\\delta\\to 0^+$, which justifies approximating proximal operators by sampling. If the results hold, global zeroth-order optimization becomes achievable with only approximate, sample-based proximal operators, and the two proposed algorithms, TT-IPP and MC-IPP, instantiate this in practice.","feed_headline":"Inexact proximal steps provably find global minima","feed_subtitle":"Square-summable estimation errors are enough for zeroth-order proximal iterations to reach the global minimum.","key_machinery":"The mechanism is the inexact proximal iteration with adaptive regularization: when consecutive iterates barely move, the parameter $t_k$ is increased toward its ceiling $T$, and this ceiling is what lets the Moreau-envelope argument push cluster points below $f_{\\min}+\\mu$; when the iterates move more, $t_k$ is decreased to encourage local refinement. To evaluate the proximal operator without gradients, the paper uses the Gibbs-measure identity $\\mathrm{prox}^{\\delta}_{tf}(x)=\\int z\\exp(-\\varphi(z)/\\delta)\\,dz\\,/\\,\\int \\exp(-\\varphi(z)/\\delta)\\,dz$, where $\\varphi(z)=f(z)+\\|z-x\\|^2/(2t)$; the quadratic term and small $\\delta$ concentrate the measure at the proximal point. The tensor-train variant builds a low-rank TT cross approximation of the discretized function $\\exp(-f/\\delta)$ on a mesh, with cost linear in the dimension, and the Monte Carlo variant samples from a Gaussian centered at $x$ with variance $\\delta t I_d$ and applies an exponentially weighted moving average for variance reduction.","core_discovery":"The central claim is that global minimization does not require exact proximal operators: an inexact proximal point iteration $x_{k+1}=\\alpha_k y_k+(1-\\alpha_k)x_k$, with $y_k$ an estimate of the true proximal point $\\hat{x}_k\\in\\mathrm{prox}_{t_k f}(x_k)$, converges to the global minimum whenever the errors obey $\\sum_{k=0}^\\infty \\|y_k-\\hat{x}_k\\|^2<\\infty$ and the damping and regularization parameters satisfy $\\alpha_{\\min}>1-\\eta_-$ and a sufficiently large ceiling $T$. Under Assumptions 1-3 (continuity, $p$-coercivity, and a no-flat-oscillation condition near the minimizer), Theorem 1 shows $\\lim_{k\\to\\infty} f(x_k)=f_{\\min}$; the proof controls accumulated errors, forces $t_k$ to its ceiling $T$, and uses a Moreau-envelope argument with $T>M/\\mu$ to show every cluster point is a global minimizer. Theorem 2 extends the conclusion to stochastic estimates under the tail condition $\\sum_k p_k<\\infty$, with convergence almost surely. The paper further derives the $O(\\delta)$ convergence rate of the Gibbs-measure estimator and the tensor-train estimation error bounds, and claims this is the first IPP method proved to converge to a global minimizer rather than only a stationary point.","pith_inferences":["If the summability condition is the true dividing line, then any sampling scheme whose squared error decays fast enough—adaptive MCMC, quasi-Monte Carlo, or learned surrogates—could be dropped into the same IPP framework; TT-IPP and MC-IPP are only the first two instantiations.","The mesh condition $h=O(\\delta^{(d+2)/(2s)+1})$ in Corollary 2 is impractical at high dimension, and the experiments' fixed $h=0.1$ with $\\gamma=1.1$ lie outside the proven regime; the paper's own heuristic choices therefore implicitly assume the worst-case quadrature bounds are pessimistic.","For functions where $\\exp(-f/\\delta)$ is approximately low-rank across iterations, warm-starting the TT approximation through Hadamard products could make subsequent proximal evaluations nearly free, pointing toward higher-dimensional global optimization than the current experiments."],"forward_implications":["Any deterministic zeroth-order estimator whose proximal errors are square-summable gives global convergence for coercive continuous functions satisfying Assumptions 1-3, so local minima are escaped by the IPP dynamics alone.","Stochastic estimators satisfying (13) converge almost surely, so adaptive sample-size schedules that shrink errors fast enough inherit the guarantee.","TT-IPP reaches the proven regime when the mesh spacing and TT error obey (31), and the experiments show function-evaluation cost growing nearly linearly with dimension on low-rank-structured integrands.","MC-IPP inherits the guarantee when the sample size grows sufficiently, and the experiments show it beats the compared solvers on most benchmarks under a fixed evaluation budget.","The $O(\\delta)$ Gibbs estimator rate makes decreasing $\\delta$ while keeping errors square-summable a valid route to global minimization, and the warm start (32) costs essentially one iteration."],"supporting_citations":[{"why":"the exact-proximal-point global optimization method whose convergence this work generalizes to inexact estimates.","marker":"[26]"},{"why":"introduces the summability inexactness condition (4) and the classical IPP convergence framework for convex problems that Theorem 1 adapts to global nonconvex convergence.","marker":"[60]"},{"why":"the prior IPP analysis for nonconvex functions that reaches only stationary points and is strengthened here to a global minimizer.","marker":"[61]"},{"why":"supplies the randomized TT-cross algorithm used by TT-IPP to approximate the discretized integrand with few function evaluations.","marker":"[50]"},{"why":"supplies the tensor-train decomposition, Hadamard product, and TT-rounding operations used to update approximations when $\\delta$ shrinks.","marker":"[51]"},{"why":"the Hamilton-Jacobi/Monte Carlo proximal operator estimation that MC-IPP builds on.","marker":"[52]"},{"why":"the Sobolev-space quadrature error bound behind Corollary 2's mesh-size condition.","marker":"[31]"},{"why":"Gibbs measure asymptotics used to justify approximating $\\mathrm{prox}_{tf}(x)$ by the expectation under the concentrated Gibbs measure.","marker":"[3]"}],"fun_headline_variants":["Inexact proximal steps still reach global minima","Small proximal errors still guarantee global convergence","Inexact proximal point converges to global minimum","Square-summable errors suffice for global minimization","Zeroth-order global opt via inexact proximal steps"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The guarantee rests on the cumulative squared errors of the approximate proximal estimates being finite, and the implemented TT and MC parameter choices do not verify the mesh-size and TT-error conditions (31) that would ensure this in high dimensions.","fun_headline_variants_meta":{"raw":{"variants":["Inexact proximal steps still reach global minima","Small proximal errors still guarantee global convergence","Inexact proximal point converges to global minimum","Square-summable errors suffice for global minimization","Zeroth-order global opt via inexact proximal steps"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000577,"raw_usage":{"total_tokens":2805,"prompt_tokens":1111,"completion_tokens":1694,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":727,"completion_tokens_details":{"reasoning_tokens":1623}},"tokens_in":727,"tokens_out":1694,"duration_ms":10414,"temperature":1.0,"reasoning_tokens":1623,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T14:56:17.436576+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Implement TT-IPP or MC-IPP on a coercive continuous nonconvex function with multiple local minima using the Section 6 parameter values ($h=0.1$, $\\gamma=1.1$ for TT-IPP; $C=1.1$, $c=0.9$ for MC-IPP) and record $\\sum_{k=0}^\\infty \\|y_k-\\hat{x}_k\\|^2$; if this sum diverges while $f(x_k)$ still approaches $f_{\\min}$, the observed convergence is not explained by Theorem 1's hypotheses, and if the iterates stall at a local minimizer whenever the sum diverges, the practical algorithms fail exactly where the theory's condition is violated.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"the Sobolev-space quadrature error bound behind Corollary 2's mesh-size condition."},{"cited_title":"Sankhy¯ a Ser","cited_arxiv_id":null,"evidence_quote":"Gibbs measure asymptotics used to justify approximating $\\mathrm{prox}_{tf}(x)$ by the expectation under the concentrated Gibbs measure."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"the exact-proximal-point global optimization method whose convergence this work generalizes to inexact estimates."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"the prior IPP analysis for nonconvex functions that reaches only stationary points and is strengthened here to a global minimizer."},{"cited_title":"Linear Algebra Appl","cited_arxiv_id":null,"evidence_quote":"supplies the randomized TT-cross algorithm used by TT-IPP to approximate the discretized integrand with few function evaluations."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the tensor-train decomposition, Hadamard product, and TT-rounding operations used to update approximations when $\\delta$ shrinks."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"the Hamilton-Jacobi/Monte Carlo proximal operator estimation that MC-IPP builds on."}],"review_version":1}