{"id":"38a6746a-4bec-443e-af9a-f406d8cc7405","arxiv_id":"2508.16468","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"LeAP-SSN is a parameter-free semismooth Newton method that provably converges globally from any starting point in Hilbert spaces, with O(1/k)-type rates and superlinear local speed.","lead":"This paper introduces LeAP-SSN, an optimization algorithm that pairs Newton-style fast steps with an automatic safeguard and is proven to converge from any starting point, even on nonconvex problems. The method needs no hand-tuned constants, and the authors report both global speed guarantees and quick local convergence, with tests on imaging, mechanics, and machine learning problems.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Superlinear claim may be conditional: Dennis-Moré is an algorithmic condition on generated iterates, not a problem assumption; without proving it from the LM/backtracking globalization, the advertised fast-asymptotics guarantee is unsupported.","rationale":"The reader assigned UNVERDICTED because only the abstract was reviewable. My concern is not a detected defect but a specific load-bearing point that a full-text audit must settle: the Dennis-Moré condition is normally an algorithmic condition, and the abstract's phrasing leaves open whether the algorithm's globalization actually verifies it. This aligns with the reader's weakest-assumption identification about the superlinear regularity assumptions, but I put more weight on the conditional nature of the Dennis-Moré condition. The well-posedness gap (boundedness, subproblem solvability, PL constant not being known) is also real and matches the reader's concern. Since the full text is unavailable, I cannot move the verdict from UNVERDICTED; the existing verdict remains appropriate.","tokens_in":1061,"tokens_out":4111,"duration_ms":57236,"concrete_test":"In the full text, locate the theorem that states superlinear convergence and inspect its hypotheses. Check whether the Dennis-Moré condition appears as an assumption on the generated step sequence or is proved from semismoothness together with the LeAP-SSN Levenberg-Marquardt/backtracking rule. If it is only assumed, the superlinear claim is conditional rather than an achieved property, and the abstract should be revised. Also confirm that the O(1/sqrt(k)) subgradient-rate theorem includes an assumption guaranteeing boundedness of the subgradient norms used in the stationarity measure.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim bundles global nonasymptotic rates with superlinear local convergence. The most load-bearing step is how the superlinear phase is connected to the globalization. The abstract says LeAP-SSN achieves superlinear convergence 'under mild semismoothness and Dennis-Moré or partial smoothness conditions.' In standard semismooth Newton analysis, the Dennis-Moré condition is a property of the generated step sequence—typically ||F(x_k)+J_k s_k|| = o(||s_k||)—not an assumption on the problem class. If the theorem merely assumes this condition on the iterates, the algorithm is not shown to achieve superlinear convergence from arbitrary starting points; the advertised bridge from global rates to Newton asymptotics is incomplete. Additionally, the global-rate branch has an unstated well-posedness burden: in Hilbert spaces an O(1/sqrt(k)) subgradient rate requires either bounded subgradients along the iterates or bounded level sets. 'Arbitrary starting point' presumably still requires such control, so the theorem statements must include assumptions (coercivity/boundedness below, existence of a solution, solvability of the proximal Newton subproblems, valid backtracking stepsizes) that the abstract omits. These are not internal contradictions, but they are the least protected premises in the advertised guarantee.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes LeAP-SSN (Levenberg-Marquardt Adaptive Proximal Semismooth Newton method), a parameter-free optimization algorithm for Hilbert spaces. The abstract claims global nonasymptotic rates: O(1/k) for convex objectives, O(1/sqrt(k)) for nonconvex subgradient stationarity, linear convergence under a Polyak-Lojasiewicz condition, and superlinear local convergence under semismoothness and Dennis-More or partial smoothness conditions, all without knowledge of problem-specific constants. The material provided for review consists of the abstract only; no full text, theorem statements, proofs, or numerical experiments are available.","tokens_in":1228,"tokens_out":4012,"duration_ms":47292,"significance":"If the full manuscript substantiates the abstract, the contribution is significant: a single parameter-agnostic algorithm that simultaneously provides worst-case global rates and superlinear local asymptotics in Hilbert spaces, including for non-isolated minimizers, would be a notable advance in nonsmooth and large-scale optimization. The emphasis on avoiding problem-specific constants is attractive and, if proven, would be a practical strength. At this stage, however, these are unverified claims; the absence of precise assumptions and derivations makes the significance untestable from the submitted material.","major_comments":[{"comment":"The claim of convergence from 'arbitrary starting points' with O(1/sqrt(k)) subgradient rates in Hilbert spaces requires well-posedness conditions that are not stated: the objective must be bounded below, a solution or stationary point must exist, the proximal Newton subproblems must be solvable, backtracking stepsizes must be well defined, and boundedness of subgradients along iterates or bounded level sets is typically needed. Without these hypotheses, iterates may fail to exist or the rate estimate may be vacuous. The theorem statements must list these conditions explicitly.","section":"Abstract - global rates"},{"comment":"The superlinear claim is conditional in a way that may undermine the advertised bridge from global to fast local convergence. In standard semismooth Newton analysis, the Dennis-More condition is a property of the generated step sequence, not an assumption on the problem. If the theorem merely assumes this condition on the iterates, then the algorithm is not shown to achieve superlinear convergence from arbitrary starting points. The abstract must clarify whether the condition is proved from the Levenberg-Marquardt/backtracking globalization or is an additional algorithmic assumption, and if proved, for which problem classes.","section":"Abstract - superlinear convergence"},{"comment":"Linear convergence under a Polyak-Lojasiewicz condition typically involves an unknown constant. The statement that the method 'does not require knowledge of problem-specific constants' needs clarification: the rate constant may depend on the PL constant, which is acceptable as long as the algorithm itself does not need to know it. The proof of parameter-agnostic linear convergence is delicate and is not visible in the abstract. The theorem must state the sense in which the method avoids this constant.","section":"Abstract - Polyak-Lojasiewicz condition"},{"comment":"Only the abstract was provided for review. No definitions, assumptions, proofs, or numerical results are available, so none of the central claims can be verified. In particular, the combination of global and local rates is the paper's main selling point, but no theorem statements or derivations are included to support it. A complete manuscript is required for a substantive assessment.","section":"Abstract - reviewability"}],"minor_comments":[{"comment":"The acronym LeAP-SSN is defined, but the phrase 'mild semismoothness' is vague; it would help to specify whether semismoothness applies to the residual map, the objective gradient, or a generalized Jacobian selection.","section":"Abstract - terminology"},{"comment":"The claim that practical efficiency is illustrated on imaging, contact mechanics, and machine learning is not supported in the abstract; brief problem descriptions or a pointer to the experimental section would improve clarity.","section":"Abstract - experiments"}],"recommendation":"uncertain","confidential_remarks":"This is an abstract-only submission, so I could not verify any of the advertised results. The central claims are plausible and would be significant if proven, but the missing full text prevents a fair mathematical assessment. If the complete manuscript is available, I recommend a full review focusing on the well-posedness assumptions for the global rates and on whether the Dennis-More condition is derived or assumed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: if the full paper delivers what the abstract advertises, this is a real result. The combination—global nonasymptotic rates plus superlinear local convergence for non-isolated minimisers, with no problem-specific constants—is not something I've seen claimed in a single method. The idea deserves attention.\n\nWhat's good: the abstract is honest about assumptions. It states PL, semismoothness, and Dennis–Moré or partial smoothness as conditions, not as conclusions. The rates are standard-shaped: O(1/k) convex objective values, O(1/sqrt(k)) nonconvex subgradients, linear under PL. The method design—Levenberg–Marquardt regularization plus backtracking—is a mature way to globalize Newton steps, and the Hilbert-space setting is ambitious but not crazy.\n\nWhere I want to see the proof before buying the headline:\n\nFirst, the Dennis–Moré condition in semismooth Newton theory is normally a property of the generated step sequence, not a problem-class assumption. If the theorem merely assumes this condition on the iterates, then the superlinear claim is conditional: IF the iterates happen to satisfy Dennis–Moré, THEN convergence is superlinear. That is not the same as showing LeAP-SSN achieves it from arbitrary starting points. The bridge from global rates to fast asymptotics is exactly the part that needs proof, and the abstract does not say how it is crossed.\n\nSecond, the global branch has an unstated well-posedness burden. In Hilbert spaces, an O(1/sqrt(k)) subgradient rate typically requires bounded subgradients along the iterates or bounded level sets. “Arbitrary starting point” presumably still needs coercivity or boundedness below, existence of a solution, solvability of the proximal Newton subproblems, and valid backtracking stepsizes. The abstract omits these, and they are the least protected premises in the advertised guarantee.\n\nNone of this is a detected mathematical error—we only have the abstract, so there are no proofs or numerics to audit. But those two items are exactly what a referee should press on.\n\nWho it's for: anyone working on nonsmooth optimization, Newton-type methods, or global complexity theory. It deserves a serious referee. My own verdict is unverdictable from the abstract alone, but the potential significance justifies the refereeing time.","headline":"A genuinely interesting claimed unification—parameter-free Levenberg–Marquardt proximal semismooth Newton with global rates and superlinear asymptotics in Hilbert space—but the proof of how the globalization feeds the fast local phase is the thing to read before believing it.","tokens_in":1834,"tokens_out":1652,"would_cite":false,"duration_ms":18466,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["90C30","90C53","65K05","49J52"],"pacs":[],"model":"deepseek-v4-flash","headline":"A single parameter-free algorithm claims to guarantee global convergence on convex and nonconvex problems and then switch to superlinear local speed, all in Hilbert spaces.","keywords":["semismooth Newton method","global convergence rates","Polyak-Lojasiewicz condition","Hilbert space optimization","Levenberg-Marquardt regularization","parameter-free algorithm","nonconvex nonsmooth optimization","proximal methods"],"falsifier":"Construct a semismooth nonconvex problem on a Hilbert space with a continuum of minimizers, choose an initial point where the backtracking line search initially stalls or takes tiny steps, and check whether the subgradient norm obeys the claimed O(1/sqrt(k)) bound and whether a superlinear phase eventually appears; a single counterexample that fails either global or local behavior would refute the central claim.","tokens_in":855,"feed_emoji":"⚡","tokens_out":3594,"duration_ms":40769,"temperature":0.7,"pith_summary":"The paper introduces LeAP-SSN, a semismooth Newton-type method that needs no problem-specific tuning. It claims to converge from any starting point to a stationary point in nonconvex settings, and to a global minimum under a Polyak-Lojasiewicz condition, all while providing explicit worst-case rates: O(1/k) on convex objective values, O(1/sqrt(k)) on nonconvex subgradients, and linear convergence under PL. The same algorithm then accelerates to superlinear convergence under mild semismoothness and Dennis-Moré or partial smoothness assumptions, even when the minimizer is not isolated.","feed_headline":"Parameter-free solver converges globally and superlinearly","feed_subtitle":"Guaranteed O(1/k) on convex problems, linear under PL, with no constants to tune.","key_machinery":"The central object is the LeAP-SSN iteration: a proximal semismooth Newton step regularized by an adaptive Levenberg-Marquardt parameter, combined with backtracking. The adaptive regularization keeps the Newton step well-defined and ensures a sufficient decrease without problem-specific constants, while the semismooth Newton term provides the fast local asymptotics. The Polyak-Lojasiewicz branch uses only function values and gradients, so the unknown PL constant never needs to be estimated.","core_discovery":"LeAP-SSN is a proximal semismooth Newton method whose Newton steps are stabilized by an adaptive Levenberg-Marquardt regularization and a backtracking line search. The paper claims this combination is globally convergent in Hilbert spaces from arbitrary initial points, with no knowledge of constants such as the PL parameter. On convex problems, the objective gap decays as O(1/k); on nonconvex problems, the subgradient norm decays as O(1/sqrt(k)); under a Polyak-Lojasiewicz condition the convergence is linear. Under additional semismoothness and either a Dennis-Moré or partial smoothness condition at the limit point, the same algorithm achieves superlinear convergence even when the minimizer","pith_inferences":["If the claims hold, similar adaptive regularized Newton schemes could be developed for stochastic or inexact settings where only noisy subgradients are available—the paper does not address noise.","The O(1/sqrt(k)) nonconvex rate matches the classic gradient descent rate, suggesting that second-order information can be added without sacrificing the worst-case global rate.","The non-isolated minimizer superlinear result points toward degenerate and rank-deficient problems, such as low-rank matrix recovery, though the authors do not mention this.","A concrete test: evaluate LeAP-SSN on standard imaging or sparse logistic regression benchmarks and measure whether the superlinear phase appears near non-isolated solutions in practice."],"forward_implications":["Users of nonsmooth optimization get a single method that requires no tuning yet matches the worst-case guarantees of first-order methods.","Convex and nonconvex problems in Hilbert spaces, including inverse problems and PDE-constrained settings, share a unified convergence theory.","Under a Polyak-Lojasiewicz condition, linear convergence is achieved without knowing the PL constant.","Superlinear convergence for non-isolated minimizers broadens the class of problems where Newton-type fast asymptotics apply.","The combination of global rates and local superlinear speed may remove the need for separate global and local optimization phases."],"supporting_citations":[],"fun_headline_variants":["No tuning, global rates, superlinear finish","LeAP-SSN: global to superlinear without constants","No tuning, global O(1/k), then superlinear","Global guarantees, local speed, zero parameters"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The superlinear local stage and even the existence of the iterates rely on semismoothness plus Dennis-Moré or partial smoothness at the limit point, and on the proximal Newton subproblems being solvable; the global rates alone do not depend on these, but the flagship speed result does.","fun_headline_variants_meta":{"raw":{"variants":["No tuning, global rates, superlinear finish","LeAP-SSN: global to superlinear without constants","No tuning, global O(1/k), then superlinear","Global guarantees, local speed, zero parameters"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000543,"raw_usage":{"total_tokens":2445,"prompt_tokens":762,"completion_tokens":1683,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":506,"completion_tokens_details":{"reasoning_tokens":1629}},"tokens_in":506,"tokens_out":1683,"duration_ms":13486,"temperature":1.0,"reasoning_tokens":1629,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T17:16:04.487681+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a semismooth nonconvex problem on a Hilbert space with a continuum of minimizers, choose an initial point where the backtracking line search initially stalls or takes tiny steps, and check whether the subgradient norm obeys the claimed O(1/sqrt(k)) bound and whether a superlinear phase eventually appears; a single counterexample that fails either global or local behavior would refute the central claim.","supporting_citations":[],"review_version":1}