{"id":"b8212795-af0d-4902-bd26-616a0b8d7671","arxiv_id":"2601.22370","paper_version":4,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":2,"one_line_summary":"The authors show how to swap exact proximal operators for Hamilton-Jacobi Monte Carlo estimates in five splitting algorithms and claim almost-sure convergence, with proof gaps that need repair.","lead":"Operator splitting algorithms can be run with Monte Carlo estimates of proximal operators instead of exact formulas, and the paper claims convergence for five standard schemes. The proof currently has missing assumptions and a gap in the convergence argument, so the headline guarantee is not yet established as stated.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.3's Monte Carlo bound presupposes a finite global Lipschitz constant, but Theorems 3.5–3.10 and Assumptions 3.4/3.7 never state one; without L, Jk and Mk are undefined and the summability machinery cannot start.","rationale":"Agree with the reader's weakest assumption: the missing Lipschitz hypothesis is the same load-bearing gap. I independently checked the statement of Theorem 3.3 and the proofs in Appendices C–G. The error decomposition in each proof is exactly epsilon_k = HJ-Prox(...) - exact proximal(...), and the only bound applied to that difference is Theorem 3.3, which requires a finite L. Nowhere is a global Lipschitz condition added for the approximated function; Assumptions 3.4 and 3.7 merely refer to 'the constants in Theorem 3.3 evaluated at (t_k, delta_k)', which silently presumes such an L exists. The quadratic counterexample makes the gap concrete and shows the hypotheses of the convergence theorems do not imply the hypotheses of the error bound. This is an internal mismatch between assumptions and theorem statements, not a disagreement with consensus. The reader also noted a second issue: the PPM and PGD cluster-point lemmas are proved only for exact iterations and are invoked for the perturbed ones. That issue is real and would compound the gap, but the Lipschitz issue is more fundamental because it blocks the error bound for all five algorithms, including those whose cluster-point argument is otherwise sound. The numerical experiments are concordant but do not repair a hypothesis mismatch; H.6 even uses an indicator function, reinforcing that the lack of Lipschitzness is not a purely technical corner case. The preprint as stated should not be accepted, but the gaps are identifiable and plausibly repairable by strengthening assumptions, so the verdict is unchanged and a revised version could become acceptable.","tokens_in":25522,"tokens_out":7480,"duration_ms":65409,"concrete_test":"Concrete check: take n = 1, f(x) = x^2, g = 0. This input satisfies the hypotheses of Theorem 3.5 (proper, LSC, convex). Try to instantiate Assumption 3.4: because f is not globally Lipschitz, L = infinity, so J_k = exp(2 L^2 t_k / delta_k) = infinity and M_k = n t_k delta_k + (2 sqrt(n t_k delta_k) + 3 L t_k)^2 = infinity for every k. Show that the requirements N_k >= 8 J_k / alpha_k and summability of sqrt(8 J_k M_k / (alpha_k N_k)) + sqrt(n t_k delta_k) are impossible to satisfy with finite N_k. If no parameter schedule works for this elementary quadratic, Theorems 3.5–3.10 must either add 'f and g are globally L-Lipschitz' or supply a different Monte Carlo bound that does not use L; without one of those, the almost-sure convergence claims are unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3.2's entire convergence argument routes through Theorem 3.3, whose hypotheses are: f convex, LSC, and L-Lipschitz. The constants in that theorem are J* = exp(2L^2 t / delta) and M* = n t delta + (2 sqrt(n t delta) + 3 L t)^2. Assumptions 3.4 and 3.7 then require Nk >= 8 Jk / alpha_k and summability of sqrt(8 Jk Mk / (alpha_k Nk)) + sqrt(n t_k delta_k), where Jk and Mk are 'the constants in Theorem 3.3 evaluated at (t_k, delta_k)'. But neither assumption nor any theorem statement imposes a finite L on the functions whose proximal is being approximated. For example, Theorem 3.5 assumes only that f and g are proper, LSC, and convex; f(x) = ||x||^2, g = 0 satisfies this but is not globally Lipschitz. For such data L = infinity, Jk = infinity, Mk = infinity, and the required Nk does not exist. The same gap affects PGD (g non-Lipschitz), DRS and DYS (f or g non-Lipschitz, and DYS's h is only assumed smooth, not Lipschitz), and PDHG (prox of g* requires g* to be Lipschitz, which fails even for simple Lipschitz g because the conjugate takes value +infinity outside a bounded set). Since the theorems are stated for the broader class, every one of Theorems 3.5, 3.6, 3.8, 3.9, and 3.10 invokes an error bound whose hypotheses are not met. The proofs in Appendices C–G explicitly write epsilon_k as the difference between the HJ-Prox and exact proximal maps and then apply Theorem 3.3 to bound it; with L absent, those bounds are vacuous. This is a hypothesis mismatch, not just a conservative constant, and it is load-bearing because Theorem 3.3 is the only bridge from Monte Carlo error to the deterministic perturbed-KM theorem. It is also reflected in the experiments: H.6 approximates the indicator of R_+^p, which is not L-Lipschitz, so the demonstrated configuration is outside the proven regime.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes replacing exact proximal operators with a Monte Carlo approximation called HJ-Prox in five standard operator splitting algorithms: proximal point (PPM), proximal gradient descent (PGD), Douglas–Rachford splitting (DRS), Davis–Yin splitting (DYS), and primal–dual hybrid gradient (PDHG). The main theoretical contribution is a set of almost-sure convergence theorems (Theorems 3.5–3.10) that model the approximation error as a summable perturbation of a perturbed Krasnoselskii–Mann iteration, using a new probabilistic bound on the HJ-Prox error (Theorem 3.3) together with a deterministic smoothing error bound (Theorem 3.2). The paper also reports numerical experiments on LASSO, sparse group LASSO, trend filtering, multitask learning, total variation denoising, non-negative LASSO, and overlapping group LASSO, claiming that HJ-Prox matches analytical solvers.","tokens_in":25893,"tokens_out":8758,"duration_ms":81321,"significance":"The idea of embedding a zeroth-order, derivative-free proximal approximation into splitting algorithms is timely and potentially impactful: it could remove the need for closed-form proximal operators in a broad class of convex optimization problems. The Monte Carlo error bound in Theorem 3.3 is a genuinely new technical result, and the paper correctly identifies splitting as a way to reduce the exponential-in-Lipschitz constant sample complexity compared with applying HJ-Prox to the composite objective. If the convergence theorems were valid under the stated assumptions, this would be a strong contribution. However, as detailed below, the central theoretical claim is currently unsupported because the error bound used throughout requires a global Lipschitz condition that is never assumed in the convergence theorems, and in the PDHG case the required assumption is not even satisfiable for standard problem data. The numerical experiments are encouraging but are run with fixed sample size N, which explicitly violates the theoretical sufficient conditions, so they do not compensate for the theoretical gap.","major_comments":[{"comment":"The convergence proofs for all five algorithms invoke Theorem 3.3, whose hypotheses require the function whose proximal is being approximated to be convex, lower-semicontinuous, and globally L-Lipschitz. Assumptions 3.4 and 3.7 reference the constants Jk and Mk of Theorem 3.3, but they never add a Lipschitz assumption on f, g, or g*. The theorem statements likewise assume only proper, LSC, and convex (plus smoothness for f or h in PGD and DYS). For example, f(x)=||x||^2 and g=0 satisfy the hypotheses of Theorem 3.5, yet for this data L=infinity, so J*=exp(2L^2 t/delta) and M* are infinite, the required sample size Nk in Assumption 3.4 does not exist, and the error bound in (7) is vacuous. This is not a minor technicality: every one of Theorems 3.5, 3.6, 3.8, 3.9, and 3.10 relies on bounding the HJ-Prox error by expressions that are undefined for non-Lipschitz functions. Adding a global Lipschitz assumption to the theorems would substantially narrow their scope (excluding, e.g., quadratic data terms) and still would not resolve the PDHG issue described below.","section":"Section 3.2, Assumptions 3.4/3.7 and Theorems 3.5–3.10"},{"comment":"The PDHG iteration (12) applies HJ-Prox to prox_{sigma g*}, the proximal operator of the Fenchel conjugate of g. For typical proper LSC convex g such as g(u)=||u||_1 or an indicator function, the conjugate g* is extended-valued, taking the value +infinity outside a bounded set. Theorem 3.3 requires the approximated function to be real-valued and globally L-Lipschitz, so it cannot be applied to g* at all. The proof in Appendix G does not introduce any additional condition on g* (or a substitute error bound), and indeed the only stated assumptions on g in Theorem 3.10 are proper, convex, and LSC. Consequently, the PDHG convergence claim is unsupported by the paper's own error analysis. This flaw cannot be repaired by a simple added Lipschitz assumption, because g* is not Lipschitz for the standard functions that motivate PDHG.","section":"Theorem 3.10 and Appendix G (PDHG)"},{"comment":"The convergence theory requires the Monte Carlo sample size Nk to grow according to Nk >= 8Jk/alpha_k in Assumptions 3.4 and 3.7, yet all experiments in Section 4 use a fixed N=1000, with the text stating that 'this is not the setting of our theorem' implicitly. This is not itself an error, but it means the experiments do not provide evidence for the claimed almost-sure convergence regime; they demonstrate only empirical stability under much weaker conditions. That gap is acknowledged in Section 5, but the paper should state more explicitly that the theorems do not explain the reported fixed-N performance.","section":"Section 4 and Assumptions 3.4/3.7 (sample-size schedule)"}],"minor_comments":[{"comment":"The notation in Theorem 3.3 is inconsistent: the statement reads 'Let [prox^delta_t f(x) denote...' with an opening bracket instead of a hat; the hat notation \\widehat{prox} is used in Appendix B. Please fix the typographical error.","section":"Theorem 3.3 and Appendix B"},{"comment":"The phrase 'Theorem 3.2assumes exact integral evaluation' is missing a space before 'assumes'; please correct.","section":"Section 3.2, first paragraph"},{"comment":"Lemma C.1 states the step sizes as t_k = 1/(k+1), but the proof then refers to a general divergent step-size assumption (sum t_k = infinity). Either state the lemma for general t_k with sum t_k = infinity and lsc convergence, or align the proof with the specific choice t_k=1/(k+1).","section":"Lemma C.1 in Appendix C"},{"comment":"The number of overlapping groups is reported as 298 in Section 4.4 but as 289 in equation (175) in Appendix H.7. Please make these numbers consistent.","section":"Section 4.4 and Appendix H.7"},{"comment":"The reference list contains both 'Mckenzie, D.' and 'McKenzie, D.' entries; the capitalization should be uniform.","section":"References"}],"recommendation":"reject","confidential_remarks":"The paper's core theoretical claim is invalid under the stated assumptions: the missing global Lipschitz hypothesis in Theorems 3.5–3.10 is load-bearing, and the PDHG theorem is not repairable by a simple added assumption because g* is extended-valued for standard problem data. The Monte Carlo error bound (Theorem 3.3) appears correct under its own hypotheses and could be a useful standalone contribution, but the splitting convergence theory built on it is the main advertised result. The experiments are interesting but do not test the theoretical regime. I would not encourage a major revision because the central framework, as presented, would need a fundamentally different error analysis for PDHG and would require much stronger Lipschitz assumptions that exclude common objectives. A fresh submission that restricts the theory to cases where the required assumptions are actually met, or that develops a separate treatment for PDHG, could be worth considering."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: useful paper with a real gap. The core idea—replace each proximal step in a splitting algorithm with the HJ-Prox Monte Carlo approximation and prove the perturbed iteration still converges—is sound, and the extension from proximal point to PGD, DRS, DYS, and PDHG is genuinely new. The Section 3.3 argument for why splitting reduces the Monte Carlo error (separating Lipschitz constants into Jf + Jg rather than Jf+g) is nice, and the experiments suggest the method works even when the theory doesn't yet cover the examples.\n\nThe problem is that every convergence theorem routes through Theorem 3.3, which requires f to be globally L-Lipschitz. Theorems 3.5–3.10 only assume proper, lsc, convex f and g, with no Lipschitz condition anywhere in Assumptions 3.4 or 3.7. For f(x)=||x||^2, L is infinite, J_k and M_k are infinite, and the sample-size requirement N_k ≥ 8J_k/α_k doesn't exist. So the statements are broader than the arguments support. This is not a cosmetic gap; it's the bridge that makes the summability machinery run. The PPM and PGD proofs also claim cluster points of the perturbed sequence lie in the solution set by citing lemmas that prove the exact sequence, so the closedness condition in Theorem 3.1 isn't actually verified.\n\nI want to be careful: the Monte Carlo error bound in Appendix B is mostly coherent, and the flaws are identifiable and likely repairable. Adding a global Lipschitz assumption (or at least specifying the class of functions with finite L^2 t/δ) and supplying a perturbed quasi-Fejér cluster-point argument would fix the main issue. But as written, the theorems don't cover the experiments—HJ-Prox is called on an indicator function in the non-negative LASSO, which isn't Lipschitz, so the demonstrated configurations are outside the proven regime. And there's no code, which makes the numerical results harder to evaluate.\n\nThis isn't a desk-reject. The framework is worth referee time, to push the authors to fix the hypotheses and supply the missing arguments. If the gaps are cleanly patched, it becomes a solid contribution to zeroth-order splitting methods. I'd send it out.","headline":"The central convergence theorems outrun the assumptions: Theorem 3.3's Lipschitz requirement is missing from every splitting theorem, but the flaws are identifiable and the framework is worth refereeing.","tokens_in":26553,"tokens_out":3348,"would_cite":false,"duration_ms":30509,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["90C25","65K05","65C05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Replacing exact proximal operators with Hamilton-Jacobi-based Monte Carlo approximations preserves almost-sure convergence in PPM, PGD, DRS, DYS, and PDHG.","keywords":["Hamilton-Jacobi proximal operator","operator splitting","convex optimization","Monte Carlo approximation","almost sure convergence","Krasnoselskii-Mann iteration","proximal point method","nonsmooth optimization"],"falsifier":"Run HJ-Prox PPM with the Assumption 3.4 schedule on the one-dimensional problem $\\min_x x^2$ (so $g=0$, $f(x)=x^2$): the functions are proper, LSC, and convex but not globally Lipschitz, making $J^*$ and $M^*$ infinite. Divergence, stagnation, or an error sequence that fails to be summable would show the stated assumptions are insufficient; convergence to $0$ would suggest the Lipschitz requirement can be relaxed.","tokens_in":1774,"feed_emoji":"🎲","tokens_out":4663,"duration_ms":101656,"temperature":0.7,"pith_summary":"Many composite convex optimization problems are solved by splitting, but each split requires a proximal operator, and most functions do not have one in closed form. This paper establishes that a derivative-free Monte Carlo approximation called the Hamilton-Jacobi proximal (HJ-Prox) can replace the exact operator inside five standard splitting algorithms—proximal point, proximal gradient, Douglas–Rachford, Davis–Yin, and primal–dual hybrid gradient—without sacrificing their convergence guarantees. Under parameter schedules that make the approximation errors summable, each method's iterates converge almost surely to a minimizer. The practical payoff is a recipe for nonsmooth composite optimization that relies on function evaluations alone, with no proximal calculus and no closed-form formulas.","feed_headline":"Five splitting solvers converge with Monte Carlo proximals","feed_subtitle":"Paper proves PPM, PGD, DRS, DYS, and PDHG stay almost-surely convergent, removing the closed-form proximal bottleneck.","key_machinery":"The HJ-Prox operator approximates $\\mathrm{prox}_{tf}(x)$ by the ratio $\\mathbb{E}_{y\\sim N(x,\\delta t I)}[y e^{-f(y)/\\delta}] / \\mathbb{E}_{y\\sim N(x,\\delta t I)}[e^{-f(y)/\\delta}]$, a smoothed version of the proximal map evaluated from function values alone. The load-bearing bound is Theorem 3.3: for convex, LSC, $L$-Lipschitz $f$, the Monte Carlo error falls below $\\sqrt{8J^*M^*/(\\alpha N)} + \\sqrt{n t \\delta}$ with probability at least $1-\\alpha$, where $J^*=\\exp(2L^2 t/\\delta)$. This bound converts into almost-sure summability, and Theorem 3.1 (perturbed Krasnosel'skiĭ–Mann) then supplies convergence once each algorithm map is known to be averaged.","core_discovery":"The central claim is that convergence theory for operator splitting does not require exact proximals; it only requires errors that vanish fast enough. The paper proves that HJ-Prox-based PPM (Theorem 3.5), PGD (Theorem 3.6), DRS (Theorem 3.8), DYS (Theorem 3.9), and PDHG (Theorem 3.10) all generate iterates that converge almost surely to a global minimizer, provided $\\delta_k$, $\\alpha_k$, $N_k$, and $t_k$ follow Assumptions 3.4 or 3.7. The proof strategy is to write each HJ-Prox iterate as an exact split iterate plus a random error, then apply a perturbed Krasnosel'skiĭ–Mann theorem pathwise after Borel–Cantelli establishes almost-sure summability.","pith_inferences":["A testable extension beyond the paper: the practical fixed-sample-size runs ($N=1000$) contradict the sufficient condition $N_k\\to\\infty$; a plausible explanation is that the $J^*$ bound is far from tight, and an adaptive $N_k$ rule could be benchmarked against the paper's fixed-$N$ performance.","If the global Lipschitz assumption required by Theorem 3.3 is genuinely necessary, then a simple quadratic $f(x)=x^2$ would lie outside the theorem's scope despite satisfying the stated convexity assumptions; checking convergence on such a function would reveal whether the stated assumptions are sufficient.","The same perturbed-KM route could be applied to other inexact proximal schemes, for instance random block-coordinate or randomized proximal updates, by replacing the Monte Carlo error there with the corresponding sampling error."],"forward_implications":["Following Assumption 3.4, HJ-Prox PPM and PGD converge almost surely for convex objectives with no closed-form proximal at all; only calls to the objective are needed.","Under Assumption 3.7, DRS, DYS, and PDHG keep a fixed step size $t$ while $\\delta_k$ decays, so existing splitting implementations can swap in HJ-Prox for whichever term lacks a proximal formula.","Because the Monte Carlo constant $J^*$ depends exponentially on $L^2 t/\\delta$, splitting beats applying HJ-Prox to the whole sum: the Lipschitz constants of separate terms enter additively rather than through the square of their sum.","Hybrid schemes that apply exact proximals where available and HJ-Prox only to the non-proximable component have smaller fixed-point residuals than fully approximate schemes, as the non-negative LASSO experiment demonstrates."],"supporting_citations":[{"why":"Provides the perturbed Krasnosel'skiĭ–Mann convergence theorem (Theorem 3.1) that every convergence proof instantiates.","marker":"Combettes, 2001"},{"why":"Source of the uniform deterministic error bound $\\sqrt{nt\\delta}$ for the smoothed HJ-Prox (Theorem 3.2).","marker":"Crandall & Lions, 1983"},{"why":"Introduces the HJ-Prox Monte Carlo ratio used throughout and its weakly convex motivation.","marker":"Osher et al., 2023"},{"why":"Supplies the self-normalized importance sampling ratio-of-means decomposition used in the proof of Theorem 3.3.","marker":"Agapiou et al., 2017"},{"why":"Provides the Tsirelson–Ibragimov–Sudakov concentration inequality used to bound the importance-weight second moment $J^*$.","marker":"Boucheron et al., 2013"},{"why":"Gives the Poincaré inequality for strongly log-concave measures used to bound $\\mathbb{E}_{\\pi^*}[\\|Z-\\mu\\|^2]$.","marker":"Balasubramanian et al., 2022"},{"why":"Establishes firm nonexpansiveness/averagedness of the Douglas–Rachford operator used in Theorem 3.8.","marker":"Lions & Mercier, 1979"},{"why":"Defines the primal-dual hybrid gradient operator and its averagedness/fixed-point characterization used in Theorem 3.10.","marker":"Chambolle & Pock, 2011"}],"fun_headline_variants":["Monte Carlo proximals keep five splitting solvers convergent","HJ-Prox: approximate proximals still ensure splitting convergence","No exact proximal? HJ-Prox lets splitting converge anyway","Five solvers converge with Monte Carlo approximations","Operator splitting works with numerical proximals"],"cache_read_input_tokens":28416,"weakest_assumption_plain":"The Monte Carlo error bound that drives the whole argument assumes the approximated function is globally $L$-Lipschitz, which yields the finite constant $J^*=\\exp(2L^2t/\\delta)$; the convergence theorems state only proper, lower-semicontinuous, convex functions, so without adding global Lipschitzness the key summability bound does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Monte Carlo proximals keep five splitting solvers convergent","HJ-Prox: approximate proximals still ensure splitting convergence","No exact proximal? HJ-Prox lets splitting converge anyway","Five solvers converge with Monte Carlo approximations","Operator splitting works with numerical proximals"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000238,"raw_usage":{"total_tokens":1474,"prompt_tokens":872,"completion_tokens":602,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":488,"completion_tokens_details":{"reasoning_tokens":528}},"tokens_in":488,"tokens_out":602,"duration_ms":6043,"temperature":1.0,"reasoning_tokens":528,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:38:48.406780+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run HJ-Prox PPM with the Assumption 3.4 schedule on the one-dimensional problem $\\min_x x^2$ (so $g=0$, $f(x)=x^2$): the functions are proper, LSC, and convex but not globally Lipschitz, making $J^*$ and $M^*$ infinite. Divergence, stagnation, or an error sequence that fails to be summable would show the stated assumptions are insufficient; convergence to $0$ would suggest the Lipschitz requirement can be relaxed.","supporting_citations":[],"review_version":2}