{"id":"962917d7-df7d-4770-b21b-cfdbc08cefd8","arxiv_id":"2502.05668","paper_version":3,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Normalized stochastic subgradient descent iterates converge, after perfect classification, to critical points of the normalized margin for homogeneous neural networks.","lead":"This paper proves that, after a homogeneous neural network correctly classifies every training example, the weight direction produced by stochastic subgradient descent converges to critical points of the classification margin. It extends a known margin-maximization result for smooth gradient descent to the noisy and nonsmooth setting of ReLU-type networks trained with SGD.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5 is conditional on the stability event E = [liminf m(u_k)>0], which the paper acknowledges but never shows to be non-vacuous for nonsmooth constant-step SGD; if E fails, the central claim makes no statement.","rationale":"We read the paper in good faith and checked the main proof chain. The reduction to a differential inclusion is sound: Dbar is a conservative field for m (Lemma 13), the direction update (13) has bounded remainders and martingale noise on E, and the adapted stochastic approximation result (Proposition 15) is applied with a valid Sard-type argument showing m(Z_s) has measure zero. The exponential-loss concentration of λ_i,k on active margin indices is correct because on E the norm diverges, so non-active margin gaps are amplified by ∥w_k∥^L. Minor presentation issues (e.g., the truncation index b∈N versus b∈Q, and the min/max typo in Remark 14) do not affect the argument. The weakest point is indeed event E: it is load-bearing for norm growth, noise vanishing, and convergence, and the paper explicitly says E is a stability assumption with no guarantee from a single positive-margin iterate. The reader identified exactly this assumption. Because the theorem is honest about its conditioning, we do not view this as an error; it is a scope limitation. We therefore recommend keeping the ACCEPT verdict, possibly with a request to add a discussion or experiment on the plausibility of E.","tokens_in":23693,"tokens_out":44243,"duration_ms":424797,"concrete_test":"Run constant-step SGD with a small ReLU MLP (e.g., two hidden layers, no biases) on a separable synthetic dataset (e.g., Gaussian clusters), for at least 100 random seeds and several step sizes, and record the empirical frequency of E = [liminf m(u_k)>0]. On trajectories where E holds, also measure dist(u_k, Z_s) and m(u_k) convergence by numerically computing Z_s. If E has positive probability and observed limits lie in Z_s, the conditional theorem is non-vacuous and the concern does not land; if E is never observed when iterates start with positive margin, the stability event is a real obstacle to the theorem's applicability.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Every component of the proof of Theorem 5 uses the event E defined in (6). Proposition 3 uses E to prove norm growth and the effective step decay γ_k→0; Proposition 4 uses E to bound the remainder r_k, to ensure ∑γ_k=∞, and to force the mixing weights λ_i,k to concentrate on the active margin indices I(u); Appendix D uses E again for the martingale-noise truncation. The authors are explicit that E is a stability assumption and that a single iterate with positive margin does not imply E for nonsmooth stochastic dynamics. The concern is not that the proof is wrong, but that the headline claim about 'late-stage training dynamics' is scoped to an event whose occurrence is not established for any nontrivial ReLU network. If E is rare or hard to verify, the theorem is vacuous in the target setting. The paper offers no example, simulation, or sufficient condition showing constant-step nonsmooth SGD satisfies E.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies constant-step stochastic subgradient descent (SGD) for binary classification with L-homogeneous nonsmooth neural networks (e.g., ReLU networks). It defines normalized iterates u_k = w_k/||w_k|| and, under an event E = [liminf m(u_k) > 0], proves that m(u_k) converges to a positive limit and dist(u_k, Z_s) -> 0, where Z_s is the set of critical points of the normalized margin on the unit sphere with respect to a conservative set-valued field. The proof interprets normalized SGD as a stochastic approximation of a differential inclusion associated with the margin, proves norm growth (Proposition 3) and an Euler-like decomposition (Proposition 4), and verifies the assumptions of a stochastic approximation result (Appendix D). The paper also discusses the relation to KKT points of the Lyu-Li max-margin problem and provides an extension with vanishing step sizes and more general losses (Appendix F).","tokens_in":23880,"tokens_out":21699,"duration_ms":198493,"significance":"If correct, the result is a substantial extension of Lyu-Li (2020) from smooth GD to nonsmooth and stochastic settings, showing that late-stage SGD converges to margin critical points. The proof is largely self-contained and makes elegant use of conservative fields and stochastic approximation for differential inclusions. The conditional nature of the result on the event E is explicitly acknowledged and is a limitation, not an internal inconsistency. The paper does not provide experiments, but for a theory contribution this is acceptable. Strengths include the explicit Euler-like decomposition, the Sard-type argument for m(Z_s), and the honest discussion of the stability assumption E.","major_comments":[],"minor_comments":[{"comment":"The main theorem is conditional on the event E = [liminf m(u_k) > 0], and the paper provides no sufficient condition or example showing that E can hold for nonsmooth constant-step SGD. Since the authors explicitly call this a stability assumption and discuss its difficulty, this is a limitation rather than a flaw; I suggest adding a short remark (or a small numerical illustration) in Section 4 or the discussion to indicate settings in which E is plausible, such as the vanishing-step-size framework of Appendix F.","section":"Section 4, Eq. (6)"},{"comment":"The example with p1(w)=min(0,w) and p2(w)=min(0,-w) does not satisfy m(w)=0 for w≠0 (for w=1, m(1)=-1); use instead p1(w)=max(0,w) and p2(w)=max(0,-w), which gives m≡0, \\bar D(0)=[-1,1], and ∂m(0)={0}, illustrating the claimed distinction.","section":"Appendix B, Remark 14"},{"comment":"The displayed identity a_i(λw)=λ^L a_i(w) should read a_i(λw)=λ^{L-1}a_i(w), consistent with the (L-1)-homogeneity of the conservative gradient.","section":"Section 4, 'On Assumption 2'"},{"comment":"Please correct 'Up to our knowledge' to 'To the best of our knowledge' and the typo 'sastify' to 'satisfy'.","section":"Abstract/Introduction"},{"comment":"In item 3, the constant is named c1 in the text and then written as c6 in the inequality; the notation should be consistent.","section":"Appendix F, Proposition 19"},{"comment":"The definition of η_{k+1} is correct, but a one-line derivation showing how it arises from rewriting the SGD update would improve readability.","section":"Section 6, Eq. (15)"}],"recommendation":"minor_revision","confidential_remarks":"The paper is technically sound and the proof is convincing. The main limitation is the conditional event E, but the authors are transparent about it; I did not treat it as a blocker. The example in Remark 14 is wrong and should be corrected. The paper is a good fit for the journal and I recommend minor revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main thing to know: this paper does what it says on the tin, but only on an event E that is assumed, not established. The result is the first discrete-time, stochastic, nonsmooth extension of Lyu–Li: under Assumptions 1–2, on the event E = [liminf m(u_k) > 0], normalized constant-step SGD iterates converge to the critical set Z_s of the margin on the sphere. That is a genuine advance, not a repackaging. The conservative-field stochastic-approximation viewpoint is a real change of tools, and the appendices do the heavy lifting honestly.\n\nCredit where it is due: the paper is explicit that E is a stability assumption, and it explains why the old descent-lemma arguments cannot rule out leaving the correctly classified region for nonsmooth stochastic dynamics. The proof of Proposition 4 is detailed, and Appendix D checks the Lyapunov and Sard-type conditions needed for the SA theorem. The general setting in Appendix F with vanishing step sizes and the softer event E1 is a good-faith attempt to widen the scope.\n\nNow the soft spots, in proportion. The stress-test concern is not a manufactured flaw: the paper gives no example, simulation, or sufficient condition showing that constant-step SGD on a ReLU network satisfies E. If E is rare or unverifiable, Theorem 5 has no force in the very setting that motivates it. The authors acknowledge this, but the acknowledgment does not make the theorem less conditional. This is the main caveat a reader should carry away. Second, Theorem 5's proof relies on 'minor adaptations' of results from Benaïm and Davis et al. Appendix D sketches the adaptation, but the details are compressed; a referee should ask whether the martingale-noise truncation argument fully goes through, especially the step indexing and the use of Proposition 15. These are referee-level issues, not obvious failures. Also minor: the 'arbitrary step-size' comment is accurate on E but should not be read as a statement about constant-step SGD in general.\n\nWho is this for? Researchers in implicit bias and nonsmooth optimization theory. It deserves a serious referee: the proof structure is sound enough to audit, the question is important, and the conditional limitation is clearly labeled. My recommendation: send it to review, and insist the authors either prove a nontrivial sufficient condition for E or present a concrete network/data setting where E holds. Without that, the result remains a conditional theorem whose applicability is unverified.","headline":"A real first extension of Lyu–Li to nonsmooth stochastic SGD, but the central theorem is conditional on a stability event that the paper never shows can actually occur for constant-step SGD on ReLU networks.","tokens_in":24352,"tokens_out":1354,"would_cite":true,"duration_ms":16377,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68Q32","90C26","62L20"],"pacs":[],"model":"deepseek-v4-flash","headline":"Constant-step stochastic subgradient descent on homogeneous ReLU networks converges in direction to the critical points of the normalized margin, extending the max-margin implicit bias to nonsmooth stochastic training.","keywords":["implicit bias","stochastic subgradient descent","homogeneous neural networks","margin maximization","conservative set-valued fields","differential inclusions","nonsmooth optimization","constant step-size SGD"],"falsifier":"Take a two-layer ReLU network, binary separable data, exponential loss, and constant step size, and run SGD from an initialization where the normalized margin stays positive; if for some such run the normalized directions do not approach any critical point of the normalized margin on the sphere (for instance, if they keep cycling or approach a point where the projected conservative field is nonzero), Theorem 5 is false. A cheaper check is to verify on a simple homogeneous nonsmooth example whether a direction u with m(u)>0 and \\bar D_s(u)≠0 can be an accumulation point of the iterates while E holds.","tokens_in":23524,"feed_emoji":"🎯","tokens_out":7929,"duration_ms":70344,"temperature":0.7,"pith_summary":"This paper claims that the implicit bias of gradient-based training—the tendency of overparameterized networks to steer toward max-margin solutions—survives when training is stochastic and the network is nonsmooth. Analyzing constant-step SGD on L-homogeneous ReLU-type networks with exponential or logistic loss, it shows that, whenever the normalized margin stays positive forever (event E), the normalized weight directions converge to the set of critical points of the normalized margin on the unit sphere. When the conservative fields are Clarke subgradients, these critical points are exactly the scaled KKT points of the max-margin problem previously characterized for gradient flow and smooth GD. The proof works by viewing the normalized iterates as a noisy Euler discretization of a conservative-field flow, with an effective step size that shrinks even though the nominal step size is constant.","feed_headline":"Nonsmooth SGD inherits max-margin bias of gradient flow","feed_subtitle":"Proof: constant-step subgradient descent on ReLU networks converges in direction to margin-critical points.","key_machinery":"The load-bearing object is the normalized margin m(u) = min_i p_i(u) on the unit sphere and its Riemannian conservative field \\bar D_s, defined as the tangent projection of the conservative field \\bar D of the margin. The paper proves that the normalized SGD recursion is an Euler-like stochastic approximation of the flow \\dot u(t) ∈ \\bar D_s(u(t)): the recursion is rewritten with an effective step size \\bar γ_k that tends to zero and a martingale noise whose cumulative effect vanishes. The convergence to Z_s then follows from the stochastic-approximation theorem for differential inclusions once one shows that −m is a Lyapunov function whose critical values have measure zero by Sard's theorem.","core_discovery":"The central discovery is that the late-stage dynamics of (stochastic) subgradient descent on homogeneous neural networks are governed by a conservative set-valued field of the normalized margin. The paper constructs the projected field \\bar D_s(u) = {v − ⟨v,u⟩u : v ∈ \\bar D(u)} on the unit sphere and proves that the normalized iterates u_k approximately follow the differential inclusion \\dot u(t) ∈ \\bar D_s(u(t)) with perturbations that vanish. Consequently, under the event E = [liminf m(u_k) > 0], the margin m(u_k) converges to a positive limit and dist(u_k, Z_s) → 0, where Z_s is the set of critical points of the normalized margin. When the conservative fields are Clarke subgradients, membership in Z_s coincides with being a scaled KKT point of the max-margin problem, so the result carries the implicit-bias characterization from smooth gradient descent to nonsmooth stochastic subgradient descent.","pith_inferences":["If the stability event E could be proven from initialization and overparameterization—a question the paper leaves open—Theorem 5 would become an unconditional statement about constant-step SGD on ReLU networks rather than a conditional one.","The automatic decay of the effective step size suggests that batch-size schedules could be treated as a tuning knob for the rate of convergence to Z_s without altering the limit; this is a testable and currently unexplored implication.","The conservative-field reformulation is not tied to backpropagation artifacts, so the same convergence statement should hold for any homogeneous objective with a min-structure, including multi-class margins or adversarial-margin variants; verifying that would confirm that the mechanism is the margin geometry rather than the specific loss.","When the limit margin is zero (the E1 regime), the limit direction can fail to be a max-margin direction, so the implicit bias is not unconditional: there should be a phase transition between stable (E) and unstable (E1-only) training where the asymptotic solution changes character."],"forward_implications":["With constant step size, the directions of both SGD and GD on ReLU-type homogeneous networks converge to critical points of the normalized margin, not just to some zero-error region; behind a single positive-margin iterate plus the stability event, stochasticity does not break the implicit bias.","Under the Clarke-subgradient selection, running nonsmooth SGD is asymptotically equivalent to running margin maximization: any limit direction is a scaled KKT direction of the min-norm problem.","The effective step size \\bar γ_k decays polynomially even though the nominal step size γ is constant, because the weight norm grows logarithmically; hence constant-step nonsmooth SGD becomes a convergent stochastic approximation.","The same proof scheme extends to broader losses such as e^{−q^a} and log(1+e^{−q^a}) and to a larger event E1 where the margin may tend to zero, in which case the directions still converge to Z_s but the KKT interpretation can fail.","A byproduct is that the normalized margin itself converges to a positive limit on E, giving a quantitative late-stage description: the norm grows like log k and the loss decays like k^{−ε c1}."],"supporting_citations":[{"why":"Defines the max-margin problem and proves that smooth GD on homogeneous networks converges to its KKT points; this is the result being extended to the nonsmooth stochastic setting.","marker":"(Lyu and Li, 2020)"},{"why":"Establishes directional convergence and alignment for gradient flow, providing the continuous-time benchmark and the comparison for the discrete result.","marker":"(Ji and Telgarsky, 2020)"},{"why":"Introduces conservative set-valued fields and the stochastic approximation framework that the proof of Proposition 4 and Theorem 5 relies on.","marker":"(Bolte and Pauwels, 2021)"},{"why":"Supplies stochastic approximation results for differential inclusions used to convert the Euler-like discretization plus vanishing noise into convergence to critical points.","marker":"(Benaïm, 2006)"},{"why":"Proves convergence of stochastic subgradient methods on tame functions; its Theorem 3.2 is adapted as Proposition 15 to control the cumulative martingale noise.","marker":"(Davis et al., 2020)"},{"why":"Establishes the dynamical-systems view of stochastic approximations and the Lyapunov-argument template for characterizing limit sets of discretizations.","marker":"(Benaïm et al., 2005)"},{"why":"Shows that constant learning-rate SGD on separable smooth linear classifiers converges in direction; this is the earlier fixed-step result generalized to nonsmooth homogeneous networks.","marker":"(Nacson et al., 2019c)"},{"why":"Sard's theorem is used to show the margin's critical set has zero measure, a necessary condition for the Lyapunov argument in the stochastic approximation theorem.","marker":"(Sard, 1942)"}],"fun_headline_variants":["Subgradient SGD reaches max-margin critical points","Stochastic subgradient descent matches gradient flow bias","Nonsmooth SGD converges to margin-critical points","Subgradient descent on ReLU nets matches gradient flow","Late-stage subgradient SGD approaches max margin"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument collapses without the event E = [liminf m(u_k) > 0]: if even one late-stage iterate misclassifies the training data, the norm-growth bound, the bounded noise, and the decaying effective step size all lose their footing, and the paper proves nothing about the directions.","fun_headline_variants_meta":{"raw":{"variants":["Subgradient SGD reaches max-margin critical points","Stochastic subgradient descent matches gradient flow bias","Nonsmooth SGD converges to margin-critical points","Subgradient descent on ReLU nets matches gradient flow","Late-stage subgradient SGD approaches max margin"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000651,"raw_usage":{"total_tokens":2967,"prompt_tokens":906,"completion_tokens":2061,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":522,"completion_tokens_details":{"reasoning_tokens":1986}},"tokens_in":522,"tokens_out":2061,"duration_ms":14600,"temperature":1.0,"reasoning_tokens":1986,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T18:26:28.125943+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a two-layer ReLU network, binary separable data, exponential loss, and constant step size, and run SGD from an initialization where the normalized margin stays positive; if for some such run the normalized directions do not approach any critical point of the normalized margin on the sphere (for instance, if they keep cycling or approach a point where the projected conservative field is nonzero), Theorem 5 is false. A cheaper check is to verify on a simple homogeneous nonsmooth example whether a direction u with m(u)>0 and \\bar D_s(u)≠0 can be an accumulation point of the iterates while E holds.","supporting_citations":[{"cited_title":"Gradient descent maximizes the margin of homogeneous neural networks","cited_arxiv_id":null,"evidence_quote":"Defines the max-margin problem and proves that smooth GD on homogeneous networks converges to its KKT points; this is the result being extended to the nonsmooth stochastic setting."},{"cited_title":"Conservative set valued fields, automatic differentiation, stochastic gradient methods and deep learning","cited_arxiv_id":null,"evidence_quote":"Introduces conservative set-valued fields and the stochastic approximation framework that the proof of Proposition 4 and Theorem 5 relies on."},{"cited_title":"Stochastic subgradient method converges on tame functions","cited_arxiv_id":null,"evidence_quote":"Proves convergence of stochastic subgradient methods on tame functions; its Theorem 3.2 is adapted as Proposition 15 to control the cumulative martingale noise."},{"cited_title":"The measure of the critical values of differentiable maps","cited_arxiv_id":null,"evidence_quote":"Sard's theorem is used to show the margin's critical set has zero measure, a necessary condition for the Lyapunov argument in the stochastic approximation theorem."}],"review_version":1}