{"id":"f23b8566-2670-4895-a02e-c61d515f7dfb","arxiv_id":"2411.11224","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Allowing negative step sizes in a Wolfe line search turns ascent directions into usable descent steps, giving SR1 quasi-Newton methods competitive neural-network training performance.","lead":"This paper shows that optimization methods can safely take a step in the opposite direction of a suggested search direction, which lets them use useful curvature information instead of throwing it away. The authors demonstrate that this simple trick can make a quasi-Newton method train small neural networks as well as or better than popular optimizers like Adam.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Wolfe± line search does not enforce the generalized Wolfe curvature condition used in Proposition 1, so the theorem as stated does not cover the proposed algorithm.","rationale":"The reader flagged the unused epsilon bound in Proposition 1. I agree that bound is not used and can be dropped, so it is a presentational flaw rather than the weakest point. The more consequential gap is that Algorithm 1's standard Wolfe search on -p does not enforce the lower-bound part of the generalized curvature condition (8), as demonstrated by the scalar counterexample. This matters because the paper claims the Wolfe± line search ensures the Zoutendijk condition, but the stated theorem and the described algorithm are mismatched. I do not claim the method diverges; the convergence proof actually needs only the upper-bound half of (8), so the theorem could be repaired by changing the stated condition or the line search. As written, the central convergence guarantee is not established for the exact algorithm proposed. This reinforces the CONDITIONAL verdict: the core idea is likely correct, but the theorem-to-algorithm connection and empirical reproducibility need tightening.","tokens_in":7566,"tokens_out":18143,"duration_ms":183990,"concrete_test":"Implement Algorithm 1 on f(α) = α + α² at x = 0 with p = 1, c1 = 0.1, c2 = 0.9, using a standard Wolfe line search on q = -p and checking whether it can return β = 0.6 (or any β with ∇f(-β)^T(-p) ≥ c2∇f(0)^T(-p) and -β + β² ≤ -c1β). If the returned α = -β yields ∇f(α)^T p < 0, then Eq. (8) is violated. More generally, instrument every ascent step in the l-SR1 experiments and record whether ∇f(xk + αk pk)^T pk is negative; any such event contradicts the claim that Wolfe± outputs steps satisfying the generalized Wolfe conditions.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 1's proof uses the generalized curvature condition (8) for ascent directions: 0 ≤ ∇f(xk + αk pk)^T pk ≤ c2 ∇f(xk)^T pk. Algorithm 1 obtains an ascent step by calling a standard Wolfe search on q = -pk and negating the result. Standard Wolfe on q guarantees ∇f(x + βq)^T q ≥ c2 ∇f(x)^T q, which translates only to ∇f(x + αp)^T p ≤ c2 ∇f(x)^T p; it does not enforce the lower bound 0. For f(α) = α + α² at x = 0, p = 1, with c1 = 0.1 and c2 = 0.9, β = 0.6 satisfies the standard Wolfe conditions on q = -1: Armijo gives -0.24 ≤ -0.06, and curvature gives 0.2 ≥ -0.9. The resulting step α = -0.6 has ∇f(-0.6)^T p = -0.2, violating Eq. (8). Therefore the Wolfe± implementation need not satisfy the assumptions of Proposition 1, and the paper's bridge from theorem to implemented method is broken as written. The fix is straightforward, either restate the theorem with the one-sided curvature bound the proof actually uses, or modify the line search to enforce the lower bound, but without that the stated convergence guarantee does not apply to Algorithm 1.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the use of negative step sizes in second-order and quasi-Newton methods for non-convex optimization. Its central theoretical claim (Proposition 1, Section 3) is that an algorithm using updates x_{k+1} = x_k + α_k p_k is globally convergent whenever p_k is not orthogonal to the gradient and the step sizes α_k satisfy a Wolfe-type condition, even when p_k is an ascent direction. The proof in Appendix B adapts Nocedal and Wright's Theorem 3.2 to ascent directions paired with negative step sizes. The paper also proposes a practical line search, Wolfe± (Appendix C), and reports experiments on neural network training showing that limited-memory SR1 with negative step sizes compares favorably with damping and with optimizers such as Adam.","tokens_in":7782,"tokens_out":2518,"duration_ms":23928,"significance":"The central idea is simple and potentially valuable: instead of modifying the Hessian or the search direction to force descent, one can take a negative step along an ascent direction and thereby retain second-order information. If the theoretical guarantee were airtight and the implementation actually matched the theorem, the paper would make a useful contribution to the optimization-for-machine-learning literature. The proof of Proposition 1 is a clean adaptation of a standard result, and the paper is honest in pointing out that the usual globalization strategies discard negative curvature. The experimental section covers many datasets and gives a concrete baseline comparison. However, as detailed below, the stated theorem does not currently cover the proposed algorithm, and the experiments lack statistical support, so the significance is not yet established to the standard required for publication.","major_comments":[{"comment":"The bridge between Proposition 1 and Algorithm 1 (Wolfe±) is broken. Proposition 1 assumes the generalized curvature condition (8), which includes the lower bound 0 ≤ ∇f(x_k + α_k p_k)^T p_k. Algorithm 1, however, obtains an ascent step by running a standard Wolfe search on q = -p_k and negating the result. The standard Wolfe curvature condition for the descent direction q only yields ∇f(x_k + α_k p_k)^T p_k ≤ c_2 ∇f(x_k)^T p_k; it does not enforce the lower bound 0. For example, take f(α) = α + α² at x = 0, p = 1, c_1 = 0.1, c_2 = 0.9. The step β = 0.6 satisfies the standard Wolfe conditions on q = -1: the Armijo condition gives f(-0.6) = -0.24 ≤ f(0) + c_1 β ∇f(0)^T q = -0.06, and the curvature condition gives -0.2 ≥ c_2 ∇f(0)^T q = -0.9. The resulting step α = -0.6 has ∇f(-0.6)^T p = -0.2 < 0, violating (8). Thus the implementation as written need not satisfy the assumptions of Proposition 1. The fix is either to restate the theorem with the one-sided curvature bound that the proof actually uses, or to modify the line search so that it enforces the lower bound; without such a change, the stated global convergence guarantee does not apply to Algorithm 1.","section":"Appendix C, Algorithm 1; equation (8)"},{"comment":"The assumption |α_k| ≥ ε > 0 is never used in the proof in Appendix B. The proof uses only α_k < 0 (for ascent directions) together with (8) and (6); the ε lower bound appears neither in inequality (11) nor in the summation argument. Moreover, such an assumption is questionable in practice: near a stationary point, a Wolfe line search will typically produce step sizes that shrink to zero, so a fixed positive lower bound may be unattainable. Since the proof suggests the ε bound is unnecessary, the proposition should either be stated without it or the proof should be modified to show where it is needed. As written, the abstract's phrase 'very general conditions' overstates the result, since the statement carries an extra condition that is both unused and potentially unsatisfiable.","section":"Section 3, Proposition 1; Appendix B"},{"comment":"The empirical claims rest on single runs. No error bars, no number of random seeds, and no variance information are reported for the training-error comparisons in Table 1 or Figure 2. In deep learning experiments, single-run comparisons are generally not sufficient to support a claim such as 'using negative step sizes is often more effective than common Hessian modification methods.' Please provide statistics over multiple seeds, or at least state explicitly that the results are from a single run and temper the conclusions accordingly.","section":"Section 4, Table 1"}],"minor_comments":[{"comment":"The sentence 'With negative step sizes, SR1 becomes the an effective optimizer' contains a typo: 'the an' should be 'an'.","section":"Section 4.1"},{"comment":"Reference [29] is listed as 'V . Ramanmurthy and N. Duffy. L-sr1: A novel second order optimization method for deep learning' but the author name is typically spelled 'Ramanurthy'. Please verify the spelling and the publication venue.","section":"References, [29]"},{"comment":"The definition of global convergence uses 'lim inf_{k→∞} ∥∇f(x_k)∥ = 0', but the proof of Proposition 1 concludes the stronger statement 'lim_{k→∞} ∥∇f(x_k)∥ = 0'. Consider aligning the definition with the proven statement, or clarify that the lim inf version is sufficient for the paper's purposes.","section":"Appendix A, equation (2)"},{"comment":"The line 'cosθ k ∥∇f(x_k)∥² → 0' is not typeset correctly; it should read 'cos² θ_k ∥∇f(x_k)∥² → 0'. Please fix the notation for clarity.","section":"Appendix B"}],"recommendation":"major_revision","confidential_remarks":"The core flaw is the mismatch between the theorem's assumptions and the proposed Wolfe± implementation. The proof itself is sound for the one-sided curvature condition in (8), and the ε assumption is clearly removable. Because the fix is local and does not require new machinery, I see major revision, not rejection, as the appropriate outcome. I would also ask the authors to strengthen the experimental section with multi-seed results before resubmission."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe core idea is simple and worth discussing: when a quasi-Newton method produces an uphill direction, take a negative step instead of damping or otherwise modifying the Hessian. That preserves the curvature information and costs almost nothing. The l-SR1 experiments are suggestive — on several small datasets, Wolfe± beats both Hessian damping and Adam. I’d send this to peer review, but only with the expectation of a substantial revision.\n\nWhat’s genuinely new is the application. The convergence result, Proposition 1, is a sign-flipped version of the standard Wolfe/Zoutendijk theorem, and the proof is correct under its stated assumptions. Credit: it’s self-contained and uses no fitted parameters. But the statement overreaches. The lower bound |αk| ≥ ε > 0 is never used in the proof and is unlikely to hold for a real line search near a stationary point. The abstract’s “very general conditions” is too strong; these are the standard Wolfe assumptions plus non-orthogonality.\n\nThe sharper issue is that the Wolfe± line search in Algorithm 1 does not enforce the full curvature condition (8) that Proposition 1 assumes. Calling standard Wolfe on the negated direction gives only the upper bound ∇f(x+αp)^T p ≤ c2 ∇f(x)^T p; it doesn’t give the lower bound 0. I checked the example in the stress-test note and it is sound. However, the proof in Appendix B uses only the upper bound, so the fix is straightforward — restate the theorem with the one-sided curvature condition, or modify the line search to enforce the lower bound. As written, the theorem doesn’t cover the implemented algorithm. That’s a real gap, but curable.\n\nThe experiments are the weak half. No seeds, no error bars, no architecture or hyperparameter details, no code. The DNF entries in the damping column confuse the comparison, and the heart row has l-SR1 and l-SR1+Wolfe± tied, which undercuts the “often more effective” headline. I can’t recommend citing the empirical results yet. The theoretical contribution, once recast with the one-sided bound, is solid but elementary.\n\nThis is for practitioners who want a cheap way to exploit negative curvature in quasi-Newton methods, and for optimizers who care about globalization strategies. As a workshop paper it fits; as a journal submission it needs experimental discipline. Send it to a referee — the idea is testable and the gaps are fixable — but the referee should ask for code and seeds.","headline":"A simple, promising trick for quasi-Newton methods, with a theory gap that is easy to patch and experiments that are too thin to trust yet.","tokens_in":8388,"tokens_out":6625,"would_cite":false,"duration_ms":57731,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["90C53","90C26"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that allowing negative step sizes in a Wolfe line search makes ascent-direction methods globally convergent, and shows that SR1 with this line search trains neural networks effectively.","keywords":["negative step sizes","second-order optimization","Wolfe line search","ascent directions","global convergence","symmetric rank-one (SR1)","limited-memory quasi-Newton","neural network training"],"falsifier":"A concrete counterexample would be an algorithm whose search direction is never orthogonal to the gradient, whose step sizes satisfy the Wolfe conditions but are allowed to shrink to zero, and which nevertheless fails to drive the gradient norm to zero—this would show the $\\epsilon > 0$ condition is genuinely needed. On the empirical side, finding a suite of neural-network benchmarks where l-SR1 with damping consistently beats l-SR1 with Wolfe± under equal iteration budgets would refute the paper's central practical claim.","tokens_in":7251,"feed_emoji":"","tokens_out":4953,"duration_ms":45021,"temperature":0.7,"pith_summary":"This paper argues that second-order optimization methods can be made globally convergent without discarding negative curvature information, by allowing the step size to be negative. It proves that any algorithm whose search direction is not orthogonal to the gradient is globally convergent when paired with a Wolfe line search extended to both positive and negative step sizes. The paper then demonstrates experimentally that limited-memory SR1 with this extended line search, called Wolfe±, often trains neural networks more effectively than common Hessian modification methods and competitively with Adam.","feed_headline":"Stepping backward can make optimizers globally converge","feed_subtitle":"A simple Wolfe line search extension lets SR1 train neural networks better than damping or Adam.","key_machinery":"The key machinery is the Wolfe± line search, a simple extension of the standard Wolfe conditions that permits negative step sizes: if the search direction is an ascent direction, a standard Wolfe search is called on $-p_k$ and the returned step is negated. The proof machinery is the Zoutendijk condition, which bounds the total decrease of $f$ by $c \\sum \\cos^2\\theta_k \\|\\nabla f(x_k)\\|^2 < \\infty$; because the direction is never orthogonal to the gradient, this forces the gradient norms to converge to zero.","core_discovery":"The central result is Proposition 1: for a twice-differentiable, Lipschitz-smooth objective that is bounded below, any algorithm following $x_{k+1} = x_k + \\alpha_k p_k$ with step sizes satisfying $|\\alpha_k| \\ge \\epsilon > 0$ and the Wolfe conditions is globally convergent in the sense $\\|\\nabla f(x_k)\\| \\to 0$, even when $p_k$ is an ascent direction. The proof extends the classical Zoutendijk argument to ascent directions paired with negative step sizes, showing that sufficient decrease still yields the summability of $\\cos^2\\theta_k \\|\\nabla f(x_k)\\|^2$, and since $|\\cos\\theta_k| \\ge \\delta > 0$, the gradient norms must vanish. The paper further claims that SR1, which can produce indefinite Hessian approximations and hence ascent directions, becomes a competitive neural-network optimizer when negative step sizes are allowed.","pith_inferences":["The paper's proof never uses the assumption $|\\alpha_k| \\ge \\epsilon > 0$; if that observation holds, the global convergence result likely remains true even when step sizes shrink to zero, covering practical line searches that approach a stationary point.","A broader principle suggested by the paper is that globalization strategies preserving curvature information—such as taking a backward step along an ascent direction—may outperform strategies that modify curvature, like damping or trust-region projection.","Wolfe± could be combined with other ascent-producing methods, such as saddle-free Newton or conjugate gradient variants, potentially improving their saddle-escape behavior.","A concrete testable extension is to apply Wolfe± to stochastic or mini-batch settings, where the search direction is a noisy estimate and the curvature information is even less reliable."],"forward_implications":["Second-order methods that produce ascent directions, such as SR1 with non-positive-definite Hessian approximations, can be made globally convergent without Hessian modification, trust regions, or cubic regularization.","Adding negative step sizes is computationally inexpensive—just a sign flip in the line search—compared with the eigenvalue decomposition required by damping.","In the paper's experiments, l-SR1 with Wolfe± achieved lower training error than l-SR1 with damping on most datasets and often outperformed Adam and l-BFGS.","The benefit of negative step sizes appears to grow with network depth, making the approach more relevant for deeper architectures."],"supporting_citations":[{"why":"Defines the Wolfe conditions that the paper extends to allow negative step sizes.","marker":"[35]"},{"why":"Provides the classical Zoutendijk convergence proof (Theorem 3.2) that Proposition 1 is built on.","marker":"[25]"},{"why":"Establishes the convergence of SR1 quasi-Newton matrices, the method used with Wolfe± in the experiments.","marker":"[8]"},{"why":"Introduces limited-memory SR1 (l-SR1), the algorithm tested with Wolfe± in the numerical results.","marker":"[4]"},{"why":"Supports the claim that SR1 variants can converge faster by more closely approximating the true Hessian.","marker":"[14]"}],"fun_headline_variants":["Negative step sizes ensure global convergence","Backward steps: the missing optimizer ingredient","SR1 with negative steps trains networks better","Wolfe line search allows negative step sizes","Negative step sizes: a simple fix for convergence"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The theorem assumes step sizes are bounded away from zero, $|\\alpha_k| \\ge \\epsilon > 0$, while satisfying the Wolfe conditions, but real line searches near a stationary point typically produce step sizes that shrink to zero, so this hypothesis may not hold in practice even though the proof appears not to need it.","fun_headline_variants_meta":{"raw":{"variants":["Negative step sizes ensure global convergence","Backward steps: the missing optimizer ingredient","SR1 with negative steps trains networks better","Wolfe line search allows negative step sizes","Negative step sizes: a simple fix for convergence"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001121,"raw_usage":{"total_tokens":4614,"prompt_tokens":841,"completion_tokens":3773,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":457,"completion_tokens_details":{"reasoning_tokens":3707}},"tokens_in":457,"tokens_out":3773,"duration_ms":26065,"temperature":1.0,"reasoning_tokens":3707,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T18:47:37.204216+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete counterexample would be an algorithm whose search direction is never orthogonal to the gradient, whose step sizes satisfy the Wolfe conditions but are allowed to shrink to zero, and which nevertheless fails to drive the gradient norm to zero—this would show the $\\epsilon > 0$ condition is genuinely needed. On the empirical side, finding a suite of neural-network benchmarks where l-SR1 with damping consistently beats l-SR1 with Wolfe± under equal iteration budgets would refute the paper's central practical claim.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the Wolfe conditions that the paper extends to allow negative step sizes."},{"cited_title":"Nocedal and S","cited_arxiv_id":null,"evidence_quote":"Provides the classical Zoutendijk convergence proof (Theorem 3.2) that Proposition 1 is built on."},{"cited_title":"Conn, N.I.M","cited_arxiv_id":null,"evidence_quote":"Establishes the convergence of SR1 quasi-Newton matrices, the method used with Wolfe± in the experiments."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces limited-memory SR1 (l-SR1), the algorithm tested with Wolfe± in the numerical results."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supports the claim that SR1 variants can converge faster by more closely approximating the true Hessian."}],"review_version":1}