{"id":"9dd3546f-3c9a-401a-a70d-782f1b58f7b9","arxiv_id":"2412.11773","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Gradient descent with the new adaptive step size reaches near-optimal or first-known convergence rates for ℓ-smooth functions, including the previously open quadratic-growth case.","lead":"This paper finds a new formula for choosing gradient descent step sizes that works for a broad class of 'generalized smooth' functions, improving known speed guarantees and covering cases where no guarantees existed. A generalist should read it because it offers a single adaptive step size rule that unifies and extends several popular clipping strategies in optimization.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Superquadratic ℓ is a real gap in the 'unified' claim: Theorem 5.1 bounds a quotient, not ||∇f||², and for ρ>2 no nonconvex gradient-norm rate follows without bounded gradients; the paper acknowledges this but it limits the title.","rationale":"The reader identified the same weakest assumption: the conversion from the quotient bound to a gradient-norm rate requires monotonicity of ψ2 or bounded gradients. I independently checked the key algebraic steps: the descent inequality in Theorem 5.1, the derivation of the step size from Lemma D.1, the ρ=2 rate in (12), and the convex arguments in Sections 7 and 8. These are correct; the quotient bound is genuinely new and the ρ=2 nonconvex rate is valid. The only substantial limitation is the superquadratic regime, where the main theorem does not by itself certify stationarity. The paper is transparent about this (Section 6 explicitly assumes A.6.1), so the limitation does not invalidate the claimed results, but it does qualify the 'unified' characterization. The stochastic section compounds the issue by defining r without requiring finiteness; for exponential ℓ the step size collapses to zero. Overall, the reader's CONDITIONAL verdict remains appropriate: the mathematics is sound, the superquadratic gap is a genuine but disclosed caveat, and the missing experimental code remains the main practical reason for conditionality.","tokens_in":26528,"tokens_out":41695,"duration_ms":379766,"concrete_test":"Take ℓ(s)=s³ and attempt to convert (9) into a gradient-norm bound. Here ψ2(x)=x²/(2x)³ = 1/(8x), so the inequality ψ2(||∇f||) ≤ 8Δ/T is equivalent to ||∇f|| ≥ T/(64Δ), meaning the set of gradients satisfying the bound is unbounded. This analytically confirms that no finite gradient-norm rate can follow from Theorem 5.1 for ρ>2 without an additional assumption such as Assumption 6.1, and that Corollary 5.2 cannot be extended to this regime.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central deterministic machinery is sound for ℓ with ψ2(x)=x²/ℓ(2x) increasing: the descent lemma and the quotient bound min_k ||∇f(x_k)||²/ℓ(2||∇f(x_k)||) ≤ 4Δ/T are correctly proven, and the conversions for ρ≤2, including the previously open ρ=2 case, follow cleanly. The load-bearing weakness is the conversion step itself. When ℓ grows faster than quadratically, ψ2 is not increasing and actually tends to 0 at infinity; e.g. for ℓ(s)=L0+L1s^ρ with ρ>2, ψ2(x) ∼ x^{2-ρ} → 0. Then the main guarantee (9) can be satisfied with ||∇f(x_k)|| arbitrarily large, because the quotient becomes small precisely when the gradient is large. Corollary 5.2 cannot be invoked, and no bound on ||∇f(x_k)||² follows. The paper's Section 6 repairs this only by assuming the strong and generally unverifiable Assumption 6.1 (||∇f(x)|| ≤ M for all x). This is not a proof error, since the section is explicitly labeled as requiring A.6.1, but it is the real restriction on the 'unified' claim: for superquadratic ℓ and unbounded gradients, the paper provides no nonconvex convergence guarantee. The stochastic extension has a parallel issue, since Algorithm 2's ratio r = sup ℓ(2s)/ℓ(s) is infinite for exponential ℓ, making the step size zero, although the paper does not explicitly exclude this case.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies gradient descent under the generalized smoothness assumption ||∇²f(x)|| ≤ ℓ(||∇f(x)||) for a nondecreasing locally Lipschitz positive function ℓ. Its main technical contribution is the step-size rule γ_k = ∫_0^1 dv / ℓ(||∇f(x_k)|| + ||∇f(x_k)|| v), which minimizes a pointwise upper bound derived from the generalized descent lemma. The central nonconvex result, Theorem 5.1, gives the monotonicity inequality f(x_{k+1}) ≤ f(x_k) − (γ_k/4)||∇f(x_k)||² and the quotient bound min_k ||∇f(x_k)||² / ℓ(2||∇f(x_k)||) ≤ 4Δ/T. Under the additional condition that ψ_2(x) = x²/ℓ(2x) is strictly increasing, Corollary 5.2 converts this into a rate on the gradient norm; this recovers classical L-smoothness, matches the best known (L0,L1)-smoothness rate, and gives the first nonconvex GD rate for (2,L0,L1)-smoothness. The paper also proves convex rates (Theorems 7.2, 8.1, 8.3), derives small-ε corollaries with dominant term ℓ(0)R²/ε, extends the analysis to stochastic gradients (Theorem 9.2), and reports numerical experiments on two one-dimensional examples.","tokens_in":26857,"tokens_out":7013,"duration_ms":66388,"significance":"If the results are taken as stated, the paper makes a solid contribution: it identifies a single adaptive step-size rule that provably recovers the L-smoothness rate, improves the existing (ρ,L0,L1)-smoothness rates for ρ<2, closes the previously open ρ=2 case, and gives new convex rates whose leading term ℓ(0)R²/ε is stronger than the earlier ℓ(||∇f(x0)||)R²/ε dependence. The proofs are unusually transparent: Theorem 5.1 is short, the descent inequality and the ψ_2 inversion in Corollary 5.2 check out, and the convex potential Lemma I.2 is correct. The main novelty is the step-size formula itself, and the paper deserves credit for presenting it as a clean, parameter-free derivation rather than as a fitted schedule. The significance is tempered by the fact that the universal quotient bound does not automatically yield a gradient-norm rate for superquadratic ℓ; the nonconvex treatment of ρ>2 and exponential ℓ requires Assumption 6.1, and the stochastic extension requires a finite ratio r = sup_s ℓ(2s)/ℓ(s). These are real restrictions on the 'unified' claim, though they do not invalidate the central mechanism.","major_comments":[{"comment":"The text in Section 1.2 says the theory 'remains applicable to virtually any ℓ functions,' but this is true only for the quotient guarantee (9), not for the gradient-norm convergence rates advertised in Table 1. For ℓ(s) = L0 + L1s^ρ with ρ>2, the map ψ_2(x) = x²/ℓ(2x) is not increasing and in fact tends to 0 at infinity, so Corollary 5.2 cannot be invoked; the paper itself acknowledges this in Section 6 and imposes the bounded-gradient Assumption 6.1 to obtain rates. The manuscript should reword the universality claim so that it refers to the quotient bound and the step-size rule, and should state clearly in the introduction that nonconvex gradient-norm rates for superquadratic ℓ are obtained only under Assumption 6.1.","section":"Section 1.2 and Section 6"},{"comment":"Theorem 9.2 is stated for the general ℓ-smoothness assumption without requiring r = sup_{s≥0} ℓ(2s)/ℓ(s) to be finite, yet Algorithm 2 divides the step size by 5r and the proof uses the bound r ≥ ℓ(||∇f(x_k)|| + ||∇f(x_k)||v)/ℓ(||∇f(x_k)||(1+v)/2). For ℓ(s) = L0 + L1s²e^s, for instance, r = ∞ and the algorithm prescribes γ_k = 0, so the theorem is vacuous in exactly the cases that the paper highlights as novel. The theorem statement and Algorithm 2 should explicitly impose r < ∞, and the text should note that the stochastic extension covers ℓ with at most exponential growth only when the ratio is finite.","section":"Section 9, Algorithm 2, Theorem 9.2"},{"comment":"The conversion of the quotient bound into the explicit rate (12) is sound for 0 ≤ ρ ≤ 2, but the paper should be more careful about the condition 'ψ_2 is strictly increasing' in Corollary 5.2. The condition is not merely a technical convenience: without it, the quotient min_k ||∇f(x_k)||²/ℓ(2||∇f(x_k)||) can be small precisely because ||∇f(x_k)|| is large, so no bound on the gradient norm follows. Placing this remark immediately after Theorem 5.1, rather than only in the later superquadratic section, would prevent the reader from overreading the main theorem as a universal gradient-norm guarantee.","section":"Corollary 5.2 and Section 5.3"}],"minor_comments":[{"comment":"There is a typo in the displayed definition of ℓ: 'ℓ(s) = ℓ(s) = L0 + L1s^ρ' repeats ℓ(s); also Remark 4.7 says 'for any p ≥ 0' where the variable is ρ.","section":"Section 5.3"},{"comment":"The condition 'for all T ≥ 1 such that 2 8∆/T ∈ im(ψ_2)' appears to contain a typo: the factor should be 8∆/T, matching the bound in (10).","section":"Corollary 5.2"},{"comment":"In the last paragraph of the proof of Theorem 9.2, the text says to sum the descent inequality and 'use Assumption 6.1'; the argument only needs the lower bound on f from Assumption 3.2, not the bounded-gradient Assumption 6.1. This should be corrected.","section":"Theorem 9.2 proof"},{"comment":"The sentence 'Overall, for any ℓ, the total number of computed stochastic gradients is Θ(B × T)' is too strong in view of the finite-r issue; it should be qualified to hold for ℓ with r < ∞.","section":"Section 9, after Theorem 9.2"}],"recommendation":"major_revision","confidential_remarks":"The mathematical core of the paper is sound and the presentation is generally careful. The main barrier to acceptance is not correctness of the key lemmas but the need to tighten the scope claims: the nonconvex gradient-norm rates are not universal, and the stochastic theorem is missing an explicit finiteness assumption on r. These are fixable within the manuscript's scope. The paper is a good fit for this venue and the related-work discussion appears fair."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nTwo things to know. First, the paper delivers a real result: under (2,L0,L1)-smoothness — the ρ=2 case that Li et al. explicitly left open — gradient descent with the step size γ_k = ∫_0^1 dv/ℓ(‖∇f(x_k)‖ + ‖∇f(x_k)‖v) converges at rate O(L0Δ/ε + L1Δ) for nonconvex functions. The same step size rule cleans up the convex theory, improving the leading term from ℓ(‖∇f(x0)‖)R²/ε to ℓ(0)R²/ε. I checked the descent lemma and the convex potential argument in the appendix; they are correct. The step size is derived by minimizing a pointwise upper bound, and no constants are fitted; the proofs use only the rule plus the smoothness assumption.\n\nSecond, the title's \"unified\" is doing some work. The main nonconvex guarantee is a quotient bound, min ‖∇f‖²/ℓ(2‖∇f‖) ≤ 4Δ/T. Converting that into a rate for ‖∇f‖² requires the map ψ_2(x)=x²/ℓ(2x) to be increasing. For ℓ(s)=L0+L1s^ρ with ρ>2, ψ_2 tends to 0, the conversion fails, and the paper recovers a rate only under Assumption 6.1, a global bounded-gradient bound that is strong and not usually verifiable. The paper labels this clearly and I don't see a proof error; but \"unified\" oversells it. The nonconvex theory is genuinely general for ρ≤2, and conditional on bounded gradients beyond that.\n\nThe stochastic extension has a smaller version of the same issue: Algorithm 2 requires r=sup ℓ(2s)/ℓ(s), which is infinite for exponential ℓ, making the step size zero. The paper doesn't exclude that case. Again not a proof error, but a reader running the motivating examples from Section 2 will hit it.\n\nMinor: the experiments have no code or data. For a theory paper's illustrative Figure 2 that's a small thing.\n\nWho is this for? Anyone working on (L0,L1)-smoothness, clipping, and generalized smoothness will want the ρ=2 rate and the convex improvement. It deserves a serious referee. I'd recommend sending it to review, with the likely outcome being acceptance after the authors either temper the \"unified\" language in the nonconvex part or state the superquadratic caveat up front and fix the stochastic r=∞ edge.","headline":"First nonconvex GD rate for ρ=2 under generalized smoothness, proved cleanly; the 'unified' claim has a real boundary at superquadratic ℓ unless gradients are bounded.","tokens_in":27485,"tokens_out":2671,"would_cite":true,"duration_ms":24739,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["90C25","90C30","65K05"],"pacs":[],"model":"deepseek-v4-flash","headline":"A single adaptive step size rule for gradient descent is proven to work under generalized smoothness, including the previously open case where the Hessian grows quadratically with the gradient norm.","keywords":["gradient descent","generalized smoothness","ℓ-smoothness","adaptive step size","(L0,L1)-smoothness","nonconvex optimization","convex optimization","stochastic gradient descent"],"falsifier":"Take the $(2, 800, 2)$-smooth function $f(x) = -\\log x - \\log(0.1 - x)$ from the paper's experiments, run Algorithm 1 from $x_0 = 10^{-7}$, and measure the iteration count $T(\\varepsilon)$ needed to reach $\\|\\nabla f(x_k)\\|^2 \\le \\varepsilon$ as $\\varepsilon \\to 0$; the claimed rate $O(800\\Delta/\\varepsilon + 2\\Delta)$ predicts $T(\\varepsilon)$ grows like $1/\\varepsilon$, so observing growth like $1/\\varepsilon^2$ would falsify the $\\rho = 2$ nonconvex claim.","tokens_in":26296,"feed_emoji":"📉","tokens_out":16427,"duration_ms":117813,"temperature":0.7,"pith_summary":"Under the assumption that the Hessian norm is bounded by a nondecreasing function of the gradient norm, $\\|\\nabla^2 f(x)\\| \\leq \\ell(\\|\\nabla f(x)\\|)$, this paper derives a new adaptive step size for gradient descent: $\\gamma_k = \\int_0^1 \\frac{dv}{\\ell(\\|\\nabla f(x_k)\\| + \\|\\nabla f(x_k)\\| v)}$. This step size is the exact minimizer of a generalized quadratic upper bound, so it plays the role that $1/L$ plays in classical smooth optimization. The paper proves a master nonconvex bound that yields explicit convergence rates for $(\\rho, L_0, L_1)$-smooth functions, including the first guarantee for plain gradient descent in the previously open quadratic-growth case $\\rho = 2$. In the convex and stochastic settings it also improves the known rates, reducing the dominant term for small $\\varepsilon$ from $\\ell(\\|\\nabla f(x_0)\\|)R^2/\\varepsilon$ to $\\ell(0)R^2/\\varepsilon$. If these results hold, one adaptive rule unifies $L$-smoothness, $(L_0, L_1)$-smoothness, and more general growth models under a single theory.","feed_headline":"New adaptive step size settles GD's quadratic-smoothness case","feed_subtitle":"One integral-based step size recovers classical L-smooth rates and covers the previously open quadratic-growth case.","key_machinery":"The central object is the $q$-function $q(s; a) = \\int_0^s dv/\\ell(a+v)$ and its inverse $q^{-1}$; under $\\ell$-smoothness, $q^{-1}(\\|y-x\\|; \\|\\nabla f(x)\\|)$ bounds the change in gradient along a segment (Lemma 4.3), and integrating it gives a generalized descent lemma (Lemma 4.5). Minimizing the right-hand side of that lemma over step direction and length yields the step size $\\gamma_k$ as the optimal point (Corollary 4.6), and the identity $q^{-1}(\\gamma_k \\|\\nabla f(x_k)\\|; \\|\\nabla f(x_k)\\|) = \\|\\nabla f(x_k)\\|$ lets the proofs sidestep computing $q^{-1}$ explicitly. The monotone function $\\psi_2(x) = x^2/\\ell(2x)$ is what converts the master nonconvex bound into an explicit gradient-norm rate; the convex theory instead uses a two-phase argument (Theorem 8.1) based on the fact that the gradient norms $\\|\\nabla f(x_k)\\|$ are nonincreasing along the iterates.","core_discovery":"With $f$ twice differentiable on an open convex domain and a nondecreasing, positive, locally Lipschitz function $\\ell$ satisfying $\\|\\nabla^2 f(x)\\| \\leq \\ell(\\|\\nabla f(x)\\|)$ for all $x$, the paper defines the step size $\\gamma_k = \\int_0^1 dv / \\ell(\\|\\nabla f(x_k)\\| + \\|\\nabla f(x_k)\\| v)$ and proves that gradient descent with this step size satisfies $f(x_{k+1}) \\leq f(x_k) - \\tfrac{\\gamma_k}{4}\\|\\nabla f(x_k)\\|^2$, hence $\\min_{0 \\le k < T} \\|\\nabla f(x_k)\\|^2 / \\ell(2\\|\\nabla f(x_k)\\|) \\leq 4\\Delta / T$. Under the additional hypothesis that $\\psi_2(x) = x^2/\\ell(2x)$ is strictly increasing and unbounded, this master bound converts into explicit rates: for $(\\rho, L_0, L_1)$-smoothness with $0 \\le \\rho \\le 2$, an $\\varepsilon$-stationary point is found in $O(L_0\\Delta/\\varepsilon + L_1\\Delta/\\varepsilon^{(2-\\rho)/2})$ iterations, and for $\\rho = 2$ the rate is $O(L_0\\Delta/\\varepsilon + L_1\\Delta)$, a case for which no plain-GD guarantee previously existed. When $\\ell$ grows superquadratically, the same bound yields rates under an additional bounded-gradient assumption $\\|\\nabla f(x)\\| \\le M$. In the convex setting, the paper improves the dominating term for small $\\varepsilon$ from $\\ell(\\|\\nabla f(x_0)\\|)R^2/\\varepsilon$ to $\\ell(0)R^2/\\varepsilon$, discovers a new min-term $L_1\\Delta^{1/2}R/\\varepsilon^{1/2}$ for $(L_0, L_1)$-smoothness, and gives a stochastic version whose small-$\\varepsilon$ complexity matches known lower bounds up to logarithms.","pith_inferences":["Because the step size is defined solely from $\\ell$ and the current gradient norm, it can be implemented by numerical quadrature without tuning $L_0$ or $L_1$; a natural testable question is whether it is competitive with tuned clipping in large-scale training with unknown smoothness constants.","The identity $q^{-1}(\\gamma_k \\|\\nabla f(x_k)\\|; \\|\\nabla f(x_k)\\|) = \\|\\nabla f(x_k)\\|$ suggests a design principle for other first-order methods—choose the step so that one step limits the gradient change to the current gradient norm—which may extend to heavy-ball or adaptive methods under the same growth model.","The master bound $\\min_k \\|\\nabla f(x_k)\\|^2/\\ell(2\\|\\nabla f(x_k)\\|) \\le 4\\Delta/T$ may itself be the meaningful stationarity measure when $\\ell$ is superquadratic; whether that measure supports a useful complexity theory without bounded gradients is left open.","A direct stress test is to measure $T(\\varepsilon)$ on the paper's own example $f(x) = -\\log x - \\log(0.1-x)$ at smaller $\\varepsilon$ than reported; if the $L_1\\Delta$ term is real, the iteration count should scale as $1/\\varepsilon$, not $1/\\varepsilon^2$."],"forward_implications":["For $(2, L_0, L_1)$-smooth nonconvex functions, plain gradient descent now has a proof of convergence with rate $O(L_0\\Delta/\\varepsilon + L_1\\Delta)$, covering a case previously listed as open in the $\\ell$-smoothness literature.","For $(\\rho, L_0, L_1)$-smooth functions with $0 \\le \\rho < 2$, the dependence on the initial gradient norm disappears: the rate improves from $O(\\ell(\\|\\nabla f(x_0)\\|)\\Delta/\\varepsilon)$ to $O(L_0\\Delta/\\varepsilon + L_1\\Delta/\\varepsilon^{(2-\\rho)/2})$.","In the convex setting, for small target accuracy $\\varepsilon$, gradient descent with the new step size behaves like classical GD with step size $1/\\ell(0)$, giving a $\\Theta(\\ell(0)R^2/\\varepsilon)$ rate that no longer involves the possibly large initial gradient norm.","The new convex term $L_1\\Delta^{1/2}R/\\varepsilon^{1/2}$ for $(L_0, L_1)$-smooth functions can be substantially better than the earlier $L_1^2R^2$ or $L_1\\|\\nabla f(x_0)\\|R^2/\\varepsilon$ terms when the distance $R$ to the optimum is large.","The stochastic extension achieves, for small $\\varepsilon$, complexity $\\tilde{O}(\\sigma^2 L_0 \\Delta/\\varepsilon^2)$ up to log factors, recovering the known lower bound for smooth stochastic nonconvex optimization."],"supporting_citations":[{"why":"Introduces the $\\ell$-smoothness assumption, defines the problem class, and supplies the baseline rates in both convex and nonconvex settings that this paper improves.","marker":"Li et al. (2024a)"},{"why":"Gives the state-of-the-art $(L_0,L_1)$-smoothness rates that the new step size matches in the linear case and improves via the new min-terms.","marker":"Vankov et al. (2024)"},{"why":"Provides the classical $L$-smoothness theory—the quadratic upper bound and the $O(LR^2/\\varepsilon)$ rate—that the generalized descent lemma extends.","marker":"Nesterov (2018)"},{"why":"Supplies Gronwall's lemma used in the proof of Lemma 4.3 to turn the differential inequality into the $q^{-1}$ gradient bound.","marker":"Gronwall (1919)"},{"why":"Analyzes clipped gradient descent, the step-size family that the new continuous rule generalizes, and provides one of the convex comparison rates.","marker":"Koloskova et al. (2023)"},{"why":"Concurrently derived convex $(L_0,L_1)$-smooth rates with clipping and adaptivity that the paper's new min-terms improve upon.","marker":"Gorbunov et al. (2024)"},{"why":"Establishes lower bounds for smooth nonconvex GD, used to state that the recovered $L$-smooth rate is optimal up to constants.","marker":"Carmon et al. (2020)"},{"why":"Provides the large-deviation inequality for light-tailed sums used in the stochastic-gradient convergence proof.","marker":"Juditsky & Nemirovski (2008)"},{"why":"Lower bound for stochastic nonconvex optimization that the small-$\\varepsilon$ complexity of the SGD extension matches up to logarithms.","marker":"Arjevani et al. (2022)"}],"fun_headline_variants":["Integral step size closes GD's quadratic gap","One adaptive step unifies GD smoothness rates","Unified GD via single integral step rule"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The explicit nonconvex rates require the map $\\psi_2(x) = x^2/\\ell(2x)$ to be strictly increasing and unbounded, which holds for polynomial growth with $\\rho \\le 2$ but fails when $\\ell$ grows superquadratically, forcing the paper to assume bounded gradients ($\\|\\nabla f(x)\\| \\le M$) for the $\\rho > 2$ and exponential cases.","fun_headline_variants_meta":{"raw":{"variants":["Integral step size closes GD's quadratic gap","One adaptive step unifies GD smoothness rates","Unified GD via single integral step rule"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001077,"raw_usage":{"total_tokens":4651,"prompt_tokens":1232,"completion_tokens":3419,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":848,"completion_tokens_details":{"reasoning_tokens":3373}},"tokens_in":848,"tokens_out":3419,"duration_ms":22060,"temperature":1.0,"reasoning_tokens":3373,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T14:36:49.733556+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the $(2, 800, 2)$-smooth function $f(x) = -\\log x - \\log(0.1 - x)$ from the paper's experiments, run Algorithm 1 from $x_0 = 10^{-7}$, and measure the iteration count $T(\\varepsilon)$ needed to reach $\\|\\nabla f(x_k)\\|^2 \\le \\varepsilon$ as $\\varepsilon \\to 0$; the claimed rate $O(800\\Delta/\\varepsilon + 2\\Delta)$ predicts $T(\\varepsilon)$ grows like $1/\\varepsilon$, so observing growth like $1/\\varepsilon^2$ would falsify the $\\rho = 2$ nonconvex claim.","supporting_citations":[],"review_version":1}