{"id":"8300cd19-a930-4688-bf10-d092ea3c680a","arxiv_id":"2505.03222","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":4,"one_line_summary":"A noisy gradient descent with adaptive Gaussian noise converges linearly to the global minimizer of nearly convex functions when a sharp lower bound is known.","lead":"This paper proves that adding a special kind of random noise to gradient descent can provably find the global minimum of a family of nonconvex functions with many local minima. It delivers a missing mathematical guarantee for this 'nearly convex' class, which could guide better noise schedules in optimization.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Assumption 4.1 is too weak for the conditional step in the proof of Theorem 4.1: unbiasedness and variance bounds are stated only for fixed x, not along the random iterate sequence, so the one-step contraction (5.6) is not justified as written.","rationale":"I read the paper in good faith and checked the main algebraic chain in Lemma 5.4 and Theorem 4.1, including the handling of β, the choice η = 2α/(5L²), s = λ/(3L), and the bounds in (5.13)–(5.15). The contraction argument is internally consistent once the stochastic oracle is assumed to be conditionally unbiased. The most load-bearing weakness is therefore not the dimension-dependent small-perturbation bound (3.2), which is a scope restriction explicitly built into the definition of nearly convex, nor the overclaim about being the first guarantee of efficient escape, which is a novelty statement. It is that Assumption 4.1, as printed, is insufficient for the proof's conditional expectation step. If the intended convention is that each call to SG uses fresh noise independent of the history, the fix is a one-line strengthening of Assumption 4.1; but as written the theorem statement is incomplete. This is why I would keep the reader's CONDITIONAL verdict rather than upgrading to ACCEPT. The reader flagged conditional unbiasedness only as a minor item in the rationale and made the small-perturbation bound the weakest assumption; I regard the conditional-oracle gap as the more load-bearing of the two, hence partial agreement. The proposed analytical check with a fixed coin flip oracle shows that without the conditional assumption the key one-step inequality can fail on a one-point strongly convex function, which is the simplest nearly convex case.","tokens_in":30413,"tokens_out":26479,"duration_ms":247718,"concrete_test":"Analytic check: take d = 1, f(x) = x²/2, so α = L = 1, η = 2/5, λ = 8/5, and β = 0. Let U ~ Bernoulli(1/2) be fixed from the start and included in F_0. Define the oracle SG(x) = x(1 − 3(1 − 2U)), so for each fixed x, E_U[SG(x)] = x = ∇f(x) and the variance is finite, satisfying Assumption 4.1 literally. On the event U = 0, E[SG(x_t) | F_t] = −2x_t and ω_t = −3x_t, so E[∥y_t − ηω_t − x*∥² | F_t] = ∥(1 − η)x_t − η(−3x_t)∥² + noise terms = 3.24∥x_t∥² + ..., while the proof's claimed bound would be at most ∥y_t − x*∥² + η²r² = 0.36∥x_t∥² + η²r². Recompute the displayed inequality before (5.2) in the proof of Theorem 4.1; it fails unless Assumption 4.1 is strengthened to conditional unbiasedness and conditional variance bounds. This settles whether the missing assumption is load-bearing.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem 4.1, the authors define ω_t = SG(x_t) − ∇f(x_t) and then use E[∥y_t − ηω_t − x*∥² | F_t] ≤ ∥y_t − x*∥² + η²r². This drops the cross term −2η⟨y_t − x*, E[ω_t | F_t]⟩. Assumption 4.1 only guarantees E[SG(x)] = ∇f(x) and E∥SG(x) − ∇f(x)∥² ≤ r² for each fixed deterministic x. It does not state that the oracle noise is independent of the past given x_t, nor that E[SG(x_t) | F_t] = ∇f(x_t) and E[∥ω_t∥² | F_t] ≤ r². Without conditional unbiasedness, E[ω_t | F_t] can be nonzero and aligned with y_t − x*, adding a positive term of order ηδ∥y_t − x*∥² to the one-step bound. For δ > ηλ/200 this term can exceed the claimed contraction factor (1 − ηλ/100), so the inequality preceding (5.2) of the proof is not derivable. The same conditional step is reused in the probability estimates and in Theorem 4.2. This is a patchable assumption gap rather than an error in the later algebra: adding to Assumption 4.1 the requirements E[SG(x_t) | F_t] = ∇f(x_t) and E[∥SG(x_t) − ∇f(x_t)∥² | F_t] ≤ r² for the induced filtration closes the proof. The paper should state this explicitly, because the linear-convergence conclusion rests entirely on this one-step contraction.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a new regularity class called nearly convex functions, defined as small perturbations of one-point strongly convex functions, where the perturbation is controlled by the dimension-dependent bound in Definition 3.3(c). For this class the authors propose the Gaussian Noise Descent (GND) algorithm, which adds an adaptive Gaussian noise whose variance depends on the difference between the current function value and a lower bound estimate f_lb. The main result, Theorem 4.1, states that with step size eta = 2alpha/(5L^2), noise factor s = lambda/(3L), and f_lb <= f*, the iterates y_t = x_t - eta grad f(x_t) satisfy E||y_t - x*||^2 <= (1 - eta lambda/100)^t ||y_0 - x*||^2 + 100b, where b measures the stochastic gradient variance and the lower-bound gap. In particular, with full gradients and f_lb = f*, linear convergence to the unique global minimizer is obtained. A double-loop variant, DL-GND, adaptively updates f_lb and is shown to reach an epsilon-neighborhood in O(ln^2(1/epsilon)) steps in the full-gradient case and O((1/epsilon) ln(1/epsilon)) steps in the stochastic case. The paper also contains examples of nearly convex functions and numerical experiments on a univariate nearly convex function and on Rastrigin functions.","tokens_in":30785,"tokens_out":8612,"duration_ms":84724,"significance":"If the main theorem is correct as stated, this is a genuine theoretical contribution: it gives the first linear-rate global convergence guarantee for a noisy gradient descent method on a nonconvex function class with multiple strict local minima. The derivation is self-contained and the constants eta and s are explicit rather than fitted to the conclusion. The proof of Lemma 5.4 is a real contraction calculation, and the double-loop lower-bound estimation in Theorem 4.2 is a nontrivial extension. The paper's claims about the breadth of the nearly convex class, however, are partly undercut by the dimension-dependent small-perturbation bound (3.2) and by the fact that the stochastic-oracle assumption is not stated in the conditional form actually used in the proof. The numerical experiments are reasonable but do not verify near convexity for the Rastrigin test functions.","major_comments":[{"comment":"Assumption 4.1 states unbiasedness and variance bounds only pointwise for fixed x: E[SG(x)] = grad f(x) and E||SG(x) - grad f(x)||^2 <= r^2. In the proof of Theorem 4.1, however, the step after the centered display following Eq. (5.16) uses E[omega_t | F_t] = 0 and E||omega_t||^2 | F_t] <= r^2 to obtain E[||y_t - eta omega_t - x*||^2 | F_t] <= ||y_t - x*||^2 + eta^2 r^2. This conditional step is not justified by Assumption 4.1 alone: E[omega_t | F_t] can be a nonzero F_t-measurable vector aligned with y_t - x*, which would add a positive cross term of order eta * delta * ||y_t - x*||^2 and can destroy the contraction factor (1 - eta lambda/100). The same conditional unbiasedness is reused in the probability estimates of Theorem 4.1 and in the proof of Theorem 4.2. This is a patchable assumption gap rather than an error in the subsequent algebra: the authors should either add to Assumption 4.1 the requirements E[SG(x_t) | F_t] = grad f(x_t) and E[||SG(x_t) - grad f(x_t)||^2 | F_t] <= r^2 for the induced filtration, or explicitly state that the oracle noises are independent of the past. Without such a strengthening, Theorem 4.1 is not derivable as written.","section":"Assumption 4.1 and the proof of Theorem 4.1 in Section 5.2"},{"comment":"The small-perturbation bound beta(f,x*,alpha) <= (1/4) sqrt(alpha^5/(d L^3)) shrinks as d^{-1/2} with dimension, and the text immediately after Definition 3.3 says the constant can be relaxed but gives no quantitative relaxation. This bound is load-bearing: Lemma 5.4's contraction factor in Eq. (5.14) is derived from the specific constant 1/4 through the estimate C1 < 19/28 lambda in Eq. (5.13). For a reader evaluating whether the class of nearly convex functions is broad enough to be significant, the dimension dependence and the lack of a relaxed version are important. The authors should either prove a quantitative relaxation of (3.2) or state explicitly that the linear global-convergence guarantee applies only to perturbations of order O(d^{-1/2}) relative to the underlying quadratic term.","section":"Definition 3.3(c), Eq. (3.2)"}],"minor_comments":[{"comment":"The inequality sigma(x) <= sqrt(eta s L/2) ||x - x*||^2 + eta s (f* - f_lb) is dimensionally inconsistent: the first term scales as length squared and the second as a function value. What is used in the subsequent derivation is sigma(x)^2 <= (eta s L/2) ||x - x*||^2 + eta s (f* - f_lb), or equivalently sigma(x) <= sqrt(eta s L/2) ||x - x*|| + sqrt(eta s (f* - f_lb)). Please correct this display.","section":"Display before Eq. (5.11)"},{"comment":"The proof of Theorem 4.1 refers to 'by using Lemma 5.4, (4.2) and (5.2)' and the proof of Theorem 4.2 repeatedly says 'we know that (5.3) holds' and 'by using (5.3)'; these references do not correspond to numbered equations in the manuscript and should be renumbered to point to the specific displayed inequalities being used.","section":"Proofs of Theorems 4.1 and 4.2, Section 5"},{"comment":"The Rastrigin experiments are presented as validating the theoretical findings, but the paper does not verify that the Rastrigin functions with the chosen parameters satisfy Definition 3.3(c), and near convexity is not established for them. The text should state explicitly that these experiments are illustrative rather than a direct verification of the assumptions of Theorem 4.1.","section":"Section 6.2, Rastrigin experiments"},{"comment":"The phrase 'first theoretical guarantee that noisy gradient descent can efficiently escape local minima and achieve global convergence' is stronger than necessary, since Ref. [5] already provides an algebraic global-convergence guarantee for an adaptive-noise scheme; the new contribution is the linear rate under the nearly convex class. Please qualify the claim accordingly.","section":"Introduction, first paragraph of Section 1"}],"recommendation":"major_revision","confidential_remarks":"The conditional-unbiasedness gap in Assumption 4.1 is the main technical issue. It is localized and patchable: adding the conditional version of unbiasedness and variance bound closes the proof of Theorem 4.1 and the steps of Theorem 4.2 that reuse it. The contraction algebra in Lemma 5.4 appears internally consistent, and the parameter choices eta = 2alpha/(5L^2), s = lambda/(3L) are not fitted to the conclusion. The dimension-dependent bound in Eq. (3.2) is a genuine limitation of the function class and should be addressed by the authors, either by proving a relaxation or by stating the restriction explicitly. The numerical section is a useful illustration but should not be described as validating the assumptions of the theorem for Rastrigin functions without verification."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe headline: this paper has a genuine new result, but you have to trim the packaging. It defines (α,L)-nearly convex functions—nonconvex, possibly with infinitely many strict local minima and saddle points—and proves that their GND algorithm, SGD plus adaptive Gaussian noise scaled by √(f(x)−f_lb), converges linearly to the global minimizer in both probability and state space. The linear rate is new; earlier adaptive-noise schemes [5,6] only got algebraic rates. The proof is real: Lemma 5.4's contraction with η=2α/5L², s=λ/3L and the β bound (3.2) checks out, and the double-loop DL-GND for estimating f_lb is a useful add-on. The experiments, though simple, do show the expected behavior.\n\nThe soft spots are two, and both are manageable. First, the introduction claims this is 'the first theoretical guarantee that noisy gradient descent can efficiently escape local minima and achieve global convergence.' That is an overclaim, since [5,6] already prove global convergence with algebraic rates; the novelty is linear convergence for this class, not the first global guarantee. Second, the stress-test note is right: Assumption 4.1 states unbiasedness and variance bounds for each fixed x, but the proof of Theorem 4.1 conditions on F_t and needs E[SG(x_t)|F_t]=∇f(x_t) and E[||ω_t||²|F_t] ≤ r². As written, that step is not justified. The fix is a one-line patch—state the conditional version or assume oracle noise is independent of the past—and it doesn't affect the contraction algebra. This is a patchable assumption gap, not a hidden flaw.\n\nThe class is narrow, and that limits the audience. The d^{−1/2} scaling in (3.2) means nearly convexity is quite thin in high dimension, and the paper admits the constant is analysis-friendly. That is a real limitation, but minor relative to the proof's contribution. Knowing f* exactly for the clean rate is addressed by DL-GND.\n\nVerdict: worth serious peer review. The derivation deserves a referee's time; with the assumption patched and the 'first' claim moderated, I'd be comfortable seeing it in a good optimization journal. I'd bring it to a reading group if anyone works on global nonconvex SGD, but I wouldn't rush to cite it unless the class broadens.","headline":"A real new result—linear-rate global convergence for a nonconvex class—but the 'first' claim overshoots and Assumption 4.1 needs a conditional-unbiasedness patch.","tokens_in":31359,"tokens_out":2882,"would_cite":true,"duration_ms":28991,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65K05","90C26","90C06"],"pacs":[],"model":"deepseek-v4-flash","headline":"Adding Gaussian noise whose strength tracks the gap between the current function value and a lower bound makes stochastic gradient descent converge linearly to the global minimum of 'nearly convex' nonconvex functions.","keywords":["stochastic gradient descent","global optimization","linear convergence","nearly convex functions","adaptive Gaussian noise","one-point strong convexity","double-loop algorithm","Rastrigin function"],"falsifier":"A direct test of the theorem's quantitative prediction: take a provably $(\\alpha,L)$-nearly convex function, for instance $J^2_{\\varepsilon,R}$ with $4\\varepsilon(1+\\varepsilon\\sqrt{1+R^2})^{3/2} = 1$ in $d = 1$, run GND with $\\eta = 2\\alpha/(5L^2)$, $s = \\lambda/(3L)$, and $f_{\\mathrm{lb}} = f^*$ over many random trajectories, and compare the empirical contraction factor of $E[\\|y_t - x^*\\|^2]$ with the predicted $1 - \\eta\\lambda/100$ and the zero asymptotic error. A second check is dimensional: determine whether any choice of $(\\varepsilon, R, \\alpha, L)$ satisfies (3.2) at $d = 10$; if the nearly convex class is empty in ten dimensions, then Theorem 4.1 does not cover the 10-dimensional Rastrigin experiments reported in the paper, and the claim that the guarantees scale to high dimension is unsupported.","tokens_in":30153,"feed_emoji":"🎯","tokens_out":18452,"duration_ms":145926,"temperature":0.7,"pith_summary":"Plain gradient descent on a nonconvex function can stop forever at a local minimum. The paper introduces a class of nonconvex functions called nearly convex functions — small perturbations of strongly convex functions that may still contain many strict local minima and saddle points — and proves that a gradient-descent scheme with adaptive Gaussian noise finds their unique global minimizer. The twist is the noise rule: at each step the noise variance is proportional to the gap between the current function value and a lower bound on the optimum, so the algorithm is noisy exactly when it needs to escape a basin and nearly quiet when it approaches the global minimum. With exact gradients and a valid lower bound, convergence to the global minimizer is linear; with stochastic gradients it is linear to a neighborhood whose size is set by the gradient variance and the lower-bound error. The paper also gives a double-loop variant that learns the lower bound on the fly, reaching an $\\varepsilon$-neighborhood in $O(\\ln^2(1/\\varepsilon))$ full-gradient steps.","feed_headline":"Adaptive noise makes SGD provably reach global minima","feed_subtitle":"On nonconvex functions with many local minima, the right noise strength gives linear convergence to the global optimum.","key_machinery":"The machinery has three parts. First, the new regularity condition nearly convex (Definition 3.3): $f$ has a unique global minimizer $x^*$, satisfies $\\|\\nabla f(x)\\| \\le L\\|x - x^*\\|$, and its perturbation distance $\\beta(f, x^*, \\alpha)$ to the family of one-point strongly convex functions (gradient pointing toward the minimizer: $\\langle \\nabla f(x), x - \\bar x\\rangle \\ge \\alpha\\|x - \\bar x\\|^2$) is at most $\\frac14\\sqrt{\\alpha^5/(dL^3)}$. That small-perturbation bound carries the argument: Proposition 3.1 sandwiches $f$ between two quadratics around $x^*$, and Proposition 3.2 caps the barrier height of any local basin, so no basin is deep enough to trap the noise-driven process for long. Second, the adaptive noise rule $\\sigma_t := \\sqrt{\\eta s\\,(f(x_{t+1/2}) - f_{\\mathrm{lb}})_+}$ injected after each gradient step: large far from the lower bound, negligible near it. Third, the contraction lemma (Lemma 5.4), which shows that one step of this update reduces the expected squared distance by the factor $(1 - \\eta\\lambda/100)$ plus an error proportional to $(f^* - f_{\\mathrm{lb}})$; the proof propagates the Gaussian expectation through the gradient map $G(x) = x - \\eta\\nabla f(x)$ and uses the perturbation bounds to control the cross terms.","core_discovery":"The central result, Theorem 4.1, states that if $f$ is $(\\alpha,L)$-nearly convex and the stochastic gradient oracle has variance at most $r^2$, then with $\\eta = 2\\alpha/(5L^2)$, $s = \\lambda/(3L)$, and any lower bound $f_{\\mathrm{lb}} \\le f^*$, the GND iterates satisfy $$E[\\|y_t - x^*\\|^2] \\le \\bigl(1 - \\tfrac{\\eta\\$\\lambda$}{100}\\bigr)^t \\|y_0 - x^*\\|^2 + 100b,$$ where $\\lambda = 2\\alpha - \\eta L^2$ and $b = \\frac{\\eta r^2}{2\\alpha-\\eta L^2} + \\frac{5\\eta\\lambda+14}{42L}(f^* - f_{\\mathrm{lb}})$. When the full gradient is used and $f_{\\mathrm{lb}} = f^*$, the error term vanishes ($b = 0$) and the iterates converge linearly to the unique global minimizer $x^*$, a regime the paper describes as the first guarantee that a noisy gradient method efficiently escapes local minima and converges globally. The variable $y_t = x_t - \\eta\\nabla f(x_t)$ is the point after the full-gradient step; the stochastic-gradient point $x_{t+1/2} = y_t - \\eta\\omega_t$ (with oracle noise $\\omega_t$) receives the Gaussian perturbation $\\sigma_t \\xi_t$, and the next tracked point is $y_{t+1} = G(x_{t+1})$, so the Gaussian convolution of $f$ is what tames the perturbation. Theorem 4.2 extends the guarantee to the double-loop DL-GND algorithm, which alternately refines the lower-bound estimate and runs GND so the user need not know $f^*$ in advance; with full gradients it reaches an $\\varepsilon$-neighborhood of $x^*$ in $O(\\ln^2(1/\\varepsilon))$ iterations with high probability.","pith_inferences":["Editorial extension: the dimension-shrinking factor $1/\\sqrt{d}$ in (3.2) is the part most likely to limit the theory's reach; if the threshold could be relaxed to a dimension-independent constant, the class would plausibly cover high-dimensional objectives of the Rastrigin type, and one can test this by checking whether the contraction condition in Remark 4 still holds for such functions.","Editorial extension: the rule 'noise variance proportional to the value gap' is a transferable design principle; the same schedule could be applied to derivative-free methods, evolutionary strategies, or reinforcement learning, where the objective value (or return) is observed but the gradient is not, suggesting a family of global optimizers beyond SGD.","Editorial extension: the paper proves PL implies nearly convexity in one dimension but leaves the higher-dimensional implication open; if it holds under mild smoothness assumptions, the nearly-convex theory would automatically cover many over-parameterized machine-learning objectives that satisfy PL-type conditions.","Editorial extension: the double-loop lower-bound update is effectively a running best-value tracker; replacing the convex combination with a soft-max over recent best values, or with a schedule that never raises $f_{\\mathrm{lb}}$ above $f^*$, could improve the constants in Theorem 4.2 while preserving the $O(\\ln^2(1/\\varepsilon))$ rate - a testable algorithmic variant."],"forward_implications":["On any $(\\alpha,L)$-nearly convex function with exact gradients, choosing $\\eta = 2\\alpha/(5L^2)$, $s = \\lambda/(3L)$, and $f_{\\mathrm{lb}} = f^*$ yields linear convergence to the global minimizer, with $O(\\ln(1/\\varepsilon))$ iterations needed to reach squared distance $\\varepsilon$ with high probability.","With stochastic gradients of variance $r^2$, the same algorithm converges linearly to a neighborhood of radius proportional to $\\eta r^2/\\lambda$ (plus a lower-bound error term), so oracle noise, not the landscape's local minima, limits the achievable accuracy.","The double-loop DL-GND reaches an $\\varepsilon$-neighborhood of the global minimizer in $O(\\ln^2(1/\\varepsilon))$ full-gradient iterations with high probability and without knowing $f^*$ in advance, because it refines the lower bound from the best function values seen along the run.","Because nearly convex functions may contain infinitely many strict local minima and saddle points, the guarantees cover a landscape class strictly outside the reach of standard global-convergence theory for first-order methods, which typically excludes non-global stationary points."],"supporting_citations":[{"why":"the adaptive sublevel-set acceptance scheme with algebraic convergence that GND builds on and improves to a linear rate","marker":"[5]"},{"why":"supplies the convolution-smoothing view of SGD and the one-point strong convexity notion that motivates the nearly convex definition","marker":"[14]"},{"why":"one of the diffusion analyses showing that noise decaying too fast traps the process in local minima, used as the baseline the authors improve upon","marker":"[3]"},{"why":"the companion diffusion result establishing the critical O(1/sqrt(ln t)) noise decay rate, cited as the reason adaptive noise is needed","marker":"[9]"},{"why":"the PL and quadratic-growth linear-convergence toolkit used to compare nearly convexity with standard regularity conditions","marker":"[13]"},{"why":"the restricted secant inequality, which the paper shows is equivalent to one-point strong convexity and hence a special case of nearly convexity","marker":"[23]"}],"fun_headline_variants":["Adaptive Gaussian noise gives SGD global convergence","SGD with tuned noise escapes local minima","Global optimization via noise-perturbed SGD","Adaptive noise provably steers SGD to global optimum"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument collapses if the perturbation bound $\\beta(f, x^*, \\alpha) \\le \\frac14\\sqrt{\\alpha^5/(dL^3)}$ fails: the paper sets this constant for analytical convenience, admits it could be relaxed but gives no quantitative relaxation, and since it shrinks as $1/\\sqrt{d}$, the class of provably covered functions thins out as dimension grows.","fun_headline_variants_meta":{"raw":{"variants":["Adaptive Gaussian noise gives SGD global convergence","SGD with tuned noise escapes local minima","Global optimization via noise-perturbed SGD","Adaptive noise provably steers SGD to global optimum"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00056,"raw_usage":{"total_tokens":2763,"prompt_tokens":1147,"completion_tokens":1616,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":763,"completion_tokens_details":{"reasoning_tokens":1558}},"tokens_in":763,"tokens_out":1616,"duration_ms":11679,"temperature":1.0,"reasoning_tokens":1558,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T23:58:17.874350+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct test of the theorem's quantitative prediction: take a provably $(\\alpha,L)$-nearly convex function, for instance $J^2_{\\varepsilon,R}$ with $4\\varepsilon(1+\\varepsilon\\sqrt{1+R^2})^{3/2} = 1$ in $d = 1$, run GND with $\\eta = 2\\alpha/(5L^2)$, $s = \\lambda/(3L)$, and $f_{\\mathrm{lb}} = f^*$ over many random trajectories, and compare the empirical contraction factor of $E[\\|y_t - x^*\\|^2]$ with the predicted $1 - \\eta\\lambda/100$ and the zero asymptotic error. A second check is dimensional: determine whether any choice of $(\\varepsilon, R, \\alpha, L)$ satisfies (3.2) at $d = 10$; if the nearly convex class is empty in ten dimensions, then Theorem 4.1 does not cover the 10-dimensional Rastrigin experiments reported in the paper, and the claim that the guarantees scale to high dimension is unsupported.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the convolution-smoothing view of SGD and the one-point strong convexity notion that motivates the nearly convex definition"},{"cited_title":"Diffusion for global optimization in Rn","cited_arxiv_id":null,"evidence_quote":"one of the diffusion analyses showing that noise decaying too fast traps the process in local minima, used as the baseline the authors improve upon"},{"cited_title":"Linear convergence of gradient and proximal- gradient methods under the polyak-/suppress lojasiewicz condition","cited_arxiv_id":null,"evidence_quote":"the PL and quadratic-growth linear-convergence toolkit used to compare nearly convexity with standard regularity conditions"}],"review_version":1}