{"id":"ecd6bfe6-a1a0-48b8-84d2-b97b735ca3a6","arxiv_id":"2505.11621","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"The paper proves high-probability, non-asymptotic bounds showing that kernel ridge regression and two-layer ReLU networks in the NTK regime can achieve both arbitrarily small training and test error without assuming the regression function lies in the kernel RKHS.","lead":"This paper argues that 'almost benign overfitting', where a model has both very low training error and very low test error at the same time, can occur even in the classical U-shaped risk curve if model complexity grows with sample size. It proves this for kernel ridge regression and for two-layer ReLU networks trained by gradient flow, assuming only boundedness of the target function and noise.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"ReLU NTK has an infinite-dimensional kernel (odd degrees h>=3), so the theorem's 'arbitrary bounded f*' claim fails: for f* = Y_3 no finite L satisfies (4.1) with lambda_epsilon > 0.","rationale":"The reader's verdict identified a real overclaim about dimension: Assumption 2(i) requires d = Omega(log(1/delta)), which is incompatible with Definition 1 for a fixed low-dimensional input. That concern is legitimate and supports a conditional verdict. However, there is a more fundamental obstruction that the reader did not flag. The paper's own eigenvalue calculation shows the ReLU NTK has zero eigenvalues for all odd spherical harmonics of degree >= 3, making the RKHS non-dense and the operator H non-injective. The spectral decomposition used throughout Section 4 is therefore internally inconsistent: a compact positive operator with infinitely many positive eigenvalues and an infinite-dimensional kernel admits no nonincreasing eigenvalue enumeration whose eigenvectors form a complete orthonormal basis. More importantly, the advertised claim that the results hold for arbitrary bounded f* is false: for f* equal to a bounded odd spherical harmonic, Equation (4.1) cannot be satisfied with lambda_epsilon > 0, so T_epsilon is undefined and the approximation theorem has no content. This is not a merely technical gap; it directly invalidates the central novelty stated in the abstract and in Theorem 11. The proof technique may be salvageable by adding an explicit assumption that f* lies in the closure of the range of H (equivalently, has no odd-degree components) and by fixing the spectral ordering, but as stated the central claim is unsupported. For this reason the appropriate verdict is REJECT rather than CONDITIONAL, despite the paper's substantial and apparently careful technical machinery.","tokens_in":68889,"tokens_out":28565,"duration_ms":297486,"concrete_test":"Analytic check: fix d = 3 and set f* to the normalized degree-3 spherical harmonic (e.g., the Legendre polynomial P_3(x_3), scaled to be bounded by 1 and unit L2 norm). Using the eigenvalues in Appendix D.2.3, verify directly that H f* = 0. Then attempt to find L_epsilon satisfying (4.1) with lambda_epsilon > 0; if no such L_epsilon exists, the assumptions of Theorem 8 are unsatisfiable for this f*. As a numerical corroboration, run population gradient flow for a wide two-layer ReLU network initialized as in Section D.2.2 with large m and track ||f_t - f*||_2; if the norm saturates above epsilon for the prescribed T_epsilon, the claimed approximation guarantee fails.","verdict_should_be":"REJECT","load_bearing_attack":"Appendix D.2.3's own spectral calculation gives mu_h = 0 for every odd h >= 3, while positive eigenvalues exist for h = 1 and for infinitely many even h >= 4. Thus the ReLU NTK operator H has an infinite-dimensional kernel. Yet Section D.2.3 claims the eigenfunctions phi_l form an orthonormal basis of L2 with lambda_1 >= lambda_2 >= ... and lambda_l -> 0 'from above'; no such nonincreasing enumeration exists when both the positive spectrum and the zero eigenspace are infinite-dimensional. The consequence is direct: take any bounded f* with a nonzero component on an odd spherical harmonic of degree >= 3, for example the normalized Y_3. Then H f* = 0. To satisfy (4.1), either Y_3 remains in the tail, leaving tail norm equal to 1 for every finite L, or Y_3 enters the 'top L' space, which forces the corresponding eigenvalue to be 0 and hence lambda_epsilon = lambda_{L_epsilon} = 0. In both cases T_epsilon = (2/lambda_epsilon) log(2/sqrt(epsilon)) is undefined or infinite and Assumption 3 cannot be satisfied. The approximation proof in Lemma 23(ii)(a) uses 4 <zeta_t, H zeta_t>_2 >= 4 lambda_epsilon ||zeta_t^L||_2^2; with lambda_epsilon = 0 this lower bound is 0 and no exponential decay follows. Therefore Theorem 8, and consequently Theorem 11, do not hold for arbitrary bounded f* as claimed in the abstract. The same denseness problem invalidates the KRR/NTK case in Section 3, since that argument also requires the RKHS of the NTK to be dense in L2(rho_{d-1}).","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes reinterpreting benign overfitting through 'almost benign overfitting,' where empirical risk and excess risk can both be made arbitrarily small by jointly increasing sample size and model complexity. The authors prove such guarantees for two case studies: kernel ridge regression with the ReLU neural tangent kernel, and least-squares regression with a two-layer ReLU network trained by gradient flow in the NTK regime. The main advertised contribution is that these results hold under essentially no assumptions on the regression function beyond boundedness, and in low input dimension, using a novel approximation/estimation decomposition of the excess risk that avoids uniform convergence. The paper also provides experiments on synthetic, Abalone, and Wine data that show risk curves consistent with the hypothesis.","tokens_in":69335,"tokens_out":5953,"duration_ms":66429,"significance":"If the main theorem were correct, the paper would represent a substantial conceptual shift: benign overfitting would no longer require high dimensionality, kernel eigenvalue decay conditions, or interpolation, and the neural-network result would be the first generalization guarantee of this type for arbitrary bounded regression functions. The proof machinery is genuinely elaborate and contains independently interesting tools, including a Hadamard-product bound for integral operators and concentration results for vector-valued U- and V-statistics. The paper is also honest about its limitations, such as only providing upper bounds. However, these strengths do not compensate for the central technical gaps described below, which affect the core claims of the paper.","major_comments":[{"comment":"The ReLU NTK on the sphere has an infinite-dimensional kernel, and this invalidates the claimed arbitrary-bounded-f* result. The paper's own spectral calculation in Appendix D.2.3 gives mu_h = 0 for every odd h >= 3, while positive eigenvalues exist for h = 1, h = 2, and infinitely many even h >= 4. Consequently, the operator H has both an infinite-dimensional positive eigenspace and an infinite-dimensional zero eigenspace, so the statement that the eigenfunctions form an orthonormal basis with lambda_1 >= lambda_2 >= ... and lambda_l -> 0 'from above' is not correct as an enumeration with nonzero eigenvalues only. More importantly, for a bounded f* with a nonzero component on an odd spherical harmonic of degree at least 3, for example the normalized Y_3, Eq. (4.1) cannot be satisfied with lambda_epsilon > 0: if Y_3 is left in the tail, the tail norm is 1 for every finite L, and if Y_3 is placed among the top L eigenfunctions, the corresponding eigenvalue is 0, so lambda_epsilon = 0 and T_epsilon in Eq. (4.2) is undefined. Hence Theorem 8, and therefore Theorems 10 and 11, do not hold for arbitrary bounded f* as claimed in the abstract and in Section 4.2.","section":"Appendix D.2.3 and Eq. (4.1)"},{"comment":"The KRR section relies on the assertion that the RKHS of the ReLU NTK is dense in L2(rho_{d-1}), writing 'By the denseness of H in L2(rho), there is an f_epsilon in H such that ||f* - f_epsilon||_2^2 <= epsilon/8.' This assertion is false for the ReLU NTK: as the spectral calculation in Appendix D.2.3 shows, the kernel misses all odd spherical harmonics of degree at least 3, so functions such as Y_3 are orthogonal to the RKHS. Therefore the approximation argument behind Theorem 3 and the benign-overfitting conclusion in Theorem 6 only hold for regression functions lying in a proper subspace, not for arbitrary bounded f*. This is a load-bearing error in the paper's central claim of no assumptions on f*.","section":"Section 3, paragraph 'By the denseness of H in L2'"},{"comment":"Definition 1 requires that for every epsilon, delta > 0 there exists a sample size n such that, with probability at least 1 - delta, both risks are at most epsilon. For a fixed problem, the input dimension d is fixed, but Assumptions 1(i) and 2(i) require e^{-d} <= delta/4 and e^{-d} <= delta/12, respectively, i.e., d >= Omega(log(1/delta)). Thus, for any fixed low-dimensional problem, the stated theorems cannot cover arbitrarily small delta, and the results do not achieve Definition 1. The paper's claim that the results 'hold on low-dimensional inputs' is therefore only true in the limited sense that d is not required to grow with n or m, but d must still grow without bound as delta tends to zero. This is a mismatch between the theorem statements and the formal definition of the object they claim to establish.","section":"Definition 1 vs. Assumptions 1(i) and 2(i)"}],"minor_comments":[{"comment":"There is a typo in the Section 4 heading: 'Bengin Overfitting' should read 'Benign Overfitting.' The same typo appears in the table of contents.","section":"Title of Section 4 and Figure captions"},{"comment":"The phrase 'lambda_l -> 0 from above' is misleading and should be replaced by a correct treatment of the zero eigenspace; the current wording suggests that all eigenvalues are positive, which contradicts the formula mu_h = 0 for odd h >= 3 appearing a few lines later.","section":"Appendix D.2.3, paragraph after spectral decomposition"},{"comment":"The experiments use gradient descent with learning rate 0.1, while the theory is developed for gradient flow; the paper should state explicitly that discretization is not covered by the proofs and that the experiments are only heuristic support.","section":"Section 4.3 and Appendix D.8"}],"recommendation":"reject","confidential_remarks":"The spectral nullspace issue is decisive and is not a matter of tightening constants: the ReLU NTK simply cannot approximate odd spherical harmonics of degree at least 3. Since the paper's flagship claim is that no assumptions on f* are needed, this is a load-bearing error that cannot be repaired within the manuscript's current scope. The fixed-dimension issue with Definition 1 is an additional and independent defect. The sophisticated proof machinery might be salvageable under substantially restricted assumptions, but that would be a different paper."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know. First, the approximation-estimation decomposition and the U/V-statistic concentration arguments are genuinely new and well executed; the real-induction control of the gradient trajectory is a nice piece of work. Second, the main theorem as stated is false, and the reason is in the paper's own appendix.\n\nD.2.3 computes the NTK eigenvalues. Odd spherical harmonics of degree h >= 3 have eigenvalue exactly zero, while h=1 and all even h >= 4 are positive. So the NTK operator has an infinite-dimensional kernel. That immediately contradicts the paper's claim that the eigenfunctions form an orthonormal basis with lambda_l -> 0 'from above' — you can't enumerate infinitely many positive eigenvalues and infinitely many zeros in a single decreasing sequence. More importantly, it kills Theorem 8 and therefore Theorem 11. Take any bounded f* with a nonzero Y_3 component. The tail norm in (4.1) can't go below 1 for any finite L unless you put Y_3 into the 'top L' space, but then lambda_epsilon = 0, T_epsilon is infinite, and the exponential decay argument in Lemma 23(ii) collapses. The approximation error simply does not vanish. The same issue invalidates the KRR/NTK case study in Section 3: the RKHS is not dense in L2(rho_{d-1}), so the approximator f_epsilon need not exist.\n\nThe reader's own d-vs-delta concern is real but minor by comparison. The claim that fixed low dimension can satisfy Definition 1 for all delta conflicts with Assumption 2(i), which forces d >= Omega(log(1/delta)). That's a fixable overstatement if the target is 'for each delta, choose d large enough.' The notational glitch around T'_epsilon is trivial.\n\nWhat's worth keeping: the decomposition of excess risk along the gradient-flow trajectory, the vector-valued Hoeffding bounds for U/V-statistics, and the general strategy of avoiding uniform convergence. These could be useful in other settings, and the experiments honestly illustrate the claimed phenomenon. But the central theorem overreaches, and the abstract's 'no assumptions on f*' claim is grounded in an operator that doesn't actually have full rank.\n\nRecommendation: send it to reviewers, because the arguments are sophisticated and a referee report can pinpoint exactly where the spectral assumption fails. But it should not be accepted in this form; the authors need to restrict the class of f* to the positive spectral subspace (or replace the NTK with a universal kernel), and the current claims about arbitrary bounded regression functions should be withdrawn.","headline":"The proof machinery is real, but the ReLU NTK's infinite-dimensional kernel sinks the 'any bounded f*' theorem; major revision needed.","tokens_in":69827,"tokens_out":3916,"would_cite":false,"duration_ms":37121,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Benign overfitting can occur in the classical U-shaped regime when sample size and model complexity grow together, the paper proves for kernel ridge regression and trained two-layer ReLU networks.","keywords":["benign overfitting","almost benign overfitting","kernel ridge regression","neural tangent kernel","two-layer ReLU networks","gradient flow","implicit regularization","generalization bound"],"falsifier":"Evaluate the theorem's dimension–confidence trade-off at the dimensions its experiments use: with $d=7$ (Abalone), Assumption 2(i) requires $\\delta\\ge 12e^{-7}\\approx 0.011$, so at $\\delta=0.001$ the theorem's hypotheses are void. Running the Abalone-style experiment at $d=7$ with $\\delta$ below that threshold — or checking whether a bounded target whose mass sits mostly on the low eigenvalues of the NTK operator still allows both risks below $\\varepsilon$ at fixed $d$ — would show whether the claim survives outside the stated scaling regime.","tokens_in":68655,"feed_emoji":"📈","tokens_out":17637,"duration_ms":150790,"temperature":0.7,"pith_summary":"The paper argues that 'almost benign overfitting' — a model that fits noisy training data to arbitrarily small error while also generalizing to arbitrarily small excess risk — can occur in the classical U-shaped regime of the risk-versus-complexity curve, provided sample size and model complexity are increased together. Prior work placed benign overfitting outside the classical regime, in the interpolation region, and typically required high-dimensional inputs or structural assumptions on the regression function. The paper proves the claim in two settings: kernel ridge regression, and two-layer ReLU networks trained by gradient flow in the neural-tangent-kernel regime, with no assumptions on the regression function or the noise beyond boundedness. The proof decomposes excess risk into approximation and estimation errors and reads gradient flow as an implicit regularizer, which lets the analysis avoid uniform convergence. The central picture is that the trough of the U-curve moves 'down and to the right' as data accumulate, so the best model at large sample size is both larger and better fitting.","feed_headline":"More data lets overfitted models generalize, proof shows","feed_subtitle":"Kernel ridge and ReLU nets can push training and test error below any epsilon at once.","key_machinery":"The load-bearing mechanism is the approximation–estimation decomposition of the excess risk along the gradient-flow trajectory, with gradient flow itself treated as an implicit regularizer. The paper compares an $m$-neuron ReLU network $\\hat f_t$ trained on the empirical risk with an oracle network $f_t$ of the same architecture trained on the population risk: the oracle gap $\\|f_t-f_*\\|_2$ is the approximation error, and the tracking gap $\\|\\hat f_t-f_t\\|_2$ is the estimation error. The implicit regularizer is the stopping time $T_\\varepsilon = 2\\lambda_\\varepsilon^{-1}\\log(2/\\sqrt\\varepsilon)$, where $\\lambda_\\varepsilon$ is the $L_\\varepsilon$-th eigenvalue of the analytical neural tangent kernel operator, chosen so that the component of $f_*$ in the remaining high-frequency eigenspace has norm at most $\\sqrt\\varepsilon/4$. Three supporting devices carry the proofs: a spectral-norm bound for Hadamard products extended from matrices to integral operators (used for the approximation error), new concentration inequalities for vector-valued U- and V-statistics that allow the estimation-error bound to be made at initialization and then iterated $U_\\varepsilon$ times with the factorial $U_\\varepsilon!$ absorbing the remainder, and a real-induction argument that keeps the weights near initialization so the minimum eigenvalue of the NTK Gram matrix stays bounded along the whole trajectory.","core_discovery":"The central discovery, stated on the paper's own terms, is that whether overfitting is benign is decided by the joint scaling of sample size and model complexity, not by model complexity alone. The paper defines benign overfitting as achieving empirical risk at most $\\varepsilon$ and excess risk $R(\\hat f)-R(f_*) \\le \\varepsilon$ simultaneously, for every $\\varepsilon,\\delta>0$ with probability at least $1-\\delta$ (Definition 1), deliberately allowing training error that is small but not exactly zero. For kernel ridge regression with the neural tangent kernel, Theorem 6 shows that when the regularization parameter $\\gamma$ is small and the sample size $n$ is large relative to $\\varepsilon$ and $\\delta$ (Assumption 1), the regularized empirical risk minimizer achieves both bounds on a single high-probability event. For two-layer ReLU networks trained by full-batch gradient flow in the NTK regime — where each neuron moves so little that the network behaves like a kernel method — Theorem 11 shows that for any essentially bounded regression function $f_*$, if input dimension, width $m$, and sample size $n$ satisfy Assumptions 2 and 3, then at stopping time $T_\\varepsilon = \\frac{2}{\\lambda_\\varepsilon}\\log\\frac{2}{\\sqrt\\varepsilon}$ the trained network satisfies both bounds on the same event of probability at least $1-\\delta$. The authors claim this as the first generalization result in this setting that assumes nothing about the regression function or the noise beyond boundedness.","pith_inferences":["I read the Section 2 narrative as a reconciliation claim the paper states but does not fully prove: earlier interpolation-regime results hold the model fixed on the upward slope of a U-curve drawn at a fixed sample size, whereas this paper moves along the moving trough; a head-to-head comparison of the two regimes under identical scaling is a natural next step.","The vector-valued U-/V-statistic concentration bounds and the iterated-remainder trick are separable tools that should transfer to other settings needing high-probability control of two coupled trajectories without uniform convergence, such as stochastic gradient descent or feature-learning dynamics.","A testable extension suggested by the assumptions: the paper notes that smooth activations would simplify the proofs, so if the mechanism is really the approximation–estimation trade-off rather than ReLU specifics, the benign-overfitting window should widen (smaller required $n$ and $m$) for smooth activations.","The conclusion flags that only upper bounds are provided and matching lower bounds remain open; I draw the consequence that the 'low-dimensional inputs' claim should be read as conditioned on $d$ growing with $\\log(1/\\delta)$, given Assumption 2(i)'s requirement $e^{-d}\\le\\delta/12$."],"forward_implications":["Benign overfitting does not require leaving the classical regime: a practitioner who adds data and model capacity together can push both training error and test error below any desired level, without high-dimensional inputs or a well-specified model class.","For two-layer ReLU networks in the NTK regime, generalization to the Bayes-optimal risk is achievable for every bounded regression function, with no RKHS-of-NTK assumption and no noise model beyond boundedness.","The KRR result is not tied to the specific kernel: the same proof works for any bounded kernel whose Gram matrix has the required minimum-eigenvalue lower bound and whose RKHS is dense in $L^2(\\rho_{d-1})$.","Because the analysis deliberately avoids uniform convergence over the parameter space, the bounds do not deteriorate as the width $m$ grows; more parameters do not hurt the generalization guarantee.","The experiments corroborate the predicted 'down and to the right' shift: on Abalone, Wine, and synthetic sphere data, the point where excess risk crosses below empirical risk moves later in training and lower in value as $n$ increases, and both risks at the crossing fall."],"supporting_citations":[{"why":"Defines the neural tangent kernel and the lazy-training regime that the gradient-flow analysis of Section 4 operates in.","marker":"Jacot et al. (2018)"},{"why":"Supplies the integral-operator technique that the kernel ridge regression proofs use to bound excess risk without uniform convergence.","marker":"Caponnetto and De Vito (2007)"},{"why":"Gives the closed-form regularized minimizer identities on which Theorems 2-5 rely.","marker":"Park and Muandet (2020)"},{"why":"Provides the V-statistic representation used in the concentration bound of Proposition 14.","marker":"Lee (1990)"},{"why":"Provides the permutation representation of U-statistics used in Proposition 13.","marker":"Serfling (1980)"},{"why":"Supplies the martingale concentration result in Banach spaces behind the vector-valued Hoeffding bound that controls the estimation error.","marker":"Pinelis (1992)"},{"why":"Provides the Gaussian singular-value and sub-Gaussian concentration results that yield the minimum-eigenvalue lower bounds for the Gram and NTK matrices.","marker":"Vershynin (2018)"},{"why":"Supplies the Matrix Chernoff inequality used to lower-bound the minimum eigenvalue of the initial NTK Gram matrix (Lemma 18).","marker":"Tropp (2012)"},{"why":"Gives the Funk-Hecke formula and spherical-harmonic machinery used to compute the NTK operator's spectrum on the sphere (Lemma 15, Appendix D.2.3).","marker":"Müller (1998)"},{"why":"Provides the spectral-norm bound for Hadamard products of positive semidefinite matrices that is extended to integral operators in Lemma 12 and used throughout.","marker":"Horn and Johnson (2013)"}],"fun_headline_variants":["Sample size, not complexity, decides benign overfitting","Benign overfitting needs enough data, new proof shows","Joint scaling of data and model yields benign overfitting","ReLU nets and kernels: benign overfitting without tight assumptions","Almost benign overfitting: train and test error shrink together"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is a set of scaling relations: the input dimension $d$ must be at least logarithmic in the inverse failure probability ($e^{-d}\\le\\delta/12$), and the width $m$ and sample size $n$ must be large enough relative to $d$, $\\varepsilon$, and the spectral gap $\\lambda_\\varepsilon$; if these fail, the high-probability minimum-eigenvalue bounds on which every theorem rests are unavailable.","fun_headline_variants_meta":{"raw":{"variants":["Sample size, not complexity, decides benign overfitting","Benign overfitting needs enough data, new proof shows","Joint scaling of data and model yields benign overfitting","ReLU nets and kernels: benign overfitting without tight assumptions","Almost benign overfitting: train and test error shrink together"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000198,"raw_usage":{"total_tokens":1454,"prompt_tokens":1115,"completion_tokens":339,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":731,"completion_tokens_details":{"reasoning_tokens":259}},"tokens_in":731,"tokens_out":339,"duration_ms":3732,"temperature":1.0,"reasoning_tokens":259,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T20:53:33.009990+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the theorem's dimension–confidence trade-off at the dimensions its experiments use: with $d=7$ (Abalone), Assumption 2(i) requires $\\delta\\ge 12e^{-7}\\approx 0.011$, so at $\\delta=0.001$ the theorem's hypotheses are void. Running the Abalone-style experiment at $d=7$ with $\\delta$ below that threshold — or checking whether a bounded target whose mass sits mostly on the low eigenvalues of the NTK operator still allows both risks below $\\varepsilon$ at fixed $d$ — would show whether the claim survives outside the stated scaling regime.","supporting_citations":[{"cited_title":"Neural T angent K ernel: C onvergence and G eneralization in N eural N etworks","cited_arxiv_id":null,"evidence_quote":"Defines the neural tangent kernel and the lazy-training regime that the gradient-flow analysis of Section 4 operates in."},{"cited_title":"Approximation T heorems of M athematical S tatistics","cited_arxiv_id":null,"evidence_quote":"Provides the permutation representation of U-statistics used in Proposition 13."},{"cited_title":"An A pproach to I nequalities for the D istributions of I nfinite- D imensional M artingales","cited_arxiv_id":null,"evidence_quote":"Supplies the martingale concentration result in Banach spaces behind the vector-valued Hoeffding bound that controls the estimation error."},{"cited_title":"High- D imensional P robability: A n I ntroduction with A pplications in D ata S cience , volume 47","cited_arxiv_id":null,"evidence_quote":"Provides the Gaussian singular-value and sub-Gaussian concentration results that yield the minimum-eigenvalue lower bounds for the Gram and NTK matrices."},{"cited_title":"User- F riendly T ail B ounds for S ums of R andom M atrices","cited_arxiv_id":null,"evidence_quote":"Supplies the Matrix Chernoff inequality used to lower-bound the minimum eigenvalue of the initial NTK Gram matrix (Lemma 18)."},{"cited_title":"Matrix A nalysis","cited_arxiv_id":null,"evidence_quote":"Provides the spectral-norm bound for Hadamard products of positive semidefinite matrices that is extended to integral operators in Lemma 12 and used throughout."}],"review_version":1}