{"id":"81fb6677-289b-4dfb-9c34-614abe5ee7d6","arxiv_id":"2411.13733","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Low-rank layers in deep networks yield Gaussian complexity bounds where the rank factor appears once, not once per layer, improving on prior norm-based bounds.","lead":"The authors derive a new generalization bound for deep networks whose weight matrices are low rank and spectrally bounded. The bound keeps rank and width factors from multiplying across layers, which may explain why low-rank training solutions and neural collapse generalize well.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 6's proof drops the supremum over the spectral-norm ball, replacing it with the Frobenius norm of a fixed weight; Theorem 7's main bound is therefore unsupported as written.","rationale":"The paper's central contribution is the Gaussian complexity bound for low-rank spectral-norm-constrained ReLU networks, with the advertised feature that rank factors do not multiply across layers. The argument's architecture is: bound the diameter D(Z_{i-1}) (Lemma 5, valid for piecewise-linear activations), bound the Gaussian-average Lipschitz quotient R(W_i,Z_{i-1}) (Lemma 6), and combine them in Maurer's chain rule (Section 4.5). Lemma 6 is the only step where the rank of the linear layer is converted into a Frobenius-norm-based bound, and it is the step whose proof is demonstrably wrong: the supremum over the spectral-norm ball is dropped. Without Lemma 6, the recursion has no valid bound for the disturbance terms, so Theorem 7 does not follow. This is not a matter of an overly strong assumption or a missing reference; it is an internal inconsistency in the proof.\n\nThe concern is not that the claim is false. The exact computation of sup_{||W||_2≤B} ⟨W,A⟩ = B||A||_* suggests a repair: ||A||_* ≤ √rank(A)||A||_F, and rank(A) for A=Γ(Z−Z') is at most 2 min_{j<i} rank(W_j). Combining with the √r factor already present in the diameter bound yields a disturbance term of order r_i ∏_{j≤i}||W_j||√h_i, which still gives the stated O(Lr) scaling. So the paper's qualitative thesis—low-rank layers prevent rank multiplication across layers—is plausible.\n\nThe limitations paragraph acknowledging the C_1^L factor is candid and not a correctness issue. The finite-class assumption in Theorem 3 is also unaddressed, but it is a standard technicality. The reader's stated weakest_assumption (piecewise linearity) is a scope restriction rather than the point where the proof breaks; the actual breakdown is Lemma 6. I therefore agree with the REJECT verdict for the current version, while noting the result may be repairable.","tokens_in":13994,"tokens_out":18845,"duration_ms":139292,"concrete_test":"Independently re-derive Lemma 6 for the class F={W∈R^{2×2}: ||W||_2≤1} with a concrete pair Z,Z' and a fixed Gaussian draw; compute both sup_{||W||_2≤1} ⟨W, Γ(Z−Z')⟩ and ||W_ℓ||_F ||Γ(Z−Z')||_F, and verify that the second displayed inequality fails when Γ(Z−Z') has unequal singular values. Then redo the recursion of Section 4.5 replacing the erroneous inequality with the identity sup_W ⟨W,A⟩ = B||A||_*. Check whether the product D(Z_{i−1})R(W_i,Z_{i−1}) still yields at most O(C_1^L ∏B_i L r √h/m); if it does, the proof needs revision but the theorem's conclusion survives, and if not, the advertised rank-depth trade-off is invalid.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 6 (Section 4.3) claims R(F, F_ℓ−1) ≤ ||W_ℓ||_F √h_ℓ for the class F={W_ℓ x : ||W_ℓ||_2 ≤ B}. Its proof starts from the definition R = sup_{Z,Z'} E_γ [ sup_{||W_ℓ||_2≤B} ⟨W_ℓ, Γ(Z−Z')⟩ / ||Z−Z'||_F ] and immediately replaces the inner supremum by ||W_ℓ||_F ||Γ(Z−Z')||_F, with W_ℓ no longer quantified. That is not valid: for a fixed random matrix A=Γ(Z−Z'), the correct value of the supremum over the spectral-norm ball is B||A||_* (nuclear norm), which can exceed ||W_ℓ||_F||A||_F for any rank-deficient W_ℓ. Example: B=1, A=diag(1,1), W_ℓ=diag(1,0); the left side is 2, the right side is √2 < 2. Since R is a class-level quantity in Maurer's chain rule, the product D(Z_{i−1})R(W_i,Z_{i−1}) in the recursion of Section 4.5 is not bounded by the claimed expression, so the derivation of Eq. (4) and the simplified O(C_1^L ∏B_i L r √h/m) bound collapses. The gap is likely repairable: using B||A||_* ≤ B√(2 min_{j<i} r_j) ||A||_F leads to a similar O(Lr) scaling after multiplying by the √r factor already present in D(Z_{i−1}), but the preprint does not supply this argument. Hence the main theorem is unproven in the current version.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes generalization bounds for deep neural networks whose weight matrices are constrained in spectral norm and rank. The authors apply Maurer's chain rule for Gaussian complexity to the class F_L = {f_W = W_L φ(...φ(W_1 x)...) : ||W_i||_2 ≤ B_i, rank(W_i) ≤ r_i} and claim a bound of the form O(C_1^L ∏_{i=1}^L B_i · L r √h / m), which would avoid the rank factor multiplying across layers that appears when prior norm-based bounds are evaluated on this class. The paper also discusses comparisons with Golowich et al., Bartlett et al., and Neyshabur et al., and argues that intermediate neural collapse can improve the bound via a rank-1 layer. The main proof is in Section 4, with key lemmas bounding the diameter of the function class (Lemma 5) and the Gaussian average of Lipschitz coefficients (Lemma 6), and a final assembly in Theorem 7.","tokens_in":14291,"tokens_out":17457,"duration_ms":160486,"significance":"If the main theorem were correct, the paper would offer a genuinely useful contribution: it would show that rank constraints control Gaussian complexity linearly in depth rather than exponentially, which is a non-obvious and potentially important improvement for the analysis of low-rank-trained deep networks. The use of Maurer's chain rule to sidestep the multiplicative rank accumulation is conceptually appealing, and the diameter bound in Lemma 5 is a solid and reusable ingredient. The paper is also clearly written in its high-level structure, with explicit comparisons in Table 1 and a self-contained derivation from standard results (Maurer's chain rule, Talagrand's contraction, Bartlett–Mendelson Gaussian complexity bounds). However, the central proof currently contains a serious flaw: Lemma 6, which is load-bearing for Theorem 7, is incorrect as stated. The significance of the contribution is therefore conditional on a successful repair of the proof.","major_comments":[{"comment":"The proof of Lemma 6 is invalid as written. The quantity R(F, F_{ℓ-1}) is defined as a supremum over the class F = {W_ℓ : ||W_ℓ||_2 ≤ B}, but the proof replaces sup_{||W_ℓ||_2≤B} ⟨W_ℓ, Γ(Z−Z')⟩ with ||W_ℓ||_F ||Γ(Z−Z')||_F for a fixed matrix W_ℓ, without any justification. The correct value of this supremum is B ||Γ(Z−Z')||_* (the nuclear norm), which is not bounded by ||W_ℓ||_F ||Γ(Z−Z')||_F for a fixed W_ℓ. A concrete counterexample is B=1, A = Γ(Z−Z') = diag(1,1), and W_ℓ = diag(1,0): the left-hand side equals 2 while the right-hand side equals √2. Since R(F, Y) appears in the chain rule (Theorem 3) and in the recursion of Section 4.5, Lemma 6 is load-bearing and its failure invalidates the proof of Theorem 7 as it currently stands.","section":"Section 4.3, Lemma 6"},{"comment":"The theorem statement and its proof conflate individual weight matrices with the function class. Eq. (4) is written in terms of ||W_i||_F and ||W_i||_2 for a particular network, but Gaussian complexity is a supremum over all f_W in F_L, so a valid upper bound must depend only on the class parameters B_i, r_i, h_i, and the data. In the recursion of Section 4.5, the chain rule's first term requires the Lipschitz constant of the class of linear maps, which is sup_{||W||_2≤B_i} ||W||_2 = B_i, not ||W_i||_2 for an arbitrary member. This is the same class-versus-instance conflation seen in Lemma 6 and is a separate obstruction to the correctness of the stated bound.","section":"Theorem 7 and Section 4.5"},{"comment":"The definition of κ_i in Eq. (4) is not consistent with the lemmas it is supposed to combine. Combining Lemma 5 with any corrected version of Lemma 6 would produce, up to universal constants, a term of the form √(r_{i-1} r_i) (∏_{j=1}^{i} ||W_j||_2) √h_i, rather than √(r_i κ_i) ||W_i||_F √h_i with κ_i ∝ ∏_{j≠i} ||W_j||_2. Moreover, the simplified bound O(C_1^L ∏_{i=1}^L B_i L r √h / m) does not follow algebraically from Eq. (4) with the printed expression for κ_i, even if the Frobenius norms are replaced by √r_i B_i. The authors should state a fully explicit final bound and verify the algebra that leads to the claimed O(L r √h) scaling.","section":"Eq. (4) and the simplified bound"},{"comment":"Lemma 10 in the appendix, which is used by Lemma 4, contains the same error: the proof bounds sup_W ⟨W, ΓX⟩ by ||W||_F ||ΓX||_F for a fixed W, but for the class {||W||_2 ≤ B} the supremum is B||ΓX||_*, and if a rank constraint is imposed it is B times the sum of the r largest singular values of ΓX. This is not bounded by the fixed-matrix Frobenius expression used in the proof. The shallow-network bound should be rederived using the correct class-level supremum, and the deep linear bound of Lemma 4 must be revisited accordingly.","section":"Appendix, Lemma 10"}],"minor_comments":[{"comment":"The manuscript contains numerous typos and OCR-style artifacts (e.g., 'F ell−1' in Lemma 6, 'radicaltp' in Lemma 5's proof, and 'd' in the definition of κ_i). A careful proofreading pass is needed.","section":"Throughout"},{"comment":"Table 1 reports the denominator √m for the proposed bound, while Theorem 7's simplified bound has m in the denominator (and the abstract also writes √h/m). These should be reconciled, since the normalized Gaussian complexity scales as 1/m.","section":"Table 1"},{"comment":"Maurer's chain rule is stated for finite Y and finite F, but the function classes considered in the paper are infinite. The authors should justify the extension to compact or otherwise well-behaved infinite classes, or state the necessary approximation argument.","section":"Theorem 3"},{"comment":"Theorem 8 is presented as 'Theorem 4 of Golowich et al. [14]' but includes a log^{3/2}(m) term and a constant cℓ; please verify the exact statement and ensure the reference is accurate.","section":"Section 5, Theorem 8"}],"recommendation":"major_revision","confidential_remarks":"The core result is not proven in the current version because Lemma 6 is incorrect and the theorem statement conflates class-level and instance-level quantities. However, the overall strategy is plausible and the diameter lemma is a useful contribution. The errors are localized and appear repairable: replacing the invalid fixed-matrix bound in Lemma 6 with a correct class-level bound (e.g., using the truncated nuclear norm under the rank constraint) and rewriting Theorem 7 with explicit class parameters could make the paper publishable. I therefore recommend major revision rather than rejection, but the revised proof must be checked carefully."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Before your next theory reading group: this is the paper that tries to get rank-dependent Gaussian complexity for deep ReLU nets by applying Maurer's chain rule, and the central idea is real. But as it stands, Theorem 7 isn't proven; Lemma 6, which is load-bearing, has a quantifier error.\n\nWhat's genuinely nice: the observation in Lemma 5 that the diameter of the function class up to layer ℓ is controlled by the smallest rank among the first ℓ layers. That is what prevents rank factors from multiplying across depth, and it's not in the prior norm-based bounds. The comparison to Golowich et al., Neyshabur et al., and Bartlett et al. is careful and fairly presented. The neural collapse discussion is speculative but not overdone.\n\nNow the problem. In Lemma 6 they start with R over Z, Z' of (1/||Z−Z'||) E sup_{||W||_2≤B} ⟨W, Γ(Z−Z')⟩. Next line they replace the sup with ||W_ℓ||_F ||Γ(Z−Z')||_F for a W_ℓ that is no longer quantified. That's wrong: sup over the spectral ball of ⟨W, A⟩ is B times the nuclear norm of A, which can exceed the Frobenius product for a particular low-rank W. The stress-test's scalar example is crisp. Since this R bound feeds into the recursion in Section 4.5, the main theorem collapses as written.\n\nI want to stress the gap looks repairable. Use B||A||_* ≤ B√(2r)||A||_F, and the extra √r is likely absorbed by the existing rank factors. But the preprint doesn't supply that argument, so a skeptical referee can't verify the bound.\n\nSmaller items: the chain rule's final term is dropped by assuming the zero function is in the class; that's fine for ReLU. Lemma 5 needs piecewise linear activations; for general Lipschitz activations the diagonal-matrix representation fails, but they state that assumption.\n\nMy take: send it to review. The core idea is worth a referee's time, and the flaw is a curable proof bug rather than a wrong concept. The right outcome is probably 'revise and resubmit,' not outright acceptance. For citing in your own work, wait until the proof is fixed.","headline":"A clever rank-aware bound, but Lemma 6 invalidates the main theorem as written; still worth a revision.","tokens_in":14921,"tokens_out":2442,"would_cite":false,"duration_ms":872917,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68T07","68Q32"],"pacs":[],"model":"deepseek-v4-flash","headline":"Deep networks with low-rank layers have generalization bounds where rank grows as Lr√h rather than compounding as r^{L/2}.","keywords":["generalization bounds","Gaussian complexity","low-rank layers","deep neural networks","chain rule","spectral norm","neural collapse","Rademacher complexity"],"falsifier":"A concrete check is to estimate the empirical Gaussian complexity of ReLU networks with all layers of fixed spectral norm and rank $1$ at depths $L=5,10,20$; after factoring out the universal constant term $C_1^L$, the theorem predicts growth like $L\\sqrt{h}$ in the normalized complexity, whereas rank multiplication across layers would predict growth like $2^{L/2}$. A direct algebraic check of Lemma 5 would also settle it: search for two low-rank ReLU networks whose output difference on a fixed sample exceeds the stated diameter bound $\\|X\\|_F\\sqrt{2\\min_j r_j}\\,2\\prod_j\\|W_j\\|_2$.","tokens_in":13713,"feed_emoji":"🧠","tokens_out":14031,"duration_ms":143668,"temperature":0.7,"pith_summary":"The paper argues that when a deep network's weight matrices are constrained in both spectral norm and rank, the generalization error can be bounded in a way that pays only once for low rank instead of once per layer. It gets this from a chain rule for Gaussian complexity that peels the network layer by layer, together with a diameter estimate showing that the intermediate feature class has diameter controlled by the smallest rank among the layers, not by the product of their ranks. If the argument is right, the empirically observed bias of trained networks toward low-rank weights becomes a provable source of tighter generalization. On the class $F_L = \\{f_W(x)=W_L\\varphi(\\ldots\\varphi(W_1x)\\ldots) : \\|W_i\\|_2 \\le B_i,\\ \\mathrm{rank}(W_i) \\le r_i\\}$, the simplified bound is $O(C_1^L \\prod_i B_i\\, L\\, r\\sqrt{h}/\\sqrt{m})$, replacing the $r^{L/2}$ factor that appears when earlier norm-based bounds are evaluated on the same class.","feed_headline":"Low-rank layers keep rank from compounding across depth","feed_subtitle":"A chain-rule bound makes the generalization gap scale with layer rank, not with rank raised to the depth power.","key_machinery":"The load-bearing object is the chain rule for expected suprema of Gaussian processes, $G(F\\circ Y) \\le C_1 L(F)\\,G(Y) + C_2 D(Y)\\,R(F,Y) + G(F(y_0))$, which is used to peel one layer at a time. Two estimates feed into it. Lemma 5 bounds the diameter $D(Y)$ of the representation class by writing the difference of two networks' outputs as $(A_i-A'_i)x$ with $A_i$ a product of weight matrices and diagonal activation matrices; because every factor has rank at most its layer rank, $\\mathrm{rank}(A_i)\\le \\min_j \\mathrm{rank}(W_j)$ and $\\|A_i\\|_2\\le \\prod_j \\|W_j\\|_2$. Lemma 6 bounds the Gaussian average of Lipschitz quotients $R(F,Y)$ by $\\|W_i\\|_F\\sqrt{h_i}$. These pieces, combined with a vector-valued Gaussian complexity that places one Gaussian variable on each output coordinate, are what prevent rank and width factors from multiplying across layers.","core_discovery":"The central result is Theorem 7, an upper bound on the Gaussian complexity of $F_L$. For a sample of $m$ points with $\\|x_i\\|_2\\le R$, a piecewise-linear $1$-Lipschitz activation $\\varphi$, and layer widths $h_i$, the bound reads approximately $$\\hat{G}_S(F_L) \\lesssim \\frac{\\|X\\|_F}{m}\\left( (\\|W_1\\|_F\\sqrt{h_1})\\prod_{i=2}^L C_1\\|W_i\\|_2 + \\sum_{i=2}^L $C_1^{{L-i}}$ $C_2^{2}$ \\sqrt{2 r_i \\kappa_i}\\, \\|W_i\\|_F\\sqrt{h_i} \\right),$$ where $r_i=\\min_{j\\le i}\\mathrm{rank}(W_j)$ and $\\kappa_i\\propto\\prod_{j\\neq i}\\|W_j\\|_2$. With uniform rank and spectral bounds this simplifies to $O(C_1^L \\prod_i B_i\\, L\\, r\\sqrt{h}/\\sqrt{m})$. The decisive point is that the rank enters through the running minimum of layer ranks in the disturbance terms, so low-rank layers stop the rank from compounding across depth, whereas earlier bounds evaluated on the same class carry $r^{L/2}$. The same chain-rule perspective shows that a rank-$1$ layer lets all layers above it be absorbed into a scalar Lipschitz map, which is the paper's explanation for why intermediate neural collapse should improve generalization.","pith_inferences":["This suggests a testable architectural principle: after training, cutting the singular values of one intermediate layer down to the minimum rank in the chain should improve out-of-sample performance without retraining, and the improvement should not depend on the ranks of the other layers.","The chain-rule treatment could plausibly be extended to Rademacher complexity using the recently proved Bernoulli-process analogue, which would bring the same rank-halting effect to architectures such as convolutional or residual networks.","If the exponential-in-depth factor $C_1^L$ is removable, as similar depth factors were removed in earlier norm-based bounds, the remaining $L\\,r\\sqrt{h}$ dependence would make these bounds practically relevant for very deep low-rank models; this is an extrapolation beyond the paper's proof.","One could also use the diameter argument to predict that generalization depends on the minimum rank along the chain rather than the average or maximum rank, a claim that empirical rank-reduction experiments could test."],"forward_implications":["Low-rank inductive bias becomes provable generalization: on $F_L$ the gap is $O(C_1^L\\prod_i B_i\\, L\\, r\\sqrt{h}/\\sqrt{m})$, so rank enters linearly and width only through $\\sqrt{h}$.","Intermediate neural collapse should tighten bounds: a rank-$1$ layer absorbs all layers above it into one scalar Lipschitz map, so the complexity is controlled by the layers below the collapsed layer.","Compared with earlier norm-based bounds on the same class, which scale as $\\sqrt{L}\\,r^{L/2}$ or $\\sqrt{L^3 r h}$, the new bound is better for deep low-rank networks but exposes a rank-depth tradeoff rather than a uniform win.","The result gives a concrete mechanism by which the observed low-rank bias of training translates into better generalization, tying optimization findings to statistical guarantees."],"supporting_citations":[{"why":"supplies the layer-by-layer chain rule for Gaussian complexes that the main bound is built on.","marker":"[1]"},{"why":"the prior norm-based size-independent bound whose evaluation on rank-constrained classes is the main comparison.","marker":"[14]"},{"why":"introduces the vector-valued contraction inequality and the coordinate-wise Gaussian complexity definition used throughout.","marker":"[28]"},{"why":"provides the Gaussian-complexity generalization bound that converts the complexity estimate into a generalization gap.","marker":"[24]"},{"why":"one of the spectrally-normalized margin bounds compared in Table 1, scaling as $\\sqrt{L^3 r h}$ on the class $F_L$.","marker":"[11]"},{"why":"documents neural collapse, the empirical phenomenon used to motivate rank-1 layers and the simplification of deep bounds.","marker":"[22]"},{"why":"the PAC-Bayesian spectral-norm bound compared in Table 1, giving the same $\\sqrt{L^3 r h}$ scaling on $F_L$.","marker":"[29]"}],"fun_headline_variants":["Low-rank layers stop rank from compounding across depth","Gaussian complexity bound: rank enters as running minimum","Chain rule yields sharper bounds for low-rank deep nets","Low-rank layers prevent rank from exponentiating with depth","New bound: rank scales as min over layers, not depth power"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof depends on the activation being piecewise linear and $1$-Lipschitz (ReLU or Leaky ReLU), because that is what lets the difference between two networks' outputs be written through diagonal activation matrices with rank bounded by the smallest layer rank; for smooth activations the diameter estimate that halts rank accumulation has no stated proof, and the function class must also contain the zero function so the base term of the chain rule drops.","fun_headline_variants_meta":{"raw":{"variants":["Low-rank layers stop rank from compounding across depth","Gaussian complexity bound: rank enters as running minimum","Chain rule yields sharper bounds for low-rank deep nets","Low-rank layers prevent rank from exponentiating with depth","New bound: rank scales as min over layers, not depth power"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000303,"raw_usage":{"total_tokens":1756,"prompt_tokens":974,"completion_tokens":782,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":590,"completion_tokens_details":{"reasoning_tokens":703}},"tokens_in":590,"tokens_out":782,"duration_ms":7754,"temperature":1.0,"reasoning_tokens":703,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T15:59:14.318819+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete check is to estimate the empirical Gaussian complexity of ReLU networks with all layers of fixed spectral norm and rank $1$ at depths $L=5,10,20$; after factoring out the universal constant term $C_1^L$, the theorem predicts growth like $L\\sqrt{h}$ in the normalized complexity, whereas rank multiplication across layers would predict growth like $2^{L/2}$. A direct algebraic check of Lemma 5 would also settle it: search for two low-rank ReLU networks whose output difference on a fixed sample exceeds the stated diameter bound $\\|X\\|_F\\sqrt{2\\min_j r_j}\\,2\\prod_j\\|W_j\\|_2$.","supporting_citations":[{"cited_title":"A chain rule for the expected suprema of g aussian processes","cited_arxiv_id":null,"evidence_quote":"supplies the layer-by-layer chain rule for Gaussian complexes that the main bound is built on."},{"cited_title":"Siz e-independent sample complexity of neural networks","cited_arxiv_id":null,"evidence_quote":"the prior norm-based size-independent bound whose evaluation on rank-constrained classes is the main comparison."},{"cited_title":"A vector-contraction inequality for r ademacher complexities","cited_arxiv_id":null,"evidence_quote":"introduces the vector-valued contraction inequality and the coordinate-wise Gaussian complexity definition used throughout."},{"cited_title":"Rademacher and g aussian complexities: Risk bounds and structural results","cited_arxiv_id":null,"evidence_quote":"provides the Gaussian-complexity generalization bound that converts the complexity estimate into a generalization gap."},{"cited_title":"Advances in neural information processing systems , 30, 2017","cited_arxiv_id":null,"evidence_quote":"one of the spectrally-normalized margin bounds compared in Table 1, scaling as $\\sqrt{L^3 r h}$ on the class $F_L$."},{"cited_title":"Prevalence of neural collapse during the terminal phase of deep learning training","cited_arxiv_id":null,"evidence_quote":"documents neural collapse, the empirical phenomenon used to motivate rank-1 layers and the simplification of deep bounds."},{"cited_title":"A pac-bayesian approach to spectrally- normalized margin bounds for neural networks","cited_arxiv_id":null,"evidence_quote":"the PAC-Bayesian spectral-norm bound compared in Table 1, giving the same $\\sqrt{L^3 r h}$ scaling on $F_L$."}],"review_version":1}