{"id":"c34ed24a-0de8-41e1-bf47-f47e42f91a21","arxiv_id":"2412.09779","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Deep supervised learners for exponential-family outcomes achieve test-error rates set by the 2β-entropic dimension of the input distribution, improving on Minkowski-dimension-based bounds.","lead":"The paper proves theoretical speed limits for deep supervised learning when labels follow an exponential family, and it shows that the error depends on the data's intrinsic dimension rather than the full input dimension. It also shows that the sample-size rate is nearly optimal when the inputs have a bounded density.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 12's entropic-dimension rate rests entirely on the unproved self-citation Chakraborty and Bartlett (2024b, Theorem 18); if that approximation theorem fails or needs extra conditions, the headline rate collapses to the ambient-dimension rate.","rationale":"The reader's weakest_assumption is precisely the load-bearing premise I would flag: Section 6.1.4 imports the entropic-dimension L2(λ) approximation theorem from the authors' own unpublished GAN preprint without proof or statement. I read the surrounding argument in good faith. The rest of Theorem 12's proof is internally consistent: Lemma 20 converts approximation error plus the pseudo-dimension term into the final bound, the balance ε ≍ n^{−β/(2β+d*)} with α = β/(2β+d*) gives the claimed exponent, and the success probability 1 − 3 exp(−n^{d*/(2β+d*)}) matches. The cited approximation theorem is also plausible: a covering argument reproduces the ε^{−d*/β} weight count, so the risk is unverified dependency rather than demonstrated falsehood. Two further issues appear, each of lesser weight. First, the minimax lower-bound construction (Section 6.2) claims the translated bumps have 'disjoint support', but centers are δ apart with support radius δ, so adjacent supports overlap; the identity ||f_ω − f_ω'||²_{L2(Leb)} = ||ω − ω'||_1 ∫ h_δ² is false as written. The construction is readily repaired by spacing centers 2δ apart, which leaves the packing size and the rate unchanged, so this is a constant-factor fix. Second, Lemma 21's stated weight bound is exponential in d: the leading factor is ⌈1/(2(η/20)^{1/β})⌉^d, and setting η = ε/(C d^{⌊β⌋}) (needed for error at most C d^{⌊β⌋} η) yields W ≈ (20 C d^{⌊β⌋}/ε)^{d/β}/2^d, which is not bounded by d^{⌊β⌋} ε^{−d/β} log(1/ε) with a d-independent constant as Theorem 8's proof assumes. The n-rate n^{−2β/(2β+d)} for fixed d is unaffected, but the advertised 'polynomial dependence on d' does not follow from the stated Lemma 21 as written. These findings do not move the verdict: the central n-rates are plausible, both gaps are reparable, and the primary blocker remains the unverified external approximation theorem, so CONDITIONAL remains appropriate.","tokens_in":26975,"tokens_out":43940,"duration_ms":402315,"concrete_test":"Extract the proof of Theorem 18 from Chakraborty and Bartlett (2024b, arXiv:2401.15801) and re-derive the exact statement used in Section 6.1.4: for f0 ∈ H^β([0,1]^d) and any d* larger than the 2β-entropic dimension, a ReLU network with O(ε^{−d*/β} log(1/ε)) weights achieves ||f̂ − f0||_{L2(λ)} ≤ ε. In particular, verify (i) the covering is the (ε, ε^{2β})-cover of Definition 11 with mass threshold ε^{2β}, (ii) the local Taylor polynomial has L2(λ) error O(ε^β) on the covered set while the uncovered tail of mass ε^{2β} contributes O(ε^{2β}), and (iii) the partition of unity over the covering balls is ReLU-implementable with size O(number of balls × poly(d, log(1/ε))) and output bounded by 2C. If Theorem 18 needs any condition beyond A1–A3 plus the entropic-dimension condition, Theorem 12 must state it; if the theorem holds as cited, the reader's blocker is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is Theorem 12: under A1–A3 and for any d* larger than the 2β-entropic dimension (Definition 11), ReLU networks trained on the Bregman loss achieve ||f̂ − f0||²_{L2(λ)} ≼ n^{−2β/(2β+d*)} (log n)^5, the claimed fastest known supervised-learning rate. The only step where the entropic dimension enters is Section 6.1.4, which states 'by Chakraborty and Bartlett (2024b, Theorem 18), we can find f* ∈ F ... such that ||f* − f0||_{L2(λ)} ≤ ε', with F = RN(L, W, 2C) and W ≼ ε^{−d*/β} log(1/ε). That theorem is neither proved nor stated in this manuscript; it is a self-citation to the authors' unpublished GAN preprint (arXiv:2401.15801), and the exact statement needed here (β-Hölder f0, L2(λ) norm, ReLU network of size ε^{−d*/β}, output bound 2C) cannot be checked from the present paper. Every other ingredient in the proof (Lemma 20 localization, the Bartlett et al. (2019) pseudo-dimension bound) is generic, so the entire improvement over the Minkowski-dimension rate of Nakada and Imaizumi (2020) reduces to the content of that citation. A direct computation supports plausibility: for radius δ, the 2β-entropic condition gives a cover of mass at least 1 − δ^{2β} by δ^{−d*} balls; Taylor approximation on each ball gives L2(λ) error O(δ^β) with O(δ^{−d*}) cells, and setting δ^β = ε recovers exactly the claimed O(ε^{−d*/β}) weight count. The nonstandard part is the ReLU implementation of a partition of unity over arbitrary covering balls with only O(ε^{−d*/β}) weights and the matching of the tail-mass parameter δ^{2β} to the smoothness β; if Theorem 18 requires any condition beyond A1–A3 plus the entropic-dimension condition, Theorem 12 needs that condition stated as an assumption. Absent a reproduced proof, the headline rate is unverifiable from this manuscript.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies supervised deep learning with exponential-family responses and analyzes the expected test error of the maximum-likelihood estimator, which is equivalent to minimizing a Bregman divergence. The main upper bound (Theorem 8) shows that under a bounded density on the covariates, ReLU networks with suitably chosen size achieve an L2(λ) error of order d^{2⌊β⌋(β+d)/(2β+d)} n^{-2β/(2β+d)} (log n)^5, and Theorem 10 gives a matching minimax lower bound under a lower-bounded density. The central contribution is Theorem 12, which replaces the ambient dimension d by an entropic dimension d^* > \\bar d_{2β}(λ) and claims the rate n^{-2β/(2β+d^*)} (log n)^5 for intrinsically low-dimensional data. The proofs combine a Bregman oracle inequality, localization-based generalization bounds, pseudo-dimension estimates, and ReLU approximation results.","tokens_in":27492,"tokens_out":13104,"duration_ms":125168,"significance":"If the results are correct, the entropic-dimension rate in Theorem 12 is a genuine improvement over existing intrinsic-dimension bounds such as those of Nakada and Imaizumi (2020), and the explicit polynomial dependence on d in Theorem 8 is a useful complement to the usual rate-in-n statements. The paper also covers classification and other exponential-family losses in a unified way, and the localization machinery in Lemmas 17–20 is presented in enough detail to be checked. The main caveats are that the proof of Theorem 12 delegates its entire entropic-dimension content to an unproved self-citation, and the proof of Theorem 10 contains an incorrect disjoint-support claim; these are load-bearing issues that need to be addressed before the results can be fully trusted.","major_comments":[{"comment":"The minimax lower bound construction places bumps h_δ of radius δ=1/m on centers spaced 1/m apart. Adjacent bumps therefore overlap on sets of positive Lebesgue measure, so the asserted identity ||f_ω – f_ω'||²_{L2(λ)} = ||ω – ω'||₁ ∫ h_δ² dx is invalid: the expansion contains cross terms δ_ξ δ_η ∫ h_δ(·–θ_ξ)h_δ(·–θ_η) dx that can have either sign. Consequently the claimed separation constant δ^{2β} is not established, and the subsequent KL bound in (11), which also uses the linear-in-||ω–ω'||₁ expression, does not follow. The argument can be repaired by taking δ = c/m with c < 1/2 so that the bump supports are disjoint, but as written the proof of the minimax rate is incorrect.","section":"6.2 (Theorem 10)"},{"comment":"The rate improvement in Theorem 12 rests entirely on the invocation 'by Chakraborty and Bartlett (2024b, Theorem 18), we can find f* ∈ F ...' for a ReLU network approximation of a β-Hölder function in L2(λ) with O(ε^{-d*/β} log(1/ε)) weights. This theorem is neither stated nor proved in the present manuscript, and it is a self-citation to an unpublished preprint (arXiv:2401.15801). The approximation lemma proved in the appendix (Lemma 21) treats only the ambient-dimension case, so the claimed entropic-dimension rate cannot be verified from the text. To make the paper self-contained, the authors should state the precise approximation theorem they need (including the bounded-output condition) and provide a proof in an appendix.","section":"6.1.4 (Theorem 12)"}],"minor_comments":[{"comment":"The proof of Lemma 7 loses the squares in the strong-convexity and Lipschitz constants: the display after (12) gives R ≤ τ2 σ1 ||f0–f̂||², but the correct bound is R ≤ τ2 σ1² ||f0–f̂||², and similarly (15) should read R ≥ τ1 σ2² ||f0–f̂||². Since the constants do not affect the rates, the lemma should be restated as R ≍ ||f̂–f0||²_{L2(λ)} or corrected to σ2²/σ1 and σ1²/σ2.","section":"A.1 (Lemma 7)"},{"comment":"The network size expressions contain typos: 'Lǫ ≼ ǫ log(1/ǫ)' and 'Wǫ ≼ ǫ ǫ^{-d⋆/β} log(1/ǫ)' have spurious factors of ǫ on the right-hand sides, and the first term in the following display reads 'ǫ n^{-2α}', which should presumably be n^{-2α}. These typos make the calibration of ε and α hard to follow.","section":"6.1.4"},{"comment":"The statement 'f ∈ Hβ(R, R, C)' should be 'f ∈ Hβ(R^d, R, C)' (or Hβ([0,1]^d, R, C)), since the proof works on [0,1]^d with the d-dimensional Lebesgue measure.","section":"Appendix B (Lemma 21)"},{"comment":"In the last display of the proof of Theorem 10, '≿ n n^{-2β/(2β+d)}' contains a stray factor n; the intended conclusion is δ^{2β} ≍ n^{-2β/(2β+d)}.","section":"6.2"},{"comment":"The text refers to the 'Bergman divergence' where 'Bregman divergence' is meant (see also Definition 5).","section":"Section 3"}],"recommendation":"major_revision","confidential_remarks":"The central concern is the dependence of Theorem 12 on the authors' own unpublished preprint for the key approximation step. If the cited theorem is unavailable or has additional hypotheses, the headline rate collapses to the ambient-dimension rate, so the editor may want to verify that the citation is accessible and that the statement used in Section 6.1.4 is a direct consequence. The error in the proof of Theorem 10 is local and easily fixed by spacing the bumps so that their supports are disjoint, but it should be corrected before publication. The paper is otherwise a reasonable fit for a statistical learning theory journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does something genuinely useful: it sets supervised deep learning in an exponential-family/Bregman framework that covers regression and classification, and it obtains rate bounds that depend on the intrinsic dimension of the covariate distribution. The equivalence between excess risk and squared L2 error (Lemma 7) is clean, and the explicit polynomial dependence on ambient dimension in Theorem 8 is new and well argued. Lemma 21, the ReLU Lp-approximation result with explicit d-dependence, is a real contribution that may have independent value. The localization proofs are careful and the exposition is organized.\n\nThe soft spots are real but not fatal. First, the minimax lower bound in Theorem 10 constructs bumps with radius δ = 1/m on centers spaced 1/m, so adjacent bumps have overlapping support. The asserted identity ||fω−fω'||² = ||ω−ω'||₁∫hδ² dx is therefore false; cross terms from overlaps contribute. This is likely repairable by choosing δ < c/m with c < 1/2, and the rate would probably survive, but the proof as written has a gap.\n\nMore important is the load-bearing dependency in Theorem 12. The entire improvement over the Minkowski-dimension rate comes from an approximation result cited from the authors' own unpublished preprint (Chakraborty and Bartlett 2024b, Theorem 18). That theorem is neither stated nor proved here, and it is nonstandard: it claims a ReLU network with O(ε^{-d*/β}) weights can approximate a β-Hölder function to L2(λ) error ε under only an entropic-dimension condition. The stress-test computation suggests the rate is plausible, but plausibility is not a proof. If that theorem has hidden extra conditions, the headline rate collapses to the ambient-dimension rate. This is the single biggest issue for the paper's significance.\n\nThere are also minor typos in Section 6.1.4 (a stray ε multiplying n^{-2α}, and a repeated ε in the network-size display), which are distractions but easy to fix.\n\nWho is this for? Statistical learning theorists working on intrinsic dimension and nonparametric rates. The framework is appealing and the ambient-dimension result (Theorem 8) is self-contained and solid. The entropic-dimension rate is a promissory note. A serious referee should be assigned, with instructions to demand that either the approximation theorem be proved in the paper or stated precisely with a full proof in an appendix. The lower-bound overlap should also be corrected. Conditional acceptance after those revisions is the right call, not rejection.\n\nYes, this deserves peer review. I would not cite the entropic-dimension rate in my own work until the dependency is resolved, but I would bring it to a reading group for discussion.","headline":"A useful framework and a solid ambient-dimension result, but the headline entropic-dimension rate rests on an unproved self-citation and the minimax lower bound has a support-overlap error.","tokens_in":27982,"tokens_out":2578,"would_cite":false,"duration_ms":28825,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62G08","62C20","68T07"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that for supervised deep learning with exponential-family responses, the test error of a ReLU network is governed by the entropic dimension of the covariate distribution rather than the ambient dimension, yielding error…","keywords":["supervised deep learning","exponential families","entropic dimension","Bregman divergence","ReLU networks","intrinsic dimension","minimax rates","Hölder smoothness"],"falsifier":"Take the approximation theorem cited from the companion paper and test it on a concrete case: let $\\lambda$ concentrate on a self-similar set with known $2\\beta$-entropic dimension, choose a $\\beta$-Hölder $f_0$, and compute the best $L^2(\\lambda)$ approximation error achievable by ReLU networks with $W$ weights. If the error decays slower than $W^{-\\beta/d^*}$ for some $d^*>\\bar d_{2\\beta}(\\lambda)$, or if the theorem needs assumptions beyond A1–A3 (for example absolute continuity or a density bound), then the proof of Theorem 12 is incomplete.","tokens_in":26768,"feed_emoji":"📉","tokens_out":16022,"duration_ms":133949,"temperature":0.7,"pith_summary":"This paper aims to show that in supervised deep learning, the relevant notion of data dimension is how the covariate mass concentrates, not the dimension of its support. Modeling the response as an exponential family with a $\\beta$-Hölder smooth mean function, the authors prove that a deep network with rectified linear unit activations trained by the maximum-likelihood (Bregman) loss achieves $L^2(\\lambda)$ error of order $\\tilde{\\mathcal{O}}(n^{-2\\beta/(2\\beta+d^*)})$ for any $d^*$ above the $2\\beta$-entropic dimension of the covariate distribution. Because the entropic dimension is never larger than the Minkowski or Wasserstein dimension, this improves known rates for deep supervised learners. The paper also shows that with a bounded covariate density the error depends only polynomially on the ambient dimension $d$, and that the sample-size exponent is nearly minimax optimal. If correct, these results unify regression and classification under a single low-dimensional learning guarantee that needs no manifold assumption.","feed_headline":"Entropic dimension sets the speed of deep learning test error","feed_subtitle":"For exponential-family supervised learning, deep ReLU nets reach near-optimal rates set by covariate mass concentration.","key_machinery":"The load-bearing objects are the Bregman divergence attached to the exponential family, the entropic dimension of the covariate measure, and an oracle inequality that splits the error into approximation and generalization terms. Maximum likelihood for an exponential family is equivalent to minimizing $d_\\varphi(y\\|\\mu(f(x)))$, where $\\varphi$ is the conjugate of the log-partition function, and the smoothness and strong-convexity assumptions let Lemma 7 compare the excess risk with $\\|f-f_0\\|^2_{L^2(\\lambda)}$ up to constants. The entropic dimension, $\\bar d_{2\\beta}(\\lambda)=\\limsup_{\\varepsilon\\downarrow 0}\\log N_\\varepsilon(\\lambda,\\varepsilon^{2\\beta})/\\log(1/\\varepsilon)$, counts how many $\\varepsilon$-balls are needed to cover all but an $\\varepsilon^{2\\beta}$ fraction of the covariate mass; this number, rather than the support dimension, enters the network approximation bound and fixes the sample-size exponent. The generalization side is controlled by pseudodimension bounds for ReLU networks together with localization lemmas, while a companion approximation result supplies $\\varepsilon$ error with $O(\\varepsilon^{-d^*/\\beta}\\log(1/\\varepsilon))$ weights whenever $d^*>\\bar d_{2\\beta}(\\lambda)$.","core_discovery":"The central claim is that minimizing the Bregman divergence over ReLU networks learns the conditional mean at a rate determined by the entropic dimension of the covariates. Theorem 12 states that under Assumptions A1–A3, for any $d^*>\\bar d_{2\\beta}(\\lambda)$, a network of depth $O(\\log n)$ and $O(n^{d^*/(2\\beta+d^*)}\\log n)$ weights satisfies $\\|\\hat f-f_0\\|^2_{L^2(\\lambda)}\\lesssim n^{-2\\beta/(2\\beta+d^*)}(\\log n)^5$ with probability at least $1-3\\exp(-n^{d^*/(2\\beta+d^*)})$. Since $\\bar d_{2\\beta}(\\lambda)\\le \\dim_M(\\lambda)$, this is faster than rates based on the Minkowski dimension of the support; for regular supports it recovers manifold-based rates without uniform-sharpness assumptions. In the bounded-density case, Theorem 8 supplies an explicit polynomial factor $d^{2\\lfloor\\beta\\rfloor(\\beta+d)/(2\\beta+d)}$ in the ambient dimension, and Theorem 10 shows the exponent $n^{-2\\beta/(2\\beta+d)}$ is minimax, so the deep learner is nearly optimal.","pith_inferences":["An extension the paper leaves implicit is that the oracle-inequality-plus-localization proof mechanism is not tied to exponential families: any loss whose population risk is sandwiched between constants times $\\|f-f_0\\|^2_{L^2(\\lambda)}$ and whose stochastic gradient terms are sub-Gaussian would plausibly inherit the same entropic-dimension rate; this conjecture is mine, not the paper's.","Because $\\bar d_{2\\beta}(\\lambda)$ measures concentration of mass, any preprocessing step that makes covariate mass concentrate more, such as feature extraction, embedding, or subsampling, would lower the rate exponent; the paper does not analyze such data-dependent transformations.","A fully self-contained version of this work would need the companion approximation theorem proved in the same document; checking that theorem on covariates that are singular or have unbounded density is the direct way to test whether Theorem 12's assumptions are sufficient.","Estimating the entropic dimension from data and plugging it into the network-size choice would turn the guarantee into an adaptive procedure, but the paper does not address estimation of $\\bar d_{2\\beta}(\\lambda)$."],"forward_implications":["In the bounded-density case, the explicit bound $\\tilde{\\mathcal{O}}(d^{2\\lfloor\\beta\\rfloor(\\beta+d)/(2\\beta+d)} n^{-2\\beta/(2\\beta+d)})$ shows that the ambient-dimension dependence is polynomial, so high-dimensional input spaces do not automatically destroy the guarantee.","The minimax lower bound of Theorem 10 means that no estimator, deep or otherwise, can improve the $n^{-2\\beta/(2\\beta+d)}$ exponent in the bounded-density case except by logarithmic factors, so the ReLU learner's rate is essentially the best possible.","Because $\\bar d_{2\\beta}(\\lambda)\\le \\dim_M(\\lambda)$, any distribution whose mass concentrates in subregions inherits a strictly faster rate than Minkowski-dimension bounds, with strict improvement possible even on simple examples.","When the covariate support is a $\\tilde d$-regular set, the rate becomes $\\tilde{\\mathcal{O}}(n^{-2\\beta/(2\\beta+\\tilde d)})$, recovering manifold-based nonparametric regression results without requiring uniform-sharpness assumptions on the manifold.","Since the setup includes classification through the Bernoulli exponential family, low-dimensional structure speeds up the binary cross-entropy learning rate just as it does for Gaussian regression."],"supporting_citations":[{"why":"Defines the entropic dimension, provides the ReLU product-network construction, and supplies the companion approximation theorem that lets Theorem 12 convert $\\bar d_{2\\beta}(\\lambda)$ into a network size.","marker":"Chakraborty and Bartlett (2024b)"},{"why":"Establishes the Minkowski-dimension-based rate for deep regression that Theorem 12 improves.","marker":"Nakada and Imaizumi (2020)"},{"why":"Defines the Wasserstein dimension and regular sets used in Lemma 15 to bound the entropic dimension on $\\tilde d$-regular supports.","marker":"Weed and Bach (2019)"},{"why":"Proves manifold-based nonparametric regression rates for deep ReLU networks, which Corollary 16 recovers as a special case without uniform-sharpness assumptions.","marker":"Chen et al. (2022)"},{"why":"Gives the classical minimax rates for estimating a smooth regression function, recovered as a special case of Theorem 10.","marker":"Yang and Barron (1999)"},{"why":"Supplies the nearly tight pseudodimension bounds for ReLU networks used in Lemmata 18–20 to control the generalization error.","marker":"Bartlett et al. (2019)"},{"why":"Provides the deep-ReLU nonparametric regression rate with ambient dimension $d$ that the exponential-family analysis generalizes.","marker":"Schmidt-Hieber (2020)"}],"fun_headline_variants":["Entropic dimension sets deep learning speed limits","Deep learning rate tied to data's entropic dimension","Near-optimal deep learning with exponential families","Entropic dimension beats Minkowski in deep learning","Polynomial dimension cost for exponential-family nets"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is a result borrowed from the authors' own companion work: for a $\\beta$-Hölder function and any $d^*$ above the entropic dimension, a ReLU network with about $\\epsilon^{-d^*/\\beta}\\log(1/\\epsilon)$ weights can approximate the function in $L^2(\\lambda)$ to error $\\epsilon$; if that result fails for the measures covered by Assumptions A1–A3, Theorem 12's low-dimensional rate collapses to the slower ambient-dimension rate.","fun_headline_variants_meta":{"raw":{"variants":["Entropic dimension sets deep learning speed limits","Deep learning rate tied to data's entropic dimension","Near-optimal deep learning with exponential families","Entropic dimension beats Minkowski in deep learning","Polynomial dimension cost for exponential-family nets"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000218,"raw_usage":{"total_tokens":1520,"prompt_tokens":1104,"completion_tokens":416,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":720,"completion_tokens_details":{"reasoning_tokens":346}},"tokens_in":720,"tokens_out":416,"duration_ms":4264,"temperature":1.0,"reasoning_tokens":346,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T16:44:33.423310+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the approximation theorem cited from the companion paper and test it on a concrete case: let $\\lambda$ concentrate on a self-similar set with known $2\\beta$-entropic dimension, choose a $\\beta$-Hölder $f_0$, and compute the best $L^2(\\lambda)$ approximation error achievable by ReLU networks with $W$ weights. If the error decays slower than $W^{-\\beta/d^*}$ for some $d^*>\\bar d_{2\\beta}(\\lambda)$, or if the theorem needs assumptions beyond A1–A3 (for example absolute continuity or a density bound), then the proof of Theorem 12 is incomplete.","supporting_citations":[{"cited_title":"and Imaizumi, M","cited_arxiv_id":null,"evidence_quote":"Establishes the Minkowski-dimension-based rate for deep regression that Theorem 12 improves."},{"cited_title":"and Bach, F","cited_arxiv_id":null,"evidence_quote":"Defines the Wasserstein dimension and regular sets used in Lemma 15 to bound the entropic dimension on $\\tilde d$-regular supports."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves manifold-based nonparametric regression rates for deep ReLU networks, which Corollary 16 recovers as a special case without uniform-sharpness assumptions."},{"cited_title":"and Barron, A","cited_arxiv_id":null,"evidence_quote":"Gives the classical minimax rates for estimating a smooth regression function, recovered as a special case of Theorem 10."},{"cited_title":"L., Harvey, N., Liaw, C., and Mehrabian, A","cited_arxiv_id":null,"evidence_quote":"Supplies the nearly tight pseudodimension bounds for ReLU networks used in Lemmata 18–20 to control the generalization error."}],"review_version":1}