Pith. sign in

REVIEW 2 major objections 4 minor 2 cited by

Genericity of Polyak-Lojasiewicz Inequalities for Entropic Mean-Field Neural ODEs

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read For an open dense set of initial feature-label distributions, the entropic mean-field neural ODE has a unique stable global minimizer, and near it the cost satisfies a local Polyak-Lojasiewicz inequality.

desk verdict Solid advance on PL genericity for entropic mean-field ResNets; the central theorem holds up, but the descent-convergence application is explicitly conjectural. read the letter →

arxiv 2507.08486 v1 pith:GQZ7AAWV submitted 2025-07-11 math.OC math.APmath.PR

Samuel Daudin , François Delarue This is my paper · ORCID
classification math.OCmath.APmath.PR MSC 49K2049K4568T07
keywords mean-fieldcontrolneuralODEResNetentropicregularizationPolyak-LojasiewiczinequalityJacobiconditionuniversalapproximationgenericity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Entropic mean-field neural ODEs—a continuum-of-layers idealization of ResNets—are studied as optimal control problems in which the control is a time-dependent probability measure over parameters and the running cost is relative entropy. The paper establishes that for an open and dense set of initial feature-label distributions, the regression cost has a unique stable global minimizer, meaning the second variation is strictly positive and the linearized optimality system has only the zero solution. Near such minimizers the cost satisfies a local Polyak-Lojasiewicz inequality, so gradient descent initialized close enough to the optimal parameter distribution converges exponentially fast in cost. The result holds for any positive entropic penalty strength, with no lower threshold.

What carries the argument

The key object is the discriminating property imposed on the drift $b$: if $\mathbb{E}[b(X,a)\cdot Z]=0$ for all parameters $a$, then $\mathbb{E}[Z|X]=0$ almost surely. This universal-approximation-style condition lets the first-order optimality formula, which gives the optimal control in Gibbs form $\nu^*_t(a) \propto \exp(-\ell(a) - \epsilon^{-1} \int b(x,a)\cdot \nabla_x u^*_t \, d\gamma^*_t)$, be read as an injectivity statement. Two optimal controls that agree at the initial time must agree for all times, so optimal trajectories cannot bifurcate. The no-bifurcation property, combined with dynamic programming, yields the Jacobi condition, and the same linearized system controls stability of the minimizer and the local Polyak-Lojasiewicz estimate. Quantitative control through log-Sobolev inequalities for the Gibbs reference measures carries the argument.

What would settle it

A concrete test: take a smooth bounded activation that is not universal, for example a truncated polynomial in a single hidden unit, so the discriminating property fails, and search for two distinct solutions of the first-order system (2.11)-(2.12) with the same initial control value. Finding such a pair would refute the no-bifurcation proposition that the open-dense conclusion rests on.

Watch

Extended reading notes

Core claim

The central claim is that ill-posedness is topologically exceptional. Let $O$ be the set of pairs $(t_0, \gamma_0)$ for which the control problem has exactly one minimizer and that minimizer is stable in the sense that the linearized forward-backward system has only the trivial solution. The paper proves that $O$ is open and dense in $[0,T] \times P_3(\mathbb{R}^{d_1} \times \mathbb{R}^{d_2})$. Along any optimal trajectory, $(t_1, \gamma^*_{t_1})$ belongs to $O$ for every later time $t_1$, which is the Jacobi no-conjugate-point condition; this is what makes $O$ dense. Moreover, for every compact subset of $O$ there exist constants $r,c$ such that any control $\nu$ within integrated relative-entropy distance $r$ of the minimizer satisfies $I \ge c(J-J^*)$, where $I$ is the Fisher-information-like functional measuring violation of the first-order optimality condition.

Load-bearing premise

The argument collapses if the drift family $\{b(\cdot,a)\}$ is not rich enough to separate conditional means: it must be that any $Z$ with $\mathbb{E}[b(X,a)\cdot Z]=0$ for all $a$ has $\mathbb{E}[Z|X]=0$ almost surely, and without that richness two optimal controls could coincide at the initial time and then split, destroying the Jacobi condition and the denseness of $O$.

Editorial extensions

If this is right

  • For initial conditions in $O$, the unique global minimizer is isolated and its Hessian is non-degenerate.
  • Every later time along an optimal trajectory is itself in $O$, so the good-initial-condition property propagates forward in network depth.
  • A local Polyak-Lojasiewicz inequality holds uniformly on compact subsets of $O$, giving a quadratic cost-to-gradient gap near the optimum.
  • Gradient descent on the parameter measure, started near $\nu^*$, converges exponentially fast in cost to the optimal value, as the paper explains follows from the PL estimate and a standard invariant-set argument.
  • The conclusions require no lower bound on the entropic penalty $\epsilon$; the set $O$ and the constants may depend on $\epsilon$ but hold for every $\epsilon>0$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the same discriminating property is satisfied by other parameterized drift families that are dense in $C_0$, the genericity result should carry over to architectures beyond the prototypical ResNet example.
  • The $\epsilon \to 0^+$ limit is left open; a moment penalty would likely need to replace entropy, and one consequence is that PL constants are not expected to be uniform as $\epsilon$ vanishes.
  • A finite-width testable analogue: for empirical initial distributions approaching a point in $O$, the optimal values and parameter distributions should converge with rates set by the stability modulus, and the local PL neighborhood should shrink at a quantifiable rate.
  • The discriminating property is exactly a universal approximation statement in the drift, so the paper's main theorem can be read as a translation of universal approximation into a robustness property of optimal control.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper studies a relaxed mean-field optimal control formulation of deep ResNets with entropic regularization. It introduces a notion of stable minimizer through the triviality of a linearized system and proves two main results: (i) an open dense set O of initial conditions (t0,γ0) for which the control problem has a unique stable minimizer, and (ii) a local Polyak–Lojasiewicz inequality on compact subsets of O. The proof combines the Gibbs form of optimal controls, a discriminating-property-based no-bifurcation argument, a Jacobi-type condition, compactness and perturbation analysis, and log-Sobolev inequalities. The paper is long and technical; Sections 4 and 5 contain the core arguments, while Sections 6–9 supply the postponed technical proofs.

Significance. If the result holds, it is a substantial contribution: it establishes genericity of uniqueness, stability, and a local PL inequality for a nonconvex mean-field neural ODE model, and it does so without imposing a lower bound on the entropic regularization parameter. The main assumptions are explicit, including a discriminating property of the vector field, and the paper gives a concrete route to verify this property for tanh/logistic activations. The proof is self-contained and structured, with no circular parameter fitting; the central claims do not assume the conclusion. The local PL inequality is the central mathematical result, while the exponential convergence of gradient descent is carefully presented as a follow-up conjecture rather than as a proved theorem.

major comments (2)
  1. [Sec. 4, Theorem 4.1; Sec. 2.5, Definition 2.25 and (2.40)] The statement of Theorem 4.1 is false as written at the endpoint t1=T. For t1=T, the cost J((T,γ0),ν) is independent of ν, so there is no unique minimizer; moreover, the notion of stable minimizer in Definition 2.25 is only defined for initial times in [0,T). Thus the assertion '(t1,γ*_t1) ∈ O for all t1 ∈ (t0,T]' must be corrected to t1 ∈ (t0,T). The density argument only uses times t1 arbitrarily close to t0, so this is a boundary artifact rather than a defect of the interior genericity statement, but the main theorem should be stated correctly.
  2. [Sec. 1.3 and Sec. 5] Identity (1.9), which links the derivative of the cost along the gradient flow to the functional I, is used to explain how the local PL inequality implies exponential convergence of the descent, but its proof is not supplied; the authors explicitly note that 'the proof ... would deserve to be expanded'. Since the abstract and introduction advertise the gradient-descent consequence, I recommend either providing a proof of (1.9), perhaps in an appendix, or stating unambiguously in the introduction that the exponential-convergence application is a conjecture whose missing ingredient is exactly (1.9). This does not affect the PL inequality itself, but it affects the advertised application.
minor comments (4)
  1. [Sec. 1.8 (Notation)] In the notation paragraph, the norm for C^k_b is written with the same symbol ∥φ∥_{C^k_{p,q}} as the norm for C^k_{p,q}; this is likely a typographical error and should be corrected to the intended C^k_b norm.
  2. [Sec. 2.3, display (2.18)] The denominator '|t2-t1]' contains a misplaced bracket; it should read '|t2-t1|'.
  3. [Sec. 4.5, Lemma 4.6 and Example 1.1] The assertion that tanh and logistic satisfy the non-compact sup-norm approximation property used in Lemma 4.6 is cited to Itô [27], but the standard universal approximation theorem is usually stated for compact sets. Since the discriminating property needs the approximation on all of R^{d1}, please add a proof or a precise reference for the non-compact statement that is strong enough for Lemma 4.6.
  4. [Sec. 5, Theorem 5.1] In the paragraph after the statement, the phrase 'stable minimum for J((t0,γ0), ·)' in the definition of O in (2.40) is slightly ambiguous; the intended meaning is 'unique global minimizer, and that minimizer is stable'. This is clear from context but could be made explicit at first occurrence.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the PL-genericity results are derived from explicit structural assumptions and do not reduce to a fit or to a self-citation chain.

full rationale

No circular step is present. The central claims, namely the open denseness of the set O of initial conditions with a unique stable minimizer and the local Polyak-Lojasiewicz inequality, are derived from the explicitly stated Assumption (Discriminating Property) and the smoothness, growth, and convexity conditions in Assumption (Regularity). The discriminating property is a universal-approximation-style injectivity condition with independent mathematical content; it is not a restatement of the existence of a unique stable minimizer or of the PL inequality. The proof chain runs through the first-order optimality system (Theorem 2.7), the no-bifurcation Propositions 4.2 and 4.3, the Jacobi condition and denseness argument in Theorem 4.1, the definition of stability via the linearized system, and the contradiction proof of Theorem 5.1. No parameter is fitted to a subset of data and then renamed as a prediction, and no equation is defined in terms of the target inequality. The few citations to the authors' own prior work (e.g., Carmona-Delarue for mean-field control background) are not load-bearing. The paper itself flags the unproved descent identity (1.9) and the omitted LaSalle convergence argument as gaps in the descent application, but these are acknowledged limitations that do not affect the local PL inequality or the genericity theorem. The endpoint-T overstatement in Theorem 4.1, where (T, gamma*_T) is claimed to lie in O although the terminal problem has no unique minimizer because the cost is independent of the control, is a boundary artifact rather than a circularity.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No numerical parameters are fitted to data. The central claim depends on the structural assumptions on b, ell, L, and on the entropic penalty, all stated as domain assumptions. No new physical or mathematical entities are postulated beyond the auxiliary measures and stable-solution concepts introduced within the proof.

assumptions (3)
  • domain assumption Regularity hypotheses: b is C^4 in (x,a) with growth bounds and b(x,0)=0; the prior measure satisfies nu_infinity proportional to exp(-ell) with ell convex and Hessian bounded below by c(1+|a|^2)I; the terminal cost L is C^3 with controlled growth.
    These are the standing structural hypotheses stated in Section 1.2. They make the control problem well posed, guarantee the Gibbs form of optimal controls, and provide the log-Sobolev and exponential bounds used throughout.
  • domain assumption Discriminating property of the vector field b: for any random variables X,Z with E|Z| finite, if E[b(X,a) dot Z] = 0 for all a, then E[Z|X] = 0 almost surely.
    This is the load-bearing universal-approximation-like condition. It is used in Propositions 4.2 and 4.3 to prove no bifurcation of optimal controls, which drives the Jacobi condition and the denseness of O.
  • standard math Standard results from PDE, measure theory, and convex analysis: well-posedness of continuity and transport equations, log-Sobolev and Holley-Stroock inequalities, Pinsker-type inequalities, Gronwall's lemma, and compactness arguments.
    These classical tools are invoked throughout Sections 2, 6, 7, 8 and the appendices. They are not specific to the authors and are used without proof.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Genericity of Polyak-Lojasiewicz Inequalities for Entropic Mean-Field Neural ODEs." pith.science (2026). https://pith.science/paper/GQZ7AAWV

@misc{pith2026250708486,
  author       = {Pith},
  title        = {Pith review of: Genericity of Polyak-Lojasiewicz Inequalities for Entropic Mean-Field Neural ODEs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GQZ7AAWV}},
  note         = {Machine review of arXiv:2507.08486}
}
read the original abstract

We address the behavior of idealized deep residual neural networks (ResNets), modeled via an optimal control problem set over continuity (or adjoint transport) equations. The continuity equations describe the statistical evolution of the features in the asymptotic regime where the layers of the network form a continuum. The velocity field is expressed through the network activation function, which is itself viewed as a function of the statistical distribution of the network parameters (weights and biases). From a mathematical standpoint, the control is interpreted in a relaxed sense, taking values in the space of probability measures over the set of parameters. We investigate the optimal behavior of the network when the cost functional arises from a regression problem and includes an additional entropic regularization term on the distribution of the parameters. In this framework, we focus in particular on the existence of stable optimizers --that is, optimizers at which the Hessian of the cost is non-degenerate. We show that, for an open and dense set of initial data, understood here as probability distributions over features and associated labels, there exists a unique stable global minimizer of the control problem. Moreover, we show that such minimizers satisfy a local Polyak--Lojasiewicz inequality, which can lead to exponential convergence of the corresponding gradient descent when the initialization lies sufficiently close to the optimal parameters. This result thus demonstrates the genericity (with respect to the distribution of features and labels) of the Polyak--Lojasiewicz condition in ResNets with a continuum of layers and under entropic penalization.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Dropout and Random Gradient Masking Are Asymptotically Equivalent in Large ResNets

    stat.ML 2026-07 conditional novelty 6.0 of 10

    As ResNets grow deep and wide with fixed dropout rate, dropout training and random-gradient-masking training converge to the same limiting dynamics, and the common masking variants collapse to one limit.

  2. From Sublinear to Linear: Local Convergence in Finite-Width Networks via Locally Polyak-Lojasiewicz Regions

    stat.ML 2025-07 conditional novelty 4.0 of 10

    Local NTK positivity plus Lipschitz stability gives a local Polyak-Lojasiewicz constant lambda0 minus L_Theta times the region radius, yielding linear gradient descent convergence whenever the iterates stay in the LQC...

Reference graph

Works this paper leans on

37 extracted references · 31 canonical work pages · cited by 2 Pith papers

  1. [1]

    Control on the manifolds of mappings with a view to the deep learning

    Andrei Agrachev and Andrey Sarychev. Control on the manifolds of mappings with a view to the deep learning. J. Dyn. Control Syst., 28(4):989–1008, 2022

    work page 2022

  2. [2]

    Lectures in Mathematics ETH Zürich

    Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré.Gradient flows in metric spaces and in the space of probability measures. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, second edition, 2008

    work page 2008

  3. [3]

    Springer, Cham, 2014

    Dominique Bakry, Ivan Gentil, and Michel Ledoux.Analysis and geometry of Markov diffusion operators, volume 348 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Cham, 2014

    work page 2014

  4. [4]

    Understanding the training of infinitely deep and wide resnets with conditional optimal transport, 2024

    Raphaël Barboni, Gabriel Peyré, and François-Xavier Vialard. Understanding the training of infinitely deep and wide resnets with conditional optimal transport, 2024

    work page 2024

  5. [5]

    Bertsekas and Steven E

    Dimitri P. Bertsekas and Steven E. Shreve.Stochastic optimal control, volume 139 ofMathematics in Science and Engineering. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. The discrete time case

    work page 1978

  6. [6]

    Weighted Csiszár-Kullback-Pinsker inequalities and applications to trans- portation inequalities

    François Bolley and Cédric Villani. Weighted Csiszár-Kullback-Pinsker inequalities and applications to trans- portation inequalities. Annales de la Faculté des sciences de Toulouse : Mathématiques, Ser. 6, 14(3):331–352, 2005. GENERICITY OF POLYAK–LOJASIEWICZ INEQUALITIES 99

    work page 2005

  7. [7]

    A measure theoretical approach to the mean-field maximum principle for training NeurODEs.Nonlinear Anal., 227:Paper No

    Benoît Bonnet, Cristina Cipriani, Massimo Fornasier, and Hui Huang. A measure theoretical approach to the mean-field maximum principle for training NeurODEs.Nonlinear Anal., 227:Paper No. 113161, 55, 2023

    work page 2023

  8. [8]

    Stable solutions in potential mean field game systems.NoDEA Nonlinear Differential Equations Appl., 25(1):Paper No

    Ariela Briani and Pierre Cardaliaguet. Stable solutions in potential mean field game systems.NoDEA Nonlinear Differential Equations Appl., 25(1):Paper No. 1, 26, 2018

    work page 2018

Show all 37 references
  1. [9]

    Springer, New York, 2019

    Amarjit Budhiraja and Paul Dupuis.Analysis and approximation of rare events, volume 94 ofProbability Theory and Stochastic Modelling. Springer, New York, 2019. Representations and weak convergence methods

    2019

  2. [10]

    Birkhäuser Boston, Inc., Boston, MA, 2004

    Piermarco Cannarsa and Carlo Sinestrari.Semiconcave functions, Hamilton-Jacobi equations, and optimal con- trol, volume 58 ofProgress in Nonlinear Differential Equations and their Applications. Birkhäuser Boston, Inc., Boston, MA, 2004

    2004

  3. [11]

    Princeton University Press, Princeton, NJ, 2019

    Pierre Cardaliaguet, François Delarue, Jean-Michel Lasry, and Pierre-Louis Lions.The master equation and the convergence problem in mean field games, volume 201 ofAnnals of Mathematics Studies. Princeton University Press, Princeton, NJ, 2019

    2019

  4. [12]

    Souganidis

    Pierre Cardaliaguet, Joe Jackson, Nikiforos Mimikos-Stamatopoulos, and Panagiotis E. Souganidis. Sharp con- vergence rates for mean field control in the region of strong regularity.arXiv, 2312.11373, 2023

    2023 arXiv

  5. [13]

    Souganidis

    Pierre Cardaliaguet and Panagiotis E. Souganidis. Regularity of the value function and quantitative propagation of chaos for mean field control problems.Nonlinear Differential Equations and Applications, 30, 3 2023

    2023

  6. [14]

    Springer, 2018

    René Carmona and François Delarue.Probabilistic Theory of Mean Field Games with Applications I : Mean Field FBSDEs, Control, and Games. Springer, 2018

    2018

  7. [15]

    On the global convergence of gradient descent for over-parameterized models using optimal transport

    Lénaïc Chizat and Francis Bach. On the global convergence of gradient descent for over-parameterized models using optimal transport. InProceedings of the 32nd International Conference on Neural Information Processing Systems, NIPS’18, page 3040–3050, Red Hook, NY, USA, 2018. C...

    2018

  8. [16]

    The feature speed formula: a flexible approach to scale hyper-parameters of deep neural networks

    Lénaïc Chizat and Praneeth Netrapalli. The feature speed formula: a flexible approach to scale hyper-parameters of deep neural networks. In A. Globerson, L. Mackey, D. Belgrave, A. Fan, U. Paquet, J. Tomczak, and C. Zhang, editors, Advances in Neural Information Processing Sys...

    2024

  9. [17]

    Asymptotic analysis of deep residual networks.ArXiv e-prints, 2212.08199, 2023

    Rama Cont, Alain Rossier, and Renyuan Xu. Asymptotic analysis of deep residual networks.ArXiv e-prints, 2212.08199, 2023

    2023 arXiv

  10. [18]

    Deep neural networks, generic universal interpolation, and controlled odes.SIAM Journal on Mathematics of Data Science, 2(3):901–919, 2020

    Christa Cuchiero, Martin Larsson, and Josef Teichmann. Deep neural networks, generic universal interpolation, and controlled odes.SIAM Journal on Mathematics of Data Science, 2(3):901–919, 2020

    2020

  11. [19]

    Overparameterization of deep resnet: zero loss and mean-field analysis.Journal of machine learning research, 23(48):1–65, 2022

    Zhiyan Ding, Shi Chen, Qin Li, and Stephen J Wright. Overparameterization of deep resnet: zero loss and mean-field analysis.Journal of machine learning research, 23(48):1–65, 2022

    2022

  12. [20]

    A proposal on machine learning via dynamical systems.Commun

    Weinan E. A proposal on machine learning via dynamical systems.Commun. Math. Stat., 5(1):1–11, 2017

    2017

  13. [21]

    A mean-field optimal control formulation of deep learning.Res

    Weinan E, Jiequn Han, and Qianxiao Li. A mean-field optimal control formulation of deep learning.Res. Math. Sci., 6(1):Paper No. 10, 41, 2019

    2019

  14. [22]

    A gradient flow on control space with rough initial condition, 2024

    Paul Gassiat and Florin Suciu. A gradient flow on control space with rough initial condition, 2024

    2024

  15. [23]

    Stable architectures for deep neural networks.Inverse Problems, 34(1):014004, 22, 2018

    Eldad Haber and Lars Ruthotto. Stable architectures for deep neural networks.Inverse Problems, 34(1):014004, 22, 2018

    2018

  16. [24]

    Deep residual learning for image recognition

    Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 770–778, 2016

    2016

  17. [25]

    Mean-field langevin system, optimal control and deep neural networks

    Kaitong Hu, Anna Kazeykina, and Zhenjie Ren. Mean-field langevin system, optimal control and deep neural networks. arXiv, 1909.07278, 2019

    1909 arXiv

  18. [26]

    A convergence result of a continuous model of deep learning via lojasiewicz–simon inequality

    Noboru Isobe. A convergence result of a continuous model of deep learning via lojasiewicz–simon inequality. arXiv preprint arXiv:2311.15365, 2023

    2023 arXiv

  19. [27]

    Approximation of continuous functions on rd by linear combinations of shifted rotations of a sigmoid function with and without scaling.Neural Networks, 5(1):105–115, 1992

    Yoshifusa Ito. Approximation of continuous functions on rd by linear combinations of shifted rotations of a sigmoid function with and without scaling.Neural Networks, 5(1):105–115, 1992

    1992

  20. [28]

    Mean-field neural odes via relaxed optimal control.arXiv, 1912.05475, 2021

    Jean-François Jabir, David Šiška, and Lukasz Szpruch. Mean-field neural odes via relaxed optimal control.arXiv, 1912.05475, 2021

    1912 arXiv

  21. [29]

    A general characterization of the mean field limit for stochastic differential games.Probab

    Daniel Lacker. A general characterization of the mean field limit for stochastic differential games.Probab. Theory Related Fields, 165(3-4):581–648, 2016

    2016

  22. [30]

    Deep learning via dynamical systems: an approximation perspective

    Qianxiao Li, Ting Lin, and Zuowei Shen. Deep learning via dynamical systems: an approximation perspective. J. Eur. Math. Soc. (JEMS), 25(5):1671–1709, 2023

    2023

  23. [31]

    Cours au collège de france, equations aux dérivées partielles et applications

    Pierre-Louis Lions. Cours au collège de france, equations aux dérivées partielles et applications. https://www.college-de-france.fr/site/pierre-louis-lions/course-2010-2011.htm, 2010-11

    2010

  24. [32]

    A mean field analysis of deep resnet and beyond: Towards provably optimization via overparameterization from depth

    Yiping Lu, Chao Ma, Yulong Lu, Jianfeng Lu, and Lexing Ying. A mean field analysis of deep resnet and beyond: Towards provably optimization via overparameterization from depth. InInternational Conference on Machine Learning, pages 6426–6436. PMLR, 2020

    2020

  25. [33]

    A mean field view of the landscape of two-layer neural networks

    Song Mei, Andrea Montanari, and Phan-Minh Nguyen. A mean field view of the landscape of two-layer neural networks. Proceedings of the National Academy of Sciences, 115(33):E7665–E7671, 2018. 100 S. DAUDIN AND F. DELARUE

    2018

  26. [34]

    Local convergence rates for wasserstein gradient flows and mckean-vlasov equations with multiple stationary solutions, 2024

    Pierre Monmarché and Julien Reygner. Local convergence rates for wasserstein gradient flows and mckean-vlasov equations with multiple stationary solutions, 2024

    2024

  27. [35]

    Rotskoff and Eric Vanden-Eijnden

    Grant M. Rotskoff and Eric Vanden-Eijnden. Neural networks as interacting particle systems: Asymptotic con- vexity of the loss landscape and universal scaling of the approximation error.CoRR, abs/1805.00915, 2018

    2018 arXiv

  28. [36]

    Neural ode control for classification, approximation, and transport

    Domènec Ruiz-Balet and Enrique Zuazua. Neural ode control for classification, approximation, and transport. SIAM Review, 65(3):735–773, 2023

    2023

  29. [37]

    Deep learning approximation of diffeomorphisms via linear-control systems.Mathematical Control and Related Fields, 13(3):1226–1257, 2023

    Alessandro Scagliotti. Deep learning approximation of diffeomorphisms via linear-control systems.Mathematical Control and Related Fields, 13(3):1226–1257, 2023. (S. Daudin) Université Paris Cité, CNRS, Sorbonne Université, Laboratoire Jacques-Louis Lions (LJLL), F-75006, Paris...

    2023

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.