Pith. sign in

REVIEW 3 major objections 5 minor 1 cited by

Rapid Bayesian Computation and Estimation for Neural Networks via Log-Concave Coupling

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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

Pith's one-line read A log-concave coupling rewrites a multimodal neural-net posterior as a mixture of samplable densities, and a discrete-prior version carries fourth-root risk bounds.

desk verdict A solid, honest representation theorem for neural-net posteriors, but the advertised 'rapid sampling' remains an unproven assumption; deserves peer review with a reframing. read the letter →

arxiv 2411.17667 v3 pith:ZSKLD4J7 submitted 2024-11-26 math.ST stat.TH

Curtis McDonald , Andrew R. Barron This is my paper · ORCID
classification math.STstat.TH MSC 62F1562M4565C05
keywords NeuralNetworksBayesianMethodsSamplingStatisticalLearninglog-concavecouplingMarkovChainMonteCarloriskbounds
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

The paper claims that the posterior distribution over the interior weights of a single-hidden-layer neural network can be rewritten as a mixture whose components are all log-concave, provided the network is wide enough in a specific sense. Because log-concave densities can be sampled in polynomial time, this turns Bayesian training into a sequence of log-concave sampling problems. The paper also proves risk bounds for a discretized uniform prior: squared-error generalization error decays like $(\log d / N)^{1/4}$, and, under Gaussian data with matched gain, expected Kullback divergence decays like $(\log d / N)^{1/3}$. The payoff would be a Bayesian training procedure with both polynomial-time sampling and statistical risk control.

What carries the argument

The central object is the log-concave coupling: a joint distribution $\pi_b(w,z) = \pi_b(w)\, \pi_b(z|w)$ that preserves the original posterior $\pi_b(w)$ as its marginal while making both the reverse conditional $\pi_b(w|z)$ and the induced marginal $\pi_b(z)$ log-concave. The forward coupling is a Gaussian $z_{i,k} \sim \mathrm{Normal}(w_k \cdot x_i, 1/\kappa)$, truncated to a set that bounds the bilinear term in the joint log-likelihood, with $\kappa$ scaled as $\sqrt{3/2}\, E_2 \, \beta \, C_N \, V / K$, large enough to force the Hessian of $\log \pi_b(w|z)$ negative definite. Log-concavity of the marginal is shown through the Hessian identity $\nabla^2 \log \pi_b(z) = \kappa\bigl(-I + \kappa\, \mathrm{Cov}_{\pi_b(w|z)}[Xw]\bigr)$, so it suffices to bound the conditional covariance; the proof uses a Hölder inequality with high-order moments of the uniform prior and a cumulant generating function bound on the tilted prior.

What would settle it

Take a small instance satisfying Theorem 1's conditions, such as $K=2$ and $d$ large enough that $Kd \ge A_3(\beta N)^2$, and evaluate the Hessian of $\log \pi_b(z)$ at several points $z$ in the constrained set using the identity $\nabla^2 \log \pi_b(z) = \kappa(-I + \kappa \, \mathrm{Cov}_{\pi_b(w|z)}[Xw])$; if any eigenvalue is positive for a valid $z$, the marginal is not log-concave and Theorem 1 fails.

Watch

Extended reading notes

Core claim

Theorem 1 establishes a log-concave coupling: for a continuous uniform prior on $(B_1^d)^K$, with $K \ge 2$, $d \ge 2$, $\beta N \ge 2$, and under $K \log(2Kd/\delta) \le \beta N$ and $Kd \ge A_3(\beta N)^2$, the posterior $\pi_b(w)$ equals $\int \pi_b(w|z) \pi_b(z) \, dz$, where each reverse conditional $\pi_b(w|z)$ is log-concave and the induced marginal $\pi_b(z)$ is log-concave (strictly, if the inequality is strict). The auxiliary variable $z$ comes from a Gaussian forward coupling $z_{i,k} \sim \mathrm{Normal}(w_k \cdot x_i, 1/\kappa)$ restricted to a high-probability set, with $\kappa$ chosen just large enough to cancel the potentially positive second-derivative terms in the Hessian of the log posterior. The score of the marginal is itself an expectation under the reverse conditional, so both levels are accessible to MCMC. Separately, for the discrete grid prior, the paper proves regret and risk bounds of order $(\log d / N)^{1/4}$, and a Kullback risk bound of order $(\log d / N)^{1/3}$ under Gaussian data with $\beta = 1/\sigma^2$; the risk control is for the discrete prior while the log-concavity construction uses the continuous prior, a gap the paper explicitly leaves for future work.

Load-bearing premise

The load-bearing premise is that one can actually sample from the two log-concave densities in the coupling—the reverse conditional $\pi_b(w|z)$ and the marginal $\pi_b(z)$, whose score is itself an expectation under $\pi_b(w|z)$—in a low-polynomial number of MCMC iterations; the paper treats this as a black box and provides no mixing-time or finite-sample error analysis.

Editorial extensions

If this is right

  • If the log-concave coupling holds, posterior means and predictive densities for the continuous-uniform-prior model can be computed by sampling only log-concave densities, so Bayesian training becomes a polynomial-time procedure whenever log-concave samplers are available.
  • The risk theorems imply that the Cesàro posterior mean estimates any function in the variational class to squared error $O((\log d)/N)^{1/4}$ with no assumptions on the error distribution beyond finite variance.
  • Under independent Gaussian noise with $\beta = 1/\sigma^2$, the predictive density converges in expected Kullback divergence at rate $O((\log d)/N)^{1/3}$, a faster rate than the fourth-root squared-error bound.
  • The parameter choices balancing the bounds require $K = O(N^{1/4})$ and $d = \Omega(N^{5/4})$ in the natural scaling; the paper notes that repeating input coordinates can artificially inflate $d$ and satisfy the width condition without changing the function class.
  • Adaptive priors over a finite grid of $K$, $d$, $\beta$, and $V$ values can be handled by running independent samplers in parallel and weighing by posterior probabilities, so the user need not know the right hyperparameters in advance.

Reading between the lines

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

  • A testable strengthening would be to verify numerically the conjecture in Section 8 that the prior covariance dominates the conditional covariance up to a constant; if true, the condition $Kd \ge C(\beta N)^2$ could drop to $Kd \ge C(\beta N)$, improving the scaling of the method.
  • The dimension-inflation trick suggests an extension where the same coupling works for the original low-dimensional inputs after lifting; one could test whether the inflated sampler's mixing time degrades gracefully with the inflation factor, which the paper does not analyze.
  • The paper's risk bounds hold for a discrete grid prior while the coupling uses the continuous prior; the coupled discrete/continuous construction sketched in Section 8, if carried through, would yield a single polynomial-time algorithm with the stated risk guarantees.
  • Because the score of the marginal is itself a conditional expectation, the two-level MCMC scheme resembles a stochastic gradient sampler; one could try replacing the inner MCMC with an unbiased estimator of the conditional mean to obtain a single-loop algorithm, though the paper does not analyze this.
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

3 major / 5 minor

Summary. This paper studies posterior sampling and risk guarantees for single-hidden-layer neural networks with ℓ1-bounded internal weights and fixed outer weights. The main structural result (Theorem 1) shows that, under a continuous uniform prior on (B_1^d)^K and the condition Kd ≥ C(βN)^2, the posterior π_b(w) can be written as ∫ π_b(w|z)π_b(z) dz, where the reverse conditional π_b(w|z) and the induced marginal π_b(z) are log-concave. The main statistical results (Theorems 2–4) are risk bounds for a discrete uniform prior on a grid: arbitrary-sequence squared regret of order O((log d)/N)^{1/4}, iid mean-squared risk of the same order, and expected Kullback divergence of order O((log d)/N)^{1/3} under Gaussian errors with β=1/σ². The paper claims that these ingredients yield rapid Bayesian computation via sampling from log-concave densities, with the sampling algorithm treated as a black box.

Significance. The log-concave coupling construction is a meaningful structural insight: it identifies an auxiliary-variable decomposition that removes the multimodality obstruction for a class of neural-network posteriors, and it is proved with a detailed, largely self-contained argument that carefully tracks constants. The discrete-prior risk bounds are also valuable and are obtained via Barron's index of resolvability together with a new grid-discretization approximation lemma. However, the advertised computational claim is not established at the theorem level. The paper proves log-concavity, not rapid mixing, and the nested score-estimation scheme is left unanalyzed. The rigorous contributions are the representation theorem and the risk bounds; the step from these to a polynomial-time Bayesian training algorithm is conditional on unverified sampling assumptions, as the paper itself acknowledges in Section 8.

major comments (3)
  1. [§4.1, Theorem 1; §5.3, Remark 2; §8] The central computational claim is not proven. Theorem 1 establishes only that π_b(w|z) and π_b(z) are log-concave; it does not provide any mixing-time, convergence-rate, or bias/variance analysis for the two-level MCMC scheme that the paper proposes. As Remark 2 notes, the score of the marginal π_b(z) is itself an expectation under the reverse conditional π_b(w|z), so every gradient evaluation in an outer MCMC step requires an inner MCMC run. The paper gives no ergodicity or finite-sample error analysis for this nested procedure, and Section 8 explicitly states that 'the further details of sampling must be worked out.' Thus the abstract's statement that neuron parameters 'can be sampled from the posterior by only sampling log-concave densities' is an algorithmic premise, not a theorem, and it is load-bearing for the title and for the claimed polynomial-time training algorithm.
  2. [§7, Theorem 1] The reverse conditional π_b(w|z) is only weakly log-concave: Section 7 notes that it is flat in directions orthogonal to the rows of the data matrix X. The paper cites Ball Walk and Hit-and-Run results [38,33] for weakly log-concave targets, but it does not verify their hypotheses for this family of densities—e.g., any bound on the condition number, a warm-start distribution, or a log-Sobolev inequality. Moreover, the regime Kd ≥ A_3(βN)^2 with βN ≥ 2 typically forces d to be much larger than N, which makes the flat directions high-dimensional. Without an iteration-complexity bound that accounts for these flat directions, the claimed reduction to log-concave sampling does not yield a polynomial-time algorithm, even granting access to a generic log-concave sampler.
  3. [§4.2, §6, §8] The paper's two main results apply to different priors and are not connected. The log-concave coupling of Theorem 1 is for the continuous uniform prior on (B_1^d)^K, while the risk bounds in Theorems 2–4 are for the discrete uniform prior on a grid. The paper acknowledges in Section 8 that bridging these results is future work. Consequently, the paper does not provide an end-to-end procedure that is simultaneously provably polynomial-time and provably risk-controlled. The contributions should be presented as separate representation and risk results, or the missing bridge should be supplied before the 'rapid Bayesian computation' claim is made.
minor comments (5)
  1. [§1] The statement 'we adopt a sampling problem we can solve instead of an optimization problem we cannot' is premature given that the sampling problem is not yet proved to be solvable in polynomial time.
  2. [§4.1] Theorem 1 and condition (4.6): the restriction parameter δ should be explicitly defined with its allowed range in the theorem statement, since it is a threshold that also appears in the definition of C_z in (5.18).
  3. [§5.3 and §8] The paper would benefit from a pseudocode description of the proposed two-level sampler (outer MCMC on z, inner MCMC for the score), so that the required verifications are concrete.
  4. [Full text] There are numerous typographical artifacts in the full-text version (e.g., equation (5.18) and the surrounding display), and a careful proofreading pass is needed.
  5. [§5.4, Remark 3] Remark 3 handles rounding of the optimal ℓ* in the proof of Theorem 7 heuristically; a formal treatment with explicit rounding terms would strengthen the proof.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: log-concave coupling is constructed and proven in-paper; cited approximation results have independent support.

full rationale

Theorem 1 is not circular: the paper explicitly constructs a Gaussian forward coupling (Sec. 5.1) and proves the reverse conditional is log-concave by choosing the coupling precision c so that its negative Hessian dominates the non-log-concave term (Theorems 5 and 6), and proves the induced marginal is log-concave via a Holder/cumulant variance bound (Theorem 7). No fitted value is relabeled as a prediction; the risk bounds for the discrete prior are derived from the index of resolvability (Lemma 9) plus an externally published approximation theorem [30], which is independent support even though one author is a coauthor. The main caveat, acknowledged in Section 8 ('The further details of sampling must be worked out'), is that the advertised rapid sampling presumes a black-box log-concave sampler and a nested MCMC estimate of the marginal score (Remark 2), which are unanalyzed; this is a completeness gap rather than a circularity. Consequently no circular step can be exhibited from the paper's equations.

Assumptions & free parameters 5 free parameters · 4 assumptions · 2 invented entities

The central log-concavity theorem relies on the bounded activation assumptions, the bounded data assumption, and the black-box log-concave sampler. The risk bounds rely on the established approximation results and the index of resolvability. The auxiliary variables and the set C_z are invented computational constructs without independent evidence outside the paper.

free parameters (5)
  • gain β = const * (log d / N)^{1/4} in risk bounds
    Chosen analytically to optimize the risk upper bounds, not estimated from data. In the coupling theorem it is a given inverse temperature.
  • number of neurons K = const * (N / log d)^{1/4} in risk bounds
    Chosen to balance the index of resolvability terms; a modeling choice.
  • grid size m = const * (N / log d)^{1/4} in risk bounds
    Controls discretization error in the discrete prior; chosen to optimize the bound.
  • auxiliary variance ρ = sqrt(3/2) E2 β C_B V / K
    Set by formula to make the reverse conditional Hessian negative definite; depends on data range C_B and network scale V.
  • restriction parameter δ = small constant (e.g., 1/300)
    Defines the set C_z where auxiliary variables are constrained; chosen small enough to satisfy the technical conditions.
assumptions (4)
  • domain assumption The activation function σ has |σ| ≤ E0, |σ'| ≤ E1, |σ''| ≤ E2 on [-1,1].
    Used throughout the Hessian and Taylor-expansion arguments; stated in Section 3 and assumed in all theorems.
  • domain assumption Input vectors x_i ∈ [-1,1]^d with x_i,1 = 1.
    Standard bounded-data assumption; used in the moment bounds and the definition of C_z.
  • domain assumption Existence of a polynomial-time log-concave sampling algorithm as a black box.
    The paper presumes access to such algorithms (Lovász-Vempala, Dwivedi et al.) and does not analyze the specific two-level MCMC procedure.
  • standard math Approximation bounds for neural networks with ℓ1-controlled weights from Klusowski-Barron [30].
    Used in Lemma 10 and Lemma 11 to show a good discrete approximator exists; cited as an established external result.
invented entities (2)
  • Auxiliary random variables z_{i,k} (Gaussian forward coupling)
    purpose: To create a joint density whose reverse conditional and induced marginal are log-concave, enabling sampling from the posterior.
    Internal computational device introduced in Section 5.1; no external falsifiable prediction.
  • Constrained set C_z for auxiliary variables
    purpose: To keep the bilinear term in the joint log-likelihood bounded and thereby prove log-concavity of the marginal π(z).
    Defined in Eq (5.18); a technical restriction that is internal to the proof.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Rapid Bayesian Computation and Estimation for Neural Networks via Log-Concave Coupling." pith.science (2026). https://pith.science/paper/ZSKLD4J7

@misc{pith2026241117667,
  author       = {Pith},
  title        = {Pith review of: Rapid Bayesian Computation and Estimation for Neural Networks via Log-Concave Coupling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZSKLD4J7}},
  note         = {Machine review of arXiv:2411.17667}
}
abstract

This paper studies a Bayesian estimation procedure for single-hidden-layer neural networks using $\ell_{1}$ controlled weights. We study the structure of the posterior density and provide a representation that makes it amenable to rapid sampling via Markov Chain Monte Carlo (MCMC), and to statistical risk guarantees. The neural network has $K$ neurons, internal weight dimension $d$, and fix the outer weights. Thus, $Kd$ parameters overall. With $N$ data observations, use a gain parameter of $\beta$ in the posterior density. The posterior is multimodal and not naturally suited to rapid mixing of direct MCMC algorithms. For a continuous uniform prior on the $\ell_{1}$ ball, we show that the posterior density can be written as a mixture density with suitably defined auxiliary random variables, where the mixture components are log-concave. Furthermore, when the number of model parameters $Kd$ is large enough that $Kd \geq C(\beta N)^{2}$, the mixing distribution of the auxiliary random variables is also log-concave. Thus, neuron parameters can be sampled from the posterior by only sampling log-concave densities. The authors refer to the mixture density as a log-concave coupling. For a discrete uniform prior restricted to a grid, we study the statistical risk (generalization error) of procedures based on the posterior. Using a gain of $\beta = C [(\log d)/N]^{1/4}$, we demonstrate squared error is on the order $O([(\log d)/N]^{1/4})$. Using independent Gaussian data with a variance $\sigma^{2} $ that matches the inverse gain, $\beta = 1/\sigma^{2}$, we show that the expected Kullback divergence has a cube root power $O([(\log d)/N]^{1/3})$. Future work aims to bridge the sampling ability of the continuous uniform prior with the risk control of the discrete uniform prior, resulting in a polynomial time Bayesian training algorithm for neural networks with statistical risk control.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

  1. Denoising growth complexity: Data geometry and certified schedules for diffusion sampling

    math.ST 2026-07 accept novelty 8.0 of 10

    A new measure, the denoising growth complexity, provides local KL error bounds for Euler diffusion samplers and yields certified, geometry-adaptive schedules.

Reference graph

Works this paper leans on

47 extracted references · 44 canonical work pages · cited by 1 Pith paper

  1. [1]

    Allen-Zhu, Y

    Z. Allen-Zhu, Y . Li, and Z. Song, A convergence theory for deep learning via over- parameterization. In International Conference on Machine Learning, pp. 242–252, PMLR, 2019

    work page 2019

  2. [2]

    Bakry and M

    D. Bakry and M. Emery, Diffusions hypercontractives. Seminaire de probabilites de Stras- bourg 19 (1985), 177–206

    work page 1985

  3. [3]

    Bakry, I

    D. Bakry, I. Gentil, and M. Ledoux, Analysis and Geometry of Markov Diffusion Operators. 103, Springer, 2014

    work page 2014

  4. [4]

    A. R. Barron, The exponential convergence of posterior probabilities wi th implications for Bayes estimators of density functions . Department of Statistics, University of Illinois Champaign, IL, 1988

    work page 1988

  5. [5]

    A. R. Barron, Neural net approximation. In Proc. 7th Yale Workshop on Adaptive and Learn- ing Systems, pp. 69–72, 1, 1992

    work page 1992

  6. [6]

    A. R. Barron, Universal approximation bounds for superp ositions of a sigmoidal function. IEEE Transactions on Information Theory 39 (1993), no. 3, 930–945

    work page 1993

  7. [7]

    A. R. Barron, Information-theoretic characterization of Bayes performance and the choice of priors in parametric and nonparametric problems. In Proc. Valencia Conference, Bayesian Statistics 6 (1998), 22–52

    work page 1998

  8. [8]

    A. R. Barron, Shannon lecture: Information theory and hi gh-dimensional Bayesian com- putation. In IEEE Internat. Symposium on Information Theory , 2024

    work page 2024

Show all 47 references
  1. [9]

    A. R. Barron and J. M. Klusowski, Approximation and estim ation for high-dimensional deep learning networks. arXiv:1809.03090 (2018)

    2018 arXiv

  2. [10]

    A. R. Barron and J. M. Klusowski, Complexity, statistic al risk, and metric entropy of deep nets using total path variation. arXiv:1902.00800 (2019)

    2019 arXiv

  3. [11]

    A. R. Barron and C. McDonald, Log concave coupling for sa mpling from neural net pos- terior distributions. In Proc. IMS-NUS Singapore Workshop on Statistical Machine Learn- ing for High Dimensional Data , 2024

    2024

  4. [12]

    Bauerschmidt and T

    R. Bauerschmidt and T. Bodineau, A very simple proof of t he LSI for high temperature spin systems. Journal of Functional Analysis 276 (2019), no. 8, 2582–2588

    2019

  5. [13]

    S. G. Bobkov and M. Ledoux, From Brunn-Minkowski to Bras camp-Lieb and to logar- ithmic Sobolev inequalities. Geometric and Functional Analysis 10 (2000), 1028–1052 Rapid Bayesian Computation and Estimation for Neural Netwo rks via Log-Concave Coupling 69

    2000

  6. [14]

    Y . Cai, J. Wu, S. Mei, M. Lindsey, and P . Bartlett, Large s tepsize gradient descent for non-homogeneous two-layer networks: Margin improvement and fast optimization. In The Thirty-eighth Annual Conference on Neural Information Pro cessing Systems, 2024

    2024

  7. [15]

    Charnock, L

    T. Charnock, L. Perreault-Levasseur, and F. Lanusse, Bayesian neural networks. InArtificial Intelligence for High Energy Physics, pp. 663–713, WORLD SCIENTIFIC, 2020

    2020

  8. [16]

    Y . Chen, S. Chewi, A. Salim, and A. Wibisono, Improved analysis for a proximal algorithm for sampling. In Conference on Learning Theory , pp. 2984–3014, PMLR, 2022

    2022

  9. [17]

    Chizat, E

    L. Chizat, E. Oyallon, and F. Bach, On lazy training in di fferentiable programming. Advances in Neural Information Processing Systems 32 (2019)

    2019

  10. [18]

    S. S. Dey, G. Wang, and Y . Xie, Approximation algorithmsfor training one-node relu neural networks. IEEE Transactions on Signal Processing 68 (2020), 6696–6706

    2020

  11. [19]

    S. Du, J. Lee, H. Li, L. Wang, and X. Zhai, Gradient descen t finds global minima of deep neural networks. In International Conference on Machine Learning, pp. 1675–1685, PMLR, 2019

    2019

  12. [20]

    Dwivedi, Y

    R. Dwivedi, Y . Chen, M. J. Wainwright, and B. Yu, Log-con cave sampling: Metropolis- Hastings algorithms are fast. Journal of Machine Learning Research 20 (2019), no. 183, 1–42

    2019

  13. [21]

    S. Frei, G. Vardi, P . Bartlett, and N. Srebro, The double -edged sword of implicit bias: Generalization vs. robustness in relu networks. In The Thirty-seventh Annual Conference on Neural Information Processing Systems, 2023

    2023

  14. [22]

    Froese and C

    V . Froese and C. Hertrich, Training neural networks is np-hard in fixed dimension. Advances in Neural Information Processing Systems 36 (2024)

    2024

  15. [23]

    Gallego and D

    V . Gallego and D. Ríos Insua, Current advances in neural networks. Annual Review of Statistics and Its Application 9 (2022), no. 1, 197–222

    2022

  16. [24]

    S. Goel, A. Klivans, P . Manurangsi, and D. Reichman, Tight Hardness Results for Training Depth-2 ReLU Networks. In12th Innovations in Theoretical Computer Science Conference (ITCS 2021) , pp. 22:1–22:14, 185, Schloss Dagstuhl – Leibniz-Zentrum f ür Informatik, 2021

    2021

  17. [25]

    R. D. Gordon, Values of Mills’ ratio of area to bounding o rdinate and of the normal prob- ability integral for large values of the argument. The Annals of Mathematical Statistics 12 (1941), no. 3, 364–366

    1941

  18. [26]

    Hanin and A

    B. Hanin and A. Zlokapa, Bayesian inference with deep we akly nonlinear networks. 2024, arXiv:2405.16630

    2024 arXiv

  19. [27]

    J. Hron, R. Novak, J. Pennington, and J. Sohl-Dickstein , Wide Bayesian neural networks have a simple weight posterior: theory and accelerated samp ling. In International Confer- ence on Machine Learning , pp. 8926–8945, PMLR, 2022

    2022

  20. [28]

    Huang, D

    X. Huang, D. Zou, Y .-A. Ma, H. Dong, and T. Zhang, Faster sampling via stochastic gradient proximal sampler. arXiv:2405.16734 (2024)

    2024 arXiv

  21. [29]

    Jacot, F

    A. Jacot, F. Gabriel, and C. Hongler, Neural tangent kernel: Convergence and generalization in neural networks. Advances in Neural Information Processing Systems 31 (2018)

    2018

  22. [30]

    J. M. Klusowski and A. R. Barron, Approximation by combi nations of ReLU and squared ReLU ridge functions with ℓ1 andℓ0 controls. IEEE Transactions on Information Theory 64 (2018), no. 12, 7649–7656 70 C. McDonald and A. R. Barron

    2018

  23. [31]

    Kook, Y .-T

    Y . Kook, Y .-T. Lee, R. Shen, and S. Vempala, Sampling wit h Riemannian Hamiltonian Monte Carlo in a constrained space. Advances in Neural Information Processing Systems 35 (2022), 31684–31696

    2022

  24. [32]

    Kook and S

    Y . Kook and S. S. Vempala, Gaussian cooling and Dikin wal ks: The interior-point method for logconcave sampling. In The Thirty Seventh Annual Conference on Learning Theory , pp. 3137–3240, PMLR, 2024

    2024

  25. [33]

    Kook and S

    Y . Kook and S. S. Vempala, Sampling and integration of logconcave functions by algorithmic diffusion. arXiv preprint arXiv:2411.13462 (2024)

    2024 arXiv

  26. [34]

    Y . T. Lee, R. Shen, and K. Tian, Structured logconcave sampling with a restricted Gaussian oracle. In Conference on Learning Theory , pp. 2993–3050, PMLR, 2021

    2021

  27. [35]

    E. L. Lehmann, J. P . Romano, and G. Casella, Testing Statistical Hypotheses. 3, Springer, 1986

    1986

  28. [36]

    C. Liu, L. Zhu, and M. Belkin, Loss landscapes and optimi zation in over-parameterized non-linear systems and neural networks. Applied and Computational Harmonic Analysis 59 (2022), 85–116

    2022

  29. [37]

    Livingstone, M

    S. Livingstone, M. Betancourt, S. Byrne, and M. Girolam i, On the geometric ergodicity of Hamiltonian Monte Carlo. Bernoulli 25 (2019), no. 4A, 3109 – 3138

    2019

  30. [38]

    Lovász and S

    L. Lovász and S. Vempala, The geometry of logconcave functions and sampling algorithms. Random Structures & Algorithms 30 (2007), no. 3, 307–358

    2007

  31. [39]

    McDonald and A

    C. McDonald and A. R. Barron, Log-concave coupling for s ampling neural net posteriors. In 2024 IEEE International Symposium on Information Theory (I SIT), pp. 2251–2256, IEEE, 2024

    2024

  32. [40]

    Montanari and Y

    A. Montanari and Y . Wu, Provably efficient posterior samp ling for sparse linear regression via measure decomposition. arXiv:2406.19550 (2024)

    2024 arXiv

  33. [41]

    R. M. Neal, Bayesian Learning for Neural Networks . Lecture Notes in Statistics 118, Springer, New Y ork, NY, 1996

    1996

  34. [42]

    Robbins, A remark on Stirling’s formula

    H. Robbins, A remark on Stirling’s formula. The American Mathematical Monthly 62 (1955), no. 1, 26–29

    1955

  35. [43]

    Srinivasan, A

    V . Srinivasan, A. Wibisono, and A. Wilson, Fast sampling from constrained spaces using the Metropolis-adjusted mirror Langevin algorithm. In The Thirty Seventh Annual Conference on Learning Theory , pp. 4593–4635, PMLR, 2024

    2024

  36. [44]

    M. K. Titsias and O. Papaspiliopoulos, Auxiliary gradi ent-based sampling algorithms. Journal of the Royal Statistical Society Series B: Statisti cal Methodology 80 (2018), no. 4, 749–767

    2018

  37. [45]

    Tsigler and P

    A. Tsigler and P . L. Bartlett, Benign overfitting in ridg e regression. Journal of Machine Learning Research 24 (2023), no. 123, 1–76

    2023

  38. [46]

    V . H. Vu, On the infeasibility of training neural networ ks with small mean-squared error. IEEE Transactions on Information Theory 44 (1998), no. 7, 2892–2900

    1998

  39. [47]

    D. Zou, Y . Cao, D. Zhou, and Q. Gu, Gradient descent optim izes over-parameterized deep relu networks. Machine Learning 109 (2020), 467–492 Rapid Bayesian Computation and Estimation for Neural Netwo rks via Log-Concave Coupling 71 Curtis McDonald Department of Statistics and ...

    2020

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.