Pith. sign in

REVIEW 4 major objections 5 minor 36 references

Approximate Message Passing for Bayesian Neural Networks

T0 review · 4 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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

Pith's one-line read The paper claims that message passing for Bayesian neural networks can scale to convolutional networks and avoid double-counting training data, yielding better-calibrated uncertainty on CIFAR-10.

desk verdict Novel MP framework for BNNs that scales to CNNs and avoids double-counting, but the calibration evidence is weaker than claimed and the prior is empirically fitted — worth a referee, not a publication as is. read the letter →

arxiv 2501.15573 v1 pith:B34HAA3R submitted 2025-01-26 cs.LG cs.CV

Romeo Sommerfeld , Christian Helms , Ralf Herbrich This is my paper · ORCID
classification cs.LGcs.CV MSC 68T0762F15
keywords Bayesianneuralnetworksmessagepassingfactorgraphsbeliefpropagationuncertaintyquantificationconvolutionalmomentmatchingcalibration
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

This paper seeks to establish that the predictive posterior of a Bayesian neural network can be computed by message passing on a factor graph, and that doing so without reusing training data fixes the overconfidence that has limited earlier message-passing methods. The authors derive Gaussian message equations for the three operations a neural network needs—weighted sums, nonlinearities, and scalar products—and combine them with batched loopy belief propagation. On CIFAR-10 with an 890k-parameter convolutional network, they report accuracy comparable to AdamW and IVON while achieving lower negative log-likelihood and expected calibration error. On synthetic regression data, they report a 0.9 correlation between credible-interval mass and actual coverage of the true function outside the training range. If correct, the paper shows that message passing is a viable third route to Bayesian deep learning, alongside Markov chain Monte Carlo and variational inference.

What carries the argument

The central object is a scalar-level factor graph in which each neural-network operation is a Dirac-delta factor: a weighted sum $\delta(s-c^\top v)$, a nonlinearity $\delta(a-\phi(z))$, and a product $\delta(z-ab)$. Messages are restricted to scaled one-dimensional Gaussian densities using natural parameters (precision $\rho=1/\sigma^2$ and precision-mean $\tau=\mu/\sigma^2$), where pointwise multiplication becomes addition and Gaussian division is well defined. Nonlinearity factors are handled by moment matching, either on the message directly or on the updated marginal in the expectation-propagation style, and product factors use variational message passing. Training proceeds by batched loopy belief propagation with an alternating forward/backward schedule; a trainer object stores aggregate messages from inactive examples and divides out the old batch contribution before multiplying in the new one, which is the mechanism that prevents data double-counting.

What would settle it

A direct falsifier is to compare the method's predictive intervals against a gold-standard sampler, such as Hamiltonian Monte Carlo, on a small regression task where the posterior is known to be multimodal or strongly correlated between weights; if the diagonal-Gaussian posterior gives coverage far below nominal across the entire input range, the central approximation fails. A cheaper check is to run the existing synthetic sine experiment with, say, 1,000 seeds and test whether the one-sigma coverage on the positive side systematically stays far below 68%.

Watch

Extended reading notes

Core claim

The central discovery claimed is that a diagonal-Gaussian approximation to the posterior over weights—meaning each weight gets its own independent normal distribution—can be propagated through an entire convolutional network by treating every scalar operation as a factor and passing moment-matched Gaussian messages, provided the training data enter the factor graph once rather than once per epoch. The paper states this is the first message-passing method that handles convolutional neural networks and avoids double-counting training data, a limitation it attributes to earlier methods such as probabilistic backpropagation and the factor-graph approach of Lucibello et al. (2022). On CIFAR-10, the claimed result is accuracy of 0.773 versus 0.783 for AdamW and 0.772 for IVON, with a negative log-likelihood of 0.997 compared to 1.736 and 1.316, and an expected calibration error of 0.029 compared to 0.046 and 0.035. The method also scales to a multilayer perceptron with 5.6 million parameters.

Load-bearing premise

The whole method depends on the assumption that approximating each weight's uncertainty by a separate bell curve, updated by moment-matched messages, gives a faithful picture of the true posterior; the paper itself notes that loopy belief propagation convergence is not guaranteed and its synthetic results show coverage below the expected level on one side of the data.

Editorial extensions

If this is right

  • Convolutional architectures can now be trained with message passing, so uncertainty quantification by belief propagation is not restricted to small multilayer perceptrons.
  • Because the batching rule divides out the previous aggregate message, data are counted once per training pass, which should remove the main known source of message-passing overconfidence.
  • The CIFAR-10 results, if they reproduce, imply that a parameter-free message-passing posterior can match variational inference on accuracy while providing better calibrated probabilities.
  • The reusable message equations for sums, nonlinearities, and products open the door to other factor-graph models beyond neural networks.
  • Scaling to a 5.6-million-parameter MLP indicates that the method's memory and runtime, though currently worse than deterministic training, are not a hard barrier to moderately large models.

Reading between the lines

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

  • If the calibration edge survives across architectures and datasets, message passing could become a preferred method for safety-critical classifiers where negative log-likelihood and calibration error matter more than raw accuracy.
  • The synthetic asymmetry in coverage—36% versus 61% for one-sigma intervals on the two sides—suggests the diagonal-Gaussian posterior may be systematically too narrow on one side, so the next test is to measure coverage separately on each side with more seeds.
  • An implication the authors leave implicit is that replacing the empirical prior-variance formula with a principled depth-aware initialization could remove the remaining unsolved prior problem and further stabilize deep networks.
  • The authors' proposed extension to division and exponential factors would make the same message-passing machinery applicable to normalization layers and softmax in transformers, so a concrete next experiment is to run the method on a ResNet-style architecture once those factors exist.
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

4 major / 5 minor

Summary. The paper proposes an approximate message passing (MP) framework for Bayesian neural networks, modeling the predictive posterior as a factor graph and approximating messages with diagonal Gaussians via moment matching. It introduces a batching scheme to avoid double-counting training data and derives closed-form message equations for linear, nonlinearity, product, and training-signal factors. The method is evaluated on a synthetic sine regression task with coverage checks, on CIFAR-10 with a roughly 890k-parameter CNN, and on a 5.6M-parameter MLP, reporting competitive accuracy and a calibration edge over AdamW and IVON.

Significance. If substantiated, the framework would be a valuable contribution to BNN inference: it provides detailed Gaussian message derivations for core operations of modern networks, scales to CNNs, and is accompanied by a public Julia implementation. The paper is also transparent about several limitations, including the lack of a convergence guarantee for loopy belief propagation and the empirical nature of the prior initialization. However, the empirical evidence for the central uncertainty-quality claims is weaker than the text suggests, and the current presentation overstates the calibration results.

major comments (4)
  1. [Section 5.1] The reported coverage rates do not support the statement that the posterior is 'reasonably well-calibrated' or that the deviations are 'slightly lower' than nominal. For extrapolation to positive x, only 36% of 1σ intervals contain the true function (nominal 68%), and 2σ coverage is 68% (nominal 95%); negative x also under-covers (61/86/93). These are large systematic deviations, especially on the positive side. The 0.90 correlation quoted in the abstract is a rank correlation between credible-interval mass and coverage rate across p and x, which is not a calibration measure and can be high even when coverage is far below nominal. The abstract and Section 5.1 need to report the actual coverage rates and either temper the calibration claim or present a proper coverage analysis that accounts for the asymmetry.
  2. [Section 5.2, Table 1] The CIFAR-10 comparison is based on a single training run per method. The reported differences in ECE (0.029 vs 0.035–0.046) and NLL (0.997 vs 1.316–1.736) could be within run-to-run variability, particularly under a 25-epoch schedule with stochastic optimization. The claim that MP has 'a notable edge' in calibration needs to be supported by multiple seeds or confidence intervals.
  3. [Section 4.3 and Appendix D] The prior variance coefficients are obtained from empirical fits to simulated message statistics rather than derived from the message equations, and Appendix D explicitly states that the variance-control goal was not achieved ('we fail to achieve this goal' and 'choosing a good prior is still an unsolved problem'). Because the same fitted prior is used in the synthetic coverage experiment, the calibration measurements have a self-tuning component: the prior was selected to keep message variances near a target, and the coverage results are not independent of that choice. The paper should either derive the coefficients, or treat them as tuned hyperparameters and analyze the sensitivity of the uncertainty results to them.
  4. [Section 3.2] The paper acknowledges that loopy belief propagation convergence is not guaranteed, but it does not report convergence diagnostics, a stopping rule, or a study of sensitivity to the damping factor and iteration schedule. Since the method's reliability rests on the loopy BP loop, this information is needed to assess whether the reported results reflect a stable fixed point of the message updates.
minor comments (5)
  1. [Section 1, Contribution 3] The statement that the method requires 'no hyperparameter tuning' is stronger than what the paper supports: the leak value, damping factor, target prior variance (1.5), bias variance (0.5), and the fitted coefficients in the prior formula are all set empirically.
  2. [Section 4.3] The formula for σ2_p is misformatted; the denominator should be clearly written as (0.8041 + 0.4496 · d1).
  3. [Appendix B.1, Building Block 3] The notation N(0; µ2, σ2_2) for a Gaussian density evaluated at 0 is nonstandard and should be replaced by, for example, φ(0; µ2, σ2_2) or a clarifying sentence.
  4. [Experiments] The 5.6M-parameter MLP experiment mentioned in the abstract is not described in the experiments section; adding a brief setup (architecture, dataset, number of runs) would strengthen the scalability claim.
  5. [References] The references contain formatting issues such as missing spaces and concatenated URLs (e.g., 'URLhttps://arxiv.org/abs/1604.07316'); the final version should correct these.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the message-passing derivation is self-contained, and the empirically fitted prior variance is not a fitted prediction.

full rationale

The paper's derivation chain is self-contained. Section 2 constructs the factor graph from the standard identity p(θ|D) ∝ p(θ)∏p(y_i|f_θ(x_i)); Section 3.1 derives Gaussian message updates from the Gaussian product identity and Dirac-delta sifting, with moment matching as the stated approximation rule; Section 3.2 gives an explicit batch update that removes the old aggregate batch message before adding the new one, which is the standard algebraic condition for avoiding double-counting; and Appendix B/E derive the ReLU, LeakyReLU, product, sum, softmax, and argmax messages from elementary Gaussian moment formulas. None of these steps is defined in terms of the CIFAR-10 or synthetic results they are used to explain. The CIFAR-10 evaluation is against external baselines (AdamW, IVON) and is not fitted. The one potentially self-tuning element is the prior-variance formula in Section 4.3, whose constants 0.8041 and 0.4496 are fitted in Appendix D to empirical LeakyReLU message statistics with a target variance of 1.5. But the synthetic coverage measurement in Section 5.1 is not forced by that fit: the paper reports 1σ coverage of only 36% on positive extrapolation and 61% on negative extrapolation, which is undercoverage, not the nominal 68% that a fit-to-coverage construction would have produced. The prior is used to control message-variance magnitude, not to match the reported coverage rates, so this is a hyperparameter-choice caveat rather than a prediction that reduces to its input. The self-citations (Stern et al. 2009; Kurle et al. 2022) are background and standard technique, not load-bearing uniqueness arguments.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central method relies on the Gaussian message approximation and the loopy BP scheduling assumption. The prior is empirically fitted, which adds free parameters. No new physical or mathematical entities are introduced.

free parameters (3)
  • prior variance formula coefficients = 0.8041 and 0.4496 (plus target variance 1.5 and bias prior variance 0.5)
    The prior variance for each linear layer is set by an empirical formula derived from fitting to sampled leak statistics in Appendix D. The target prior predictive variance of 1.5 and the bias prior variance of 0.5 are hand-chosen.
  • leak value for LeakyReLU = 0.1
    The leak is fixed at 0.1 in all experiments and used in the empirical fitting of the prior parameters.
  • damping factor (exponential moving average on batch messages) = not specified numerically
    The paper mentions light message damping on aggregated batch messages to stabilize training, but the coefficient value is not reported.
assumptions (5)
  • domain assumption Messages and marginals are approximated by scaled one-dimensional Gaussian densities
    Adopted in Section 3.1; all message equations rely on the Gaussian family being closed under multiplication and moment matching. If this approximation is poor, the posterior approximation is inaccurate.
  • domain assumption The factor graph decomposition at scalar level with Dirac delta factors faithfully represents the neural network integrand
    Section 2 states that the scalar-level factor graph with Dirac deltas multiplies to a function proportional to the integrand. This requires the decomposition to be exact before approximations are applied.
  • domain assumption Loopy belief propagation with the alternating forward/backward schedule converges to a reasonable posterior approximation
    Section 3.2 acknowledges convergence is not guaranteed under the message-passing schedule; the practical success depends on this assumption holding.
  • domain assumption Prior means sampled from spectral parametrization improve feature learning
    Section 4.3 adopts spectral parametrization (Yang et al., 2024) to break symmetry, which is an external result assumed to hold in this setting.
  • domain assumption The batching scheme prevents double-counting by dividing old aggregate messages before multiplying new ones
    Section 3.2 describes the divide-and-multiply update; this is a design assumption that the math correctly prevents information reuse across batches.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Approximate Message Passing for Bayesian Neural Networks." pith.science (2026). https://pith.science/paper/B34HAA3R

@misc{pith2026250115573,
  author       = {Pith},
  title        = {Pith review of: Approximate Message Passing for Bayesian Neural Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B34HAA3R}},
  note         = {Machine review of arXiv:2501.15573}
}
read the original abstract

Bayesian neural networks (BNNs) offer the potential for reliable uncertainty quantification and interpretability, which are critical for trustworthy AI in high-stakes domains. However, existing methods often struggle with issues such as overconfidence, hyperparameter sensitivity, and posterior collapse, leaving room for alternative approaches. In this work, we advance message passing (MP) for BNNs and present a novel framework that models the predictive posterior as a factor graph. To the best of our knowledge, our framework is the first MP method that handles convolutional neural networks and avoids double-counting training data, a limitation of previous MP methods that causes overconfidence. We evaluate our approach on CIFAR-10 with a convolutional neural network of roughly 890k parameters and find that it can compete with the SOTA baselines AdamW and IVON, even having an edge in terms of calibration. On synthetic data, we validate the uncertainty estimates and observe a strong correlation (0.9) between posterior credible intervals and its probability of covering the true data-generating function outside the training range. While our method scales to an MLP with 5.6 million parameters, further improvements are necessary to match the scale and performance of state-of-the-art variational inference methods.

Figures

Figures reproduced from arXiv: 2501.15573 by the authors.

Figure 1
Figure 1. Conceptual vector-valued factor graph for a simple MLP. Each training example has its [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. A full FactorGraph models all messages for one batch of training examples. To iterate the [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Fitting MLPs of width 16 with increasing depth. Between any linear layers we apply [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: A deterministic factor corresponding to the LeakyReLU [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

36 extracted references · 15 canonical work pages

  1. [1]

    Weight uncertainty in neural networks, 2015

    Charles Blundell, Julien Cornebise, Koray Kavukcuoglu, and Daan Wierstra. Weight uncertainty in neural networks, 2015. URL https://arxiv.org/abs/1505.05424

    arXiv 2015

  2. [2]

    Jackel, Mathew Monfort, Urs Muller, Jiakai Zhang, Xin Zhang, Jake Zhao, and Karol Zieba

    Mariusz Bojarski, Davide Del Testa, Daniel Dworakowski, Bernhard Firner, Beat Flepp, Prasoon Goyal, Lawrence D. Jackel, Mathew Monfort, Urs Muller, Jiakai Zhang, Xin Zhang, Jake Zhao, and Karol Zieba. End to end learning for self-driving cars, 2016. URL https://arxiv.org/abs/1604.07316

    arXiv 2016

  3. [3]

    Wide Mean-Field Bayesian Neural Networks Ignore the Data

    Beau Coker, Wessel P. Bruinsma, David R. Burt, Weiwei Pan, and Finale Doshi-Velez. Wide mean-field bayesian neural networks ignore the data, 2022. URL https://arxiv.org/abs/2202.11670

    work page Pith review arXiv 2022

  4. [4]

    Laplace redux -- effortless bayesian deep learning, 2022

    Erik Daxberger, Agustinus Kristiadi, Alexander Immer, Runa Eschenhagen, Matthias Bauer, and Philipp Hennig. Laplace redux -- effortless bayesian deep learning, 2022. URL https://arxiv.org/abs/2106.14806

    arXiv 2022

  5. [5]

    Assumed density filtering methods for learning bayesian neural networks

    Soumya Ghosh, Francesco Delle Fave, and Jonathan Yedidia. Assumed density filtering methods for learning bayesian neural networks. Proceedings of the AAAI Conference on Artificial Intelligence, 30 0 (1), Feb. 2016. doi:10.1609/aaai.v30i1.10296. URL https://ojs.aaai.org/index.php/AAAI/article/view/10296

    work page doi:10.1609/aaai.v30i1.10296 2016

  6. [6]

    Practical variational inference for neural networks

    Alex Graves. Practical variational inference for neural networks. In J. Shawe-Taylor, R. Zemel, P. Bartlett, F. Pereira, and K.Q. Weinberger (eds.), Advances in Neural Information Processing Systems, volume 24. Curran Associates, Inc., 2011. URL https://proceedings.neurips.cc/paper_files/paper/2011/file/7eb3c8be3d411e8ebfab08eba5f49632-Paper.pdf

    work page 2011

  7. [7]

    Benchmarking uncertainty estimation methods for deep learning with safety-related metrics

    Maximilian Henne, Adrian Schwaiger, Karsten Roscher, and Gereon Weiss. Benchmarking uncertainty estimation methods for deep learning with safety-related metrics. In SafeAI@ AAAI, pp.\ 83--90, 2020

    work page 2020

  8. [8]

    José Miguel Hernández-Lobato and Ryan P. Adams. Probabilistic backpropagation for scalable learning of bayesian neural networks, 2015. URL https://arxiv.org/abs/1502.05336

    work page Pith review arXiv 2015

Show all 36 references
  1. [9]

    Loopy belief propagation: Convergence and effects of message errors

    Alexander Ihler, John III, and Alan Willsky. Loopy belief propagation: Convergence and effects of message errors. Journal of Machine Learning Research, 6: 0 905--936, 05 2005

    2005

  2. [10]

    The bayesian learning rule, 2024

    Mohammad Emtiyaz Khan and Håvard Rue. The bayesian learning rule, 2024. URL https://arxiv.org/abs/2107.04562

    2024 arXiv

  3. [11]

    Bayesian dark knowledge, 2015

    Anoop Korattikara, Vivek Rathod, Kevin Murphy, and Max Welling. Bayesian dark knowledge, 2015. URL https://arxiv.org/abs/1506.04416

    2015 arXiv

  4. [12]

    Kschischang, B.J

    F.R. Kschischang, B.J. Frey, and H.-A. Loeliger. Factor graphs and the sum-product algorithm. IEEE Transactions on Information Theory, 47 0 (2): 0 498--519, 2001. doi:10.1109/18.910572

    2001 doi

  5. [13]

    On the detrimental effect of invariances in the likelihood for variational inference, 2022

    Richard Kurle, Ralf Herbrich, Tim Januschowski, Yuyang Wang, and Jan Gasthaus. On the detrimental effect of invariances in the likelihood for variational inference, 2022. URL https://arxiv.org/abs/2209.07157

    2022 arXiv

  6. [14]

    A convnet for the 2020s, 2022

    Zhuang Liu, Hanzi Mao, Chao-Yuan Wu, Christoph Feichtenhofer, Trevor Darrell, and Saining Xie. A convnet for the 2020s, 2022. URL https://arxiv.org/abs/2201.03545

    2022 arXiv

  7. [15]

    SGDR: stochastic gradient descent with restarts

    Ilya Loshchilov and Frank Hutter. SGDR: stochastic gradient descent with restarts. CoRR, abs/1608.03983, 2016. URL http://arxiv.org/abs/1608.03983

    2016 arXiv

  8. [16]

    Fixing weight decay regularization in adam

    Ilya Loshchilov and Frank Hutter. Fixing weight decay regularization in adam. CoRR, abs/1711.05101, 2017. URL http://arxiv.org/abs/1711.05101

    2017 arXiv

  9. [17]

    Deep learning via message passing algorithms based on belief propagation

    Carlo Lucibello, Fabrizio Pittorino, Gabriele Perugini, and Riccardo Zecchina. Deep learning via message passing algorithms based on belief propagation. Machine Learning: Science and Technology, 3 0 (3): 0 035005, jul 2022. doi:10.1088/2632-2153/ac7d3b. URL https://dx.doi.org/...

    2022 doi

  10. [18]

    Thomas P. Minka. Expectation propagation for approximate bayesian inference. In Proceedings of the Seventeenth Conference on Uncertainty in Artificial Intelligence, UAI'01, pp.\ 362–369, San Francisco, CA, USA, 2001. Morgan Kaufmann Publishers Inc. ISBN 1558608001

    2001

  11. [19]

    Eunnet: Efficient un-normalized convolution layer for stable training of deep residual networks without batch normalization layer

    Khanh-Binh Nguyen, Jaehyuk Choi, and Joon-Sung Yang. Eunnet: Efficient un-normalized convolution layer for stable training of deep residual networks without batch normalization layer. IEEE Access, 11: 0 76977--76988, 2023. doi:10.1109/ACCESS.2023.3244072

    2023

  12. [20]

    Turner, Rio Yokota, and Mohammad Emtiyaz Khan

    Kazuki Osawa, Siddharth Swaroop, Anirudh Jain, Runa Eschenhagen, Richard E. Turner, Rio Yokota, and Mohammad Emtiyaz Khan. Practical deep learning with bayesian principles, 2019. URL https://arxiv.org/abs/1906.02506

    2019 arXiv

  13. [21]

    Approximate blocked gibbs sampling for bayesian neural networks, 2023

    Theodore Papamarkou. Approximate blocked gibbs sampling for bayesian neural networks, 2023. URL https://arxiv.org/abs/2208.11389

    2023 arXiv

  14. [22]

    Osborne, Tim G

    Theodore Papamarkou, Maria Skoularidou, Konstantina Palla, Laurence Aitchison, Julyan Arbel, David Dunson, Maurizio Filippone, Vincent Fortuin, Philipp Hennig, José Miguel Hernández-Lobato, Aliaksandr Hubin, Alexander Immer, Theofanis Karaletsos, Mohammad Emtiyaz Khan, Agustin...

    2024 arXiv

  15. [23]

    Sam 2: Segment anything in images and videos, 2024

    Nikhila Ravi, Valentin Gabeur, Yuan-Ting Hu, Ronghang Hu, Chaitanya Ryali, Tengyu Ma, Haitham Khedr, Roman Rädle, Chloe Rolland, Laura Gustafson, Eric Mintun, Junting Pan, Kalyan Vasudev Alwala, Nicolas Carion, Chao-Yuan Wu, Ross Girshick, Piotr Dollár, and Christoph Feichtenh...

    2024 arXiv

  16. [24]

    Variational learning is effective for large deep networks, 2024

    Yuesong Shen, Nico Daheim, Bai Cong, Peter Nickl, Gian Maria Marconi, Clement Bazan, Rio Yokota, Iryna Gurevych, Daniel Cremers, Mohammad Emtiyaz Khan, and Thomas Möllenhoff. Variational learning is effective for large deep networks, 2024. URL https://arxiv.org/abs/2402.17641

    2024 arXiv

  17. [25]

    Filter response normalization layer: Eliminating batch dependence in the training of deep neural networks

    Saurabh Singh and Shankar Krishnan. Filter response normalization layer: Eliminating batch dependence in the training of deep neural networks. CoRR, abs/1911.09737, 2019. URL http://arxiv.org/abs/1911.09737

    1911 arXiv

  18. [26]

    Expectation backpropagation: parameter-free training of multilayer neural networks with continuous or discrete weights

    Daniel Soudry, Itay Hubara, and Ron Meir. Expectation backpropagation: parameter-free training of multilayer neural networks with continuous or discrete weights. In Proceedings of the 27th International Conference on Neural Information Processing Systems - Volume 1, NIPS'14, p...

    2014

  19. [27]

    Matchbox: Large scale bayesian recommendations

    David Stern, Ralf Herbrich, and Thore Graepel. Matchbox: Large scale bayesian recommendations. In Proceedings of the 18th International World Wide Web Conference, January 2009. URL https://www.microsoft.com/en-us/research/publication/matchbox-large-scale-bayesian-recommendations/

    2009

  20. [28]

    Gomez, Lukasz Kaiser, and Illia Polosukhin

    Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is all you need, 2023. URL https://arxiv.org/abs/1706.03762

    2023 arXiv

  21. [29]

    Schoenholz, and Jeffrey Pennington

    Lechao Xiao, Yasaman Bahri, Jascha Sohl-Dickstein, Samuel S. Schoenholz, and Jeffrey Pennington. Dynamical isometry and a mean field theory of cnns: How to train 10,000-layer vanilla convolutional neural networks, 2018. URL https://arxiv.org/abs/1806.05393

    2018 arXiv

  22. [30]

    Simon, and Jeremy Bernstein

    Greg Yang, James B. Simon, and Jeremy Bernstein. A spectral condition for feature learning, 2024. URL https://arxiv.org/abs/2310.17813

    2024 arXiv

  23. [31]

    Advances in variational inference, 2018

    Cheng Zhang, Judith Butepage, Hedvig Kjellstrom, and Stephan Mandt. Advances in variational inference, 2018. URL https://arxiv.org/abs/1711.05597

    2018 arXiv

  24. [32]

    Beyond recommendations: From backward to forward ai support of pilots' decision-making process

    Zelun Tony Zhang, Sebastian S Feger, Lucas Dullenkopf, Rulu Liao, Lukas S \"u sslin, Yuanting Liu, and Andreas Butz. Beyond recommendations: From backward to forward ai support of pilots' decision-making process. arXiv preprint arXiv:2406.08959, 2024

    2024 arXiv

  25. [33]

    @esa (Ref

    \@ifxundefined[1] #1\@undefined \@firstoftwo \@secondoftwo \@ifnum[1] #1 \@firstoftwo \@secondoftwo \@ifx[1] #1 \@firstoftwo \@secondoftwo [2] @ #1 \@temptokena #2 #1 @ \@temptokena \@ifclassloaded agu2001 natbib The agu2001 class already includes natbib coding, so you should ...

  26. [34]

    \@lbibitem[] @bibitem@first@sw\@secondoftwo \@lbibitem[#1]#2 \@extra@b@citeb \@ifundefined br@#2\@extra@b@citeb \@namedef br@#2 \@nameuse br@#2\@extra@b@citeb \@ifundefined b@#2\@extra@b@citeb @num @parse #2 @tmp #1 NAT@b@open@#2 NAT@b@shut@#2 \@ifnum @merge>\@ne @bibitem@firs...

  27. [35]

    , " * write output.state after.block = add.period write newline

    @open @close @open @close and [1] URL: #1 \@ifundefined chapter * \@mkboth \@ifxundefined @sectionbib * \@mkboth * \@mkboth\@gobbletwo \@ifclassloaded amsart * \@ifclassloaded amsbook * \@ifxundefined @heading @heading NAT@ctr thebibliography [1] @ \@biblabel @NAT@ctr \@bibset...

  28. [36]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 gl...

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.