Pith. sign in

REVIEW 2 major objections 4 minor 1 cited by

Averaging hypotheses over a symmetry group shrinks the KL-divergence term in PAC-Bayes bounds and cannot increase true risk, extending symmetry-based generalization guarantees to non-compact groups and non-invariant data.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · deepseek-v4-flash

2026-08-04 09:04 UTC pith:347BXMGP

load-bearing objection Original idea undermined by a false base bound for unbounded losses; needs revision. the 2 major comments →

arxiv 2510.17303 v2 pith:347BXMGP submitted 2025-10-20 cs.LG stat.ML

Symmetries in PAC-Bayesian Learning

Armin Beck , Peter Ochs This is my paper
classification cs.LG stat.ML
keywords PAC-Bayesgeneralization boundsequivariancenon-compact symmetrynon-invariant dataKL divergenceaveraging operatortranslation symmetry
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper tries to show that symmetry can be converted directly into tighter generalization guarantees: take any hypothesis distribution, average each hypothesis over the symmetry group to make it equivariant, and the KL-divergence term in a standard PAC-Bayes bound can only decrease. In addition, if the data share the symmetry—an equivariant target function and a group-invariant convex loss—the averaged hypothesis has true risk no larger than the original. If correct, this moves the theory of symmetric models beyond compact groups and perfectly invariant distributions, covering practically important cases such as translations and partially symmetric data. The authors demonstrate the mechanism on a standard PAC-Bayes bound and confirm on rotated and translated MNIST that equivariant networks attain both lower test error and a tighter generalization bound than a non-symmetric baseline.

Core claim

The central claim is a KL-divergence decomposition for the averaging operator Q: for any two distributions on a hypothesis class closed under Q, DKL(μ∥ν)=DKL(Q∗μ∥Q∗ν)+∫ log((dμ/dν)/(dQ∗μ/dQ∗ν)) dμ, so pushing both measures through the equivariant projection can only reduce the divergence. Inserting this into a standard PAC-Bayes bound gives a bound whose complexity term is no larger for the equivariant posterior (Theorem 2.6). When the data come from an equivariant target function plus independent noise and the loss is group-invariant and convex in its first argument, averaging a hypothesis also cannot increase its true risk (Proposition 3.4); and for equivariant hypotheses the risk and empi

What carries the argument

The averaging operator Q (Definition 2.2): it sends a measurable hypothesis f to πG(x)·∫ g^{-1}·f(g·πXφ(x)) κ(πXφ(x), dg), where πXφ selects the representative of the orbit of x, πG gives the group element taking that representative to x, and κ is the disintegration kernel of the input distribution over orbit representatives. Q is a measurable projection onto the subspace of equivariant functions. Its role is to feed into the KL-decomposition lemma for pushforward measures: pushing two measures through a measurable map cannot increase their KL divergence, and Q isolates the equivariant component, leaving a nonnegative remainder that describes the non-equivariant difference. This is the mecha

Load-bearing premise

The construction assumes the symmetry group acts freely on the input space—no non-identity transformation fixes any input—so every input decomposes uniquely into a group element and an orbit representative; natural symmetries such as rotations on images with rotational symmetry (or the all-zero image) violate this.

What would settle it

Rerun the paper's rotated-MNIST experiment with rotation angles spanning the full circle, including 180° where digits like '0' and '8' are fixed by the rotation (a non-free action), and check whether the equivariant model's PAC-Bayes bound is still tighter and whether the averaging operator is well-defined for inputs with nontrivial stabilizers. If the bound degrades or Q depends on the chosen orbit representative, the extension to general non-compact symmetries is limited to free actions.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Convolutional (translation-equivariant) models fall under PAC-Bayes guarantees, since translations are non-compact; the paper claims these are the first such bounds.
  • When data are symmetric, equivariant models are preferable to non-equivariant ones: their generalization bound is no worse and their true risk is no larger.
  • For equivariant hypotheses, the empirical risk can be estimated on one representative per orbit, so training and evaluation can be restricted to a reduced input set without weakening the guarantee.
  • The KL-decomposition argument is not tied to the particular baseline bound; it transfers to other PAC-Bayes formulations that use a KL complexity term.
  • In the special case of compact groups and invariant data, the averaging operator reduces to classical Haar-measure group averaging, so prior results are contained as a special case.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper: if the free-action assumption were relaxed by quotienting out stabilizers, the construction would likely extend to symmetries with fixed points, such as rotations acting on digits like '8' or '0'; the abstract KL inequality only needs measurability of Q, not freeness, so the obstruction is the representative-based definition rather than the bound itself.
  • Beyond the paper: the representative-set reduction suggests a concrete training recipe—train an equivariant model on one sample per orbit and evaluate the PAC-Bayes bound there; this could be tested by comparing generalization gaps on quotient datasets.
  • Beyond the paper: because the improvement comes only from the KL term, the empirical-risk term is untouched; a testable prediction is that the bound improvement shrinks continuously as the data distribution becomes less symmetric, disappearing when the remainder term vanishes.
  • Beyond the paper: the same pushforward-KL trick could be applied to other complexity measures in PAC-Bayes bounds, but the remainder term would have to be re-derived for each.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The manuscript develops a PAC-Bayesian framework for equivariant models under general (possibly non-compact) group actions and non-invariant data distributions. The main technical tool is an averaging operator that maps arbitrary hypotheses to equivariant ones via a disintegration of the data distribution over orbit representatives. Theorem 2.5 proves a KL decomposition for this operator; Theorem 2.6 derives a PAC-Bayes bound with a reduced KL term; Proposition 3.4 states that, under symmetric data, averaging does not increase the true risk; Theorem 3.7 gives a bound using empirical risk on orbit representatives. The paper validates the approach on rotated and translated MNIST with cross-entropy loss.

Significance. If the results were correct, the paper would extend symmetry-based PAC-Bayes analysis beyond compact groups and invariant distributions, providing theoretical support for the practical success of equivariant models. The KL-decomposition lemma (Thm 2.5) is conceptually clean and general, and the appendix contains detailed measure-theoretic proofs and a fully worked Gaussian toy example. However, the central theorems rely on a McAllester-type bound that is false for unbounded losses, and the free-action assumption excludes the rotation symmetries used in the experiments. These issues must be resolved before the contribution can be considered sound.

major comments (2)
  1. [§1.2, Theorem 1.2 (and Theorems 2.6, 3.7)] Theorem 1.2 is stated for 'any measurable loss function ℓ:Y×Y→[0,∞)' but uses the square-root McAllester bound with denominator 2n−1. This bound requires a loss bounded in [0,1] (or a sub-Gaussian tail condition). Without boundedness the theorem is false: take H={f≡0}, squared loss ℓ(a,b)=(a−b)^2, Y=0 w.p. 0.99 and Y=100 w.p. 0.01, n=100, δ=0.05, and Q=P_H. Then KL=0 and the RHS is below 0.22 whenever all samples have Y_i=0, an event of probability 0.99^100≈0.366>δ, while the true risk is 100. Thus the claimed high-probability inequality fails. Since Theorems 2.6 and 3.7 are direct consequences of Theorem 1.2, they inherit the defect. The experiments use cross-entropy, which is unbounded, so the reported bounds are not justified by the stated theory. The authors should either restrict all statements to bounded losses (e.g., [0,1]) and re-run experiments with a bounded surrogate, or repla
  2. [§2, Assumption 2.1.2] The entire construction requires the G-action on X to be free. This is not satisfied for the rotation group acting on images: any rotationally symmetric image — e.g., a blank image or a symmetric digit — has a non-trivial stabilizer. The paper's experiments use rotated MNIST, so Assumption 2.1.2 is violated in exactly the empirical setting that is claimed to validate the theory. The assumption is introduced only 'for simplicity' and no relaxation is discussed. To make the experimental claims covered, the authors must either handle actions with stabilizers (e.g., by passing to the quotient by stabilizers) or restrict the data support to an open free subset, which is not done.
minor comments (4)
  1. [§3.1] Typo: 'the cAllester’s PAC-Bayesian boun' should be 'McAllester's PAC-Bayesian bound'.
  2. [Theorem 3.7] The theorem states the bound in terms of S_n^φ but the final sentence defines only S_n. Define S_n^φ explicitly as n i.i.d. copies of (X_φ, Y_φ).
  3. [Appendix C, Example C.1] The pushforward covariance Σ is singular; the KL computation is only valid after restricting the measures to the subspace U. This should be stated explicitly to avoid confusion about Gaussian KL on R^2.
  4. [Remark 2.3] The statement that the kernel equals the Haar measure 'PX-almost surely' is ambiguous. Clarify that this holds for the conditional kernel under the invariant-data assumption.

Circularity Check

0 steps flagged

No circularity: the central derivation applies external lemmas (KL decomposition and McAllester's PAC-Bayes bound) to a newly defined averaging operator; no fitted value is relabeled as a prediction.

full rationale

The derivation chain is not circular. Theorem 2.5 is explicitly a specialization of the external Lemma 1.3 (from [20], by Lyle et al., not the present authors) to the measurable averaging operator Q: the proof says 'The statement follows directly from Lemma 1.3 and the measurability of the average operator.' Theorem 2.6 is McAllester's external PAC-Bayes bound (Theorem 1.2, [23]) applied to the pushforward Q∗Q, and contains no fitted constants. Proposition 3.4 is derived from convexity, G-invariance of the loss, and the disintegration theorem, not from the theorem being proved. Theorem 3.7 is a direct combination of Theorem 2.6 with the distributional identity in Corollary 3.6. The numerical section optimizes the PAC-Bayes right-hand side and then evaluates the resulting upper bound; because a PAC-Bayes bound holds for any posterior, this is a legitimate computation rather than a circular prediction. There are no self-citations by the present authors, and neither the KL lemma nor the McAllester bound is internal to the paper. Concerns about Theorem 1.2's validity for unbounded losses or about the free-action assumption are correctness and robustness issues, not circularity, so they do not affect the circularity score.

Axiom & Free-Parameter Ledger

1 free parameters · 8 axioms · 0 invented entities

The theory introduces no new physical entities or parameters; it relies on standard measure theory, the external KL lemma, and a set of modeling assumptions that restrict applicability. The only hand-chosen constant is the prior sigma in the experiments.

free parameters (1)
  • Prior standard deviation sigma = 0.05
    Hand-chosen for the PAC-Bayes prior in the experiments; affects the reported bound but not the central theorems.
axioms (8)
  • domain assumption Standard Borel spaces and measurable group actions (Assumption 2.1.1)
    The entire measure-theoretic framework requires the input/output spaces and group to be standard Borel, and actions measurable.
  • domain assumption Free group action on X (Assumption 2.1.2)
    Needed for the bijection G×Xφ→X that defines the averaging operator; many natural group actions are not free.
  • domain assumption Hypothesis class closed under averaging (Assumption 2.4)
    Needed so the pushforward Q*Q remains a distribution on H; reasonable for equivariant model classes but not automatic.
  • domain assumption Data generated by G-equivariant f* (Assumption 3.1)
    Central to Proposition 3.4 and Lemma 3.5; assumes the true labeling is equivariant under the group.
  • domain assumption Loss convex in first argument and G-invariant (Assumption 3.3)
    Convexity is needed for the Jensen step in Proposition 3.4; G-invariance is used throughout. Excludes 0-1 loss and multi-class cross-entropy with vector outputs.
  • standard math Disintegration Theorem (Theorem 1.1)
    Standard measure-theoretic result used to define the kernel κ and to decompose integrals over orbits.
  • standard math McAllester's PAC-Bayes bound (Theorem 1.2)
    The baseline bound that the paper adapts; taken as given from the literature.
  • standard math KL decomposition lemma from [20] (Lemma 1.3)
    External result stating D_KL(µ||ν) = D_KL(α*µ||α*ν) + positive remainder; used as the engine for reducing the complexity term.

pith-pipeline@v1.3.0-alltime-deepseek · 13793 in / 19130 out tokens · 144320 ms · 2026-08-04T09:04:09.168175+00:00 · methodology

0 comments
read the original abstract

Symmetries are known to improve the empirical performance of machine learning models, yet theoretical guarantees explaining these gains remain limited. Prior work has focused mainly on compact group symmetries and often assumes that the data distribution itself is invariant, an assumption rarely satisfied in real-world applications. In this work, we extend generalization guarantees to the broader setting of non-compact symmetries, such as translations and to non-invariant data distributions. Building on the PAC-Bayes framework, we adapt and tighten existing bounds, demonstrating the approach on McAllester's PAC-Bayes bound while showing that it applies to a wide range of PAC-Bayes bounds. We validate our theory with experiments on several datasets with non-uniform and non-compact transformations, where the derived guarantees not only hold but also improve upon prior results. These findings provide theoretical evidence that, for symmetric data, symmetric models are preferable beyond the narrow setting of compact groups and invariant distributions, opening the way to a more general understanding of symmetries in machine learning.

Figures

Figures reproduced from arXiv: 2510.17303 by Armin Beck, Peter Ochs.

Figure 1
Figure 1. Figure 1: The figure displays the distribution of the test errors for the baseline CNN and the equivariant [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 1 Pith paper

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

  1. PAC--Bayes Bounds on Quotient Parameter Spaces: Geometry-induced Implicit-Bias Priors

    cs.LG 2026-07 conditional novelty 5.0

    Identifying parameters that define the same predictor reduces the PAC-Bayes KL complexity term, and a geometry-tilted 'implicit-bias' prior can tighten the certificate when it is closer to the learned posterior.

Reference graph

Works this paper leans on

30 extracted references · 4 canonical work pages · cited by 1 Pith paper

  1. [1]

    User-Friendly Introduction to PAC-Bayes Bounds

    Pierre Alquier. “User-Friendly Introduction to PAC-Bayes Bounds”. In:Foundations and Trends®in Machine Learning17.2 (2024), pp. 174–303.doi:10.1561/2200000100.url:http://dx.doi.org/10. 1561/2200000100

    work page doi:10.1561/2200000100.url:http://dx.doi.org/10 2024

  2. [2]

    Stronger Generalization Bounds for Deep Nets via a Compression Approach

    Sanjeev Arora et al. “Stronger Generalization Bounds for Deep Nets via a Compression Approach”. In: Proceedings of the 35th International Conference on Machine Learning (ICML 2018). Vol. 80. Proceedings of Machine Learning Research. PMLR, July 2018, pp. 390–418. arXiv:1802 . 05296.url:https : / / proceedings.mlr.press/v80/arora18b/arora18b.pdf

    2018

  3. [3]

    Convolutional Neural Networks Are Not Invariant to Translation, but They Can Learn to Be

    Valerio Biscione and Jeffrey S. Bowers. “Convolutional Neural Networks Are Not Invariant to Translation, but They Can Learn to Be”. In:Journal of Machine Learning Research22.229 (2021), pp. 1–28.url: http://jmlr.org/papers/v22/21-0019.html

    2021

  4. [4]

    Probabilistic Symmetries and Invariant Neural Networks

    Benjamin Bloem-Reddy and Yee Whye Teh. “Probabilistic Symmetries and Invariant Neural Networks”. In:Journal of Machine Learning Research21 (2020), pp. 1–61. arXiv:1901.06082.url:http://jmlr. org/papers/v21/19-322.html

    Pith/arXiv arXiv 2020

  5. [5]

    PAC-Bayesian Supervised Classification: The Thermodynamics of Statistical Learning

    Olivier Catoni. “PAC-Bayesian Supervised Classification: The Thermodynamics of Statistical Learning”. In:IMS Lecture Notes Monograph Series. Vol. 56. Institute of Mathematical Statistics, 2007, pp. 1–163. doi:10.1214/074921707000000391.url:http://dx.doi.org/10.1214/074921707000000391

    work page doi:10.1214/074921707000000391.url:http://dx.doi.org/10.1214/074921707000000391 2007

  6. [6]

    Group Equivariant Convolutional Networks

    Taco Cohen and Max Welling. “Group Equivariant Convolutional Networks”. In:Proceedings of the 33rd International Conference on Machine Learning. Ed. by Maria Florina Balcan and Kilian Q. Weinberger. Vol. 48. Proceedings of Machine Learning Research. PMLR, June 2016, pp. 2990–2999.url:https : //proceedings.mlr.press/v48/cohenc16.html

    2016

  7. [7]

    A General Theory of Equivariant CNNs on Homoge- neous Spaces

    Taco S. Cohen, Mario Geiger, and Maurice Weiler. “A General Theory of Equivariant CNNs on Homoge- neous Spaces”. In:Advances in Neural Information Processing Systems 33 (NeurIPS). Curran Associates, Inc., 2019, pp. 9142–9153

    2019

  8. [8]

    Steerable CNNs

    Taco S. Cohen and Max Welling. “Steerable CNNs”. In:CoRRabs/1612.08498 (2016). arXiv:1612.08498. url:https://arxiv.org/abs/1612.08498

    Pith/arXiv arXiv 2016

  9. [9]

    Computing Nonvacuous Generalization Bounds for Deep (Stochastic) Neural Networks with Many More Parameters than Training Data

    Gintare Karolina Dziugaite and Daniel M. Roy. “Computing Nonvacuous Generalization Bounds for Deep (Stochastic) Neural Networks with Many More Parameters than Training Data”. In:Proceedings of the 33rd Conference on Uncertainty in Artificial Intelligence (UAI 2017). AUAI Press, 2017, pp. 231–240. arXiv:1703.11008.url:https://auai.org/uai2017/proceedings/p...

    Pith/arXiv arXiv 2017

  10. [10]

    Group Symmetry in PAC Learning

    Bryn Elesedy. “Group Symmetry in PAC Learning”. In:Proceedings of the ICLR 2022 Workshop on Geometrical and Topological Representation Learning. Spotlight presentation. 2022.url:https : / / openreview.net/forum?id=HxeTEZJaxq

    2022

  11. [11]

    Provably Strict Generalisation Benefit for Equivariant Models

    Bryn Elesedy and Sheheryar Zaidi. “Provably Strict Generalisation Benefit for Equivariant Models”. In: Proceedings of the 38th International Conference on Machine Learning. Ed. by Marina Meila and Tong Zhang. Vol. 139. Proceedings of Machine Learning Research. PMLR, July 2021, pp. 2959–2969.url: https://proceedings.mlr.press/v139/elesedy21a.html

    2021

  12. [12]

    PAC-Bayesian Learning of Linear Classifiers

    Pascal Germain et al. “PAC-Bayesian Learning of Linear Classifiers”. In:Proceedings of the 26th Annual International Conference on Machine Learning (ICML 2009). ICML ’09. New York, NY, USA: Association for Computing Machinery, 2009, pp. 353–360.doi:10.1145/1553374.1553419.url:https://doi.org/ 10.1145/1553374.1553419

    arXiv 2009

  13. [13]

    Olav Kallenberg.Foundations of Modern Probability. 2nd. Probability and Its Applications. New York: Springer-Verlag, 2002, pp. xx+638.isbn: 0-387-95313-2.doi:10.1007/978-1-4757-4015-8

    doi:10.1007/978-1-4757-4015-8 2002

  14. [14]

    Kechris.Classical Descriptive Set Theory

    Alexander S. Kechris.Classical Descriptive Set Theory. Graduate Texts in Mathematics. Springer-Verlag, 1995.isbn: 9780387943749.url:https://link.springer.com/book/10.1007/978-1-4612-4190-4

    doi:10.1007/978-1-4612-4190-4 1995

  15. [15]

    Adam: A Method for Stochastic Optimization

    Diederik P. Kingma and Jimmy Ba. “Adam: A Method for Stochastic Optimization”. In:Proceedings of the International Conference on Learning Representations (ICLR 2015). 2015, pp. 1–13.url:https : //arxiv.org/abs/1412.6980

    Pith/arXiv arXiv 2015

  16. [16]

    On the Generalization of Equivariance and Convolution in Neu- ral Networks to the Action of Compact Groups

    Risi Kondor and Shubhendu Trivedi. “On the Generalization of Equivariance and Convolution in Neu- ral Networks to the Action of Compact Groups”. In:Proceedings of the 35th International Conference on Machine Learning (ICML 2018). Vol. 80. Proceedings of Machine Learning Research. PMLR, July 2018, pp. 2747–2755.doi:10.5555/3295222.3295394.url:https://proce...

    arXiv 2018

  17. [17]

    PAC-Bayes & Margins

    John Langford and John Shawe-Taylor. “PAC-Bayes & Margins”. In:Proceedings of the 16th International Conference on Neural Information Processing Systems (NIPS 2002). NIPS’02. Cambridge, MA, USA: MIT Press, 2002, pp. 439–446.url:https://papers.nips.cc/paper/2968618-pac-bayes-margins.pdf

    arXiv 2002

  18. [18]

    An Empirical Evaluation of Deep Architectures on Problems with Many Factors of Variation

    Hugo Larochelle et al. “An Empirical Evaluation of Deep Architectures on Problems with Many Factors of Variation”. In:Proceedings of the 24th International Conference on Machine Learning (ICML 2007). ICML ’07. New York, NY, USA: Association for Computing Machinery, 2007, pp. 473–480.doi:10.1145/ 1273496.1273556.url:https://doi.org/10.1145/1273496.1273556

    arXiv 2007

  19. [19]

    An Analysis of the Effect of Invariance on Generalization in Neural Networks

    Clare Lyle et al. “An Analysis of the Effect of Invariance on Generalization in Neural Networks”. In: Proceedings of the Understanding and Improving Generalization in Deep Learning Workshop. 2019.url: https://oatml.cs.ox.ac.uk/publications/201906_Lyle2019Understanding.html

    2019

  20. [20]

    On the Benefits of Invariance in Neural Networks

    Clare Lyle et al. “On the Benefits of Invariance in Neural Networks”. In:CoRRabs/2005.00178 (2020). arXiv:2005.00178.url:https://arxiv.org/abs/2005.00178

    Pith/arXiv arXiv 2005

  21. [21]

    A Note on the PAC-Bayesian Theorem

    Andreas Maurer. “A Note on the PAC-Bayesian Theorem”. In:CoRRcs.LG/0411099 (2004). arXiv: cs.LG/0411099.url:http://arxiv.org/abs/cs.LG/0411099

    arXiv 2004

  22. [22]

    Simplified PAC-Bayesian Margin Bounds

    David McAllester. “Simplified PAC-Bayesian Margin Bounds”. In:Learning Theory and Kernel Machines. Ed. by Bernhard Schölkopf and Manfred K. Warmuth. Berlin, Heidelberg: Springer Berlin Heidelberg, 2003, pp. 203–215.isbn: 978-3-540-45167-9.url:https://link.springer.com/chapter/10.1007/978- 3-540-45167-9_16

    doi:10.1007/978- 2003

  23. [23]

    PAC-Bayesian Stochastic Model Selection

    David A. McAllester. “PAC-Bayesian Stochastic Model Selection”. In:Machine Learning51.1 (2003), pp. 5–21.doi:10.1023/A:1021764713219

    work page doi:10.1023/a:1021764713219 2003

  24. [24]

    Some PAC-Bayesian Theorems

    David A. McAllester. “Some PAC-Bayesian Theorems”. In:Machine Learning37.3 (1999), pp. 355–363. doi:10.1023/A:1007618624809.url:https://doi.org/10.1023/A:1007618624809

    work page doi:10.1023/a:1007618624809.url:https://doi.org/10.1023/a:1007618624809 1999

  25. [25]

    Ab initio Solution of the Many-Electron Schrödinger Equation with Deep Neural Networks

    David Pfau et al. “Ab initio Solution of the Many-Electron Schrödinger Equation with Deep Neural Networks”. In:Physical Review Research2.3 (Sept. 2020), p. 033429.doi:10.1103/PhysRevResearch. 2.033429.url:http://dx.doi.org/10.1103/PhysRevResearch.2.033429

    doi:10.1103/physrevresearch 2020

  26. [26]

    PAC-Bayesian Generalisation Error Bounds for Gaussian Process Classification

    Matthias Seeger. “PAC-Bayesian Generalisation Error Bounds for Gaussian Process Classification”. In: Journal of Machine Learning Research3 (Mar. 2003), pp. 233–269.doi:10.1162/153244303765208386. url:https://doi.org/10.1162/153244303765208386. 14

    doi:10.1162/153244303765208386 2003

  27. [27]

    PAC-Bayes–Empirical–Bernstein Inequality

    Ilya O. Tolstikhin and Yevgeny Seldin. “PAC-Bayes–Empirical–Bernstein Inequality”. In:Advances in Neural Information Processing Systems 26 (NeurIPS 2013). Ed. by C. J. Burges et al. Curran Associates, Inc., 2013, pp. 109–117.url:https://proceedings.neurips.cc/paper_files/paper/2013/file/ a97da629b098b75c294dffdc3e463904-Paper.pdf

    2013

  28. [28]

    General E(2)-Equivariant Steerable CNNs

    Maurice Weiler and Gabriele Cesa. “General E(2)-Equivariant Steerable CNNs”. In:Advances in Neural Information Processing Systems 32 (NeurIPS 2019).Ed.byH.Wallachetal.CurranAssociates,Inc.,2019, pp.14334–14345.url:https://proceedings.neurips.cc/paper/2019/hash/45d6637b718d0f24a237069fe41b0db4- Abstract.html

    2019

  29. [29]

    3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data

    Maurice Weiler et al. “3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data”.In:Advances in Neural Information Processing Systems 31 (NeurIPS 2018).2018,pp.10381–10392. doi:10.5555/3327546.3327700. arXiv:1807.02547.url:https://arxiv.org/abs/1807.02547

    arXiv 2018

  30. [30]

    Non-Vacuous Generalization Bounds at the ImageNet Scale: A PAC-Bayesian Com- pression Approach

    Wenda Zhou et al. “Non-Vacuous Generalization Bounds at the ImageNet Scale: A PAC-Bayesian Com- pression Approach”. In:Proceedings of the 7th International Conference on Learning Representations (ICLR 2019). OpenReview.net, May 2019.url:https://openreview.net/forum?id=BJgqqsAct7. 15

    2019