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

Fabio Anselmi; Nicola Aladrah

REVIEW 3 major objections 5 minor 15 references

This paper proves that PAC-Bayes generalization bounds can be computed on the quotient of parameter space by predictor-preserving symmetries, which removes a nonnegative KL term caused by redundant parameterizations and never worsens the ce

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

2026-08-01 15:27 UTC pith:Z6BGWMAS

load-bearing objection Quotient PAC-Bayes reduction is correct but standard; the geometry-induced prior misses the slice Jacobian and the experiments don't test the headline claim. the 3 major comments →

arxiv 2607.18422 v1 pith:Z6BGWMAS submitted 2026-07-20 cs.LG stat.ML

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

Nicola Aladrah , Fabio Anselmi This is my paper

classification cs.LG stat.ML

keywords PAC-Bayes boundsparameter symmetryquotient spaceimplicit biasKL divergencegeneralization boundsstochastic gradient descentoverparameterization

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.

This paper argues that PAC-Bayesian analysis should be performed on the quotient predictor space, where parameters that define the same predictor are identified. It proves that pushing a prior and posterior through the quotient map preserves the empirical and population Gibbs risks while reducing the KL complexity term by the conditional within-orbit divergence, so the certificate is never larger than the original parameter-space bound. To choose a prior on the quotient, the paper constructs an implicit-bias prior from the geometry of symmetry breaking, tilting a neutral reference prior by the volume of equivalent parameterizations; it shows the certificate improves exactly when this prior is closer in KL to the learned posterior. In experiments, this prior reduces mean quotient-space KL by 40.69% and the mean certificate by 21.40% in a Fourier-Hadamard model, and shows a smaller, scale-dependent improvement in Query-Key attention.

Core claim

The paper establishes two main results. First, Theorem 1: for any predictor-preserving parameter symmetry, applying the same PAC-Bayes inequality on the quotient space yields a certificate no larger than the parameter-space certificate, because the KL divergence decomposes as KL(Q_u||P_u) = KL(Q_b||P_b) - Delta_phi with Delta_phi nonnegative, while the risk terms are exactly preserved. Second, Proposition 3: a data-independent implicit-bias prior, defined as the neutral prior tilted by exp(-L_chi) where L_chi is half the log-determinant of the constraint Gram matrix of a transverse slice, improves the certificate precisely when the prior-improvement criterion E_{Q}[L_chi] + log Z_chi <= 0 ho

What carries the argument

The central objects are the quotient map phi that collapses each symmetry orbit to a single predictor, and the chain rule of relative entropy that identifies the removed KL contribution as the integral over the quotient of the conditional KL between fiber distributions. The implicit-bias prior is built from the symmetry-breaking geometry: a transverse slice chi and the Gram matrix G_chi of its constraint gradients. The stationary density of symmetry-reduced SGD, stated as Theorem 2 from the authors' companion work, contributes a factor det(G_chi)^{-1/2}, which becomes the tilting prior P_IB proportional to exp(-L_chi) P_0. The prior-improvement criterion Delta_KL = E_{Q}[L_chi] + log Z_chi d

Load-bearing premise

The load-bearing premise is Theorem 2 from the authors' companion work, which states that the stationary density of symmetry-reduced SGD is proportional to exp(-beta/sigma^2 L) det(G_chi)^{-1/2}; it is cited and not re-derived here, so if that theorem fails, the implicit-bias prior is an arbitrary hand-picked tilt and the reported KL reductions lack geometric grounding.

What would settle it

Two checks would settle the claims. For Theorem 1, compute exact KL values in a finite model with a single symmetry and verify that KL(Q_u||P_u) equals KL(Q_b||P_b) minus the conditional within-orbit KL. For the implicit-bias prior, simulate plain SGD on the Hadamard model and compare the empirical distribution of w = a⊙c on the slice to the predicted stationary density proportional to |w|^{-1/2} exp(-beta/sigma^2 L(w)); a systematic mismatch would falsify Theorem 2 and with it the prior's geometric interpretation.

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

If this is right

Any PAC-Bayes bound with a complexity term nondecreasing in KL can be applied on the quotient space with no loss in risk fidelity and no larger certificate, so symmetry reduction is a generally safe preprocessing step for PAC-Bayes analysis.
The exact reduction in KL is interpretable: it isolates predictor-level complexity from representational complexity, making explicit what portion of the bound is caused by redundant parameterizations.
If the learned posterior aligns with the geometry-induced prior (Delta_KL <= 0), the implicit-bias prior provably yields a tighter certificate than a neutral prior, without any change to the training objective.
The improvement is conditional and scale-dependent: in the Query-Key experiment, the implicit-bias prior helps for reference-prior scales at or above 1 in the tested grid, and hurts at the smallest scales.
The Fourier-Hadamard experiment shows the effect can be substantial, reducing the certificate by 21.40% on average, while the Query-Key experiment illustrates that the magnitude depends on how many symmetry directions the correction acts on.

Where Pith is reading between the lines

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

The quotienting argument is independent of the implicit-bias prior, so even a generic neutral prior on the quotient never worsens the bound; the practical advice to identify symmetries before applying PAC-Bayes holds regardless of how the prior is chosen.
The prior-improvement criterion depends only on E_Q[L_chi] and the normalizer Z_chi, so it can be computed cheaply and might serve as a model-selection signal for choosing among parameterizations, provided the choice is made independently of the certification sample.
The implicit-bias prior approximates the ideal posterior-matched prior; a natural extension is to invert the construction and design parameter symmetries whose induced L_chi matches a target posterior family, as the paper's conclusion hints.
The experiments fit an isotropic Gaussian posterior; the same criterion applies to structured posteriors, but the reported KL reductions should not be read as intrinsic to the Hadamard parameterization, since they depend on the posterior used.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

Quotient PAC-Bayes reduction is correct but standard; the geometry-induced prior misses the slice Jacobian and the experiments don't test the headline claim.

read the letter

The quotient part is the cleanest thing here. Pushing a prior and posterior through a predictor-preserving quotient map leaves Gibbs risks unchanged and decomposes the KL divergence into a quotient term plus a nonnegative within-orbit term. Props 1-2 and Thm 1 are correct; the proof is just the chain rule for KL in a disintegration. That is a genuinely useful framing, though not a new principle. Prop 3 is also an exact identity and the experiments correctly isolate the prior-dependent KL term.

The second contribution - the geometry-induced implicit-bias prior - does not hold up as stated. Theorem 2 is imported from the authors' own preprint and asserted without proof or independent verification. More concretely, the step from that theorem to Eq. (23) appears to drop the slice Jacobian. The stationary density in Theorem 2 is with respect to the induced surface measure on the slice. When you pull that back to the quotient coordinates, you must multiply by sqrt(det G_slice). In the Hadamard case det G_slice = 1/det G_chi, so the density with respect to Lebesgue is proportional to det(G_chi)^{-1}, not det(G_chi)^{-1/2}. The paper's P_IB uses exp(-L_chi) = det(G_chi)^{-1/2}, so it is a different measure. That is a load-bearing gap, not a minor one. The 40.69% KL reduction in the Fourier experiment is exactly what an arbitrary tilt toward small |w| would produce, so the observed numbers do not validate the geometric story.

The experiments also never test the quotienting claim itself; they compare two priors on the quotient. That is fine for Prop 3, but the abstract and title advertise quotienting as the main contribution and it remains empirically untested. The math says it works, so this is a presentation issue rather than a fatal one.

What the paper needs: a correct derivation of the slice density with the Jacobian, or an explicit statement that P_IB is a heuristic with no well-defined geometric origin; independent verification of Theorem 2; and public code/data.

Net: the quotient reduction is worth refereeing and could be a useful short paper on its own. The implicit-bias prior as presented is unsupported. I would not accept the central claim about geometry-induced priors until the measure theory is fixed. But this deserves a serious referee, not a desk rejection, because catching this required a careful look and the quotient idea has value.

Recommendation: send to peer review, with a clear request for the Jacobian calculation and for a direct test of the quotient bound.

Referee Report

3 major / 5 minor

Summary. This paper proposes to run PAC-Bayes analysis on the quotient of parameter space by predictor-preserving symmetries. It proves (Props 1–2, Thm 1) that pushing prior and posterior through the quotient map preserves empirical and population Gibbs risk while reducing the KL term by the conditional within-orbit divergence. It then constructs an 'implicit-bias prior' P_IB by tilting a reference prior with the geometric factor det(G_chi)^{-1/2} from a companion paper's Theorem 2, and derives an exact prior-improvement criterion (Prop 3). Experiments on Fourier–Hadamard and Query–Key attention compare neutral and implicit-bias priors on the quotient, reporting a 40.69% KL reduction in the Fourier case.

Significance. The quotient decomposition is clean and appears correct under the stated standard-Borel and absolute-continuity assumptions; it gives a general way to separate representational complexity from predictor-level complexity. Prop 3 is an exact algebraic identity and provides a falsifiable, posterior-dependent criterion for prior comparison. These are valuable contributions. However, the second contribution—the geometric prior—has a serious measure-theoretic gap in translating the slice density to a prior on the quotient, and the experiments do not validate the quotienting theorem. The headline empirical numbers therefore do not yet support the geometric-implicit-bias interpretation.

major comments (3)

[§3, Eq. (23) and §3.1, Eq. (33)] The construction of P_IB does not follow from Theorem 2. Theorem 2 gives a density with respect to the induced surface measure on S_chi, not with respect to Lebesgue measure on U. To express it as a measure on U—the space on which P0^u and the PAC-Bayes KL are defined—one must include the slice volume factor. In the Hadamard case, dσ_slice(w) = ∏_i (2|w_i|)^{-1/2} dw_i, so the correct pullback of the stationary density is proportional to det(G_chi)^{-1} = ∏(2|w_i|)^{-1}, not det(G_chi)^{-1/2}. This density is not normalizable (∫_0^ε w^{-1} dw = ∞), so Eq. (33)'s prior is an arbitrary tilt, not the geometry-induced prior claimed. The 40.69% KL reduction in Table 1 may be an artifact. This is a load-bearing gap for the second contribution.
[§4 (Table 1) and §5] The experiments do not test Theorem 1. Table 1 compares two priors on the quotient (neutral vs implicit-bias) for the same quotient posterior; no parameter-space KL(Qb||Pb) is computed for the Hadamard model, so the claimed reduction from parameter-space to quotient-space certificate is not demonstrated empirically. The conclusion admits that the experiments evaluate prior comparison, but the abstract's 'We test this prediction' overstates the support. Add a parameter-space baseline or clearly label the quotient theorem as proof-only.
[§3, Theorem 2] P_IB is constructed by citing Theorem 2 of [1], which is not proved or independently verified in this manuscript. Because that theorem is the sole source of the 'implicit bias' interpretation and its translation to a measure on U already involves nontrivial Jacobian choices, the paper should supply a self-contained derivation of the density on U (including the slice volume factor) rather than taking Theorem 2 at face value. Without this, the geometric meaning of P_IB is unsupported.

minor comments (5)

[Eq. (4)] The complexity term constant should be checked against the cited McAllester bound; as written, log n + 2 appears slightly different from standard forms and the exact provenance should be given.
[Appendix B.2, Eq. (B.5)] The experiments use the regularized loss L_chi,ε with ε=10^{-3}, but the exact prior in Eq. (33) is defined with L_chi = 1/2 log(2|w_i|). State explicitly that the experiments test a regularized version of the prior and how ε affects the normalizing constant.
[§3.1, Eq. (28)] The branch convention a_i>0 does not fully specify the slice: c_i = ±a_i gives two components for each coordinate. Clarify how the quotient identification handles signs and why this does not affect the computed Gram matrix.
[§1 and §3] The 'ideal posterior-matched prior' is referenced but never defined formally, and no approximation bound is given. Either define it precisely or remove the claim that P_IB approximates it.
[Data and Code Availability] The statement that data and code are 'available from the authors upon reasonable request' is not sufficient for reproducibility. A public repository would greatly strengthen the paper.

Circularity Check

0 steps flagged

No significant circularity: the quotient KL decomposition and prior-improvement criterion are derived in-text, and the implicit-bias prior is a fixed geometry-defined quantity compared with a separately fitted posterior.

full rationale

The derivation chain is self-contained for its central results. Proposition 2 and Theorem 1 derive KL(Q_u||P_u) = KL(Q_b||P_b) - Delta_phi by the standard relative-entropy chain rule under disintegration (Eqs. A.4-A.6); this is an identity, not an assumption of the conclusion. Proposition 3 is also algebraic: with dP_IB/dP0 = e^{-L_chi}/Z_chi, the computation gives Delta_KL = E_Q[L_chi] + log Z_chi (Eqs. A.12-A.17), and the condition Delta_KL <= 0 is the definition of 'P_IB has smaller KL' rearranged; no fitted parameter is relabeled as a prediction. The experiments hold the fitted posterior Q and empirical Gibbs risk fixed and compare a fixed geometry-defined prior P_IB with a neutral prior P0, so the reported 40.69% KL reduction is not forced by construction. The construction of P_IB does rely on Theorem 2 of the authors' prior work [1], which is stated but not re-proved here; however that theorem is a parameter-free geometric statement about an SGD stationary density, does not itself assert the PAC-Bayes improvement, and is used as a stated external premise rather than an output-equivalent input. A possible missing slice-Jacobian factor is a correctness concern about the application of that theorem, not a circularity. No load-bearing step equates its conclusion with its input by definition.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 1 invented entities

The paper's logical skeleton is mostly self-contained (quotient KL reduction, Proposition 3), but the prior construction leans on Theorem 2 from the authors' own earlier preprint [1], which is neither proven nor independently verified here. The reference-prior scale s0 and the regularization ε are hand-chosen and affect the experimental results, though they are swept/reported rather than fitted to maximize the headline numbers.

free parameters (3)

s0 (reference prior scale) = swept 0.25–20; headline at s0=20
Prior width for the neutral isotropic Gaussian P0; the comparison of P0 vs PIB is scale-dependent, and the headline improvements are at s0=20.
epsilon (regularization in L_chi) = 1e-3
Added inside the log in the regularized implicit-bias loss (Eq. B.5, B.10) to keep the prior integrable; affects the value of L_chi and hence ΔKL.
posterior variance estimator = isotropic scalar variance
The posterior is restricted to an isotropic Gaussian fitted to trajectory snapshots; the certificate depends on this posterior choice.

axioms (5)

domain assumption Parameter-space symmetry: f_{g·b}=f_b and a measurable predictor map on U exists (Eq. 5–6)
Required for risks to be constant on orbits; assumed for the group actions considered.
standard math B^0 and U are standard Borel spaces and Q_b ≪ P_b (Prop 2)
Needed for disintegration and absolute continuity to ensure KL decomposition.
ad hoc to paper Theorem 2 of [1]: the stationary density of symmetry-reduced SGD is ∝ exp(-β/σ² L) det G^{-1/2} (Eq. 19)
The implicit-bias prior is built from this self-cited result; it is not proven or independently verified in this paper.
domain assumption The group acts smoothly, freely, properly and the slice S_chi intersects each orbit once (Theorem 2)
Assumed for the geometric construction to give a well-defined volume factor; inherited from [1].
standard math Normalizability 0<Z_chi<∞ (Eq. 22)
Assumed so P_IB is a probability measure; checked numerically in the experiments.

invented entities (1)

ideal posterior-matched prior no independent evidence
purpose: Conceptual benchmark that minimizes KL by using training data; used to motivate P_IB as an approximation.
The paper states it is inadmissible as a PAC-Bayes prior; no evidence is provided that P_IB approximates it.

pith-pipeline@v1.3.0-alltime-deepseek · 12846 in / 13426 out tokens · 115619 ms · 2026-08-01T15:27:21.258249+00:00 · methodology

0 comments

read the original abstract

Overparameterized models often have continuous parameter symmetries, so different parameters define the same predictor. We show that PAC--Bayesian analysis should be performed on the quotient predictor space: pushing a prior and posterior to the quotient preserves the empirical and population Gibbs risks while removing the nonnegative KL contribution caused solely by how the two distributions differ among parameterizations of the same predictor. Quotienting alone does not determine which prior to use. We construct a canonical choice of one parameterization for each predictor and account for the geometric volume of its equivalent parameterizations. This transforms a neutral reference prior into a data-independent prior that reflects the model's implicit bias. It approximates the ideal but inadmissible posterior-matched prior, which would minimize the KL term by depending on the training data. The resulting certificate is tighter exactly when this geometry-induced prior has smaller KL divergence from the learned quotient posterior than the neutral prior. We test this prediction in Fourier regression with a Hadamard parameterization and in Query-Key attention, using ordinary SGD without an explicit regularizer. The implicit-bias prior reduces the mean quotient-space KL by \(40.69\%\) and the mean PAC--Bayes certificate by \(21.40\%\) in the Fourier-Hadamard experiment. The smaller, prior-scale-dependent improvement in Query-Key attention confirms the predicted conditional nature of the effect.

discussion (0)

Reference graph

Works this paper leans on

15 extracted references · 6 linked inside Pith

[1]

Understandingandinversedesign of implicit bias in stochastic learning: a geometric per- spective

NicolaAladrahetal.“Understandingandinversedesign of implicit bias in stochastic learning: a geometric per- spective”. In:arXiv preprint arXiv:2601.06597(2026)

Pith/arXiv arXiv 2026
[2]

Parameter symmetry potentially unifies deep learning theory

Liu Ziyin et al. “Parameter symmetry potentially unifies deep learning theory”. In:arXiv preprint arXiv:2502.05300(2025)

Pith/arXiv arXiv 2025
[3]

Some pac-bayesian theorems

David A McAllester. “Some pac-bayesian theorems”. In:Proceedings of the eleventh annual conference on Computational learning theory. 1998, pp. 230–234

1998
[4]

A PAC-Bayesian tutorial with a dropout bound

David McAllester. “A PAC-Bayesian tutorial with a dropout bound”. In:arXiv preprint arXiv:1307.2118 (2013)

Pith/arXiv arXiv 2013
[5]

PAC-Bayesian stochastic model selection

David A McAllester. “PAC-Bayesian stochastic model selection”. In:Machine Learning51.1 (2003), pp. 5–21

2003
[6]

Onthebenefitsofinvarianceinneural networks

ClareLyleetal.“Onthebenefitsofinvarianceinneural networks”. In:arXiv preprint arXiv:2005.00178(2020)

Pith/arXiv arXiv 2005
[7]

PAC-Bayesian generalisation error boundsforGaussianprocessclassification

Matthias Seeger. “PAC-Bayesian generalisation error boundsforGaussianprocessclassification”.In:Journal of machine learning research3.Oct (2002), pp. 233– 269

2002
[8]

PAC-Bayesian supervised classifica- tion: the thermodynamics of statistical learning

Olivier Catoni. “PAC-Bayesian supervised classifica- tion: the thermodynamics of statistical learning”. In: arXiv preprint arXiv:0712.0248(2007)

Pith/arXiv arXiv 2007
[9]

On the properties of variational approximations of Gibbs posteriors

Pierre Alquier, James Ridgway, and Nicolas Chopin. “On the properties of variational approximations of Gibbs posteriors”. In:Journal of Machine Learning Research17.236 (2016), pp. 1–41.url:http://jmlr. org/papers/v17/15-290.html. 7

2016
[10]

Improved PAC-Bayesian bounds for linear regression

Vera Shalaeva et al. “Improved PAC-Bayesian bounds for linear regression”. In:Proceedings of the AAAI Conference on Artificial Intelligence. Vol. 34. 04. 2020, pp. 5660–5667

2020
[11]

Symmetry in neural network parameter spaces

Bo Zhao, Robin Walters, and Rose Yu. “Symmetry in neural network parameter spaces”. In:arXiv preprint arXiv:2506.13018(2025)

arXiv 2025
[12]

Geometry of the loss landscape in overparameterized neural networks: Symmetries and invariances

Berfin Simsek et al. “Geometry of the loss landscape in overparameterized neural networks: Symmetries and invariances”. In:International Conference on Machine Learning. PMLR. 2021, pp. 9722–9732

2021
[13]

Improving convergence and general- ization using parameter symmetries

Bo Zhao et al. “Improving convergence and general- ization using parameter symmetries”. In:International Conference on Learning Representations. Vol. 2024. 2024, pp. 55008–55035

2024
[14]

On the symmetries of deep learning models and their internal representations

Charles Godfrey et al. “On the symmetries of deep learning models and their internal representations”. In: Advances in Neural Information Processing Systems35 (2022), pp. 11893–11905

2022
[15]

Symmetries in PAC-Bayesian Learning

Armin Beck and Peter Ochs. “Symmetries in PAC-Bayesian Learning”. In:arXiv preprint arXiv:2510.17303(2025). A Proofs This appendix proves the quotient-risk identities, the KL de- composition along symmetry orbits, and the prior-comparison identity used in the main text. The proofs are presented in the order of the corresponding propositions and theorem. A...

Pith/arXiv arXiv 2025

[1] [1]

Understandingandinversedesign of implicit bias in stochastic learning: a geometric per- spective

NicolaAladrahetal.“Understandingandinversedesign of implicit bias in stochastic learning: a geometric per- spective”. In:arXiv preprint arXiv:2601.06597(2026)

Pith/arXiv arXiv 2026

[2] [2]

Parameter symmetry potentially unifies deep learning theory

Liu Ziyin et al. “Parameter symmetry potentially unifies deep learning theory”. In:arXiv preprint arXiv:2502.05300(2025)

Pith/arXiv arXiv 2025

[3] [3]

Some pac-bayesian theorems

David A McAllester. “Some pac-bayesian theorems”. In:Proceedings of the eleventh annual conference on Computational learning theory. 1998, pp. 230–234

1998

[4] [4]

A PAC-Bayesian tutorial with a dropout bound

David McAllester. “A PAC-Bayesian tutorial with a dropout bound”. In:arXiv preprint arXiv:1307.2118 (2013)

Pith/arXiv arXiv 2013

[5] [5]

PAC-Bayesian stochastic model selection

David A McAllester. “PAC-Bayesian stochastic model selection”. In:Machine Learning51.1 (2003), pp. 5–21

2003

[6] [6]

Onthebenefitsofinvarianceinneural networks

ClareLyleetal.“Onthebenefitsofinvarianceinneural networks”. In:arXiv preprint arXiv:2005.00178(2020)

Pith/arXiv arXiv 2005

[7] [7]

PAC-Bayesian generalisation error boundsforGaussianprocessclassification

Matthias Seeger. “PAC-Bayesian generalisation error boundsforGaussianprocessclassification”.In:Journal of machine learning research3.Oct (2002), pp. 233– 269

2002

[8] [8]

PAC-Bayesian supervised classifica- tion: the thermodynamics of statistical learning

Olivier Catoni. “PAC-Bayesian supervised classifica- tion: the thermodynamics of statistical learning”. In: arXiv preprint arXiv:0712.0248(2007)

Pith/arXiv arXiv 2007

[9] [9]

On the properties of variational approximations of Gibbs posteriors

Pierre Alquier, James Ridgway, and Nicolas Chopin. “On the properties of variational approximations of Gibbs posteriors”. In:Journal of Machine Learning Research17.236 (2016), pp. 1–41.url:http://jmlr. org/papers/v17/15-290.html. 7

2016

[10] [10]

Improved PAC-Bayesian bounds for linear regression

Vera Shalaeva et al. “Improved PAC-Bayesian bounds for linear regression”. In:Proceedings of the AAAI Conference on Artificial Intelligence. Vol. 34. 04. 2020, pp. 5660–5667

2020

[11] [11]

Symmetry in neural network parameter spaces

Bo Zhao, Robin Walters, and Rose Yu. “Symmetry in neural network parameter spaces”. In:arXiv preprint arXiv:2506.13018(2025)

arXiv 2025

[12] [12]

Geometry of the loss landscape in overparameterized neural networks: Symmetries and invariances

Berfin Simsek et al. “Geometry of the loss landscape in overparameterized neural networks: Symmetries and invariances”. In:International Conference on Machine Learning. PMLR. 2021, pp. 9722–9732

2021

[13] [13]

Improving convergence and general- ization using parameter symmetries

Bo Zhao et al. “Improving convergence and general- ization using parameter symmetries”. In:International Conference on Learning Representations. Vol. 2024. 2024, pp. 55008–55035

2024

[14] [14]

On the symmetries of deep learning models and their internal representations

Charles Godfrey et al. “On the symmetries of deep learning models and their internal representations”. In: Advances in Neural Information Processing Systems35 (2022), pp. 11893–11905

2022

[15] [15]

Symmetries in PAC-Bayesian Learning

Armin Beck and Peter Ochs. “Symmetries in PAC-Bayesian Learning”. In:arXiv preprint arXiv:2510.17303(2025). A Proofs This appendix proves the quotient-risk identities, the KL de- composition along symmetry orbits, and the prior-comparison identity used in the main text. The proofs are presented in the order of the corresponding propositions and theorem. A...

Pith/arXiv arXiv 2025