A Symmetry-Aware Exploration of Bayesian Neural Network Posteriors
Authors: Olivier Laurent
Emanuel Aldea
Gianni Franchi
Venue: ICLR 2024 poster (2024)
Keywords: bayesian posterior
bayesian neural networks
dataset
ensembles
uncertainty quantification
weight-space symmetries
computer vision
Primary area: general machine learning (i.e., none of the above)
TL;DR: We explore the Bayesian posterior of modern deep neural networks, highlighting the impact of weight-space symmetries.

Abstract: The distribution of modern deep neural networks (DNNs) weights -- crucial for uncertainty quantification and robustness -- is an eminently complex object due to its extremely high dimensionality. This paper presents one of the first large-scale explorations of the posterior distribution of deep Bayesian Neural Networks (BNNs), expanding its study to real-world vision tasks and architectures. Specifically, we investigate the optimal approach for approximating the posterior, analyze the connection between posterior quality and uncertainty quantification, delve into the impact of modes on the posterior, and explore methods for visualizing the posterior. Moreover, we uncover weight-space symmetries as a critical aspect for understanding the posterior. To this extent, we develop an in-depth assessment of the impact of both permutation and scaling symmetries that tend to obfuscate the Bayesian posterior. While the first type of transformation is known for duplicating modes, we explore the relationship between the latter and L2 regularization, challenging previous misconceptions. Finally, to help the community improve our understanding of the Bayesian posterior, we release the first large-scale checkpoint dataset, including thousands of real-world models, along with our code.

Published as a conference paper at ICLR 2024
A SYMMETRY-AWARK EXPLORATION OF BAYESIAN NEURAL NETWORK POSTERIORS
Olivier Laurent," Emanuel Aldea' & Gianni Franchi-i
SATIE, Paris-Saclay University,' U2IS, ENSTA Paris, Polytechnic Institute of Paris
ABSTRACT
The distribution modern deep neural networks (DNNs) weights crucial for uncertainty quantification and robustness is eminently complex object due extremely high dimensionality. This paper presents one of the first large-scale explorations of the posterior distribution of deep Bayesian Neural Networks (BNNs), expanding its study to real-world vision tasks and architectures. Specifi- cally, investigate the optimal approach for approximating the posterior, analyze the connection between posterior quality and uncertainty quantification, delve into the impact of modes the posterior, and explore methods for visualizing the posterior. Moreover, we uncover weight-space symmetries as a critical aspect for understanding the posterior. To this extent, we develop an in-depth assessment of the impact of both permutation and scaling symmetries that tend to obfuscate the Bayesian posterior. While the first type of transformation is known for duplicating modes, we explore the relationship between the latter and L. regularization, challenging previous misconceptions. Finally, to help the community improve our understanding of the Bayesian posterior, we release the first large-scale checkpoint dataset, including thousands rorldworld models, along with our code.
3
Original Posterior
After Scaling Remnvvl
After Scaling & Peroutation Remova
Figure 1: Weight-space symmetries greatly impact the estimated Bayesian posterior. Permuta- tion symmetries clearly increase the number of modes of the posterior distribution in the case of the last layer of 2-hidden neuron perceptron, detailed in Section 3.1.
INTRODUCTION
Despite substantial advancements in deep learning, Deep Neural Networks (DNNs) remain black box models. Various studies have sought to explore DNN loss landscapes (Li et al., 2018; Fort & Jastrzebski, 2019; Fort & Scherlis, 2019; Liu et al., 2022) achieve deeper understanding of these models. Recent works have, for instance, unveiled the interconnection of the modes obtained with Stochastic Gradient Descent (SGD) via narrow pathways that link pairs of modes, or through tunnels that connect multiple modes simultaneously (Garipov et al., 2018; Draxler et al.,
corresponding author gianni franchi @ensta connsa-ppriss

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2018). This mode connectivity primarily arises from scaling and permutation invariances, which imply that numerous weights can represent the same exact function (e.g., Entezari et al. (2022)). Several studies have delved into the relationship between these symmetries and the characteristics of the loss landscape (Neyshabur et al., 2015; Brea et al., 2019; Entezari et al., 2022). Our work investigates the connections between these symmetries and the distribution of DNN weights, crucial aspect for uncertainty quantification. These connections are highlighted in Figure 1.
Uncertainty quantification plays a pivotal role in high-stakes industrial applications such as au- tonomous driving (Levinson et al., 2011; McAllister et al., 2017; Sun et al., 2019) whhre reliable predictions and informed decision-male making are paramount. In such critical domains, understanding and effectively managing uncertainties, particularly the model-related epistemic uncertainties (Hora, 1996) arising from incomplete knowledge, is essential. Amongst the various methods introduced to address these challenges, Bayesian Neural Networks (BNNs) (Tishby et al., 1989) offer a principled and theoretically sound approach. BNNs quantify uncertainty by probabilistically modeling beliefs about parameters and outcomes (Tishby et al., 1989; Hinton & Van Camp, 1993). However, this perspective faces significant hurdles when applied to deep learning, primarily related to scalabil- ity (Izmailov et al., 2021) and the precision of approximations (MacKay, 1995). Due to their very high dimension, BNNs struggle to estimate the posterior distribution, i.e., the probability density that any set of model parameters/hyyotthsii: w generated the observed data D with given prior.
Diverging from methods such as the Maximum Likelihood Estimate or Maximum Posteriori (also Tishby al. (1989)), which we typically derive through gradient descent optimization of cross-entropy (with L2 regularization for the latter), BNNs assign a probability to each possible model (or hypothesis) and offer predictions considering the full extent of possible models. In mathematical terms, denoting the target as y. the input vector as œ, and the weight space as Q, we can express this approach through the following intractable formula, often referred to as the marginalization on the parameters of the model (Tishby et al., 1989; Rasmussen et al., 2006):
ply xz D) = / ply x w))( DDdw.
(1)
WER
The posterior distribution (D) assumes a central and arguably the most critical role in BNNs and many successful methods for quantifying uncertainty can be viewed as attempts to approximate this posterior, each with its own trade-offs in terms of accuracy and computational efficiency, as illustrated in previous research (Blundell et al., 2015; Gal & Ghahramani, 2016; Lakshminarayanan et al., 2017). While prior work (Kuncheva & Whitaker, 2003; Fort et al., 2019; Ortega et al., 2022) has established the importance of achieving diversity in the sampled DNNs drawn from the posterior -particularly when dealing with uncertain input data permutation and scaling symmetries amongst hidden units in neural networks may lead to an increased number of local minima (Zhao et al., 2023) with diversity. In the context of BNNs, this phenomenon could result in proliferation of functionally equivalent modes within the posterior distribution reducing the diversity within the inevitably limited number of samples, and degrading the quality of the uncertainty estimates.
This paper delves into the impact of weight symmetries on the posterior distribution. While there have been numerous efforts to visualize the loss landscape, we explore the possibility of conducting similar investigations for the posterior distribution. Additionally, we introduce a protocol for assess- ing the quality of posterior estimation and examine the relationship between posterior estimation and the accuracy of uncertainty quantification. Specifically, our contributions are as follows:
(1) We build a new mathematical formalism to highlight the different impacts of the permutation and scaling symmetries on the posterior and uncertainty estimation in DNNs. Notably, we explain the seeming equivalence of the marginals in Figure 1. We also perform the first in-depth exploration of the existence of scaling symmetries and their overlooked effect.
(2) We evaluate the quality of various methods for estimating the posterior distribution on real-worle applications using the Max- imum Mean Discrepancy, offering a practical benchmark to assess their performance in capturing uncertainty. (3) We release Checkpoints, a new dataset including the weights of thousands of models across various computer vision tasks and architectures, ranging from MNIST to Tiny This dataset is intended to facilitate further exploration and collaboration in the field of uncertainty in deep learning. (4) Our investigation delves into the proliferation of modes in the context of posterior symmetries and exhibits the capacity of ensembles to converge toward functiniionllly equivalent modes. Furthermore, we discuss the influence of symmetries in the training process.
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2 RELATED WORK
Epistemic uncertainty, Bayesian inference, and posterior. Epistemic uncertainty (Hora, 1996; Hüllermeier & Waegeman, 2021) plays a crucial role in accurately assessing predictive model reliability. However and despite ongoing discussions estimating this uncertainty remains challenge. BNNs (Goan & Fookes, 2020) predominantly shape the landscape of methodologies that tackle epistemic uncertainties (Gawlikowski et al., 2023). Given the complexity of dealing with posterior distributions, these approaches have mostly been tailored for enhanced scalability.
For instance, Hernández-Lobato & Adams (2015) introduced an efficient probabilistic backpropa- gation, and Blundell et al. (2015) developed BNNs by backpropagation to learn diagonal Gaussian distributions with the reparametrization trick. Similarly, Laplace methods (MacKay, 1992) estimate the posterior distribution, thanks to an approximation of the local curvature of the loss. They often focus on the final layer (Ober& & Rasmussen, 2019; Watson et al., 2021), again for scalability.
On different approach, Monte Carlo Dropout, introduced by Gal & Ghahramani (2016) and Kingma et al. (2015), is a framework that, applied to fully-connected layers, models the posterior as a mixture of Dirac distributions. Broadening the spectrum, deep ensembles (Lakshminarayanan et al., 7). arguably along with their more computationally efficient alternatives (Wen et al., 2019; Maddox et al., 2019; Franchi al., 2020; 2023; Havasi et al., 2021; Laurent et al., 2023), have been interpreted by Wilson & Izmailov (2020) as Monte Carlo estimates of Equation 1.
Markov-chain-based Bayesian posterior estimation. Neal et al. (2011) introduced Hamiltonian Monte Carlo (HMC)- - based on Monte Carlo Markov Chains (MCMC) - as an accurate method for estimating distributions, but its application large-scale problems, such as the posterior of modern DNNs, remains challenging due to its exceptionally high computational demands.
In response to these challenges, stochastic approximations of MCMC have gained attention for their ability to provide computationally feasible solutions. A prominent example is the stochastic version of Langevin dynamics (Roberts Tweedie, 1996) by Welling & Teh (2011). By adding noise into the dynamics, stochastic Langevin allows for more practical implementation on large datasets. In addition, other stochastic gradient-based methods have been introduced to improve the efficiency of MCMC sampling. Chen et al. (2014) presented Stochastic Gradient Hamiltonian Monte Carlo (SGHMC), and Zhang et al. (2020) designed C-SGLD and C-SGHMC (Cyclic Stochastic Gradient Langevin Dynamics), introducing controlled noise via cyclic preconditioning.
While stochastic approximation methods offer computational convenience, they come with the trade-off of slowing down the convergence and potentially introducing bias into the resulting inference (Bardenet et al., 2017: Zou & Gu, 2021). As such, the suitability of these approaches depends the specific application and the level of acceptable bias in the analysis.
Izmailov et al. (2021) estimated the Bayesian posterior scaling full-batcl HMC to CIFAR- 10 thanks to 512 TPUv3. While we also estimate posteriors, select another more scalable method supported and compared to HMC in Appendix B. Furthermore, we bring a novel focus that remained mostly uncharted: theoretically and empirically quantifying the impact of symmetries the posterior.
Symmetries in neural networks. The seminal work from Hecht-Nielsen (1990) established a foundational understanding by investigating permutation symmetries and setting lower bound on symmetries in multi-layer perceptrons. Albertini et al. (1993) extended this work and studied flip-sign symmetries in neural networks with odd activation functions. These works were further generalized to a broader range of activation functions by Kūrková & Kainen (1994), who suggested symmetry removal streamline evolutionary algorithms.
Recent advancements have generalized symmetries to modern neural architectures. Neyshabur et al. (2015) explored the scaling symmetries that arise in architectures containing non-negative homogeneous activation functions, including Nair & Hinton (2010)'s ubiquitous Rectified Linear Unit (ReLU). This perspective extends our understanding of symmetries to ReLU-powered archi- tectures, e.g. AlexNet (Krizhevsky et al., 2012), and ResNet architectures (He et al., 2016). This paper focuses on scaling and permutation symmetries, but other works, such as Rolnick & Kording (2020); Grigsby et al. (2023), unveil less apparent symmetries. Closer our work, Wiese et al. (2023) demonstrated that taking weight-space symmetries into account could reduce the support of the Bayesian posterior and improve MCMC posterior estimation.
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3 SYMMETRIES INCREASE THE COMPLEXITY OF BAYESIAN POSTERIORS
We study scales and permutations, the most influential weight-space symmetries and their prop- erties related to the Bayesian posterior. Since there is no posterior without prior, we advise the reader that we will work on maxima a posteriori and take the most common weight-prior amongst practitioners, the Gaussian prior on the weights, which is equivalent to 1.2 regularization. We detail the role of the priors in Appendix D.4. Now, let us start with a definition: weight-space symmetries transform the parameters of the neural networks while keeping the networks functionally invariant,
Definition 3.1. Let fw be a neural network of parameters w taking n-dimensional vectors inputs. We say that the transformation T modifying w is a weight-space symmetry operator, iff
f(( fw: or Vo R", fT( = fw (x:).
With the notation fT( (z), we apply the symmetry operator T on the weights w, resulting set of modified weights. In the following, we show that scaling and permutation symmetries have different impacts on the posterior of neural networks. They can, for instance, complicate Bayesian posteriors, creating artificial functionally equivalent modes.
3.1 AN INTRODUCTORY EXAMPLE OF ARTIFICIAL SYMMETRY-DRIVED POSTERIOR MODES
To illustrate the considerable impact of symmetries the Bayesian posterior, showcase small- scale classification example in Figure We generate this example by training wwo-hddeen-neue neuron perceptrons on linearly separable data. The figure presents the estimation of bivariate marginal of the posterior the output weights with 10,000 independently-trainn successively removing scaling (center) and then permutation symmetries (right). This figure shows that the scaling symmetries seem to disperse the points from the modes and that the most important mode duplicated due the (here, unique) permutation symmetry, which symmetrizes the graph. We detail this toy experiment in Appendix A. In the following, we develop a new mathematical framework tailored help understand the impact of these symmetries on the posterior, devise mathematical insights explaining these intuitions, and explore more empirical dimensions.
samples (left) when
3.2 BACKGROUND AND DEFINITIONS
The full extent of this formalism (including sketches of proofs, other definitions, properties, and propositions) developed in Appendix E. Here, we summarize the minimal information to under- stand the impact the Bayesian posterior of the two main symmetries scales and permutations. This part summarizes the most important results for multi-layer perceptrons, but we provide leads for generalizing our results modern DNNs such convolutional residual networks in Appendix D. 1.
3.3 SCALING SYMMETRIES
For clarity, the following definitions and properties are provided for two-layer fully connected perceptrons, without loss of generality. We first denote the line-wise and column-wise as v and D>, respectively (see Definition E.2). Given that the rectified linear unit is non-negative homogeneous i.e., for all non-negative X, r(Ax) Ar(z)= - we have the following core property for scaling symmetries (Neyshabur et al., 2015), trivially extendable to additive biases:
Property 3.1. For all E RR-XX w E RRmxn, A E (R>a)"
E (A
x r wx) = x r(wx).
Denoting the transformation of Equation 3 by T. in the case of a two-layer perceptron the core property directly follows, with the set of parameters A {A}:
Property 3.2. For any usual neural network with non-negative homogenous activations fwr the scaling operation T, with a set of non-negative parameters is a symmetry, i.c. Var R" (T)) (xx
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3.4 PERMUTATION SYMMERRIES
We also present an intuitive formalism for permutation symmetries, multiplying the weights by per- mutation matrices. For two-layer perceptrons, with Pm. the of permutation matrices, have that: Property 3.3. For all 0 E IR*x". [Rmxx,, and permulation matrices E Pm.
Vn E R", ON x ( wa)
r(wæ)).
The left term of Equation is the definition of the permutation symmetry operator of parameter == { for a network including two layers. In general, we have the following property:
Property 3.4. For any usual neural network Sw. the permutation operation Tp with set of parameters II is a symmetry, i.e., Vaz E RR" (T)lw.t) = (x)
3.5 THE BAYESIAN POSTERIOR AS MIXTURE OF DISTRIBUTIONS
With this formalism, we can establish the following proposition, a formalization and extension of Kurle et al. (2021), clarifying the impact of weight-space symmetries the Bayesian posterior.
Proposition 1. Define fw a neural network and fa its corresponding identifiable model network transformed for having sorted unit-normed neurons. Let us also denote I and & the sets of permutation sets and scaling sets, respectively, and 5t the random variable of the sorted weights with unit norm. The Bayesian posterior of a neural network Iw trained with stochustic grudient descent can be expressed as a continuous mixture of a discrete mixture:
= = w D)= f |0| 1 = (T) II),A D)dA.
(5)
AEA
IEn
Proposition 1 provides expression of the Bayesian posterior that highlights the redundancy of the resulting distribution, explaining the symmetry in Figure 1 (left). Interestingly, a direct corollary of this formula is that layer-wise, all marginal inbound posteriors are identical. This has practical consequences: in Appendix B, we show that HMC-based posterior estimation breaks this corollary. In Equation 5, the permutations play transparent role, being independent of w (except for strongly quantized spaces for w that we leave for future works). On the other hand, the part played by scaling symmetries is more complex, and diseuss their impact in the following section.
3.6 ON THE EFFECTIVE IMPACT SCALING SYMMETRIES
While the equiprobability of permutations in Equation 5 leads to simple balanced mixture of [M]] permuted terms, we have such result on scaling symmetries since the standard L2-regularized loss is not invariant to scaling symmetries (and the initialization is not "uniform").. This absence of invari- ance obscures the impacts of scaling symmetries, which mostly remains be addressed, although the "reduction" of their effeec due regularization the best our knowledge, we provide the first analysis of the tangibility of scaling symmetries and their impact on the Bayesian posterior. With this objective in mind, we define the following problem.
mentioned in, e.g., Godfrey et al. (2022). To
Definition 3.2. Let fw be neural network and w its weights without the biases. We define the scaled network representation cost problem (or the scaled repressttrrrestniiooonn problem) as the minimiza- tion of the 12-regularization term of fw (the "mass") under scaling transformations. In other words,
m
min IT (w,
This problem restriction of the representation cost minimization, egg. (Jacot, 2022) has interesting properties. Notably, we show in Appendix E.5 that the scaled-representation problem is log-log strictly convex (Boyd & Vandenberghe, 2004; Agrawal al., 2019):
Proposition 2. The scaled-representation problem is log-log strictly convex: it is equivalent strictly convex problem on RIAl and admits a single global minimum attained at A".
follows from Proposition 2 that, if not already optimal at convergence, there is an infinite number of equivalent networks networks with training loss lower than the original network. We put the proposition into practice in Figure 2 using trained OptuNets (see Appendix C.2.1): we measure their mass distribu- tion and compare them to the masses of the optimal networks found with convex optimization. The
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Miniman L2 inxes Bgg netwwrkk
300
Probobiitty density
0.03
Max. weight opt.
200-
6.07-
100
6,01
Coee
2nd Coer
Tireur
0.00
Network L2 mass at convergence
Excess of mass (56)
Max. weight at convergence
Figure 2: OptuNets trained with weight decay never converge to the minimum scaled rep- resentation. We also note that the maxima of the weights of scaled OptuNets at the minimum scaled representation referred to as "at opt." in (right) tend to be greater for layers with fewer parameters than in the original networks.
effect of scaling symmetries remains even with weight decay: neural networks seem to persist in be- ing subject scaling symmetries as shown in Figure (left). Figure 2 (center) depicts that the ratios of the mass at convergence on the minimum representation are non-negligible, the converged net- works being consistently heavier. Figure 2 (right) displays the values of the largest elements of each layer: the minimization of the mass tends to increase the heaviest weights of the layers with fewer pa- rameters (here, the convolutional layers) but does not seem to promote unfeasible values. Finally, we provide loss landscape interpretation of this property in Appendix D.5 and explain the inability of DNNs to converge to the minimum mass by its corresponding gradient being lower than SGD noise. We generalize this result to ResNet-18 CIFAR-100 in Appendix D.3 and Figure 11. In this case, the networks at minimal representation costs have extremely low mass weights due to the sequences of convolution and batch normalization layers (see Section D.11.3). Until now, we provided theoret- ical insights on the contrasted impacts of both scaling and permutation symmetries. In the following, develop more empirical studies and explore the link between posterior quality and performance.
COMPARING POSTERIOR ESTIMATIONS BY APPROX. BAYESIAN METHODS
In this section, leverage symmetries compare popular single-mode methods, namely, Monte Carlo Dropout (Gal & Ghahramani, 2016), Stochastic Weight Averaging Gaussian (SWAG) by Maddox al. (2019), variational inference BNNs (viBNNs) (Blundell al., 2015), and Laplace methods (Ritter et al., 2018). We also include their multi-modal variations, corresponding to the ap- plication of these methods on ten different independently trained models, as well as SGHMC (Chen et al., 2014), preconditioned SGLD (Li et al., 2016) and deep ensembles (DE) highlighted by Hansen & Salamon (1990) and Lakshminarayanan et al. (2017). compare these methods on three image classification tasks with different levels of difficulty, ranging from MNIST (LeCun et al., 1998) with (Deng et al., 2009)
OptuNet (392 parameters) to CIFAR- 100 (Krizhevsky, 2009) and with ResNet- 18 (He et al., 2016). To this extent, we leverage maximum mean discrepancies (MMD) to estimate the dissimilarities between the high-dimensional posterior distributions. We estimate the target posterior using 1000 independently trained neural networks and compare it to 100 samples from all previously mentioned techniques. This choice compared to the more theoretically- grounded HMC (Neal al., 2011; Izmailov al., 2021) is supported by theoretical aspects on the estimation of high-dimensional distributions (Wild et al., 2023), by computational constraints that would shatter mini-batcced HMC's guarantees in practice, and by experiments showing that full-batch HMC's theoretical performance may not be fully achieved in our real-world settings. However, we stress that sampling from independently trained checkpoints to estimate the posterior remains imperfect and can be debated. We discuss these limitations in-depth in Appendix B.
4.1 EVALUATING THE QUALITY OF THE ESTIMATION OF THE BAYESIAN POSTERIOR
One approach to assess the similarities between distributions involves estimating the distributions and subsequently quantifying the distance between these estimated distributions (Smola et al.,

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Table 1: Comparison of popular methods approximating the Bayesian posterior All scores are expressed in %, except the ACEs and the MMDs for ResNet-18 networks, expressed in %c. Acc stands for accuracy, and IDMI and OODMI are in-distribution and out-of-distributinn mutual information. NS is the MMD computed after removing the symmetries, and DE stands for Deep Ensembles. Multi-mode methods are based on ten independently trained models.
Method
MMD
NS
Ace 1
ECB
ACE
Brier
AUPR 1
FPR95
IDMI
OODMI 1
Dropout
15.0
14.3
83.3
26.1
60.0
33.4
96.4
98.6
26.1
22.2
MNIST OpturNt
One Mode
viBNN
18.8
17.1
78.
7.4
17.6
30.9
67.9
93.7
0.1
0.1
SWAG
16.0
88.3
4.9
11.9
8.7
Laplace
10.6
9.5
87.9
4.8
15.1
18.1
48.2
74.6
6.2
5.9
SGHMC
16.7
17.7
95.1
2.8
3.2
7.6
73.7
98.4
4.3
14.5
PSGLD
15.1
88.1
3.8
17.7
49.2
75.5
1.0
0,9
Multi Mode
Dropout
2.1
2.1
92.1
36.8
67.5
29.2
97.2
78.2
36.6
52.5
viBNN
2.8
2.5
86.5
17.5
31.3
24.4
27.2
21.1
$2.3
SWAG
1.8
1.3
95.0
17.5
27.6
13.1
88.7
24.6
27.6
62.2
Laplace
1.8
0.8
94.8
15.8
24.5
12.8
95.4
32.1
21.1
52.2
DE
0.0
95.3
10.9
21.0
13.5
95.7
12.8
19.3
62.6
CIFARION ResNet18
One Mode
Dropout
viBNN
4.5
9.0
10.2
57.9
74.2
24.6
14.7
3.2
63.7
38.8
76.4
60.9
79.1
47.7
5.7
2.7
4.2
9.1
SWAG
6.7
7.2
70.9
2.3
1.2
38.9
86.2
48.0
2.4
6.3
Laplace
5.7
7.0
75.1
0.9
0.9
34,6
81.3
42.4
27.6
63.3
SGHMC
73.7
4.9
36.2
79.4
62.3
0.2
0.5
Multi Mode
Dropout
0.7
79.5
4.3
1.0
29.2
78.2
48.1
20.5
46.3
viBNN
6.1
5.6
66.5
2.8
2.0
45.3
71.9
71.7
45.5
81.1
SWAG
5.0
5.4
72.8
1.5
1.1
36.9
89.1
50.6
6.5
19.7
Laplace
0.6
4.3
78.9
6.9
0.8
30.3
82.9
41.3
44.1
98.5
79.5
1.6
81.1
45.6
22.5
58.0
Tiny DinyImageNet, RReNNet88
One Mode
Dropout
63.2
8.3
8.4
viBNN
/
/
/
/
SWAG
9.1
3.9
66.4
10.5
0.7
46.2
61.9
57.7
3.0
4.5
Laplace
5.5
6.1
33.1
6.0
3.6
77.1
48.8
77.7
200.7
228.0
SGHMC
58.3
2.6
1.0
56.3
72.7
0.24
0.30
Multi Mode
Dropout
70.2
9.9
1.2
58.2
34.1
60.0
viBNN
/
/
/
/
/
SWAG
6.7
5.4
69.3
3.6
0.6
41.3
96.5
55.9
17.6
32.1
Laplace
0.5
3.1
37.0
10.9
3.3
75.1
48.4
72.5
219.5
254.7
0.0
70.3
6.5
86.3
50.2
38.4
83.4
2007; Sriperumbudur et al., 2010). However, these methods can become impractical when dealing with distributions extremely high-dimensional spaces, such as the posterior of modern DNNs. An alternative solution is to embed the probability measures into reproducing kernel Hilbert spaces (RKHS) (Bergmann, 1922; Schwartz, 1964). Within this framework, a distance metric, the maximum mean discrepancy (MMD) (Song, 2008) defined the distance between the respective mean elements within the RKHS used to quantify the dissimilarity between the distributions.
Appendix C.4 formalizes and explains implementation of the MMD reported in Table 1: we fol- low Schrab et al. (2023) and use multiple Gaussian and Laplace kernels. For tractability, we report the mean weighted by the number of parameters of each layer of the median over twenty MMD kernels between the layer-wise DE-based posterior estimation and the approximation provided by each method. The NS metric corresponds to the MMD computed after a posteriori-symmett removal using the algorithms detailed in Appendix D.2. Appendix C gathers all details concerning these experiments (including the means and maxima over the kernels of the MMDs).
4.2 PERFORMANCE METRICS AND OOD DATASETS
On top of the MMD quantifying the difference between the posterior estimations, we measure several empirical performance metrics. evaluate overall performance of the models using the accuracy and the Brier score (Brier, 1950; Gneiting et al., 2007). Furthermore, we choose the binned expected calibration error (ECE) (Naeini et al., 2015) and adaptive calibration error (ACE) (Nixon al., 2019) for tabelabel calibration and measure the quality of the out-of- distriibbbt (OOD) detec- tion using the area under the precision-recal curve (AUPR) and the false positive rate at 95% recall (FPR95), recommended by Hendrycks & Gimpel (2017). We expect the OOD detection abilities

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300
0.3
Probability Deneity
Probudit Donelty
200
OODMI
0.75
10
100
0.33
81
0.103
0.11 IDMI
0.115
0.1
816
0.30
0.75
0.31
0.35
0.
0.56
011
0.15
OODMI
IDMI
Figure 3: Experiments show no hint of a functional collapse between couples of independently trained ResNet-18 on CIFAR-100. Moreover, the in-distribution and -utoo--distrbbution mutual information (IDMI, resp. OODMI) exhibit different variances but do not seem correlated.
of the models to correlate with the quality of the estimated posterior. Finally, we report the mean di- versity of the predictions each ensemble through the mutual information (MI) (e.g. Ash (1965)), often used to measure epistemic uncertainty (Kendall & Gal, 2017; Michelmore et al., 2018). We use FashionMNIST (Xiao al., 2017), SVHN (Netzer et al., 2011), 1), and Textures (Cimpoi et al., 2014) as OOD datasets for MNIST, CIFAR-100, and TinylmageNer, respectively.
4.3 RESULTS
Table I demonstrates that multi-mode techniques consistently exhibit superior performance in terms of MMD when compared to their single-mode counterparts. This trend holds true in accuracy, negative log-likelihood, and calibration (ECE and ACE) for ResNet architectures. We provide further details on the calibration performance for OptuNets and Laplace methods in Appendix C.3.
Turning attention the assessment of epistemic uncertainty, quantified by AUPR and FPR95, multi-mode techniques, notably multi-SWAG and DE, consistently outperform other methods. This underscores the strong connection between posterior estimation and the accuracy of epistemic uncertainty quantification. However, we note that the quality of aleatoric uncertainty quantification does not steadily correlate with that of the posterior distribution estimation.
The final two columns of the table shed light on the diversity of the models sampled from the posterior. The objective is minimize in-distribution mutual information (IDMI) while maximizing out-of-distribution mutual information (OODMI). An analysis shows that mono-mode methods yield lower values than multi-mode methods, suggesting inferior diversity for the former.
DISCUSSIONS
We develop further insights on the posterior of Bayesian neural networks in relationship with sym- metries. Notably, we evaluate the risk of functional collapse, i.e., of training very similar networks in Section 5. and discuss the frequency of weight permutations in 5.2. We showcase "Checkpoints" dataset in Section D.1. In Appendix B, discuss using independent checkpoints for the pos- terior estimation. Appendix D expands these discussions, discusses the impact of the chosen prior in D.4, links the posterior recent works on the loss landscapes (Section D.5), adds observations on the number of modes of the posterior in D.8, and provides visualizations (Section D.10).
5.1 FUNCTIONAL COLLAPSE IN ENSEMBLES: STUDY OF ID AND OOD DISAGREEMENTS
Given the high number of equivalent modes due to permutation symmetries (see Section 3.5), we support broadening the concept of collapse the parameter-space e.g., in D' D'Angelo Fortuin (2021) functional collapse to account for the impact of symmetries the posterior. Parameter- space collapse is more restrictive and may not be formally involved when ensemble members lack diversity. is also harder characterize it would require an analysis of the loss landscape, at least.
We quantify functional collapse as a potential ground for the need for more complex repulsive en- sembling methods (Masegosa, 2020; Rame & Cord, 2021). We take 1000 ResNet-18 trained to esti- mate the Bayesian posterior in Section and compute the mean over the test set of their pairwise mu- tual information (see Section D.12), quantifying the divergence between the single models and their
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Meaa Kendall's r (%)
100
Mean Kendall's T (%)
100
85-
85-
85
55
40
75
Epochs OptuNet on MNIST
Epochs ResNet-18 on CIFAR 100
Figure 4: Evolution during training of the mean Kendall's r correlation between the permuta- tions towards the identifiable model for all successive steps: The correlation between successive permutations increases when the learning rate decreases.
average. We measure these values on in-distribution and OOD data with CIFAR-100 and SVHN, respectively. In Figure (left), we see that the in-distribution MI between any two networks has very low variance. Given that the usual diversity between models is satisfactory (Lakshminarayamn al., 2017), there an extremely low probability of training two similar networks, despite the huge number of potential symmetric models. These results hint that the complexity of the posterior is orders of magnitude greater than what we grasp with symmetries: large DNNs seem to empirically never fall into these numerous symmetric local minima. This may be explained by the high com- plexity of the network (here, ResNet-18), and we refer to Appendix D.9 for results on a smaller architecture. Interestingly, nificant correlation between the in-distribution and the OOD MI. This highlights that measuring the in-distribution diversity may be very poor indicator of the OOD detection performance of a model.
note in Figure (right) that in contrast to intuition, we have no sig-
5.2 FREQUENCY OF WEIGHT PERMUTATIONS DURING TRAINING
We devise new protocol to evaluate if network tends to permute during training. Given a DNN fw. we compute, for each step 8 of the training, the permutation set II, sorting its weights (and removing the corresponding symmetries). If the DNN tends to permute during the training, this implies a variation in the II,. We measure the extent of the variations using Kendall's T correlation between successive permutations II, and IIg+1. We plot the variation of the mean training instances, and the Kendall's 7 of each element of the permutation sets in Figure 4. We see that on MNIST (left), the variations of the permutation sets are scarce and gathered around points of instability. These instabilities are due to the sorting mechanism, which is based on the maximum values of the neurons' weights; we have tried other statisties on the values of the weights, but taking the maximum remains the most stable. Moreover, the weights nearly never permute in the last train- ing phase. The analysis differs for ResNet-18 18 (right) since the number of degrees of freedom much greater. We see lot of variation during the phases with a high learning rate (reduced after 25 and 50 epochs). However, as for the first case, we do not see any particular sign of permutations in the last part of the training. This evidence is in favor of weight-averaging methods such as SWAG (Maddox et al., 2019), which, therefore, have only very limited risks of averaging symmetric networks.
several
CONCLUSION
In this study, we have examined Bayesian neural network posteriors, which are pivotal for understanding uncertainty. Our findings suggest that part of the complexity in these posteriors can be attributed to the non-identifiability of modern neural networks viewing the posterior as a mixture of permuted distributions. To explore this further, we introduce the scaled representation problem and investigate the real impact of scaling symmetries. Using real-world applications, we design method to assess the quality of the posterior distribution and study its correlation with model performance, particularly regarding uncertainty quantification. While considering symmetries has provided valuable insights, our discussions hint more profound complexity going beyond these weight-space symmetries. In future work, we plan to continue our exploration of this intriguing area.

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7 ACKNOWLEDGEMENTS
This work was performed using HPC [AD011011970R3).
resources from GENCI-IDRIS (Grant 2023-
REPRODUCIBILITY STATEMENT
To ensure transparency and accessibility, use publicly available datasets, including MNIST, FaShionMNIST, CIFARIOO 100, SVHN, ImageNet-2000, and Textures. Please refer to Appendix C.2.2 for details on these datasets. Our detailed experimental methods are outlined in Appendices and C, and the proofs for our theoretical results are provided in Appendix E.
To help replicate our work, we share the source code of our experiments on GitHub, notably including code to remove symmetries from neural networks a posteriori. For our experiments in Section 4, we rely exclusively on open-source libraries. Most of our experiments are performed with Tncettnnertaintt, including the training of the standard and dropout models, but also the evaluation of their Deep Ensembles versions. For the rest, we use the GitHub repository for SGHMC, BLiTZ for variational Bayesian neural networks, and Laplace Redux (Daxberger et al., 2021) for Laplace evaluations. We also use the publicly available code from the original paper (Maddox et al., 2019) for the SWAG method.
Finally, we estimate the maximum mean discrepancies with homemade torch version of the code from Schrab et al. (2023) and solve convex optimization problems (see Definition E,6) with evxpy (Diamond & Boyd, 2016). The statistical experiments, such as Pearson's 's and Kendall's T, performed with SciPy,
ETHICS
Our primary goal in this paper is to improve our comprehension of the Bayesian posterior, which we argue is fundamental element to understand to contribute to the reliability of machine-learning methods.
We note that training a substantial number of checkpoints for estimating the posterior, especially in the case of the thousand models trained on TinylmageNet, was energy intensive (around 3 Nvidia V100 hours per training). We opted for the Jean-Zay supercomputer, a carbon-efficient cluster to mitigate the environmental impact of our research.
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TABLE OF CONTENTS SUPPLEMENTARY MATERIAL
Details on the starting example
17
A.1 Training the two-layer perceptrons
17
A.2 Removing symmetries
17
On the estimation of the posterior with independently trained checkpoints
18
B.1
Bibliographic and theoretical aspects
18
B.2 Computational aspects
19
B.3 Comparing estimations from HMCs and independently trained checkpoints
19
Details on the experiments of section 4
C.I Experimental details
21
C.2 Detailed on the architectures and datasets
23
C33 On the top-label calibration results
24
C.4 Computing Maximum Mean Discrepancies
25
C.5 Using a Wasserstein distance on the marginals
26
C.6 Full MMD Table
26
D Other discussions and details
28
The Checkpoints dataset
28
D.2 Algorithms for removing symmetries a posteriori
28
D.3 Minimum scaled representation costs of ResNet-18
28
D.4 On priors and their impact on the posterior
29
D.5 On the links between posteriors and loss landscapes
30
D.6 Does solving the scaled-repressentation problem improve your network?
31
D.7 Putting Proposition in practice
31
D.8 Estimating the number of permutation- generated modes
31
D.9 Details the functional collapse in ensembles
32
D.10 Visualization
D.11 Symmetries in modern networks
................................
34
D.12 On mutual information
36
Detailed formalism, propositions, and proofs
37
E.1
Extended formalism on symmetries
37
E.2
Permuuation-qquivariince of the gradient of the loss and SGD
................................
40
E.3
Equiprobability of the permutations
42
E.4
Theoretical insights the Bayesian posterior
43
E,5
The minimum scaled representation cost problem
45
Notations
48
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Class Class B
Feature
0r
8
Feature X
Figure 5: The training data the models whose posterior represented in the introduction, on Figure 1. The data is separable to ensure efficient training of simple 9-parameter perceptrons.
A DETAILS ON THE STARTING EXAMPLE
A.1 TRAINING THE TWO-LAYER PERCEPTRONS
To create the introductory example in Figure classes from two different normal distributions of means μ^ = (-2, 22) and μ4 = (2, and of identity covariances. We ensure that the samples (200 points each) training as simple as possible. We plot the training data in Figure 5. We then train 10,000 two-layer input,
generate the data corresponding to the two
fully separable to make
perceptrons with with SGD and a binary cross-entropy loss. We separate the first and second layers with a ReLU activation function for the posterior to exhibit scaling symmetries. We use a batch size of ten data points and a learning rate of two. Finally, we select networks with a sufficiently low loss on the training set, removing the few outliers (less than 1%) that had not been trained successfully.
hidden, and one output neurons with early stopping for epochs
A.2 REMOVING SYMMETRIES
Figure 1 shows the density of weights of the last layer (excluding the last bias). The first Figure shows the unaltered projection of the posterior on these weights, which is then transformed to guar- antee three-norm per neuron. We chose three the (whose posterior is plotted in Figure 1) is not directly normalized but rather subject to the normaliza- tion of the previous layer. Finally, we remove the permutation symmetries by ordering the weights. Contrary to the formalism developed in the following sections, we decide here to order the weights starting from the last layer (not the first) convey the message more efficiently. We check with ran- dom inputs for each network that the wymmetry-remova operations do not alter the networks. Note- books are available in the supplementary material of the ICLR submission and detail the process for generating the Figure. We provide more detail on the symmetry removal algorithms in Section D.2.
for visual purposes since the last layer
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B ON THE ESTIMATION OF THE POSTERIOR WITH INDEPENDENTLY TRAINED CHECKPOINTS
Here, we detail the different methods for estimating Bayesian posteriors of DNNs. We provide small literature review of the strengths and weaknesses of each method, briefly describe the computational complexities, and develop an in-depth study on the comparison of the posterior estimation thanks the HMC checkpoints gracefully released by Izmailov al. (2021). Notably, we highlight elements of comparison between HMC and deep ensembles-like generated checkpoints for extremely high dimensional posterior estimation.
B. BIBLIOGRAPHIC AND THEORETICAL ASPECTS
Various methods have been suggested for estimating the Bayesian posterior of DNNs, each with its own strengths and limitations. Noteworthy approaches include variational inference BNNs (Blundell al., 2015), MCMC (Neal al., 2011), Laplace methods (MacKay, 1992), and ensembles (Lak- shminarayanan et al., 2017; Wilson & Izmailov, 2020). However, all these methods face challenges and criticisms, especially the context of posterior estimation for high-dimensional modern computer vision networks the focal point of this paper. Here are key concerns for each method:
Variational inference methods. Variational inference BNNs are notoriously unstable and difficult to scale to large networks (Dusenberry al., 2020). Furthermore, they often provide simplistic approximation of the posteriors as diagonal Gaussian distributions (Blundell et al., 2015). Moreover, most of these mono-mode approaches, leading to even coarser posterior estimations.
Monte-Carlo Markov Chains. MCMC methods, including HMC (Neal et al., 2011) sometimes referred to as gold standard -, are reported some of the best algorithms to estimate distributions, although extremely computationally expensive when applied on deep neural networks (Izmailov al., 2021). MCMC's efficiency is reported to highly dependent on their hyperparameters (e.g., this course) which may need tuning. incurring supplemental costs. Moreover, MCMCs are also known for their difficulties in modeling multi-modal distributions (Mangoubi et al., 2018; Izmailov et al., 2021; Park, 2021). This could be problematic, given that the target distributions are very multi- modal (see Wilson & Izmailov (2020) for the multi-modality.. However, Izmailov al. (2021) have shown that these algorithms also demonstrated good properties in practice when applied to estimat- ing the posterior of DNNs. Finally, their faster stochastic approximations like Stochastic Gradient Hamiltonian Monte Carlo have been reported to be sometimes less reliable (Bardenet et al., 2017).
Laplace methods. Laplace (MacKay, 1992) methods model the posterior around a single maximum posteriori. Moreover, they often approximate the posterior around the weights of the last layer. Similarly to variational inference methods, these approaches are constrained in their ability to estimate posterior distributions with single modes.
Ensembles. While efficiently handling the posterior's multimodality, theoretical considerations suggest they require repulsive terms for full Bayesian consistency (D' Angelo & Fortuin, 2021). Indeed, ensembles tend to favor sampling local minima and avoid their surroundings. Ensembles also face criticism when used with limited number of inner estimators.
In Section 4, we suggest estimating the Bayesian posterior using an unrivaled number of inde- pendently trained checkpoints. This approach is chosen to address the multimodality inherent in DNN posteriors. A recent work (Wild et al., 2023) demonstrates that - in a problem with small number of local minima - the robustness of ensembles is theoretically less convincing than that of regularized methods like Langevin ensembles, although still better than variational inference. However, the other methods struggle to compete with ensembles practice, even with small UCI-regression datasets. We emphasize that the datasets and architectures in this study are complex and extremely high-diment dinensinnnionnl, exhibiting a theoretically infinite number of "local minima". We argue supported by Wild et al. (2023)'s conclusion that prioritizing local minima surroundings when the space cannot be sufficiently sampled may not be clearly detrimental to the estimation. The mechanisms to enhance algorithm convergence (D' Angelo & Fortuin, 2021; Wild
their
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250
Number of occurences
Number of occarences
150
4
50
100
12
14
5
11 13
DE Index of the maximmmm weight
HMC with 3 Chains Iudex of maximarn weight
Figure 6: The number of occurrences of the indices of the maximum weights of each first hidden convolutional layer amongst 750 samples of ResNet-20s estimated via both DE and HMC; left: the estimation based on independently trained networks leads to a smooth uniform distribution of the indices of the maxima; right: the distribution of the maxima of the samples from HMC is sharp, despite using three independent chains and discarding the first 40 checkpoints of each chain for best fairness. Please note that the figures have different scales.
al., 2023) in the sense of Wild et al. (2023) seem more relevant when the number of local minima is small, a condition not met in the real-world applications that we consider in this work.
B.2 COMPUTATIONAL ASPECTS
Our choice of deep ensemble-style checkpoints for estimating the posterior is also guided by computational constraints. While HMCs are appealing from a theoretical point of view and even arguably from practical point of view in some contexts given the inspirational work of (Izmailov et al., 2021) - its computational cost was out of reach for this study. Indeed, Izmailov et al. (2021) benefit from 512 TPUv3 and claim that their study was extremely compute-expensive. Training HMC indeed requires putting the full batch in memory, which may not be feasible for real-world datasets. Additionally, conducting hyperparameter search for HMC be time-consuming and resource-intensive. We confidently estimate the cost of their training to be two orders of magnitude greater than ours, and the VRAM requirements counting the ablation studies, and for moderate architecture, ResNet-20, which has around 275,000 parameters, 40 times less than the ResNet- 18 that we trained on TinyImageNet, whose more numerous images are also four times bigger than CIFARIO (in number of pixels).
simply of reach. All of this is without
To summarize, we can say that independently-triann trained checkpoints and HMC scale equivalently in terms of hard disk storage assuming that we get the same number of checkpoints yet exhibit much more scalable behavior compared to HMC concerning the VRAM used. Consequently, DE may be more computationally suitable for deeper and wider architectures and larger datasets.
B.3 COMPARING ESTIMATIONS FROM HMCS AND INDEPENDENTLY TRAINED CHECKPOINTS
We show that despite its numerous qualities, the estimation of the distribution by HMC raises ques- tions and that even in its best case with independent chains and discarding multiple burn-in samples.
B.3.1 IS BASED POSTERIOR ESTIMATION PERMUTATION INVARIANT?
The permutation aspect of Proposition is intuitive and boils down to the non-identifiability of the inner neurons of any neural network. As such, this property is basic and holds in the general setting studied in this work (and beyond since it extends to other activation functions). However, we show that HMC (Neal et al., 201 1), breaks this proposition, most likely due to remains of dependence be- tween successive samples. To show this phenomenon, et al. (2021), namely three full-batch HMC chains of ResNet-20s trained on CIFAR- 10, and compare them to the same number of networks that train and release in dataset (see Section D.
use the checkpoints provided by Izmailov
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60-
Number of occurenots
Number of oocurentess
-0+
30-
25-
20
30
75
20
10-
S1
63
31
63
DE Index of the maximum weight.
HMC with Chains Index of maximam weight
Figure 7: The number of occurrences of the indices of the maximum weights of each final linear layer amongst 750 samples of ResNet-22s estimated via both deep ensembles and HMC; The interpretation remains the same as for Figure although DE's p-value is lower than 5%.
More specifically, we show that the HMC-based estimation does not respect Proposition I's corollary. To recall, this corollary states that given layer, the marginals of the posterior are exactly similar. Let us consider a simple statistic, such as the index of the channel of the maximum weight. More precisely, start by averaging the X 3 kernels of the convolutional layer to recover two-dimensional tensors to recover a setting similar to the linear layer. Corollary implies that the number of occurrences of the indices should follow uniform categorical distribution. Intuitively, there is no reason for any non-identifiable neuron to contain the maximum weight more often than the others. Figure (left) shows that ensemble of independently trained samples seems qualitatively close to this result. However, it seems evident in Figure 6 (right) that it is not the case for the HMC-based samples, even the most favorable setting: using an ensemble of independent chains and discarding numerous burn-in iurn-in samples.
We strengthen this empirical evidence with another experiment in Figure 7, which shows the equiva- lent distribution for the final layer of the residual network. This layer has more input parameters and, therefore, more possible indices, but the sentence remains the same. The observation data contradicts the expectations set forth by Proposition I that should theoretically hold in the considered setting. As such, the HMC-estimated posterior is inconsistent with the theory. We limit ourselves to these two layers as they have the smallest number of indices, but release the code that generates these figures.
To be more quantitative, we perform x2 test to compare the distributions of the indices of the max- imum weights to the uniform categorical distribution. For deep ensembles, obtain 94. 1 1% and 2.3% chance of mistake when rejecting that the two distributions are equivalent, for the convolutional and linear layers, respectively. On the other hand, we obtain negligible p-values for HMC (of the order of 10 -300 in both cases. Please note that we use here a simplistic statistic, and sample com- patible with a uniform categorical distribution would be the bare minimum respect Proposition 1.
We can argue that this could be a favorable point for HMC. Indeed, the previous point suggests that HMC is likely not waste samples on symmetric modes. However, recall that independently trained checkpoints do not suffer from functional collapse either (see Section 5.
B.3.2 COMPARING THE ESTIMATIONS OF THE POSTERIOR BY APPROXIMATE BAYESIAN METHODS
Experimental setup. We test the performance of our ResNet-20 (He et al., 2016) trained on CIFAR-10 (Krizhevsky, 2009) with the SiLU activation (Ramachandran et al., 2017) and filter sponse normalization (Singh & Krishnan, 2020), and compare them to the checkpoints provided by Izmailov et al. (2021). To this end, we report the MMD and the NS, the MMD without permutation symmetries (there are no exact-scaling symmetries with SiLU activations) using both independently- trained checkpoints target distribution (MMD/ / D, NS D) and HMC (MMD/ / H, NS H).
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Table 2: Comparison of popular methods approximating the Bayesian posterior. All scores are expressed in %, except for the ACEs, expressed in %c. Acc stands for accuracy, and IDMI and OODMI are in-distribution and out-of-distribution mutual information. NS is the MMD computed after removing the symmetries, and DE stands for Deep Ensembles. MMD D and NS / D are the MMD and NS computed with a target distribution based on DE and MMD and NS are based on HMC. Multi-mode methods are based on ten independently trained models except for HMC which is based on three independent chains.
Method
MMD/D
MMD/H
NS/D
NS/H
Acc f
ECF.]
ACE.
Brier
ALIPR
FPR95
IDMI J.
OODMI 1
(One Mode
Dropout
5.7
5.5
5.7
5.4
89.9
2.0
7.3
14.7
64.9
81.6
7.8
5.0
BNN
6.7
8.5
6,6
88.0
20.0
10.3
19.9
78.1
38.6
0.1
SWAG
5.7
3,9
4.5
89.7
0.8
5.5
14.7
82.7
34.2
2.0
7.5
Laplace
4.9
6.6
5.4
90.4
5.5
79.8
42.8
1.1
Multi Mode
Dropout
BNN
0.7
2.8
2.7
0.7
2.9
2.6
2.7
93.7
92.7
2.3
2.7
9.7
91.0
81.2
31.7
26.4
11.5
15.6
70.7
SWAG
1.1
2.7
0.9
2.5
92.0
3.3
6.3
11.9
83.3
31.4
7.1
24.2
Laplace
3.6
2.2
92.7
HMC
2.3
0.0
2.2
0.0
90.3
5.7
8.1
15.0
90.0
23.7
29.7
90.5
DE
2.3
0.0
2.2
94.3
2.5
8.7
92.0
20.1
13.3
67.1
Results. Table 2 shows that deep ensembles consistently outperform HMC in all metrics except the out-of-distribution diversity. These results confirm Izmailov et al. (2021)'s claim that HMC can find very diverse samples in the function space. This may be related to HMC's hyperparameter A (the step size) that controls the trade-off between the likelihood and the acceptance rate of the samples. Indeed, the accuracy and the Brier score of HMC are below what could expect from an ensemble of checkpoints and lie between the monomodal estimations made by SWAG and Laplace.
On posterior discrepancies. First, Table 2 shows that posterior discrepancy metries are improved for multi-mode estimations when compared to mono-mode estimations. Interestingly, the MMDs estimated using HMC as reference seem slightly lower and more uniform than with deep ensem- bles'. Moreover, the MMDs based on DE may correlate slightly more with the OOD detection quality since ranks the multi-modal MC-Dropout, the second best AUPR, with a lower NS value than its counterparts. On the other hand, the distance between HMC and the other multi-mode methods uniform despite differences OOD detection quality. However, we acknowledge that these correlations not very clear, and we study another metric in Section C.5.
B.3.3 VISUALIZATION OF THE MARGINALS
We plot the histograms of the marginal posteriors as estimated by the ensemble of HMC and the ensemble of independent checkpoints in Figure 8. This provides small qualitative analysis of the estimations of the posterior.
These plots tend to confirm that the posterior estimated by the ensemble is more complex and more multi-modal than the estimation provided by the ensemble of HMC chains. We keep the same number of samples for the ensembles and HMC for the faimess of the comparison despite the greater number of checkpoints available with the ensemble. We also provide another interpretation of these Immiiol al. (2021) indicate that HMC successfully moves on iso-loss paths that link different modes in a given basin. We could hypothesize that HMC over-samples the low-loss regions along this path, leading to more mono-modal distributions with some amount of parameter correlation. We leave checking this hypothesis for future works.
C DETAILS ON THE EXPERIMENTS OF SECTION 4
C. EXPERIMENTAL DETAILS
In this section, we develop the training recipes of the different models and the parameters used for all the posterior estimation methods. We train models with Py Torch (Paszke et al., 2019) on V100 clusters. We refrain from performing any model selection to avoid biasing our posterior with the validation set: we always keep the final checkpoint.
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1.50
1.50
1.25
1.25
1.00
1.00
0.73
0.5
0.50
0.00
0.25
0.25
Dransity
0.00
Density
0.00
1.50
1.50
3.25
1.25
1.00
1.00
0.75
0.75
0.50
0.50
0.25
0.00
2.0
-1.
-10
0.01
1.0
1.5
2.0
220
-1.5
-10
-05
0.5
1.0
1.5
2.0
Value of 1.
[5]
Value of 45
Figure 8: Estimations by independently trained samples (orange) and 3-chain HMC (blue) of the posterior for two non-cherry-picked weights of the linear layer of ResNet-20. These qualitative results tend to confirm that the posterior estimated by the independently trained samples is "more multi molti- As specified in the indices and explained Corollary 1, these two figures correspond to two non-identifiable neurons, explaining their similarity.
For all the variational inference Bayesian Neural Networks (viBNN) of Blundell et al. (2015), use the default priors from Bayesian layers in torch zoo (Blitz).
OptuNet MNIST. We train OptuNet for 60 epochs with batches of size 64 using stochastic gradient descent (SGD) with a start learning rate of 0.04 and weight decay of 2 10-4 We decay the learning rate twice during training, at epochs 15 and 30, dividing the learning rate by 2.
We train the viBNNs with the same number of epochs, albeit using Monte Carlo estimates of the Evidence lower bound (ELBO) at each step and using Kullback-Leibler divergence (KLD) of 10 10-5, 5_ As usually done, we disable weight decay for the training of viBNNs.
full Hessian Laplace optimization. We use a dropout rate of 0.2 the last layer (both for training and testing) for the dropout version and perform SWAG using 20 different models (also set the maximum number to keep low-rank matrix), continuing the training with the classic high-leand learning rate schedule (starting with a linear increase) for twice as long. We collect the models 20 times with 10-epoch intervals between each of them. We sample the models with scale of 0.
For OptuNet,
Concerning the PSGLD Bayesian method, we train the neural network for 220 epochs with starting learning rate of 4 x 10 decayed linearly during 20 epochs of burn-in towards 4 x 10-10; 10 We use a diagonal bias of 10-5 with precondition decay rate of 0.99. The size of the batches remains 64, and the optimizer is a PSGLD version of RMSProp. We select a sample for the posterior and the predictions every two epochs.
We normalize the dataset as usual and perform random crop of size 28 and padding four during training. We use the normalized test images for testing.
For more details on the architecture of OptuNet, the reader may refer to Section C.2.1.
ResNet-20 CIFAR-10. We train our modified Resnet-20 with Filter Response Normaliza- tion (Singh & Krishnan, 2020) and SiLU activation (Ramachandran et al., 2017) for 200 epochs with batches of size 128, SGD with starting learning rate of 0.1 decayed twice by 10 at epochs 100 and 150, a momentum of 0.9 and a weight decay of x 10-4, We perform the classic normalization and random crop with four-pixel padding and random horizontal split.
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ViBNNs are also trained for 200 epochs with the same learning rate schedule albeit with base learning rate of 0.05. We weight the KLD with a coefficient 2 x 10-9.
We use last-layer Kronecker factorization (LLKFAC) to fit the Laplace methods (Laplace, 1774; Ritter et al., 2018). We manually average the predictions of the Monte Carlo estimation of the Laplace posterior with 100 samples. We apply dropout on the last layer only with Bernoulli coefficient of = 0.5 for the dropout models. As for MNIST, we train SWAG models twice the time of the original training recipe with the usual settings (start learning rate with linear ramp) and collect 20 models with 10-epoch intervals. We sample the models with scale of 0.1.
When training ResNet-20 and ResNet-18 18 models, we perform last-layer approximations of Laplace and Dropout. For more information on last-layer approximation, the reader may refer to Brosse al. (2020).
ResNet-18 CIFAR-100. We train the ResNet-18 for 75 epochs with batches of size 128 using SGD with Nesterov (Nesterov, 1983; Sutskever et al., 2013), with base learning rate of 0.1, momentum of 0.9, and weight decay of 5 x 10-4, Similarly MNIST, we decay the learning rate twice during training, here at epochs 25 and 50, by a factor of 10.
We train the variational BNNs with SGD for 150 epochs, with starting learning rate of 0.01, which we decay once after 80 epochs. We weight the KLD with coefficient 2 x 10- and perform three ELBO samples at each step.
As for ResNet-20, we use last-layer Kronecker factorization (LLKFAC) to fit the Laplace methods: the last layer ResNet-18 too large for its full Hessian fit memory. Moreover, Kronecker is not available for the whole network as it is not implemented for batch normalization layers, and the low-rank version is too long to train. We manually average the predictions of the Monte Carlo estimation of the Laplace posterior with 100 samples. For SWAG and dropout, parameters as for MNIST and CIFAR-10.
As for CIFAR-10, perform the classic normalization and random crop with a four-pixel padding and a random horizontal split.
ResNet-18 Tiny ImageNe... We train the ResNet-18 for 200 epochs with batch size of 128 using SGD with start learning rate of 0.2, weight decay of 10 10-4, and a momentum of 0.9. We use a cosine annealing scheduler until the end the training.
We did not manage to scale viBNNs to TinylmageNet. The other methods use precisely the same hyperparameters and methods as for R8sNet-18 on CIFAR-100.
For TinylmageNet, we perform the same preprocessing for CIFAR- 100, except that we keep the original resolution of 64 x 64 pixels and use the proper dataset statistics.
C.2 DETAILED ON THE ARCIITECTURES AND DATASETS
C.2.1 DETAILS ON OPTUNET
For OptuNet, we aimed to create the smallest number of parameters possible while keeping decent performance on MNIST. To this extent, we performed an architecture search with tree-structured Parzen estimators in Optuna (Akiba et al., 2019). To reduce the number of parameters, OptuNet uses grouped convolutions, introduced by Krizhevsky et al. (2012) see egg. Laurent al. (2023) for their formal definition in its second convolutional layer.
C.2.2 DETAILS ON THE DATASETS
This part provides some details on the different datasets used throughout the paper.
C.2.3 IN-DISTRIBUTION DATASETS
In this work, used the following datasets for training and testing.
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Conv2D
oul,timm2 Jomd, groupami
Max Pooling
stridum
Conv2D out_di- groupsm2
Average Pooling
RelU
Linear
10x10
Figure 9: Detailed architecture of OptuNet. OptuNet includes only 392 parameters yet achieves between 85% and 90% accuracy on MNIST. Only the last fully connected layer contains biases; out_ch the number of output channels, and ks is the kernel size.
MNIST.
The MNIST dataset LeCun et al. (1998) comprises 70,000 binary images of handwritten digits, each of size 28x28 pixels. These images are divided into a training set with 60,000 samples and testing set with 10,000 samples.
CIFAR-100. CIFAR-I00 Krizhevsky (2009) consists of 100 object classes and contains a training set with 50,000 images and testing set including 10,000 images. Each of the images is RGB and of size 32x32 pixels.
Imag-En
TinyImageNet is subset of ImageNet-Ie Ik (Deng et al., 2009). It consists of 200 object classes, each with 500 training and 50 validation images. Additionally, there are 50 test images per class for evaluating models. The images are RGB and have a size of 64x64 pixels.
C.2.4 OUT-OF-DISTRIBUTION DATASETS
The objective of the following -uttof-distribution datasets is to evaluate the quality of the posterior means to quantify the epistemic uncertainty. This work does not consider these datasets fully representative of complex real-world fuitof-distribution detection tasks.
dataset (Xiao et al., 2017) is set 28x28 pixel- grayscale images consisting of 60,000 training and 10,000 test images. We kept the test set is for the out-of-distibbttion tasks with MNIST.
FaShion-MNIST.
The
SVHN. The Street View House Numbers (SVHN) dataset (Netzer et al., 2011) large-scale dataset of 600,000 images of digits obtained from house numbers in Google Street View. In our work, we kept a fixed set of 10,000 images that we cropped to squares of 32x32 pixels.
The Describable Textures Dataset (Cimpoi et al., 2014) is a dataset containing 5640 images of textures divided into three subsets. Considering that the number of images is limited compared to the other testing sets, we use the concatenation all the subsets for OOD detection. We resize all the images to 64 pixels to stick to the size of TinylmageNet.
Textures.
C.3 ON THE TOP-LABEL CALIBRATION RESULTS
Considering the criticism received by the ECE (Naeini al., 2015) for instance in Nixon et al. (2019) - we also report the ACE in Table 1. The ACE (Nixon et al., 2019) is based on adaptive binning and avoids putting too much weight bins including very small numbers of samples.
may be surprising to the reader that Table depicts single-mode methods based on OptuNet having better top-label calibration than ensembles for OptuNet. We recall that OptuNets
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Success Rate (%)
Success
20
20-
102
20
Top-class Confidence (96)
Top-class Confidence (%)
Figure 10:
Calibration plots of OptuNeetbbsed variational inference BNN (left) and Multi- viBNN (right) trained on MNIST. Both models output underconfident predictions that worsen in the multi-mode setting: averaging underconfident predictions is often more underconfident.
very shallow and do not exhibit overconfidence (Guo et al., 2017). Quite the contrary, we provide two calibration plots on Figure 10 that show the underconfidence of the variational inference BNNs (Blundell et al., 2015), which worsens method. Note that we expect the confidence to decrease when resorting to ensembles since we provide an average of multiple predictions that may involve an amount disagreement.
Multi-BNN, its corresponding multi-mode
Regarding ResNet-18, only Laplace sees its calibration worsen. A hypothesis explaining this phenomenon is that Laplace optimizes its calibration post-hoc and that averaging optimized solutions may not lead calibrated ensemble but rather to under-confident meta-estimator.
Finally, we recall that comparing calibrations of mono and multi-mode methods is not completely fair, given that there is non-negligible difference in accuracy between the different methods.
C.4 COMPUTING MAXIMUM MEAN DISCREPANCIES
We formalize the computation of MMDs as used in our paper following (Schrab et al., 2023). For simplicity, keep their formalism unchanged. In work, kernel-based Maximum Mean Discrepancy (Gretton et al., 2012), that given a Reproducing Kernel Hilbert Space (RKHS) H4, of kernel is computed as:
MMD (p, q; H4)
=
sup SEHxilI Ius $1
E Ef
(7)
We use the estimator MMD of the kemne-bbaedd MMD developed by Gretton et al. (2012) and formerly implemented in Matlab, now ported to JAX by Schrab et al. (2023) and PyTorch by the authors this document. We replace by the most precise posterior approximated by indepen- dently trained checkpoints (see Appendix B) HMC (in Appendix B) and by the estimation provided by the other methods. Specifically, we leverage 20 different kernels: 10 Gaussian kernels supplemented by 10 Laplace kernels, each with different parameters, and report statistics the obtained values. Denoting 5) the set of the kernel- based RKHS, p the supposedly most precise posterior estimation by DE HMC, and q the coarser posterior estimation, report in Table 4,
the median:
median(MMD. (p. 4)) median MMD (p, q; Hx))
HeES)
the maximum:
max(MMDs (p.q)) max MMD (p, q: Ho)
and
(8)
HeEd
the mean:
MMD (p. = Σ MMD (p. g: H4).
H&ED
For computational scalability, and are not the full posterior estimations but rather Pt and gt, the ftyer-wise marginals. We report their weighted mean, with n; the number of parameters of layer I;
(eedaa((MDD)((.0))
my median(MMDD (pr g:))
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Table 3: Comparison of the Wasserstein distances on the marginals. All scores are expressed in %, except the ACEs expressed in %rand the WDs are kept unchanged. WD is the Wasserstein distance and WNS is the Wasserstein distance after the removal of symmetries and M-Mode stands for Multi-Mode.
Method
MMD
NS
WD
WNS
f
ACE
Brier
AUPR
FPR95
OODMI
MNIST OptuNet
M-Mode One Mode
17.6
SWAG
16.0
14.6
1.3
88.3
11.9
17.7
73.4
68.6
8.7
Laplace
10.6
9.5
0.3
0.6
87.9
15.1
18.1
48.2
74.6
5,9
SGHMC
16.7
17.7
0.3
2.0
95.1
3,2
7.6
73.7
98.4
14.5
BNN
2.8
2.5
0.3
0.4
86.5
17.5
24.4
96.9
27.2
52.3
SWAG
1.8
1.3
0.1
0.1
95.0
27.6
13.1
88.7
24.6
62.2
Laplace
1.8
0.8
0.0
0.1
94.8
24.5
12.8
95.4
32.1
52.2
DE
0.0
0.0
0.0
0.0
95.3
21.0
13.5
95.7
12.8
62.6
Contrary to Schrab et al. (2023), we do not perform two-sample tests, which power we would try maximize. Hence, we prefer using robust statistic, the median, the maximum and report it as MMD in Table 1. The other metric, NS, corresponds to the MMD applied on p and g after removing symmetries using the algorithms detailed Appendix D.2.
C.5 USING WASSERSTEIN DISTANCE ON THE MARGINALS
Given that the correlations between the MMDs of the posterior estimations and the OOD detec- tion performance are not completely satisfying, we test another distance, the one-dimensional Wasserstein distance (WD), the binned marginals. Although, by definition, somewhat limited, the Wasserstein distance may also be more interpretable. We provide the results on MNIST with Optunet in Table along with the most important metrics that recall from Table
Table 3 show that this distance favors, just as the MMD, multi-modal distribution estimations versus mono-modal ones. Furthermore, there good correlation between the Wasserstein distances and the MMDs in relatively small dimensions. Regarding the removal of symmetries, the WDs of SGHMC change drastically as opposed
corresponding MMDs that remain constant. However, just as for the MMDs, WDs are unable to predict the good performance in OOD detection of the BNNs and do discriminate the multi-modal Laplace from DE and Multi-SWAG despite its lower cut-of-distribution functional diversity.
C.6 FULL MMD TABLE
In Table 4, we report the weighted average over the layers of the medians, means, and maxima the Maximum Mean Discrepancies of each layer, computed in Section 4. We recall that the maximum is used to obtain the best discriminative power of two-sample tests, for instance, in Gretton et al. (2012); Schrab al. (2023). We report the median in the main table to improve the representativity of the values. However, we note that the order of the values does not seem to change, particularly between the means, medians, and maxima.
We see that, after Deep Ensembles that have perfect MMD as expected - Lapplae methods perform best on all measures, be it with or without symmetries, as well as for types of architectures. However, as shown in Table 1, they are inferior to SWAG on performance metrics. This hints that the correlation between the estimated quality the posterior and the real-world metrics may not be fully correlated. Please note that for Deep Ensembles, we used different checkpoints for the second sample than in the sample corresponding to the posterior estimation.
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Table 4: Comparison of popular methods approximating the Bayesian posterior. Maximum Mean Discrepancies for OptuNet are expressed in %. For ResNet-18 networks, MMDs are ex- pressed in %o. NS MMDs the MMDs computed after the removal of the symmetries. Multi-mode methods use ten independently trained models. DE stands for Deep Ensembles (Lakshminarayanan al., 2017) and TinyIN for TinylmageNet.
MMD
NS MMD
Method
Median
Mean
Max.
Median
Mean
Max.
Dropout
14.9
14.6
22.9
14.3
14.3
22.5
One Mode
BNN
18.8
18.3
29.5
17.1
17.0
27.4
OptuNet
SWAG
15.9
15.8
25.9
14.6
13.8
21.6
Laplace
10.6
10.5
19.1
9.5
9.4
16.4
SGHIMC
16.7
16.6
27.8
17.7
18.3
32.2
PSGLD
14.9
14.8
25.1
17.2
17.8
30.4
MNIST:
Multi Mode
Dropout
2.1
2.3
4.1
2.1
2.0
3.5
BNN
2.8
2.9
5.8
2.5
2.4
4.3
SWAG
1.8
2.0
1.3
2.5
Laplace
1.8
2.0
4.1
1.5
DE
0.0
0.0
0.0
0.0
0.0
0.0
CIFAR-100 ResNet-18
One Mode
Dropout
4.5
4.6
7.2
7.4
7.5
12.0
BNN
9.0
9.2
14.6
10.0
10.2
16.0
SWAG
6.7
7.0
11.2
7.2
12.3
Laplace
5.7
6.0
9.7
7.0
7.2
11.7
SGHMC
7.5
7.3
11.0
7.9
8.0
13.0
Multi Mode
Dropout
0.7
0.7
1.2
4.4
4.5
7.5
BNN
6.1
5.6
5.8
10.0
SWAG
5.0
5.2
8.4
5.4
5.6
9.2
Laplace
0.6
0.6
1.0
4.3
4.4
7.2
DE
0.0
0.0
0.0
0.0
0.0
0.0
One Mode
Dropout
9.5
9.6
7.8
TinyIN ResNNr--B 18
BNN
/
/
/
SWAG
9.1
9.5
15.6
5.4
5,6
9.5
Laplace
5.5
5.8
9.6
6.1
6.3
10.1
SGHMC
9.8
10.0
15.7
5.3
5.2
7.9
Multi Mode
Dropout
4.3
4.4
3.0
BNN
/
/
/
/
/
SWAG
6.7
6.9
11.3
3.9
4.2
7.3
Laplace
0.6
0.6
9.9
3.1
3.2
5.3
DE
0.0
0.0
0.0
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D OTHER DISCUSSIONS AND DETAILS
D.1 THE CHECKPOINTS DATASET
We release "Checkpoints", our extensive dataset of deep neural network checkpoints on multiple tasks and architectures on Hugging Face. Specifically, this dataset contains:
100,000 15-neuron two-layer perceptrons trained on regression task on the energy efficiency dataset (see Section C.2.2),
10,000 small time-series forecasting LSTMs (Hochreiter Schmidhuber, 1997), trained on the Exchange-rate dataset,
10,000 OptuNets (see Section C.2.1) trained on MNIST (LeCun et al., 1998),
1,024 ResNet-20 (He et al., 2016) with a SiLU activation (Ramachandran et al., 2017) and filter response normalization (Singh & Krishnan, 2020) trained on CIFAR-10 (Krizhevsky, 2009)
2,048 ResNet-18 8 (He et al., 2016) trained on CIFAR-10
9,216 ResNet- 18 trained on CIFAR- (Krizhevsky, 2009), and
2,048 ResNet-18 trained on ImageNet-200 (Deng et al., 2009).
D.2 ALGORITIMMS FOR REMOVING SYMMETRIES A POSTERIORI
Removing permutation symmetries. Similar to Pourzanjani et al. (2017), we remove permuta- tion symmetries by ordering neurons according to the value of their first corresponding parameter. In dense layers, this first parameter is the weight of the first input neuron, and convolutional layers, it corresponds to the top-left weight of the kernel of the first channel. This solution more general than that of Pourzanjani et al. (2017) since neural network layers do not always have biases. Specif- ically, practitioners often remove biases from layer when followed by batch normalization (loffe & Szegedy, 2015) to reduce the number of inference and backpropagation steps computations.
this paper, we use two different algorithms to remove the scal- ing symmetries of a neural network a posteriori. Neyshabur et al. (2015) designed the first algorithm to remove these symmetrics by normalizing the norm of the weights of each neuron in all the layers except the last. In most of the paper, including Table I, use simple posteriori normalization of the weights. We also implement this algorithm to scale the standard deviation of the weights to one.
Removing scaling symmetries.
Furthermore, define in Section 3.6 restriction of the representation cost problem that enables the deletion of scaling symmetries. We show the existence and uniqueness the solution to this problem and provide code to obtain it using convex optimization solver. It takes around 10 minutes to find the solution for ResNet-18, in which case the dimension of the scaling symmetries (and therefore the number of variables in the convex optimization problem) is equal to 7680.
While scalable to current architectures, the convex optimization is slower than the normalization of the weights, and its behavior may be more challenging grasp. Hence, stick the normalization of the weights when many computations are needed.
Removing softmax additive symmetries. To remove this type of symmetry, it sufficient to scale the sum of the biases of the last layer to some constant, for instance, to 1. This dimension is not studied in this paper.
D.3 MINIMUM SCALED REPRESENTATION COSTS OF RESNET-18
The minimum scaled representation costs of Resnet-18 (He et al., 2016) are very different from those obtained with OptuNet in Section 3.6. A key changing element between the two architectures the use of batch normalization layers. As stated in Section D.11.3, we do not use the inverse of the outbound scaling coefficients when updating the batch normalization layers. Instead, just update the mean and the variance. This means that the cost using extremely small outbound coefficients is minimal: only reduces the variance of the output. This translates in practice very
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Scaled represent. cost
800
600
400
200
500
1000
1500
2000
2500
3000
Network original representation cost
Figure 11: The distribution of the minimal scaled representation cost ("mass") vs. the representation cost at convergence. The minimum mass much lower compared OptuNets in Section 3.6 due to the batch normalization layers.
low minimum representation costs, as seen in Figure 11. Further inspection shows that the mass concentrated in the last layers of the networks.
D.4 ON PRIORS AND THEIR IMPACT ON THE POSTERIOR
In this section, we detail the priors used in our theory and experiments and provide leads on their impact on our results and analyses.
D.4.1 THE BAYESIAN INTERPRETATION WEIGIIT DECAY
Practitioners recognize the crucial role of weight regularization when employed with cross-entropy, as it introduces bias in optimizing neurons toward networks characterized by lower representation cost or "mass". This bias promotes emergence of potentially simpler and more generalizable functions within deep learning architectures (Goodfellow et al., 2016).
Additionally, weight regularization has a Bayesian interpretation. It encourages the network to prefer simpler models when faced with multiple plausible explanations for the observed data. This is akin to Ockham's 's razor: given some data, the simplest explanation is the most likely, and there need to add unnecessary hypotheses. Indeed, L.2 weight regularization corresponds exactly to setting a centered Gaussian prior on the networks' weights.
More formally, in classification and given sample (z. y/) from the joint distribution Px,y, try to fit model of parameters w to predict p(y: w.,ww,, This can be done by minimizing the likelihood of the predicted distribution: this typically corresponds the cross-entropy loss, which practitioners often minimize with stochastic gradient descent to obtain maximum likelihood estimates:
(wes(w) = Σ log p(y: w).
(10)
(D.D)))
Going further, the Bayesian framework enables the incorporation of prior knowledge regarding denoted as the distribution p(w) that complements the likelihood. It leads to the research of the maximum a posteriori (MAP) via the minimization the following loss function:
Σ log x.,ww log(w). (mL.Ds))D
(11)
The i.i.d. normal prior is the standard choice for p(w), leading to the omnipresent L2 weight regularization (or weight decay). This paper aims to tackle real-world applications and, therefore, follows the practitioners in this regularization, de facto assuming Gaussian priors weights.
D.4.2 THE IMPACT OF THE PRIORS ON SYMMETRIES & THE POSTERIOR
Most priors do not impact the results linked to permutations. Indeed, priors mostly constant for the entire network and, therefore, constant within the layers. This means that the neurons remain non-identifiable and that permutations continue occur in the posterior.
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However, priors do have important impact scaling symmetries as they are closely related the minimum scaled representation cost. Indeed, increasing the value of the variance the Gaussian prior will most likely change the outcome of the training and lead to networks that are further from their corresponding network of minimal representation cost. More specifically, in Equation 5, we expect the posterior associated to À that are further from the minimum to decrease faster. We intend to study this relationship in future works.
Changing the prior distribution could also completely change the analysis of scaling symmetries. Indeed, choosing a Laplace prior, available on most deep-learning frameworks, would make the scaled representation cost problem non-convex and likely hinder any hope of easily computable solutions. However, this type of priors is very rare in the community.
Finally, priors are key to understanding the complexity of the relationship between the loss landscape and the posterior, but we leave this point to the next section.
D.5 ON THE LINKS BETWEEN POSTERIORS AND LOSS LANDSCAPES
The impact of symmetries the loss landscape has been studied in several works (Li et al., 2018; Fort et al., 2019; Fort & Jastrzebski, 2019; Liu et al., 2022). However, their effect on the posterior much less chartered. In this discussion, argue that this is due to the inherent difficulty of this problem. Indeed, contrary to the loss landscape, it is not possible to estimate quickly the density of a Bayesian posterior in some discretized bin of the space. Consequently, measuring the impact of the weight-space symmetries on the posterior is difficult. We argue that large part of this difficulty comes from the fact that the posterior is determined both by the loss landscape and the prior in the weight space, which hinders direct links between the first two.
D.5.1 (ALSO) A PROBLEM OF PRIORS
Let us take a classic standard Gaussian distribution as prior. We know, thanks to scaling symmetries (see Section 3.1), that we can find pnareguulrzzed-loss-nvriii line (for simplicity) between any local minimum and its corresponding scaled networks with arbitrary representation cost. In doing so, move get a completely different story when considering the prior. Indeed, the prior will be (linearly) increasing with the representation cost of the scaled networks leading to linearly decreasing Bayesian posterior density. This small example hints that the interaction between the loss landscape and the prior contributes to the complexity of the posterior.
flat surface and keep the network functional unchanged. However,
However, please note that the initialization in not scaling invariant. This implies that even without prior, one wouldn't get "equiprobable" scaling symmetries. This would be impossible anyway as the uniform distribution cannot be defined on continuous unbounded spaces.
D.5.2 A LOSS POINT OF VIEW ON THE SEEEED-REEEEENTATIION PROBLEM
Property 2 establishes the strict convexity of the minimum scaled representation cost minimization after change of variables. In this section, we study the impact of this property on the loss landscape.
Given that the minimum scaled representation cost log-log strictly convex, we know that there exists a single global minimum of the problem, i.e., global minimizer of the regularization term on the scaling symmetries. This means that we take some neural network fo the end of its training, there exists a linear path of decreasing regularization loss towards the global minimum. Moreover by definition of the symmetries (see Definition 3.1), and in particular the scaling symmetries (Definition 3.1) - we know that moving in the continuous space of permutation symmetries keeps the function unchanged. Hence, it also keeps the sogative--o--lkkelhheee two results, we deduce that there exists a linear path of decreasing loss towards the minimum of the scaled-representatiin problem. We show the loss profiles for 200 trained OptuNets on this path in Figure 12 (left).
constant. Combining these
This raises a question: if a path of decreasing loss exists towards a region of lower loss, why can't 't the neural network minimize it? We argue that this may be due to the iotllarizzaiinnnnnueee gradi- ent's 's negligibility compared to the stochastic gradient descent oscillations. Indeed, Figure 12 (right) shows that the gradient and the direction towards the minimum of the scaled-representation mini-
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0..0
900
Nimber of eqoechs
Network
150
140-
120-
in tae comkination X) -00AA Aral
Dot product between the grad. and the miniram's dirvetionn
Evolution of the mass of OptuNets the path between the maximum posteriori WMAR and the global minimum of the minimum scaled representation problem Wopt. At first glance, the regularization term seems smooth and easy to optimize. Right: Evolution of the scalar product between the gradient and the unit vector to the minimum. The mean of the dot-product 0.002, compared to the mean of the full gradient's norm of around 1.302.
Figure 12: Left:
mization do not align much. Their dients. We obtain these results post-training an OptuNet for 5 epochs with weight decay of 10 10-1
is only very small compared to the norm of the gra-
D.6 DOES SOLVING THE SCALED-REIHE REPRESENTATION PROBLEM IMPROVE YOUR NETWORK?
As written in Section D.4, it seems clear that Gaussian priors (L2 regularization) are ben- eficial for generalization. Therefore, it is legitimate wonder whether the solution of the scaleddrepressentatinn problem improves over the starting network.
We recall that solving the relleeerprrsnnntooonn problem done independently from the likelihood maximization and on the space of the scaling symmetries. As such, by definition of the symmetries (see Definition 3.1), the network will not move the function space: the function will not be altered. Consequently, there will be no impact on the evaluation performance, whether in-distribution or out-of-disttriuutin.
Nevertheless, this may impact post-training, such fine-tuning or weight-averaging (Maddox et al., 2019), but we leave this for future works.
D.7 PUTTING PROPOSITION IN PRACTICE
We develop small toy example to put Proposition in practice, check our results small dimensions, and show the impact of symmetries on the estimated posterior.
To this extent, we train 100,000 2-layer MLPs on a small circle classification dataset. We plot the data in Figure 13. This small dataset forces the MLPs benefit from non-linearities allowed by their activation functions. We suggest using two-layer perceptions with two input neurons, three hidden neurons. and one output neuron. In total, the MLPs have 13 parameters. To enable exact scaling symmetries, we use ReLU activations and train for 300 epochs with batch size of 8, constant learning rate of 0.03, and a Gaussian prior corresponding an r.2-regularization of 0.003.
Figure 14 shows the projection of 10,000 trained models after dimensionality reduction by principal component analysis (PCA) (left) and T-distributed stochastic neighbor embedding (TNSE) (right). It confirms the multi-modality of the posterior distribution even for such simple model, with 13 parameters, on a very basic dataset.
D.8 ESTIMATING THE NUMBER OF FERMUTATION-GENERR.C MODES
For simplicity, suppose that the layginma posterior of the identifiable model is an n- dimensional Dirac distribution of coordinates 0 (or distribution small bounded domain around
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1,5
Claes
Claes H
1.0
Festure Y
0.0
-0.5
1.5
-1.0
-0.5
1.0
1.5
Feature X
Figure 13: The two-circles classification training dataset used as the basis to train toy models. We train the small 2-layer perceptrons separate and classify the samples from A and B. The mean accuracy after 300 epochs is around 98% on the test set sampled from the same distribution.
80
60
40
2
TSNE Rahure
20
0-
PCA
-20
-2-
-40
-60
-80
100
-75
-0
-25
100
PCA Rarmme X
TSNE Feature
X
Figure 14: Projections of the trained models. PCA (left) and TSNE (right) provide two comple- mentary views on the estimation of the posterior, which seems clearly multi-modal. The explained variance of components X and Y of the PCA are 18% and 15%, respectively.
0). The number of modes of the resulting layer-marginal posterior will be (n-k)!, with i the number of (nearly-)identical terms in 0. To recall, n of the order of 10 for the last layer of ResNet-13
This result seems difficult to extend to other distributions. While there exist some results the number modes a mixture of Gaussians, to the best of our knowledge, they either treat the case with only two elements (Eisenberger, 1964) conjectures of an upper bound (Carreira-Perpinán Perpinán & Williams, 2003).
On experimental basis, we tried to use the MeanShift algorithm (Comaniciu & Meer, 2002) to timate the number modes. Despite the number checkpoints, Mean Shift remained very sensitive to the bandwidth. It was, expected, failing to detect modes in high dimensions but also providing unsatisfactory results simpler cases, such as for the MLP on energy-efficiency (see Section D. 1).
D.9 DETAILS ON THE FUNCTIONAL COLLAPSE IN ENSEMBLES
Functional collapse on MNIST/FashionMNISS with OptuNet. this paragraph, we study the potential functional collapse of OptuNet MNIST and FashionMNIST, depicted in Figure 15. First, we see that the dispersion of the mutual information both in the training domain (right) and out of the domain (center) is greater than that of the ResNet-18. Indeed, in the left plot of Figure 15,
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0.5
Probubility Density
Probabiitt Denstty
OODMI
0.04
0.12
0.36
022
0.5 24 05
066
0.01
0
0.32
IDMI
OODMI
IDMI
Figure 15: Histograms of the in-distribution (left) and outoof-distribution distribution (center) functional diversities between couples of checkpoints as computed with mutual information; and joint plot (right) of OptuNets MNIST. There is slightly more important functional collapse between couples of independently trained OptuNet on MNIST. Contrary to ResNet- on CIFAR- 100, the in-distribution and out-of-distribution mutual information (IDMI, resp. OODMI) seem correlated, at least for small values.
2001
0.55
Probabaility Density
Probubility Deusity
OODMI
120
100
0.40
0.145
0.150
0.29
0.31
0.20
0.445
0.154
0.056
IDMI
OODMI
IDMI
Figure 16: Histograms of the in-distribution (left) and out-of-distribution distribution (center) functional diversities between couples of checkpoints as computed with Mutual Information; and joint plot (right) of ResNet-18s Tmmy-ema As for ResNet-18 CIFAR- --00, the in-distribution and out-of-distribution mutual information (IDMI, resp. OODMI) seem only very slightly correlated.
there is non-negligible probability of obtaining mutual information twice lower than the expected value. We note that the dispersion of the out-o--distribution values on FashionMNIST is also greater than for ResNets. Finally, in the case of OptuNet and contrasting with the results obtained with ResNets, there seems to be some positive correlation between the in-distribution MI and OOD MI, least when considering the lowest values. We can conclude that there is a high chance of functional collapse in small networks, which is expected considering that their posterior simpler.
Functional collapse Tiny ImageNt and Textures. Figure 16 shows the mutual information on both the in-distribution (left) and out-of-distribttion (center) datasets and their correlation (right). The plots are much closer Figure 3, which was expected they share the same architecture. However, we see that the OOD dataset (here, Textures) potentially impacts the values of the mutual information. The dispersion (center) lower than for ResNet- on SVHN, which may be more diverse. We measure the Pearson's 's correlation coefficient and obtain a statistically significant value of se 0.06. The two values are, therefore, positively correlated with very low coefficient.
D.10 VISUALIZATION
In this section, visualize the marginals of the posterior of neural networks. including OptuNets. Figure provides the marginal distributions the weights of a multi-layer perceptron trained on energy efficiency. We check with two-sample Golmogorov-Smimot tests that cannot rule
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Probubility densities
1-val=14.0
p-val 2000000 55.7
p-val-21.1 21.1
p-val 11.5
1-
2.5
0.0
2.5
225
0.0
2.5
22.5
0.0
2.5
2.5
0.0
2.5
Fully-connectedd weight [i]]
Probubility densities
1
0
2-.
5-14l=3344
p-val=472 47.8
9.33
1-
2.5
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2.5
-255
0.0
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225
0.0
2.5
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Probubility densitics
1
0
p-val-10
0.0
p-val-0.0 0.0
p-val 0.0
D-10l 0.0
2
2.5
0.0
2.5
0.0
2.5
2.5
-2.5
0.0
Staled Fully-connected weight [1.]]
Figure 17: Marginal posteriors of the last weights from multi-layer perception: The top rows the original posterior, the middle rows have been scaled to unit norm, and the bottom rows have been scaled and permuted. We indicate the p-values (in %) of a Kolmogorov-Smingg two-sample between the marginals of the pair of neurons.
that the distributions of input neurons from any layer are the same, indicating the corresponding p-values. Indeed, as long as the permutations are not removed, we keep the equivalence property of all the marginals.
Another objective is to provide visualizations of the posterior, can be done on the loss landscape (see for instance (Garipov et al., 2018)). However, this task is extremely complex given that no means exist to estimate the discretized density of the posterior distribution. To circumvent this problem, we reduce the dimension the posterior with principal component analysis with scikit-lear (Pedregosa et al., 201 1). However the explained variance increases very slowly with the number elements, which prevents meaningful interpretations.
D.11 SYMMETRIES IN MODERN NETWORKS
This section provides details how compute symmetries throughout the paper.
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D.11.1 CONVOLUTIONAL LAYERS
In Appendix E, we generalize the rowewise and column-wise products to enable their use between vectors and four-dimensional tensors. We also detail how to understand the product between permutation matrices and dinnsiiinniinall tensors.
On higher level, we see that the number of symmetries in convolutional layers is smaller than the number of parameters in a linear layer. Indeed, in our formalism, a linear layer's number of permutations and scaling degrees of freedom (sDoF) corresponds to its number of output neurons. In contrast, the number of permutations of a convolutional layer equals its number of output channels. The ratio of the number of permutations/SDoF over the number of parameters between linear and convolutional layers, therefore, equals the surface the kernel of the convolutional layer.
D.11.2 GROUPEDD-CONOOLLTTOONA LAYERS
The number of permutations of a grouped convolutional layer (Krizhevsky et al., 2012) reduces the number of permutations of the previous layer. Indeed, we have to account for the impossibility of fully reversing the permutation applied on the output neurons/channels of the preceding layer. This implies that the permutations of the previous layer need remain intra-group. Say that the previous layer has Cout output channels and that the current layer has y groups. Instead of Cout!
On the other hand, the number of output permutations does not change (provided that the next layer does not have groups). Likewise, the number of scaling degrees of freedom remains unchanged scaling symmetries modify the values of the weights channel per channel.
possible permutations, they are reduced to Camu
D.11.3 BATCH NORMALIZATION LAYERS
Counting permutation symmetries on batch normalization layers (loffe & Szegedy, 2015) is quite straightforward. Regarding permutation symmetries, batch normalization layers are seamless, as one only needs to permute the weights, bias, running mean, and running variance. As such, they do not increase nor decrease the number of permutations.
On the other hand, batch normalization layers increase the number of degrees of freedom of scaling symmetries and make the minimum scaled representation cost degenerate. Indeed, the running variances and means to the input scaling coefficients and then choose new output coefficients for the weights and biases. They increase the number of sDoF by their number of features. Moreover, they allow the weights of the next layer to be extremely small as their weights are not multiplied by the inverse of the input scaling coefficient. Denoting with a tilde the updated weights, Xin and Aout the inbound and outbound scaling coefficients, the weights of the layer, b its biases and μe, a² the mean and variance, we have
scale
j μ Ain
/² σ² X in!
(12)
W Aouti
Aout-
This confirms that the inverse of Xin is not taken into account in the computation of the rep- resentation cost. This allows convolutions values as small as wanted. This involves that the minimum representation cost is degenerate (see Definition E.7) and we consider that the weights of the layers followed by batch normalization are null. Please note that we set the batch normalization epsilon (for training numerical stability) to 0 as it not scaling invariant.
use extremely low Aout reducing their weights to
D.11.4 RESIDUAL CONNECTIONS
Residual connections tend to reduce the number of permutation and scaling symmetries (Kurle et al., 2021), and this is important aspect in ResNets. Indeed, residual connections need the output coefficient of the last layers and the permutation matrices from all inner nodes in the computation graph to be equal. This potentially reduces both the number of possible permutations
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and the number of sDoF. In the case of ResNets, this effect is counterbalanced by the shortcut convolutional layers that can create permutations and sDoF, albeit less.
D.12 ON MUTUAL INFORMATION
In our work, discrete mutual information the main criterion for out-of-distribution de- tection and quantify the functional diversity between the predictions in the ensemble. Denoting the entropy function with H. we compute the discrete mutual information with the following formula:
M
MI H
Pum. Pw.. (0],, D)
Σ 17=1
H(PP... (0 D)),
(13)
M m==1
M
where M the number of predictors, each of weights Wm- Mutual information is often used to quantify the epistemic uncertainty since it can be expressed as the difference between the total uncertainty and the aleatoric uncertainty (Smith & Gal, 2018).
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E DETAILED FORMALISM, PROPOSITIONS, AND PROOFS
This section details the definitions, properties, and propositions and provides sketches of proofs.
Throughout the paper, use intuitive definition of neural networks that we formalize in the following definition:
Definition E.1 (Neural network). Let E Rd be input datum, r the rectified linear unit, and an almost everywhere differentiable activation function. We define the L-layer neural network fe with = (Wl),, bl1 hE(1,1)] as follows:
d =
VI [1, 2 w 0 + 6
(14)
E [1, L 1], al = r(x19)
fo(x)) s(2"1)
This definition of neural networks is limited to standard multilayer perceptions. However, it can be easily extended to a mix of convolutional and fully connected layers to the cost of a heavier formalism. To do this, we should replace the relationship between z!!l and !!'-1] in Equation 14 by the kernels of the cross correloornlntion operator. It would also be possible to account for residual connections using the computation graph
Property E.1 (Gradients and backpropagation). Let denote the training loss f(w) of the L-layer neural network fo and (Will b10 hee.... with respect to I!!!, we huve,
With, for all I [1, L.J, 5! the gradient of L(w)
L V 2214) ((w)
4/ [1, 5 =V 2(w) (wi++i) (otH)))))(t+++++ o VV.. Venal!
(15)
VI E []. V Vwn ((w) = a 2-11T
vI E [1, LJ. [(w) = 8!97
E.I EXTENDED FORMALISM ON SYMMETRIES
In this section, we present several definitions and properties to extend the formalism quickly described in the main paper.
E.1.1 SCALING SYMMETRIES
We start by recalling the definitions of the line-wise and column-wise products between vector and matrix.
Definition E.2. We denote the line-wise product v and the column-wise product as D. Let w E [2" xm R"" and μ E RR" with n, m 2 1,
Vi [1, nl,j E [1, w)i Aywas and (μ t w). = poway
(16)
To handle biases, extend these definitions to the product of vectors, considering the right vector as a one-dimensional matrix. follows that for vectors X, IR"
A a AD a a.
(17)
The line-wise and column-wise products between vectors and matrices huve priority over the classic matrix product "x" but not over the element-wise product between vectors ".". We will include parentheses when deemed necessary for the clarity of the expressions.
Non-negative homogeneity of the rectified linear unit. The rectified linear unit (Nair & Hinton, 2010), used as the standard activation function in most networks with the notable exception of vision transformers (Dosovitskiy et al., 2021) - is non-negative homogenous and therefore authorizes scaling symmetries (Neyshabur et al., 2015).
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Property E.2. Denote the rectified linear unit such that r(æ) x4- For all vectors X E (R+)a and x E R r is positive-homogeneouu. i.e.
r(A x) = X r(x)).
(18)
Some properties on line-wise and column-wise products. We provide few simple properties to help understand these notations.
Property E.3. For vectors A x E RR" 0 E R xn and w E Rux.
(A
(A 1(2)) w (A -
(19)
Proof. Let denote (A 7 Ww)
Ar. Σ
(1) b
1 Wk.j Σ
f A Wk. (0 (1(), ( ) w)is
(1) wiik
(20)
(1) 8 Bin ((A 22) 7 0 w)y
(21)
However, please note that we do not have (12) (A D w) = (A A(2)) in general.
Property E.4. For all vectors ( (Rya)", A (2) R" 0 F xm and E Rax-
(A (1) 0) x x x (1111))1(1))) 4 w
2 (2)
(22)
Proof. Let us denote B = (A1 0 X X D w)
Bij =
k (1 (2)W))))
1
(23)
Σ ( Ax wk.)
(1) (2)
(24)
0 r( 2(2) D w))ls
(25)
This property is crucial for scaling symmetries, which are one of its special cases. Indeed, for all vectors À (R>o)",
(A
w) x r(w).
(26)
Finally, we can gather the previous results to extend Equation 26 to weights and biases in the following property.
Property E.5. For all vectors À E (R>o)". be R" O xn, and RTx-
(A
x x (b+
(27)
Proof.
(A (0)X (b+w)) (A ( x
(28)
- ( b b+ -
(29)
0 exxrbbuw
(30)
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This property is provided for the simplest case with only two matrices. However, it can be trivially extended to an arbitrary number of matrices, chaining the different line-wise and golumn-wisse mul- tiplications. It follows that we can extend the definition of scaling symmetries to deeper networks.
Moreover, we can also extend the scaling symmetries to mixes of convolutional and linear networks. To do this, first extend the definition of the line-wise and column-wiss products to convolutional layers through the following definition.
Definition E.3. Let the 4-dimensional tensor w E RCam xCix Xky be the weight a convo- lutional layer of input and output channels Cin and Cout, and kernel kip x kn. Denote E RR and l E RCone We extend the operators >> and v to products between vectors and 4-dimensional tensors and have,
Yoot E [1, Couell, Cin E [[, Cml, n[, E [1,m],
(A 7 () Deout Xeg, f and (p i> w consiftnn MMcoe Woott
(31)
With this definition, we keep the previous properties and enable chaining transformations on both convolutional and fully connected layers.
Equivalence with standard linear algebra. design the line-wise (V) and column-wise
(D>) notations for their intuitiveness. However, when applied to fully connected layers, they cquivalent to more common notations from linear algebra. Indeed, sticking to w E Riim À E R"" and μ R" with n, m 1, have that
V diag(A) and μa
diag(ı))
(32)
Furthermore, we can also write, with the Hadamard (elementwise) product between matrices,
(p w) = (AT x (1) ow.
(33)
These equivalences are important for implementing the log-log convex problem to minimize the L2 regularization term the space of the scaling symmetries (see Section 3.6).
E.1.2 PERMUTATION SYMMETRIES
Definition E.4. We define Pa as the set of permutations of vectors of size n. Pn. contains [Pxl = n! elements. The elements of Pa are doubly-stochastic binary matrices from {0, 1jxxn, Finally, for all πy E PR. T1 x ा. = 1n-
From this definition and property, we can directly derive the following result:
Property E.6. For all 0 E xm w E Rmmxn, and permutation matrix π Pass
Vo E RR", OTT x (π x ww) x r(ww)).
(34)
Proof. Let us take m.n N, Rmxn b RT x and π Pm permutation matrix corresponding to the bijective mapping in [1, m]. We have that,
( + w) x xx) = bo +
(35)
A=
Hence,
I x x x x)); b + Σ
(wo 21)
(36)
k=
= [r((b + x x));
(37)
and it comes directly that,
θTT x (π x wx) = x r(wx).
(38)
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This property is provided for the simplest case with only two matrices. However, it can be trivially extended to an arbitrary number matrices, chaining the different permutations. Just like for scaling symmetries, it follows that can extend the definition of permutation symmetries to deeper networks.
Moreover, for scaling symmetries in definition E.3, symmetries to convolutional layers and their 4-dimensional tensors. To do this, we consider that the permutation matrices the output and input channels and permute the whole kernels.
extend permutation operations and
E.1.3 SOFTMAX ADDITIVE SYMMETRY
For the sake of completeness, we also recall the additive softmax symmetries in this section. We left this symmetry apart in our analysis as it involves, at most, one degree of freedom.
Definition E.S. We recall that the softmax function o is defined by the following equation, given the logits a R
e
Vi E [L,n),((a):
(39)
e
Property E.7. For all a, R", R, and + the point-wise sum when applied between vectors and scalars,
a(a - b A)
a(a b).
(40)
Proof. Denote a(a + b X) and take in [1,n],
eatthit
S.
(41)
Σ
eaytbyt
f=1
S exarth
(42)
Σ Eagthy
S (0 + b));
(43)
E.2
THE GRADIENT THE LOSS AND SGD
Before coming the equivariance of the training operator, we start by proving the following lemma.
Lemma Let fa be a neural network trained with a loss L(O), the gradient of the loss V L is permutation equivariant, i.c.,
VIT n, VC(T,(0, (T
(44)
Please be aware that the notation T,(VL(0)) II) signifies that we apply permutation operation to the gradient of the loss function with respect to the weight 8.
Proof. Let fe be a neural network trained with a loss C(0) and II E П. We prove the case with II containing only Identity matrices except for the i-th matrix, that we denote π.
From property E.1, 8!'l( 2 C()) and al)= the activation after the layer |
have that 4Z
II, V WwgL((0)
5 (0) with We add the notation
of dependence of 8!l and alll in express that they correspond the gradient and the activation
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of the network of weights 0. We can adapt this result to the layer i and T(8.
V mwu ((T)(0, (T (1),0),)) 11))a(="1' (T)(0,11))
(45)
((w(it g (Tp I1)) o xxl Ta (0) al=-11' (O, II)
(46)
(Wii+2) g (T(O, Ii) o क V 2n) a a--11" a0'11"(Tp(00 (T)(O
(47)
(TW (0) πV x a (0) al=-1!"
(48)
(0)a--
(49)
= VVw1
(50)
This proof can be extended backward any II II as the action of π² and of cancel out for all the following weights.
We proved Lemma with an MLP, but this result extends convolutional layers. To do this, we need to define that the permutation matrices apply on the channels of the weight tensors.
To simplify the notations, we use neural networks and weights indifferently as arguments for the optimization and symmetry operators and 'T. For instance, we will simplify *(fo) by *()), implicitly assuming the architecture of fo.
Proposition 3. Let fog be randomly initialized neural network and *, the stochastic gradient descent operator applying times the update rule of learning rate For all steps E N, is permutation equivariant. In other words,
N, +. (T,(0,)) = (0),
(51)
We provide the proof below for multi-layer perceptrons, but it extends to convolutional neural networks.
Proof. Let E n be permutation set applied on a neural network fo trained with an objective & of gradient V.. Let define P(s) as follows:
P(s) "T, (+o (0o), x, (Tp (0,, 11))
(52)
We give proof that P(s) is verified for all E N by induction on Base case: P(0) is trivially true as *o (fer) fng hence
*0 (Tp (9o, II)) Tp (6o. II) Tp (+o (0o) II).
(53)
Induction step: Assume the induction hypothesis that for a particular k, the single case s = & holds,
meaning P(k) is true:
Tp (** (0o), II) = * (Tp (oo, IT))
(54)
Let us start with (+ (9o).II) II). We have that
(t)++11000,111
=Tp(0x+1,11)
(55)
=Tp(0k 7o VLVx)())
(56)
-Tp(xa(0)), II) - -7, (VL(((())))))))) II)
(57)
= =*x (Tp(0o), 11) 7o (T(x II))
(58)
=**+1 (T,(
(59)
Equation 58 uses the permutation-equiiaant operator until step k. Furthermore, equation 59 is exactly P(k: + 1), establishing the induction step.
of both the gradient (see Lemma 1) and the training
Since both the base case and the induction step have been proven, by induction, the statement P(s) holds for every natural number
Conclusion:
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We provide the proof of Proposition 3 with SGD for simplicity. However, it intuitively extends to SGD with momentum since the momentum is also permutation-equivare and updated in a permutation-eqqiiarian way. Furthermore, it also generalizes to (mini-)batch stochastic gradient descent, provided that the batches contain the same images.
While this also extends to the Adam optimizer (Kingma & Ba, 2015), this is most likely not true for new meta-learned optimizers such as VeLO (Metz et al., 2022).
E.3 EQCTPRORABILITY OF THE PERMUTATIONS
We build upon the results from the previous sections to devise the following proposition. Proposition 4. Let fe be neural network with initial weights independently and layer-wise identically distributed. The prabability of converging towards any symmetrically permuted network [o,So, II) given the dataset and training kyperparameters does not depend on the permutation set II. In other words,
p(O, (T( MID) = = = (D)
(60)
Proof. For simplicity, we assume an optimization scheme with early stopping and final weights 0g the end of step 8. Given Os, probability to converge to 0g after steps.
denote Og-s the space of initializations with non-zero
Let there be permutation set IT leaving fo, invariant. We want prove Equation 60, ie., that the probability to converge to ( 0,, II) given DD is the same as for 0, for any permutation set Il E IT.
We can inject information about the potential initialization points, defining 00-2 = {0) *, [0] = 0z)
as the pre-image of 0₂ by the optimization procedure with steps. It comes that
p(O, D) = p(0 E Po-s, O2 = =0. D).
(61)
Indeed, as we use the notation On E 0 for
p(0. E l0-s)= p(00 0)d0,
(62)
DEA TM
we can get, with the set of possible weights.
( p(e. = 0|0,- = 0,, D) = p(0) - 0 0(00= 00, D)d0 + p(0p = =00, = -8, D)d0 (63)
BEn
000
(64)
The left term is equal to and so is the middle term as, by definition of 0 it is impossible to have initialized the weights in Ω \ Po- and converge to 0g given the dataset D. Since
p(O, E 60- G, = 0.. D) = p(O, E f 0,, D)p(O, = 0,, D), we have Equation 61.
Furthermore, the definition of conditional probabilities implies that the probability of converging to 0, given D can be rewritten as
p(O.- 0,(D) p(0. P5ls, O, = D) p((O0 00-ss)
(65)
((D) p(00 E 00)-s)
p(O, Go ss O G., D) p(O0 B0mss)
[independence of O₂ and D)
p00 00-2; D)
(66)
= = 0, E 0 D)p(On E 00-s).
(67)
We can also apply Equation 67 for permuted network. Let there be permutation set II: Equation 67 is also valid for the permuted network (,, II) with its pre-image the set (0 II) (defined as (T, hoee ..). Indeed, Proposition 3 exactly proves that training a network starting from the permuted weights (6,. II) converges towards the permutation of the original weights (,, II).
p(O, (p(,,, M))) r(O, (O,,,, m) TT(00-s; IT), P)p(O (,000-s, M)) (68)
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Deterministic Training: Suppose the optimization algorithm is fully deterministic, given the initial- ization state. practice, the dataloader seed would be set to some value, and the backpropagation kernels would be chosen deterministic. In that case, the convergence to 0, is fully determined by the initialization O and the dataset DD ie.,
p - 0, E 0p- D) =
(69)
Non-deterministic Training: In the non-deterministic setting, corresponding in practice the order of the inputs in the batches as well as the stochasticity of backpropagation algorithms. We can marginalize the (eventually multivariate) sources of stochasticity & regain deterministic relationship.
p(0, 0. E 00 D) p(O, 0,100 E E
(70)
SEE
p(0) = Tp(O.. IT) E
(71)
EEE
= p(O) = (p.. m E (,(00-,,11)
(72)
In the case of deterministic training, the left term of Equation 68 is equal to 1 from Equation 69 with the permuted weights (0,, IT) and (0 IT).
For the right term, plO, E Tp M)) = p0 E fo-.), since the initial weights are layer-wise identically distributed and the permutations act separately inside the layers.
Finally, we have
p(O, = Tp(0,,11)/D) = p(0, = -D)
(73)
which is Proposition 4.
E.4 THEORETICAL INSIGHTS ON THE BAYESIAN POSTERIOR
We start with the following proposition, independent from Proposition 1, but still deemed interesting for the reader.
Proposition Let fw be neural network, æ an input vector, and y a scalar. We also denote Ω the space of sorted weights, and 01 remains the set of possible permutations for fw. We have
pyy x. D) = [
ply x. w(w D)dw.
(74)
WE
Proof. Let fao be neural network. We recall the marginalization on the network's weights:
ply x. D) = /
plu x w)w D)dw.
(75)
wEn
We can split the space Ω and get Ω Ω U U Ωz.. with 9z, the space of the weights ordered
{E[L,|]]]
and permuted according to and 22 the space including corner-cases: the networks with least twice the same weights in layer. In our case, Ω is theoretically of measure zero. In practice, we note that the reals are discretized on computer, and the edge cases may happen with an extremely small probability, decreasing with the chosen precision. Denote Ω the space with descending ordered weights, we have
[n] Σ
/ WER
p(y x. D)
ply x. Tp (T)pp(TT,,) D)dw.
(76)
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By definition of the symmetry operator, know that it does not affect the prediction of the transformed network:
[E ply a. (T(,w,T))) = p(y a, w)
(77)
Moreover, Proposition 4 provides
Vi,J E [n] P(T((w, D) = p(w (D)
(78)
It follows that
/ WEN>
ply x. D)
[n|
ply x,w)p(w D)dw.
(79)
Then, recall Proposition I of the main paper:
Proposition. Define fw neural network and w its corresponding identifiable model a network transformed for having sorted unit-normed neurons. Let us also denote In and A, respectively, the sets of permutation sets and scaling sets, and Q the random variable of the sorted weights with unit norm. The Bayesian posterior of neural network fw trained with stochastic gradient descent can be expressed as a continuous mixture of discrele mixture:
p(2 = w | = 1171-1 / Σ
p(Ω () D)dA.
(80)
AEA ner
Proof. Let fu be a neural network and start with the right term. We can denote by and Á the only couple of permutations and scale such that the weights Tp(Ta Å), Iī) are sorted and unit-normed. Given that Ω is also sorted and unit-normed, II and Á are the only supports of the respective sum and integral. Therefore, we have that
n
Σ NEm
A), D)dA |n] AT, D). (81)
AEA
In parallel, with ت the sorted unit-normed weights associated to w, i.e., i = ( (A,
p( w|D) p(- = ão, A\D)
(82)
=p(Ô = pp p ش A()).
(83)
(84)
Indeed, IT and are independent, and p(II) is determined by the initialization of the neural network that we suppose layer-wise identically and independently distributed. We obtain the Proposition directly when replacing w by a (A,
We then establish the following corollary in the case of network of fully connected layers. This extends to convolutional networks, paying attention that the identical distributions are multidimensional (and of dimensions the surface the convolutional kernels).
Corollary 1. Let fw be aneural network containing L layers. For any layer E [], for any input neuron and oulput neuron j.
p(w (D) = p(w]!)))) DD).
(85)
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Proof. Let us take input neuron and output neuron j. We have, from Proposition and integrating over the whole space except w!!) and switching integrals,
p( D)
E Mtem
p(O TT(T((w,A),)) D)dAdw
(86)
AEA
p(w!)
D)
p(î A., D)dAdw
(87)
AEA
TTe
WELtWO
p(w!) D) = p(woi DD).
(88)
E.5 THE MINIMUM SCALED REPRESENTATION COST PROBLEM
In this section, we develop the formalism leading to the proof of the log-log strict convexity of the minimum scaled representation cost problem (or scaled-repressnntttin problem). We start by recalling its definition.
E.5.1 DEFINITIONS
Definition E.6 (Minimum scaled representation cost problem). Let fo neural network and its weights without the biases. We define the minimum scaled representation cost problem or scaled-representatiin problem as the minimization of fo 's L2 regularization term (the "mass") under invariant scaling transformations, In other words,
m
min AE
AO).A)
(89)
We also denote the mass of the neural network fo as m m(fe) = 10
An extension of this problem, referred as "representation cost" has been tackled by e.g. Jacot (2022). However, our slightly more limiting context allows us to derive more powerful results, holding for real-world neural networks.
In this work, we are especially interested in the versions of this problem that are not degenerate. This condition was not given the main paper for conciseness, but please note that is not restrictive in real-world cases and could be appropriately handled in the theory with some caution.
Definition E.7. We say that the craled-representtrin non-degenerate if, taking any A A, we have that for all scales À A, at least one term multiplied by each X, that is non-zero.
problem of the network neural network fo is
Degenerate problems will tend to setting the weights of a layer to the smallest value possible, meaning that their is no solution to the corresponding scaled representation cost minimization. In this case (notably for the ResNet-18) we remove these layers from the computation of the cost and set the corresponding weights to O.
E.5.2 COMPUTING THE MASS OF THE NETWORK
For a neural network without skip-connections and constituted of linear and convolutional layers only, first provide definition and
devise an important property:
Definition E.8. Let W be the weight of linear convolutional layer. We denote M(W), the ma- trix representation of the mass of the layer. For linear layers, we have, with o, the Hadamard product,
M(W)= = W o W,
(90)
and for convolutional layers of kernel size (h, v).
M(W) Σ (W W) [,
(91)
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Property E.8. Let fe be L-layer neural network of weights 0 = -[Wl),b hee. The mass of the neural network scaled by A E A with A = (A1hee and Xo Xr. can be expressed as, with ( the sum of all elements of W.
m (fT,(ON))
Σ sum AP V
\ \ M (wl
(92)
This result can be extended to networks with batch normalization layers and residual connections. To build the convex problem out of this formula, we use the connection between the line-wise and column-wise products and linear algebra, as presented in E.I.I.
E.5.3 LOG-LOG CONVEXITY
Finally, we have the following core property on craleddrepresentattion problems:
Proposition 6. Let So be a neural network. The corresponding minimum scaled network represen- lation cost problem is log-log strictly convex, provided it is non-degenerate. As such, the problem is equivalent to a strictly convex problem on RAA and admits a global minimum attained in a single point denoted A
Proof. Let Sa be an L-layer neural network of weights 0 = {Wi1, 6D) hEIL,L We recall the corresponding represtrprrssnntation problem:
P min (fT.(OA)))
(93)
AE
With Property E.8, we have, for A E A with A = (Acheet) and Ao AL =1
sum A v ( D M (w"))))
P= min AEA
(94)
Let us now write Uy = log(A,) for all l [1,L l1. This possible because due to the non-negative homogeneity of the ReLU activation, the A already chosen with elements in Rzo- Please note that the need for this change of variables is why P will be referred to as log-log strictly convex. We have,
(2 v (22u M (wil))
P min NEA
sum
(95)
We can convert our operators products of diagonal matrices and Hadamard products using the formulae from E.1.1. It comes that,
sum diag (eawx) M (wil))
P min AEA
(96)
sum (( (uir) M (w!li)
(97)
-sun (M (wlr)) diag (e (c-2me-1)
(98)
In Equation 98, we see that is equivalent sum of convex and strictly convex functions. Indeed, the product ?uT 2u, can be expressed a matrix containing elements of the form (((w)). Therefore, P is equivalent to a strictly convex problem. It comes that P has, most, single solution. Moreover, since the elements of the ui are defined on R and the inner function is infinite at infinity (as a sum of positive terms with at least one of them diverging due as the problem non-degenerate), we have that has a solution, which unique.
This proof can be easily extended to batch normalization layers that include mne-dimnnsionl weights. The extension to residual networks is a bit more complicated, as one should probably resort to graphs write the equations properly. We then prove the corollary from the main paper.
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Corollary 2.
Let fo be neural network trained with L2-regularization Suppose the mass of fe is not optimal in the sense of the minimum scaled network representation cost problem. In that case, there is an infinite number of equivalent networks Tp(fo:A) A) with training loss lower than that of the original network.
Proof. Let fo be a neural network and denote the training loss by L(0) Lm (0) + (L) with (0) = m([o) the L2-regularization term. We have, by definition of the symmetry operator Ta
AA E A, (0) = L. (T,(0, A))
(99)
Take A as the optimal scaling set, LL)( is continuous everywhere in as a sum of products of continuous functions and hence continuous in A E A. Therefore, around A", where AA E L, Cm (0, A) C. (T A*). Hence the corollary.
find an open ball B
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F NOTATIONS
We summarize the main notations used in the paper in Table 5.
Table 5: Summary of the main notations of the paper.
Notations
Meaning
8
is element the possible weights/ttyoothesss !
fw
a network weights
D [(x, 9))
the set of [D] data samples and the curresponding labels
7 (ww)
transformation of the weights that leaves the corresponding neural network functionally invariant the column- wise product between veclors and matrices (see E.2)
V
the line-wise product between vectors and matrices (see B.2)
the non-linear ReLU activation (the positive part function)
the softmax activation function
ACACA
À is vector belonging to set of scaling parameters
among all possible sets of scaling parameters A for a given neural network architecture
T((w)
scaling symmetry of the weights of parameter A
x Illl
is a matrix belonging set of permutation matrices 1l among all possible sets of permutation matrices A for given neural network architecture
Th(w)
permutation symmetry of the weights of parameter II
P.
the set of permutation matrices of size
C(w)
training loss of neural network weights
m(fa) or m(w)
"mass" representation of hypothesis (scc E.6)
W.b
weights biases of layer.
diag
a function that maps vector to corresponding square diagonal matrix the entropy
H
48


Reviews:
[{"review_id": "A83hkkkQGW", "reviewer": "Reviewer_cm11", "summary": "this paper conducts a large-scale exploration of the posterior distribution of deep bayesian neural networks (bnn), focusing on real-world vision tasks and architectures. it highlights the critical role of weight-space symmetries, particularly permutation and scaling symmetries, in understanding the bayesian posterior. the paper introduces new mathematical formalisms to elucidate these symmetries' impacts on the posterior and uncertainty estimation. it evaluates various methods for estimating the posterior distribution, using maximum mean discrepancy (mmd) to assess their performance in capturing uncertainty. a significant contribution is the release of a large-scale dataset comprising thousands of models across various computer vision tasks, facilitating further research in the field of uncertainty in deep learning. the study investigates the proliferation of modes due to posterior symmetries and the tendency of ensembles to converge towards non-functionally equivalent modes. it also discusses the influence of symmetries during the training process. the findings suggest the complexity of bayesian posterior can be attributed to the non-identifiability of modern neural networks, viewing the posterior as a mixture of permuted distributions​.", "questions": "1. regarding originality in symmetry analysis: the paper provides an original analysis of weight-space symmetries in bayesian neural networks. can you elaborate on how these findings might extend or differ from the insights provided by existing studies on dnn loss landscapes, such as those by li et al. (2018) and fort & jastrzebski (2019)? specifically, how do these symmetries manifest differently in bayesian networks compared to traditional dnns?\n\n2. on comprehensive methodological approach: could you provide practical examples or case studies demonstrating how your results can be applied to actual neural network design or optimization problems?\n\n3. regarding empirical validation with a large-scale dataset: the release of a dataset including thousands of model weights is a significant contribution. however, the focus seems to be on vision tasks primarily. how might the findings and applicability of your research differ when applied to other types of neural network architectures or non-vision tasks?\n\n4. on insights into posterior symmetries and model performance: your paper suggests the complexity of bayesian posteriors can be attributed to the non-identifiability of modern neural networks. how might this insight impact the current practices in model evaluation, particularly in fields where high certainty and reliability are crucial?\n\n5. regarding advancing uncertainty quantification: could you discuss the limitations or challenges you faced using maximum mean discrepancy (mmd) to evaluate posterior estimation methods? are there scenarios where mmd might not be an effective measure, and if so, what alternatives would you recommend?\n\n6. on limited scope in empirical validation: given the paper's focus on specific architectures and vision tasks, are there plans to extend this research to include a wider variety of neural network types and tasks? how do you foresee the findings changing with different architectures or in different application domains?\n\n8. on need for further exploration of symmetries in training: the paper touches upon the influence of symmetries during the training process. could you detail any specific strategies or methodologies that could be employed during training to either leverage or mitigate these symmetries?\n\n9. regarding lack of broader contextualization: how do the findings of your paper align or contrast with other approaches in deep learning and uncertainty quantification? can you provide insights into how your research contributes to or diverges from the broader landscape of deep learning research?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. originality in symmetry analysis: the paper's exploration of weight-space symmetries, particularly permutation and scaling symmetries, in deep bayesian neural networks is highly original. it builds upon existing studies of dnn loss landscapes, like those by li et al. (2018) and fort & jastrzebski (2019), and extends these concepts to the bayesian context. the distinction between permutation and scaling symmetries and their unique impacts on the posterior is a novel contribution (section 1)​​.\n\n1. comprehensive methodological approach: the authors leverage a mathematical formalism to highlight symmetries' impacts on the posterior and uncertainty estimation in deep neural networks, adding theoretical depth (section 1). it facilitates a more nuanced understanding of the bayesian posterior, bridging a gap in current literature​​.\n\n1. empirical validation with a large-scale dataset: the release of a comprehensive dataset including the weights of thousands of models is a substantial contribution. it not only validates the paper's findings but also enables the community to conduct further research and empirical validation across various computer vision tasks (section 4). this dataset is a practical resource that can catalyze future advancements in uncertainty quantification in deep learning​​.\n\n1. insights into posterior symmetries and model performance: the paper delves into the complexity of bayesian posteriors, attributed to the non-identifiability of modern neural networks. this perspective, viewing the posterior as a mixture of permuted distributions, is a significant insight. it aligns with recent works that discuss the intricacies of dnn architectures and their implications on model performance (e.g., entezari et al. (2022))​​.\n\n1. advancing uncertainty quantification: the study's focus on uncertainty quantification, specifically how symmetries affect uncertainty estimation, is a major strength. the use of maximum mean discrepancy (mmd) to evaluate the quality of various posterior estimation methods on real-world applications is a rigorous approach, contributing to a better understanding of uncertainty in deep learning (section 4)​​.\n\np.s. really nice paper! was a joy to read.", "weaknesses": "1. limited scope in empirical validation: while the dataset released is extensive, the paper's empirical validation primarily focuses on vision tasks and specific architectures like resnet-18. this raises questions about the generalizability of the findings across different types of neural network architectures and tasks. expanding the empirical validation to include a broader range of architectures would strengthen the findings.\n\n1. need for further exploration of symmetries in training: the paper discusses the influence of symmetries during the training process but does not delve deeply into this aspect. further exploration of how these symmetries manifest and can be leveraged or mitigated during the training process could provide valuable practical insights for model development.\n\n1. lack of broader contextualization: while the paper does an excellent job of situating its contributions within the immediate field of bayesian neural networks, it could benefit from a broader contextualization. this includes discussing its findings in relation to other approaches in deep learning and uncertainty quantification, and how these approaches diverge from the paper’s findings.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "pikIwWJjAF", "reviewer": "Reviewer_QYDw", "summary": "The paper presents a study of weight-space symmetries in Bayesian neural networks (BNNs). The authors propose mathematical definitions of permutation and scaling symmetries and define the BNN posterior as a mixture of these. They further present a new way to define a unique network among all scaling-symmetrical candidates. \n\nThe paper further contains an empirical study of BNN posteriors. Here, the authors evaluate a selection of posterior approximation methods for their approximation quality on classification and OOD detection tasks, study the functional diversity among ensemble members, as well as investigate how often the BNN parameters permute during training.", "questions": "1. What is the link between the formalism of section 3 and the experiments in sections 4 and 5?\n2. I don't fully understand figure 2. What is on the second axis in the left plot, what does the middle plot show, and what is the difference between \"opt.\" and \"convergence\" in the right plot?\n3. In section 4.1, you write \"For better efficacity, MMDs are computed on RKHS that allow for comparing all of their moments.\" What does this mean?\n4. If I understand the paper correctly, the estimates of the BNN posteriors come from the parameters of a 1000-member ensemble trained using maximum likelihood. Can we be sure this is a faithful representation of the posterior? For instance, do we know anything about the higher-order moments of those modes?\n5. Related to the previous question, if the posterior is indeed approximated by 1000 ML solutions, does the MMD metric even make sense for any of the single-mode methods? One could argue that single-mode methods aren't meant to describe the posterior adequately, but the hope is that their predictive distributions aren't too affected by the single-mode limitation.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "1. I find the mathematical treatment of the scaling and permutation symmetries, as well as their contribution to the posterior, quite exciting. It may not be overly novel, but the exposition is interesting and inspiring.\n2. The empirical analysis is exciting. The study of the different posterior approximations provides a good overview of the effects of the approximations, and it feels like there's even more to be gained from Table 1 than what the authors discuss. Furthermore, section 5 appears quite novel and contains some fascinating insights.\n3. The authors promise to release a dataset of estimated posteriors for a large number of models, which could be of great value to the community.\n4. The paper overall appears original and of high technical quality and should be of great interest to the ICLR community.", "weaknesses": "1. The paper seems to lack a specific focus. While the theoretical and experimental parts of the paper are both exciting, they seem largely disconnected. The developed mathematical formalism does not appear to be used for anything, and it is even unclear what these definitions buy us in terms of understanding BNN posteriors. The experiments in section 4 do not appear to consider symmetries at all, and whereas those of section 5 do, the link to the definitions in section 3 is unclear. To me, the paper seems to be compiled from two distinct papers, each of which would be interesting to read.\n2. The clarity of the paper could be improved. The figures (1 and 2 in particular) are not explained in sufficient detail to understand what they show, and the text itself is sometimes unclear.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ZKrqX2p4ZM", "reviewer": "Reviewer_Mjnv", "summary": "This work explores Bayesian neural networks and how their geometry is affected by weight space symmetries such as permutations and re-scaling. The authors show that the posterior can be expressed as a mixture over such symmetries, highlighting the redundant structure in such distributions. The authors then explore the effect of the scaling symmetry in more detail, showing that even with weight decay, the symmetry persists. Finally, using the gained insights, a rich exploration of different approximation methods is performed, and several metrics are studied to assess the quality of the resulting posterior. Multi-modal approximations generally offer the closest approximation in terms of maximum-mean discrepancy and best test performance, while surprisingly, single-mode approximations offer the best uncertainty predictions.", "questions": "1. Why is your MNIST performance so low? As far as I know even a simple MLP with 2 hidden layers can easily reach $\\approx 97\\%$ performance.\n2. Are you employing data augmentation in these experiments? And relatedly, are you tempering the posteriors, i.e. performing a grid search over temperatures? I’m asking since the cold posterior effect [6] could be a potential confounder here for test performance, as different approximations might be influenced differently.\n\n\n[6] Wenzel et al., How Good is the Bayes Posterior in Deep Neural Networks Really?", "limitations": "", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. Connecting Bayesian neural networks with the recently obtained insights into the symmetries of the loss landscape of standard neural networks is a timely contribution and highly relevant to the field. The work is very well-written and mostly easy to follow. \n2. The experiments are impressive and on a very large-scale, which is often missing in the BNN literature. I especially appreciate the different datasets investigated in this work. I also find the separation into single and multi-mode approximations very interesting, highlighting once more that deep neural networks really possess the multi-modal structure that they’re typically associated with.\n3. I also appreciate the more in-depth investigation of the scaling symmetry, something that usually receives less attention compared to the more prominent permutation symmetry. I find it surprising that this symmetry persists to a good degree, even if weight decay is employed.", "weaknesses": "1. I think the first part of the paper regarding the symmetries is a bit extensive and insights from it are somewhat limited. To the best of my knowledge, it has been known for a while that the BNN posterior effectively consists of a mixture over such symmetries, so I don’t see much novelty in equation (8). While it sets up the subsequent discussion very nicely, I wouldn’t advertise this as one of the core contributions of this work. The in-depth study of the role of the scaling symmetry on the other hand is novel to my knowledge and fosters a better understanding.\n2. An important aspect of the experimental setup, which is however almost not discussed at all, is the choice of “gold standard” for the posterior, to which all other approximations are compared against. This choice is of course crucial for the empirical evaluation! If I understand correctly, the authors chose a deep ensemble consisting of 1000 members. I’m a bit unsure regarding this choice, as it is quite controversial to what degree deep ensembles really implement an approximation to the Bayesian predictive distribution. There are several works arguing for yes [1, 2], while others oppose this view (to some degree) [3, 4]. [4] even argues that repulsive terms are needed to ensure that deep ensembles perform proper Bayesian inference asymptotically. I don’t think that this questions the validity of the experimental setup too strongly, but a proper discussion of this choice and differing view points **have to be discussed** such that readers can make up their own mind regarding this discussion.\n3. While [2] is cited, the work really lacks a comparison to it. [2] also performs an empirical exploration of Bayesian posteriors and how other approximate methods perform, where the gold standard is full batch HMC. They only evaluate on one dataset (CIFAR10) but I still think this should be compared carefully, and how this work distinguishes itself from [2] should be properly stated. [2] also published their HMC checkpoints, I would really be interested in seeing whether the produced deep ensemble in this work produces a similar posterior. This could also clarify the discussion raised in point 2. \n4. The work is not very careful at introducing a lot of concepts formally. As maximum-mean-discrepancy comparisons are one of the core contributions of this work, I think it would be worth it to formally introduce it in the main text. You also report a metric called NS, which is only described in the caption of Table 1, and I could not find a description in the main text nor a pointer to the Appendix, detailing how the “removal of symmetries” is actually performed. I find this quite an interesting point and I would like to see more discussion regarding how this removal enables a better comparison between different posteriors. Glancing at the numbers, there seems to be no consistent pattern (i.e. MMD sometimes goes up, and sometimes down after removing the symmetries) which is surprising as I would have expected to always see a decrease. Could you elaborate more on this? The multi-modal extension of some of these algorithms is also not clear. Are you simply averaging over all the samples coming from the different modes?\n5. The uncertainty estimation results are surprising as the single-mode posteriors consistently perform better. The employed measure ECE has seen a lot of criticism recently [5] so I’m wondering whether a similar problem is at the origin here. Comparing multi-modal and single-modal is maybe also a bit unfair, as the testing accuracies vastly differ.\n\n\n[1] Wilson and Izmailov, Bayesian Deep Learning and a Probabilistic Perspective of Generalization\n\n[2] Izmailov et al., What Are Bayesian Neural Network Posteriors Really Like?\n\n[3] D’Angelo and Fortuin, Repulsive Deep Ensembles are Bayesian\n\n[4] He et al., Bayesian Deep Ensembles via the Neural Tangent Kernel\n\n[5] Ashukha et al., Pitfalls of In-Domain Uncertainty Estimation and Ensembling in Deep Learning", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "qFc5f0tF2Y", "reviewer": "Reviewer_24pF", "summary": "The paper sets out to study the Bayesian posterior of deep neural networks. It hypothesizes that the quality of uncertainty quantification made possible by a Bayesian treatment is reduced due to the presence of scaling and permutation symmetries in the weights of the network. It presents a formal definition of permutation and weight symmetry. It evaluates the quality of typical Bayesian methods for estimating the posterior distribution on computer vision tasks and presents experimental results on “functional collapse” and weight permutation events in the process of training.", "questions": "- What do you mean by ‘regroup some points around the modes’ in section 3.1? What causes this effect and how is it problematic?\n- How is the “target distribution with 1000 checkpoints” for table 1’s MMD scores established? What method is used to determine this distribution?\n- I am not sure what to take away from Table 1 and section 4.1-4.3. The study of uncertainty quantification performance of these methods has been done before, how does this experiment connect to the formal study of scale and permutation symmetries?\n- Similarly, how does the study of ‘functional collapse’ in Figure 3 and section 5.1 suggest that  “complexity of the posterior is orders of magnitude higher than what we understand, taking symmetries into account.” I don’t follow how symmetries were taken into account in this study, and what to take away from this to improve the Bayesian treatment of neural networks. It appears to me that the symmetries in the neural networks studied here were left untreated, and no interventions were made to reduce the complexity of the Bayesian posterior through these symmetries.\n- For section 5.2, what is the insight gained from the study of weight permutation? Is “a variation in the Πs” somehow problematic? How does this inform the design of Bayesian deep learning methods?\n\nTypo/nits:\n- “non-functionnally”  (bottom page 2)\n- “influencial” (start section 3)", "limitations": "", "rating": 3, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 1, "strengths": "The paper introduces a formal treatment of scaling symmetries, and introduces the notion of a `minimum scaled network`. The work demonstrates that the optimization problem to reach this minimum-scaled network is convex. They then demonstrate that on their specific small mnist model (OptuNet), a specific neural network optimization scheme with weight decay does not reach the optimum (in terms of L2 loss) achieved by further convex optimization of the L2 norm under scaling transformations.", "weaknesses": "The paper as a whole presents a meandering set of concepts and results, without drawing conclusions or presenting insights into further study of Bayesian method design. This is apparent from the conclusion section, which is vague and raises questions: what are the actual insights provided by considering symmetries? What questions and hypotheses would you consider in future work? What was the “real impact of scaling symmetries”?\n\nThe core experiment demonstrating that L2 regularisation does not remove weight scale symmetries is presented in Figure 2, but this involves a single, custom network architecture under a single specific optimization scheme. It would be insightful to study this under various benchmarks and models to demonstrate that this effect is consistent and that the difference between the minimum reached by SGD with L2 regularisation and the minimum reached by further convex optimization is significant and actually affects the quality of Bayesian posteriors.\n\nThe experiments presented do not provide evidence for the claim presented in the method section: \"SYMMETRIES INCREASE THE COMPLEXITY OF BAYESIAN POSTERIORS\" (see questions below). Nor do they study interventions informed by the theory to evaluate the benefit of the formal treatment of scale and permutation symmetries.\n\nThe reported accuracies in Table 1 are low. While this is understandable for the purposefully restricted OptuNet, a vanilla single-mode baseline resnet-18 model on cifar-100 can reach 75% accuracy without any Bayesian treatment, and with some tuning 79%. (https://paperswithcode.com/sota/image-classification-on-cifar-100 https://huggingface.co/edadaltocg/resnet18_cifar100). It seems the results and the presented checkpoint dataset represent an artificially underfitted model space. If the aim is to improve Bayesian deep learning for a real-world setting, it might be beneficial if the models studied are optimized to a reasonable performance.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "YaPfjTd9KW", "reviewer": "Reviewer_hdLm", "summary": "The paper formalizes permutation and scaling symmetries in Bayesian neural networks (BNNs), proposing to express the posterior as a mixture density with components corresponding to symmetric copies of the underlying identifiable model. Investigating scaling symmetries, the authors find the minimization of an L2 term over scaling transformations to be a convex problem. In an extensive benchmark study, they shed light on the benefit of diversity realized by multi-mode methods, the opacity of the link between in-distribution diversity and out-of-distribution (OOD) performance, and the frequency of weight permutation during training. A large codebase is to be released.", "questions": "* [Q1] Could you elaborate on the role of $\\theta$ in Property 3.1, Property 3.3?\n* [Q2] What kind of objects are $\\Pi$ and $\\mathbb{\\Pi}$?\n* [Q3] Does the posterior expression in Eq. 8 allow for other types of symmetries?\n* [Q4] What is your intuition on scaling symmetries being always present even with weight decay? Could this be because an infinite number of scaling transformations exists that leaves the (optimal) L2 norm unchanged?\n* [Q5] Why are single-mode methods competitive in terms of ECE? Is there a tendency towards either over- or underconfidence?\n* [Q6] Could you provide some intuition about the similarity of two networks having a MI of, say, 0.1?\n* [Q7] What is your idea behind connecting functional collapse to OOD performance?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "* [S1] **Originality**. Expressing the BNN posterior as a mixture w.r.t. two types of symmetries seems novel and compelling, as does posing the minimization problem of L2 regularization over scaling transformations. The empirical study of the frequency of weight permutations also adds an interesting new perspective.\n* [S2] **Formalization**. Introducing symmetry operators and the mixture posterior is beneficial to the field of BNN symmetries that so far lacks a comprehensive language.\n* [S3] **Contextualization**. The authors relate established concepts like functional collapse and OOD generalization to the quality of the posterior approximation (in the light of symmetries).\n* [S4] **Empirical evidence**. Substantive experimental results are provided and the authors are to release a broad benchmark dataset.", "weaknesses": "* [W1] **Lack of clarity**. There are some statements whose scope/explanation is not quite clear to me:\n  * “The scaling operation (...) is a symmetry for all neural networks” (Property 3.2); “neural networks seem to always be subject to scaling symmetries” – I feel this should be restricted to ReLU-type nonlinearities. Why does this hold in general?\n  * More generally about the *min-mass* problem: What, exactly, is the implication? Do we impose a second, ex-post minimization problem over the solution of standard training? How comes the problem is convex if existing literature on [representation costs](https://openreview.net/pdf?id=6iDHce-0B-a) (which tackles the same problem and is not cited?) turns out not to be a convex problem?\n  * “All the layer-wise marginal distributions are identical” (Proposition 1) – how so?\n* [W2] **Conflating Bayesian and risk-minimization perspectives**. The paper is supposed to be about BNN posteriors, yet most claims and analyses focus on standard network training. It would be useful to elaborate more on both perspectives, or reconcile them. E.g., what do Definition 3.3 / Proposition 2 mean for a Gaussian prior equivalent to L2 regularization? As-is, the impact of the paper for actual Bayesian inference remains unclear.\n* [W3] **Unclear baseline / omissions in experiments**. Somewhat related to [W2]: I don’t quite understand how the “true” posterior distribution, w.r.t. which the MMD is reported, is computed. How, exactly, is the RKHS constructed? Also, I feel that the Bayesian perspective is quite under-represented with only one MCMC method (the other methods present have no guarantees of retrieving the true posterior).\n* [W4] **Minor points**. Not relevant for my score, just to point out:\n  * Abbreviations are not always handled consistently (e.g., HMC not introduced in 2; “deep neural networks” in 3.2 despite abbreviation introduced before).\n  * Figure 1 should have axis labels.\n  * Figure 3 (right) has very low contrast.\n  * P. 2, second-to-last line: “functionnally” (should be “functionally”)\n  * P. 4, first line: “influencial” (should be “influential”)\n  * P. 4, Property 3.1: “all neural network” (should be “all neural networks”)\n  * P. 5, second line: “perceptions” (should be “perceptrons”)\n  * P. 7, second line: “efficacity” (should be “efficacy”)\n  * P. 8, third line under 5: “weights permutations” (should be “weight permutations”)\n  * $\\omega$ is not always bold.\n  * Describing Laplace methods as “characterizing the posterior distribution” is pretty vague.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (poster)