On Divergence Measures for Bayesian Pseudocoresets
Authors: Balhae Kim
Jungwon Choi
Seanie Lee
Yoonho Lee
Jung-Woo Ha
Juho Lee
Venue: NeurIPS 2022 Accept (2022)
Keywords: Bayesian pseudocoresets
Dataset distillation
Bayesian inference
Divergence measures

Abstract: A Bayesian pseudocoreset is a small synthetic dataset for which the posterior over parameters approximates that of the original dataset. While promising, the scalability of Bayesian pseudocoresets is not yet validated in large-scale problems such as image classification with deep neural networks. On the other hand, dataset distillation methods similarly construct a small dataset such that the optimization with the synthetic dataset converges to a solution similar to optimization with full data. Although dataset distillation has been empirically verified in large-scale settings, the framework is restricted to point estimates, and their adaptation to Bayesian inference has not been explored. This paper casts two representative dataset distillation algorithms as approximations to methods for constructing pseudocoresets by minimizing specific divergence measures: reverse KL divergence and Wasserstein distance. Furthermore, we provide a unifying view of such divergence measures in Bayesian pseudocoreset construction. Finally, we propose a novel Bayesian pseudocoreset algorithm based on minimizing forward KL divergence. Our empirical results demonstrate that the pseudocoresets constructed from these methods reflect the true posterior even in large-scale Bayesian inference problems.

On Divergence Measures for Bayesian Pseudocoresets
Balhae Kim', Jungwon Choi', Seanie Lee', Yoonho Lee", Jung-Woo Ha Juho Lee
KAIST', Stanford University?, NAVER AI Lab", AITRICS (balhaekim, jungwon. choi, Asnfamily02))kaiii 0kaist. ac ac.kr,
yoonho@stanford edu,, jungwoo hafnavercorp com, juholee@kaist. acc ac.kr
Abstract
A Bayesian pseudocoreset is small synthetic dataset for which the posterior over parameters approximates that of the original dataset. While promising, the scalability of Bayesian pseudocoresets is not yet validated in realistic problems such as image classification with deep neural networks. On the other hand, dataset distillation methods similarly construct small dataset such that the optimization using the synthetic dataset converges to a solution with performance competitive with optimization using full data. Although dataset distillation has been empirically verified in large-scale settings, the framework is restricted to point estimates, and their adaptation to Bayesian inference has not been explored. This paper casts two representative dataset distillation algorithms approximations methods for constructing pseudocoresets by minimizing specific divergence measures: reverse KL divergence and Wasserstein distance. Furthermore, provide unifying view such divergence measures in Bayesian pseudocoreset construction, Finally, we propose novel Bayesian pseudocoreset algorithm based minimizing forward KL. divergence. Our empirical results demonstrate that the pseudocoresets con- structed from these methods reflect the true posterior even in lige-dimensional Bayesian inference problems.
Introduction
One of the main ingredients for modern statistical machine learning is large-scale data, which enables the learning of powerful and flexible models such as deep neural networks when handled correctly [3, 32]. However, training a neural network on larger datasets usually requires many nontrivial choices, ranging from architecture choice, hyperparameter optimization, and infrastructure challenges [33]. Aside from requiring enormous computing resources for training. large-scale data often contains sensitive information that must not be shared publicly [11]. This motivates the construction of coresets [29, 14], a sparse subset of large-scall scale dataset that summarizes essential features of the original dataset. In the context of Bayesian inference, the posterior conditioned on an essential coreset should be similar the exact posterior conditioned on the full dataset.
Coreset construction becomes more difficult with high-dimensional data. A previous work [19] shows that the best coreset becomes suboptimal at large scales: the divergence between the posterior conditioned on the optimal coreset and the exact posterior grows with the dimension of the data. Motivated by this weakness of strict coresets, Manousakas et al. [19] proposes to construct Bayesian pseudocoreset: a synthetic dataset represented as a set of learnable parameters and trained to minimize the divergence between the coreset posterior and the full-data posterior. Compared to coresets, pseudocoresets scale much better with data dimension and achieve significantly lower posterior approximation error. However, the KL divergence minimization in pseudocoreset learning requires the construction of approximate posteriors on the fly during the learning process, making this approach hard apply high-dimensional models such as deep neural networks.
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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On the other hand, dataset distillation [28] aims to distill a large-scale dataset into small synthetic dataset such that the test performance of a model trained on the distilled data is comparable to that of model trained on the original data. Such methods have achieved remarkable performance in real-world problems with high-dimensional data and deep neural networks. Although dataset distillation methods have a similar motivation to pseudocoresets, the two methods optimize different objectives. Whereas pseudocoresets minimize the divergence between coreset and full-data posteriors, dataset distillation considers a non-Bayesian setting with heuristically set objective functions such as matching the gradients loss functions [36] or matching the parameters obtained from optimization trajectories [8]. Such objectives have direct Bayesian analogue due to their heuristic nature, and more theoretical grounding can advance our understanding of dataset distillation and the empirical performance of Bayesian pseudocoreset methods.
In this paper, we provide a unifying view of Bayesian pseudocoresets and dataset distillation to take advantage of both approaches. We first study various choices of divergence measures as objectives for learning pseudocoresets. While Manousakas al. [19] minimizes the reverse KI. divergence between the pscudocoreset posterior distribution and the full data posterior distribution, we show that alternative divergence measures such as forward KL divergence and Wasserstein distance are also effective in practice. Based on this perspective, we re-interpret existing dataset distillation algorithms [36, 8] approximations to the Bayesian pseudocoresets learned by minimizing reverse KL and Wasserstein distance, with specific choices of variational approximations for coreset posteriors. This connection justifies the heuristically chosen learning objectives dataset distillation algorithms, and at the same time, provides thcoretical background for using the distilled datasets obtained from the such procedure for Bayesian inference. Conversely, Bayesian pseudocoresets benefit from this connection by borrowing various ideas and tricks used in the dataset distillation algorithms to make them scale for complex tasks. For instance, the variational posteriors we identified by recasting dataset distillation algorithms as Bayesian pseudocoreset learning can be used for Bayesian pseudocoresets with various choices of divergence measures, Also, we can adapt the idea of reusing pre-computed optimization trajectories [8] which have already been shown to work large scales.
We empirically compare pseudocoreset algorithms based on three different divergence measures on high- -igh-nimensional image classification tasks. Our results demonstrate that can efficiently and scalably construct Bayesian pseudocoresets with which MCMC algorithms such as Hamilto- nian Monte-Carlo (HMC) [10, 22] or Stochastic Gradient Hamiltonian Monte-Carlo (SGHMC) [9] accurately approximate the full posterior.
Background
2.1 Bayesian Pseudocoresets
Denote observed data as x = (o 1N Given a probabilistic model indexed by a parameter 0 with some prior distribution To. we are interested in the posterior conditioned on the full data x,
1 exp Σ f(zn, 0) To( = Z(x)
(0)= = Z(x)
exp (x]f(x,0)) ())()))
(I)
where f(x, 0) = (5 [.....,,(NN, 0))T. N is the N-dimensional one vector, and (xx)
0) is partition function. The posterior тx is usually intractable to compute due to (x), so we employ approximate inference algorithms. Bayesian pseudocoreset methods [19] aim to construct synthetic dataset called a pseudocoreset u {um M with M < N such that the posterior of f conditioned on approximates the original posterior Tx. The pseudocoreset posterior is written !
M Σ m==1
Tu (8)
exp
f(um: 0) To(0)= = Z(u) exp(1I ())=()))))((()
Z(u)
Manousakas et al. [19] considered two sets of parameters, the coreset elements u and their weights {wm M We empirically found that the weights w have negligible impact on performance, so we only learn the coreset elements.

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where f(u, 8), 1,, and (u) are defined similarly. Manousakas et al. [19] suggests to learn by minimizing the KL divergence between Tu and πx:
u arg min DKL(Tul|Tx)
(3)
and show that the gradient of this objective is computed as
V Um DKL Ful [x] Cov Vuf(um, 0), 1f 1 [f,(
(4)
The expectation over the pseudocoreset posterior Tn in the above equation is further approximated with the Monte-Carlo estimation with an approximate posterior Qu R Nu. for instance, from the Laplace approximation.
2.2
Dataset Distillation
With a similar motivation as Bayesian pseudocoresets, dataset distillation methods construct a small synthetic dataset u, but from the perspective of matching optimal parameters. That is, in dataset distillation, we find u such that
arg min d(6x: (i),, 8 = arg min ((x, 0), arg min f(u,
arg
(5)
with a loss function l(., 0) and a distance function d(...) comparing two parameters. In general, optimizing d(0 02) is intractable since it is a bilevel optimization problem requiring 6 and 0 so existing works relax the problem in several ways, such as using kernel-based approaches [23, 24] solving only for short horizon in inner optimization [36, 34, 8]. Below, we briefly review two representative methods cethods-Dataaee Condensation (DC) [36] and Matching Training Trajectories (MTT) [8].
Dataset Condensation Instead of fully unrolling the inner optimization process, DC instead
computes the matching objective at every inner step starting from some random parameter ((0)
T
u = arg min Egros Σ deos (Vel(x, ((0)) Vol(u, (10)
(6)
((t) = g(--1) 1 Vol(u, Vol(u, for tort
where deus denotes the cosine similarity between two gradients. To implement this objective, we build trajectory of parameters starting from a random ((0) with the current distilled dataset u. We then compute the gradient matching objective for each layer each step of the trajectory to update Despite its appealing simplicity, this algorithm may suffer from short-horizon bias due to the limited trajectory length.
Matching Training Trajectories MTT extends the single-step gradient matching in DC and
considers longer inner optimization trajectories. Starting from a randomly initialized parameter ((0) MTT takes multiple inner optimization steps with both and and minimizes the normalized L2 distance between two parameters at the end the learning trajectories.
u = arg min Egro) "1
d2 (o(Lx), ( o((mw)]
-0(0) -000 ((0)
(8)
@(0) = 1(- 17x V (((xxx(((-10 (((-1)) for t = 1, 1.... Lx 000 0 7ha Vol(u, (((-1) for I. = 1,..., T.n.
(9)
(10)
where de(-.-) is the normalized L2 distance between parameters. Unlike DC, this algorithm starts from a common initial parameter (()), keeps track of two separate optimization trajectories (one with and another with x), and the outer-update for done only after Lx and L, inner steps. To reduce the heavy computational cost for the inner update with the full dataset x, MTT employs mini-batch SGD for the inner steps, in addition saving and re-using precomputed SGD trajectories to train u. The use of precomputed SGD trajectories is similar to that of replay buffers for reinforcement
learning [17, 20. 1J, and allows MTT to handle long inner-loop sequences. While MTT significantly improves over DC, presumably due to its longer inner optimization horizon, the algorithm requires large memory for even a moderate number of inner-steps T,, because requires backpropagation through all inner steps with u.
3

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3 On Divergence Measures for Bayesian Pseudocoresets
While the Bayesian pseudocoreset originally obtained by minimizing the reverse KL divergence, in principle we can minimize an arbitrary divergence measure D(-,)) to achieve the goal:
u' arg min ( Tu)-
(11)
In this section, we study three different choices for such divergence measures along with their relation to dataset distillation methods.
3.1 Reverse KL Divergence
Manousakas et al. [19] proposed minimize reverse KL divergence with Laplace approximation to obtain the pseudocoreset posterior Tn. Here, we show that it is possible to recast Dataset Condensation (DC) as reverse KL divergence minimization with an alternative choice for the variational posterior.
Let g(9) be parameter in a learning trajectory with loss function Éuu 0) := 1, (u, 0). Provided
that ((1) is sufficiently close local optimum, we can construct naïve approximation of mu
Yu (0) (0; 0 E), ((*) 0(t V 0(--1)).
(12)
Under this setting, we can actually show that one gradient descent step with respect to reverse KL divergence DKL [uul [ [x] is equivalent to one step of gradient matching in DC when the parameter being optimized is close to local optima. We note that while the Gaussian posterior is a simple approximation class, it is commonly used due to its demonstrated usefulness in several literatures [18], such as the Bernstein-von Mises theorem [16, 27J. However, we also note that these Gaussian approxinations are not necessary for establishing the connection between the dataset distillation and Bayesian pseudocoresets; Gaussian approximations are just special case which has the benefit of simplifying the analysis.
Proposition 3.1.
Let (u) set Eq. 12. Assume ((t-1) is close enough to local optima, so that have []0(t=1] - || << 1. Then have
V DuLu(Full x Vu Vol(xx0(t-1) (o
(13)
The proof is given in Appendix A. In other words, when sufficiently close to local optima, DC can be interpreted as a special instance of Bayesian pseudocoreset via reverse KL minimization with Gaussian approximation for the pseudocoreset posterior Tu. Throughout the paper, we will refer to the Bayesian pseudocoreset with reverse KL minimization as BPC-rKL. To simply implement the Gaussian distribution. Please
Algorithm 1 in [19] for the image dataset, we approximated refer to Appendix B.2 for the detailed algorithm.
3.2 Wasserstein Distance
We can instead choose (::,)) to be the Wasserstein distance between Tu and πx. While the Wasser- stein distance intractable in general, by approximating both nu and मx with Gaussian distributions, we can attain a closed-form expression for this distance metric. Specifically, let 0g and 0x be two parameters sufficiently close local optima. Then we can construct a crude approximation for mu and Txx as,
Tu (0) qu
(0) (0) N' On, En), (0) (0) NN 0x, Ex).
(14)
Then the Wasserstein distance between these two distributions is computed as
Wa 9x) ||0, 0x11 Tr Ex + (
(15)
Now consider a single outer-step of the MTT. Starting from a common initial 8g. the algorithm takes the gradient steps both with and u to get (1x)) and (wm) If we set (0) = =0 (L)) E) and
axe (0) (0; 0((x)) Σ) with a common covariance Σ, then the Wasserstein distance reduces to | - (Lw) 12 which is proportional to the L2 objective being optimized in the MTT. Similarly the reverse KI. divergence and BPC-rKL,, we refer to the Bayesian pseudocoreset with Wasserstein distance minimization as BPC-W.

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3.3 Forward KL Divergence
Finally, we consider an alternative Bayesian pseudocoreset algorithm based on minimizing forward KL divergence. The forward KL minimization is known to encourage a model to cover the entire target distribution while the reverse KL minimization favors mode capturing models, indicating that the forward KI. may be better choice for the Bayesian pseudocoreset. Under our setting, the forward KI. computed as
DKL [xx]||uul+ log (u)) - log (xx) ET,, [ 0)] Ex, [f 0))
(16)
Then the gradient w.r.t. the pseudocoreset u is computed as,
[xx V log (() VuL..Iiiff( 0)]
= Ex (u (u, 0)) - Vulli, [
(17)
Again, we should approximate the expectation w.r.t and Tx to obtain this gradient. We introduce similar variational posteriors,
qu No(:: ((f.)) Eu), (x NO: f( Ex).
(18)
The endpoint ((r is function of u, so in principle, the expectation w.r.t Yu involves unrolling through the parameter trajectories ((0) 0 + ( as for the BPC-W. We employ
truncated backpropagation to reduce memory cost due to this unrolling, i.e. we block the gradient flow through g(Lu) to obtain
VuL [x Tul E. Val 0)) V V. f
= VEe.[Vi( [n Ty sg((!(")) + 2.))l - [ 0(1x)) + EX((xx)
= Vu [ee. (E.e..li
fu, ( sg((!( 2 - E,, [1 Ollx) + ال 2x)
(19)
where sg(.) denotes the sop-grad operation. We further approximate this via Monte-Carke estimation,
Σ fu (g(O(())) E1/3 MN O2 E1/26(2)
(20)
5
(S) lid. (0, I). The resulting algorithm starts from common initial parameter ((0), keeps track of two parameter trajectories, and minimizes the difference in the log- likelihood 1 J (u, 0) evaluated at the Gaussian neighbors of the endpoints of the two trajectories. Note that even though we block gradient flow through (((,)) the pseudocoreset u still influences each likelihood term f(u, 0), so the truncated gradient includes meaningful learning signals. In contrast, if
with ()
(S)
we apply similar trick to the BPC-W, the matching objective would be |sg(e("")) is constant w.r.t. u. We can additionally borrow an idea from MTT, where we utilize the set of expert trajectories to efficiently evaluate the parameter trajectory with x. We call this overall algorithm BPC-fKL, and provide summary of the overall procedure in Algorithm 1.
< [|, which
Relation to contrastive divergence BPC-IKL closely related to contrastive divergence [13, 7], learning algorithm for approximating the maximum likelihood estimator. Suppose we want to train a model p=(x)= z(0) (x)) by maximizing the log-liieldhood function of training examples { J. Pdata (z).
N
L(
N Σ
lo
(21)
Then the gradient of the log likelihood is
L'()) 1 Vofo(zN)) - Eps(e) Vofo(z)] Epant (x) [ofo(z)] 2 Epo(a) [efe(z)). (22)
N
Contrastive divergence approximates this gradient by approximating the second term with sampling methods such Langevin dynamics HMC. By doing this we can obtain the MLE for log-likeldhood
5

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Algorithm 1 Bayesian Pseudocoresets with Forward KL
Require:
Set of expert parameter trajectories {r} trained with x, each parameter trajectory saves parameters at the end of training epochs.
Require:
Number of updates with the pseudocoreset (full data) Lu(Lx), total training steps K, maximum start epoch 7 number Gaussian samples S. variances Eu, Ex. learning rate 7).
Require: Diffcrentiable augmentation function A (Optional).
Initialize the pseudocoreset u by randomly selecting subset of size M from X.
for As = 1,..., K do
Sample an expert trajectory 7 = -1000) Jr=o-
Randomly choose an epoch to start S T and initialize 0 = 0x = =
Ohtain 02 xx) ) = 0((+11x)
for =
L. do
Update the network parameter 0 + 1 + (V)))((( o.
end for
Sample random Gaussian noises (cx") eu Ja-1
p ud NNO, l).
Update the pseudocoreset with the gradient using Eq. 20.
end for
without knowing the exact potential function. Our proposed BPC-IKL performs the same function as the contrastive divergence if the model parameter 0 and training data x change to the Bayesian pseudocoresets u and the parameters from the true posterior distribution which are approximated by the points near the expert trajectories. In this point of view, BPC-fKL is the maximum likelihood estimator that maximizes the -igeeiili of the parameters from the posterior distribution of the original dataset.
4 Related Works
Bayesian coresets As rumning algorithms such as MCMC and VI on large datasets is challenging due to expensive computational cost, Bayesian coresets [14], methods that represent the entire dataset as sparse and weighted sum of subsets, have been studied. Campbell and Broderick [6] interprets coreset construction as sparse approximation of vector sums and generalizes it to sparse regression problem in a Hilbert space. Campbell and Broderick [5] demonstrates that Hilbert coresets scale the coresets log- lilkeiihhodd sub-optimally, and presents modified algorithm. As Hilbert coresets include the choice of weighted 7.2 inner product or finiteddimensional projection and it is difficult to choose them optimally, another method that minimizes the KI. divergence between the coreset posterior and true posterior has also been studied recently [4]. Bayesian coresets have the advantages of simple and theoretically guaranteed quality of the posterior approximations. However, for high-dimensional data, considering only subsets of the dataset fails as even an optimal coreset has KL divergence that increases with data dimension. Additionally, privacy concerns make subset-based coresets difficult to apply to real-world conditions. Bayesian pseudocoresets [19], constructed from learned synthetic datapoints, can avoid these shortcomings of coresets, but have been empirically validated only in relatively easier oow-dimensional settings such as logistic regression. To our best knowledge, this paper is the first to experimentally demonstrate the viability of Bayesian pseudocoresets for high-dimensional real data.
Dataset distillation The goal of dataset distillation [28] is create small synthetic dataset that allows the model to have similar test performance to the original dataset even after training with the smaller synthetic dataset. Such synthetic datasets have potential applications in many subfields of machine learning such as continual learning and neural architecture search. However, dataset distillation typically involves bilevel optimization structure in the learning process, which suffers from high computational costs or training instability. Existing methods alleviate these challenges by matching one-step gradients between synthetic and real data [36, 34], solving using closed form solution of kernel ridge regression [23, 24], or reducing the normalized distance between parameters at the end of synthetic and real training trajectories [8]. However, because these methods focus only on high test accuracy, they do not consider uncertainty or the degree to which the posterior distribution on the synthetic dataset matches the true posterior distribution. While Zhao and Bilen [35] matches the distributions of embedding features for many random networks of synthetic and real data, this

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Table 1: Performance of each Bayesian pseudocoreset method with {1, 10, 20) images per class (ipe)
on the CIFARIO test dataset. We present results with HMC and A-SGHMC. The SGHMC results for the entire dataset are 0.7383 0.0052 accuracy and 0.9387 0.0152 nll, Aug denotes image augmentation during training. All values are averaged over ten random seeds.
HMC
HMC (+Aug)
A SGHMC (+Aug)
Acc (f)
NLL (U)
Acc (f)
NL)
Acc (t)
NLL (L)
Random
0.1745
2.4507 $0.0557
0.1745
2.4507=00.008
0.0014
2.9798:0.0r25
BPC +L
0.2472.9.0100
2. 2.165510.0977
0.241610.0:33
2.168810.0186
0.2162 0083
2.4617
ipe
BPC-W (MTT)
334355+0.0007
1.9911 +0.0020
+.0008
1.9589
0.2994-0.0121
2.1400=0.0ass
BPC-IKL
0.335410.00m
2.025310.011
1.346880.00
1.9574
0.2823:10.016
2.1426:000045
Random
1.2807:t0.0000
2. 0..078
0,2807 4:00000
2.1070
0. 20.00T
10 ipc
BPC-KKI
0.3253
1.9627
0.301711.00
0.3789 20
1.94922.00000
BPC-W (MTT)
0.3535+0.0127
1. 1.9450 0220
0.36-16:0 0062
1 1.9307 #0.0171
0. 4890000.177
1 1.6971=0000 0012
BPC/IKL
0..09410.0001
N.T2992..0000
0.4252 10.0001
1.7334.0.00
0 1..880
1.6762_00099
Random
0.8202210.0007
2.00622
0.32522::0.00m
2.0014:0.0012
0 0.38022000000
28999
BPC-KKI
1.3686610.0015
1 10.488
1.351811..o
1.9 1.95-40..00000
0.4307 1077000073
1 188556 0963
20 ipc
BPC-W (MTT)
0.45-16+:.00166
L.7336+a..co 0325
0.4606+0.oom 0200
#0.000
0.5691=0.00 0167
1.5439=00. 040s
BPC-IKL
0.4910s0.0088
1. (1.27790.0000
0. 1..1333..000
1. 6..4410
0.5133=0.0099
1.6701=0.0250
BPC-rKL
6.0
BPC-IKL
BPC-W
BPC-rKL
Log ampltudd
4.0
BPC-W
3.0
1.0
BPC-fKL
0.0
0.2
0.0m
Frequency
Figure 1: Examples of Bayesian pseudocoresets.
Figure 2: Log amplitude in frequency domain.
work similarly does not consider posterior distributions. This paper re- in-teereseets and extends scalable dataset distillation methods to the problem of matching posterior distribution matching.
5
Experiments
5.1 Experimental Setup
Datasets and model architecture We use the CIFARIO dataset [15] to generate Bayesian pseu- docoresets, and evaluate on the test split of CIFARIO in addition to the CIFARIO-C dataset [12], which imposes additional uncertainty through image corruptions. Following the experimental setup of previous works [36, 34], we use 3-layer ConvNet the network architecture.
Evaluation We obtain pseudocoresets from three Bayesian pseudocoreset construction methods: BPC-fKI, BPC-W, and BPC-rKI. We run two Markov chain Monte Carlo methods (HMC [22] and SGHMC [9]) and report the 1 accuracy and negative log- likelihooood (NLL) with respect to ground- truth labels. Note that pseudocoresets are small enough to run full-batch HMC in our experimental setting, and SGHMC which originally subsamples mini-batches from the full data not a favorable option in our situation. Instead, at each iteration, while holding the batch (pseudocoreset) fixed, we apply random data augmentation to the batch. As result, the gradients computed from the randomly augmented batches differ at each iteration, and we run SGHMC with these stochastic gradients. To distinguish this setting from the typical mini-batch SGHMC, we denote this as Altered-SHMMC (A (A-SGHMC). As our results were not sensitive to the choice of hyperparameters, we used a single set hyperparameters that performed best in initial experiments. Please refer Appendix for detailed evaluation settings including hyperparameters.
7

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5.2 Main Results
experimentally evaluate the effectiveness of the three pseudocoreset construction methods previ- ously discussed. We consider three different settings corresponding to different memory budgets for the pseudocoresets: n E (1, 10, 20} images per class (ipc). In each setting, we consider a random coreset baseline. in which randomly subsample images from the original dataset. To further eval- uate the effectiveness of data augmentation in pscudocoresct training, we show performance each coresct method with and without augmentations. We additionally examine the role of augmentations at test time through A-SGHMC.
Results in Table 1 show that all Bayesian pseudocoresets have higher accuracy and lower negative log- likelihood compared to random coresets. BPC-rKL slightly worse than BPC-W and BPC-fKL, which we interpret as the reverse KL suffering from the property that covers only one major mode, making the BPC-rKL posterior somewhat sub-optimal. In the ipc (image per class) setting, BPC-W and BPC-fKI. have comparable results, and as the pseudocoreset size increases, BPC-fKL is better if there is no augmentation test time and BPC-W is better if not. seems that the strength BPC-W W with higher performances with test time augmentations is due to the training method that exactly matches the pseudocoreset training trajectories and expert training trajectories that also be trained with the data augmentations. However, because the learning method of A-SGHMC is not accurate Bayesian learning, and is known that data augmentation causes a cold posterior effect [30], the HMC results without augmentations are preferred.
To qualitatively examine the pseudocoresets trained with each divergence measure, we plot the learned pseudocoreset images in Fig. 1. We see that BPC-fKL. looks the most smooth while BPC-rKL. is the most noisy. The noisiness of the images learned by BPC-rKL. may be due to the nature of the reverse KI. divergence, which makes it difficult to escape the mode captured by the initial pseudocoreset images. In contrast, BPC-W and BPC-IKL seem to learn better pseudocoreset as desired to include high-level shapes semantic features representative of each class. As more quantitative measure of image noisiness, we plot the log amplitude of the diagonal components the 2D Fourier transform of pseudocoresets in Fig. 2: BPC-rKL has the most fighffreueecc noise, as reflected in the qualitative examples. This result is consistent with previous studies showing that CNNs are not robust to high-frequency noise [26, 25]. Please refer to Appendix D for all pseudocoreset images.
5.3 Computational Cost
We show that BPC-fKL, in addition to achieving high performance in terms of accuracy and NLL, requires substantially lower computational costs compared to other pseudocoreset training methods. We measure the computational costs of each method on CIFARI 10 with fixed gradient steps In 30. We use 32 cores Intel Xeon CPU Gold 5 5120 and Tesla V100s. We note that BPC-W with many pseudocoresets per class cannot be trained on single GPU due the memory limitations. Therefore, we used a parallel implementation using 4 GPUs in all the methods for a fair comparison. To compare the training time, we measure the time required for single iteration by averaging across 100 iterations and repeat this process three times to select the largest value among them. Fig. 3a shows that the memory usage is significantly lower for BPC-fKL compared to BPC-W. This is because, unlike BPC-fKL, BPC- W must hold all gradient flows in memory for trajectory matching. As shown
2250
BPC-KKL
80
GPU memory (GB)
BPC/W
Iteration time (ms)
2000
BPC-W
BPC-rKL
BPC-KKL
60
1750
1500
1250
20
5 10 20
30
40
50
60
70
80
90 100
1
5 10 20
30 40 50 60 70 80 90 100
image per class (ipc)
image per class (ipc)
(a) GPU memory usage
(b) Iteration time
Figure 3: Comparison of computational costs of GPU memory and training time.
8

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HMC
A-SGHMC
A-SGHMC
SP
5N
SP
SN
SP
SN
SN
JPFG
JPFG
JPFC
POX
D8
POX
09
PIX
D8
PIX
03
ET
ET
ET
G8
G8
85
M6
M6
85
MB
85
M6
FT
SW
2M
SW
SWI
2M
SW
2M
Figure 4: HMC and A-SGHMC results CIFARIO-C. Each direction represents each type of corruption. Blue and orange represent BPC-IKL and BPC-W, respectively. All values are averaged over 30 runs with different random seeds. Acc higher the better, NLL - lower the better.
in Fig. 3b, BPC-IKL also has the advantage faster training time. Pscudocoreset training time increases roughly linearly with pseudocoreset size, but BPC-W's rate of increase is much faster than that BPC-IKL. Combining the two results and Table the pseudocoreset training using forward KL divergence is not only comparable in terms of accuracy and NLL but also remarkably efficient in both memory and training time, compared to BPC-W.
5.4 Robustness to Out-of-Distributin Inputs
Since one of merits of the Bayesian approach is Bayesian model averaging which has been shown to improve robustness to distributional shift and calibration, we evaluate the performance of each pseudocoreset method on an out-of-distribution dataset CIFARIO-C [12]. Fig. 4 shows accuracy and negative oggliikelhhod of each corrupted dataset for BPC-W and BPC-fKL. Note that we optimize each pseudocoreset with the clean CIFAR1O 10 datasct and evaluate them on CIFARIO with 14 corruptions. For A A-SGHMC results, apply differentiable augmentations to both the original dataset and coreset for training. but we do not use it for running HMC. As shown Fig. 4, for both HMC and A-SGHMC, BPC-fKL achieves higher comparable accuracy and lower negative log-likeling likelihood than the BPC-W W for all but one corruption, suggesting that the forward KL divergence indeed effective divergence measure for learning Bayesian pseudocoresets.
Conclusion
In this paper, we explored three divergence measures for Bayesian pseudocoresets: reverse KLD, Wasserstein distance, and forward KLD. We showed that existing dataset distillation methods can be linked to the Bayesian pseudocoresets with reverse KL.D and Wasserstein distance, and further proposed novel algorithm for learning Bayesian pseudocoresets by minimizing forward KL di- vergence. We empirically validated all three methods in terms of their ability to approximate the posterior distribution for real-world image datasets. Bayesian pseudocoresets with both Wasserstein distance and forward KL divergence can approximate the true posterior well and BPC-fKL is more effective terms of computational cost and robustness
distrribuiti data.
Limitation Despite showing promising results for the first time on Bayesian pseudocoresets for real datasets, there still exists a substantial performance gap between stochastic gradient MCMC on pseudocoreset and the original dataset. Thus, a promising future direction is to analyze whether such pseudocoresets are useful for stochastic gradient MCMC algorithms when they are used together with mini-batches of the original dataset.
Societal Impacts Our work is hardly likely to bring any negative societal impacts. Nevertheless, we should be careful while learning pseudocoresets because any bias present in the original dataset can be transferred to the pseudocoresets. On positive note, pseudocoresets can alleviate data privacy concerns by eliminating the need for access to the original dataset during downstream task learning.

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Acknowledgments
This work was partly supported by KAIST-NAVER Hypercreative Center, Korea Foundation for Advanced Studies (KFAS), Institute of Information & communications Technology Planning Evaluation (ITP)) grant funded by the Korea government (MSIT) (No.2019-0-00075 Artificial Intelligence Graduate School Program (KAIST), No. 2021-0-02066 Artificial Intelligence Innovation Hub, No.2002-0-00713), and National Research Foundation Korea (NRF) funded by the Ministry of Education (ARF-20113355D005053)
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Reviews:
[{"review_id": "zVt9vEI3sUu", "reviewer": "Reviewer_6N6Z", "summary": "This paper reinterprets heuristic dataset distillation methods as approximate Bayesian pseudocoreset procedures, with different choices of divergence between the true posterior and the pseudocoreset approximation. Motivated by this analysis, they propose a new Bayesian coreset procedure based on an approximation to the forward KL divergence, and show empirically that it outperforms alternatives.", "questions": "Is there a general recipe for constructing Bayesian pseudocoresets using Gaussian approximations? In particular, can you enumerate the different choices of how the trajectory is managed, what divergence to pick, etc. Can you quantify the various sources of error in this approximation?\n\nSmall Suggestions:\nEquation 1 has a typo, the comma in the middle (which makes it look as if the prior pi_0 is defined to be the posterior). Same typo appears in Eqn. 2\n\nEquation 4: I believe u_m should not be bold, since it’s one datapoint, and the gradient on the RHS should be with respect to u_m not u, since the overall gradient should have dimensions the size of a single datapoint.\n\nLine 67, the index should run to capital N not lowercase n. Might want to do the same in the contrastive divergence section, for consistent notation.\n\nLine 89 - Typo: Traning -> Training\n\nLine 180 - Check the grammar here and in the following lines (extends -> extend, etc.)\n\nLine 193 - In fact, [16] and other Bayesian coreset methods demonstrate applications to real data (some real data is low dimensional, believe it or not).\n\nLines 215 - this paragraph is very unclear to me.\n", "limitations": "The authors are honest about some serious limitations of the method’s performance. I think it may be worth discussing privacy in the societal impacts section; pseudocoresets may potentially help increase data privacy while enabling downstream analysis on the one hand, while on the other hand their privacy properties have not be thoroughly studied", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "This paper offers an insightful and unifying new view on heuristic dataset distillation techniques, which leads to improved practical methods. It is original (to my knowledge) and clearly written, and a significant contribution both to the dataset distillation and the Bayesian coresets literatures.\nIt suffers from two main weaknesses in terms of quality. First, the theoretical analysis is extremely rough, involving severe approximations (essentially treating the posterior as a Gaussian with zero variance, i.e. a delta function). There is a rich literature on Gaussian approximations to the posterior, including in particular Bernstein von Mises theorems. Moreover, SGD can be viewed as providing a Gaussian approximation to the  Bayesian posterior (Mandt, Hoffman, Blei, Stochastic Gradient Descent as Approximate Bayesian Inference). It would be valuable to establish more general and rigorous results in terms of Gaussian posterior approximations, and then derive the heuristic approximations as limiting cases when the covariance goes to zero (i.e. when the number of datapoints goes to infinity). The second major weakness is the empirical analysis. Despite motivating the work by Bayesian inference, there is not a rigorous evaluation of the quality of the posterior approximation (e.g. is the uncertainty correct), only evaluation of accuracy and log likelihood. ", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "k-FHeUUZ5qx", "reviewer": "Reviewer_knYM", "summary": "The authors discuss Bayesian pseudocoreset methods that uses reverse KL, KL and Wasserstein metrics special cases. They link these algorithms to data distillation as different posterior choices. Their main method is the Bayesian pseudocoreset algorithm that replaces reverse KL with a forward KL algorithm. They test these algorithms with Cifar-10 dataset with random coresets baseline.", "questions": "Have you compared your method with another baseline? At least another coreset algorithm a bit more intelligent than random, such as weighted summation or using input points distances? \n\n", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n\nFocusing/unifying on the Bayesian coresets and distillation methods is a good direction and influence for the academia. The paper might have a good potential to be proceeded with more future works.\n\nWeaknesses:\n\nThe baseline is a random coreset which is a very weak baseline. However given that they apply their algorithm on an image dataset, this might be acceptable, and to be improved as a future work.\n\nThe paper needs to be organized and motivated better. The authors might more clearly mention the relations and related works for data distillation and bayesian coresets method, then unify them. There are some optimality requirements for these relations, it could be organized better. It would be good to emphasize more on the reasons that they were able to make the Bayesian algorithm tractable with a divergence perspective.\n\nThe ideas and proofs are too straightforward and the experiments are too preliminary. \n\nIt would be good to do one more pass for proofreading.", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "hNvig7OzjC5", "reviewer": "Reviewer_RC9c", "summary": "The paper unifies the Bayesian Pseudocoresets and existing dataset distillation methods in certain settings. It investigates multiple choice divergence measurements and proposes an algorithm using forward KL-divergence to construct Bayesian Pseudocoresets.", "questions": "1. The assumption of proposition 3.1 should be defined clearer that $\\Vert \\theta_{t} - \\theta_{t-1}\\Vert$ should be sufficiently small.\n2. Results shown in figure 1 is not very illustrative due to their size. Could you pick some representatives and move the rest to the appendix? Could you demonstrate the differences quantitively rather than qualitatively? \n3. Could the performance gap between the using psuedocoreset and the entire dataset be indicated by the divergence?\n4. Could you clarify the bias amplification on the downstream task mentioned in line 299? The Bayesian Pseudocoreset algorithm [Dionysis et al. 2020] claims weight w could potentially help with reducing the computation burden when nonzero entries are small. Is there any reference or empirical results substantiating the claim that m doesn't help with the performance?", "limitations": "The limitation is well discussed. The negative societal impact is not applied.", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "- ***Strengths***:\n1. The direction of unifying existing data condensation and Bayesian Pseudocoresets is well-motivated.\n2. The effectiveness of the proposed method is well illustrated.\n\t\n\t\n- ***Problems***: \n1. Proposition 3.1 relies on two factors: (1) $\\Vert \\theta_{t} - \\theta_{t-1}\\Vert$ is sufficiently small so that the expansions in eq 24 and eq 25 hold; (2) the gaussian approximation. The second approximation might be commonly used in practice, yet this choice should be carefully elaborated on since it plays a key role throughout the paper. The first assumption actually limits the efficacy of the proposition to large t when \\theta_t is close to local optima instead of the whole trajectory as in Data Condensation (eq 6). Then the connection between BPC-rKL and the existing Dataset Distillation method is not as exact as claimed. \n2. In section 3.2, the discussed connection between BPC-W and MTT accredits mostly to the simplified setting using the gaussian approximation. As the distance of two Gaussians is minimized when the variance is shared and the distance of the mean is minimized.\n3. Theoretical-wise, the paper mainly leverages the gaussian approximation to unify the Bayesian Pseudocorset with data distillation methods. The strong reliance on the gaussian approximations without adequate elaboration downgrades its contribution.\n4. In section 5, since BPC-rKL and BPC-W are not exactly equivalent to existing methods, the dataset distillation methods and the original Bayesian Pseudocoresets algorithm are expected as baselines while missing. Especially when the implementations of these algorithms are available.\n\n***Minor Issues***:\n1. Line 422, I think it should be $\\approx$ rather than =.\n2. Line 419, the expectation seems to be over $\\pi_x$ rather than $\\pi_u$.", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "an6PijyU88E", "reviewer": "Reviewer_Pvtm", "summary": "The paper studies learning Bayesian pseduocoresets, which is the task of a small set of \"data\" by conditioning on which the posterior is close to the posterior conditioned on the full dataset.\nIt establishes the framework in which two important ingredients are (1) the choice of divergence measure and (2) the choice of variational approximation.\nFollowing this, it first recasts/reinterprets two existing, successful dataset distillation methods are Bayesian pseduocorsets learning with reverse KL/Wasserstein distance with Gaussian approximations to the posterior.\nThen it proposes to use the forward KL which in theory doesn't suffer form the mode-seeking behaviour of reverse KL.\nOn a few real-world datasets, the proposed choice of forward KL is better than existing choice of divergence measures in terms of NLL and accuracy.", "questions": "- There are some discussions regarding contrastive divergence in section 3.3. I wonder if the divergence from [1] is related there or maybe a good choice of divergence in this framework.\n\n[1] Ruiz, Francisco, and Michalis Titsias. \"A contrastive divergence for combining variational inference and mcmc.\" In International Conference on Machine Learning, pp. 5537-5545. PMLR, 2019.", "limitations": "Limitations and potential negative societal impact are addressed.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Pros\n- The paper studies an important problem of scaling Bayesian methods to large datasets by learning Bayesian pseudocoresets. This approach can benefit a wide range of Bayesian inference methods.\n- The paper establishes the framework of this problem, which is novel, and studies the relation to existing dataset distillation methods. The paper is significant as it (1) gives justification of some existing heuristics-based methods and (2) paves the way of further research in this direction.\n- The paper is clearly written with good quality.\n- Related works are discussed adequately.\n\nCons\n- The experiments might be improved a bit.\n  - I'd like to see at least one synthetic dataset where the posterior is tractable and see how the posterior each method learns differs from the true one.\n  - Some classic Bayesian inference datasets could be used so that naive HMC results could be provided as a reference.\n\nWriting\n- Equation (1) is broken: I assume the $\\pi_0$ in the middle shall be removed.", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept