Understanding Instance-based Interpretability of Variational Auto-Encoders
Authors: Zhifeng Kong
Kamalika Chaudhuri
Venue: NeurIPS 2021 Poster (2021)
Keywords: Influence function
TracIn
instance-based interpretability
unsupervised learning
VAE

Abstract: Instance-based interpretation methods have been widely studied for supervised learning methods as they help explain how black box neural networks predict. However, instance-based interpretations remain ill-understood in the context of unsupervised learning. In this paper, we investigate influence functions [Koh and Liang, 2017], a popular instance-based interpretation method, for a class of deep generative models called variational auto-encoders (VAE). We formally frame the counter-factual question answered by influence functions in this setting, and through theoretical analysis, examine what they reveal about the impact of training samples on classical unsupervised learning methods. We then introduce VAE- TracIn, a computationally efficient and theoretically sound solution based on Pruthi et al. [2020], for VAEs. Finally, we evaluate VAE-TracIn on several real world datasets with extensive quantitative and qualitative analysis.


Understanding Instance-based Interpretability of Variational Auto-Encodes
Zhifeng Kong
Kamalika Chaudhuri
Computer Science and Engineering University of California San Diego La Jolla, CA 92093 z4kongßeng.o uesd. edu
Computer Science and Engineering University of California San Diego
La Jolla, CA 92093 kamal ikadcs. uesd. edu
Abstract
Instance-basee based interpretation methods have been widely studied for supervised learning methods they help explain how black box neural networks predict. However, instance-based interpretations remain ill-understood the context unsupervised learning. In this paper, we investigate influence functions [Koh and Liang, 2017], a popular instance-based interpretation method, for class deep generative models called variational auto-encoders (VAE). We formally frame the counter-factual question answered by influence functions in this setting, and through theoretical analysis, examine what they reveal about the impact of training samples on classical unsupervised learning methods. We then introduce VAE- TracIn, computationally efficient and theoretically sound solution based on Pruthi et al. [2020]. for VAEs. Finally, we evaluate VAE-TracIn TracIn on several real world datasets with extensive quantitative and qualitative analysis.
Introduction
Instance-based interpretation methods have been popular for supervised learning as they help explain why model makes certain prediction and hence have many applications [Barshan et al., 2020, Basu et al., 2020, Chen et al., 2020, Ghorbani and Zou, 2019, Hara et al., 2019, Harutyunyan et al., 2021, Koh and Liang, 2017, Koh al., 2019, Pruthi et al., 2020, Yeh al., 2018, Yoon et al., 2020]. For classifier and test sample an instance-based interpretation ranks all training samples æ according an interpretability score between and Samples with high (low) scores are considered positively (negatively) important for the prediction of 2.
However, in the literature of unsupervised learning especially generative models, instance-base interpretations are much less understood. In this work, we investigate instance-basee interpretation methods for unsupervised learning based influence functions [Cook and Weisberg, 1980, Koh and Liang, 2017]. In particular, we theoretically analyze certain classical non-parametric and parametric methods. Then, look canonical deep generative model, variational auto-encoders (VAE, [Higgins et al., 2016, Kingma and Welling, 2013]), and explore some of the applications.
The first challenge is framing the counter-factual question for unsupervised learning. For instance- based interpretability supervised learning, the cunter-facture question "which training samples most responsible for the prediction of test sample?" which heavily relies the label information. However, there is no label in unsupervised learning. this work, we frame the counter- factual question for unsupervised Icarning "which training samples most responsible for increasing the likelihood (or reducing the loss when likelihood is not available) of a test sample?" We show that influence functions answer this counter-facti question. Then, examine influence functions for several classical unsupervised learning methods. We present theory and intuitions on how influence functions relate likelihood and pairwise distances.
35th Conference on Neural Information Processing Systems (NcurIPS 2021).

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The second challenge is how to compute influence functions in VAE. The first difficulty here is that the VAE loss of a test sample involves an expectation over the encoder, so the actual influence function cannot be precisely computed. To deal with this problem, we use the empirical average of influence functions, and prove concentration bound of the empirical average under mild conditions. Another difficulty is computation. The influence function involves inverting the Hessian of the loss with respect to all parameters, which involves massive computation for big neural networks with millions of parameters. To deal with this problem, we adapt first-order estimate of the influence function called TracIn [Pruthi et al., 2020] to VAE. We call our method VAE-TracIn. is fast because (i) it does not involve the Hessian, and (it) it can accelerate computation with only a few checkpoints.
We begin with a sanity check that examines whether training samples have the highest influences over themselves, and show VAE-TracIn passes it. We then evaluate VAE-Trach on several real world datasets. We find high (low) self influence training samples have large (small) losses. Intuitively, high self influence samples are hard to recognize or visually high-contrast, while low self influence ones share similar shapes or background. These findings lead to an application on unsupervised data cleaning, as high self influence samples are likely to be outside the data manifold. We then provide quantitative and visual analysis on influences over test data. We call high and low influence samples proponents and opponents, respectively. We find in certain cases both proponents and opponents are similar samples from the same class, while in other cases proponents have large norms.
We consider VAE-TracIn as a general- purpose tool that can potentially help understand many aspects in the unsupervised setting, including (i) detecting underlying memorization, bias or bugs [Feldman and Zhang, 2020] in unsupervised learning, (ii) performing data deletion [Asokan and Seelaman- tula, 2020, Izzo et al., 2021] in generative models, and (iis) examining training data without label information.
We make the following contributions in this paper.
We formally frame instance- based interpretations for unsupervised learning.
We examine influence functions for several classical unsupervised learning methods.
instance-based interpretation method for VAE. We provide both theoretical and empirical justification to VAE-TracIn.
We present
We evaluate VAE- on several real world datasets. We provide extensive quantitative analysis and visualization, as well as an application on unsupervised data cleaning.
1.1
Related Work
There are two lines of research instance-based based interpretation methods for supervised learning.
The first line of research frames the following counter factual question: which training samples are most responsible for the prediction of a particular test sample 2? This is answered by designing an interpretability score that measures the importance of training samples over z and selecting those with the highest scores. Many scores and their approximations have been proposed [Barshan et al., 2020, Basu et al., 2020, Chen et al., 2020, Hara et al., 2019, Koh and Liang, 2017, Koh et al., 2019, Pruthi et al., 2020, Ych et al., 2018]. Specifically, Koh and Liang [2017] introduce the influence function (IF) based on the terminology in robust statistics [Cook and Weisberg, 1980]. The intuition is removing important training sample of should result in huge increase of its test loss. Because the IF is hard to compute, Pruthi et al. [2020] propose TracIn, a fast first-order approximation to IF.
Our paper extends the counter-factie factual question to unsupervised learning where there is no label. We ask: which training samples are most responsible for increasing the likelihood (or reducing the loss) of test sample? In this paper, we propose VAE-TracIn, an instance-based interpretation method for VAE [Higgins et al., 2016, Kingma and Welling, 2013] based on TracIn and
The second line of research considers a different counter- question: which training samples are most responsible for the overall performance of the model (e.g. accuracy)? This is answered by designing an interpretability score for each training sample. Again many scores have been proposed [Ghorbani and Zou, 2019, Harutyunyan al., 2021, Yoon et al., 2020]. Terashita et al. [2021] extend this framework to specific unsupervised model called generative adversarial networks [Goodfellow
'There are different names in the literature, such as helpful/harmful samples [Koh and Liang, 2017]. excitatory/inhiiioryy points [Yeh et al., 2018], and proponents/ooponents [Pruthi et al., 2020j.

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et al., 2014]. They measure influences of samples on several evaluation metrics, and discard samples that harm these metrics. Our paper is orthogonal to these works.
The instance-based interpretation methods lead to many applications in various areas including adversarial learning. data cleaning, prototype selection, data summarization, and outlier detection [Barshan et al., 2020, Basu et al., 2020, Chen et al., 2020, Feldman and Zhang, 2020, Ghorbani and Zou, 2019, Hara et al., 2019, Harutyunyan et al., 2021, Khanna et al., 2019, Koh and Liang, 2017, Pruthi et al., 2020, Suzuki et al., 2021, Ting and Brochu, 2018, Ye et al., 2021, Yeh el al., 2018, Yoon et al., 2020]. In this paper, we apply the proposed VAE-Trachi to an unsupervised data cleaning task.
Prior works on interpreting generative models analyze their latent space via measuring disentangle- ment, explaining and visualizing representations, or analysis in interactive interface [Alvarez-M] Levarez-Meliis and Jaakkola, 2018, Bengio et al., 2013, Chen et al., 2016, Desjardins et al., 2012, 2, Kim and Mnih, 2018, Olah et al., 2017, 2018, Ross et al., 2021]. These latent space analysis are complementary the instance-based interpretation methods in this paper.
2 Instance-based Interpretations
Let X {2 E Fdd be the training set. Let A be algorithm that takes X as input and outputs a model that describes the distribution of X. AA can be a density estimator or a generative model. Let AA x E {(z;; A(X)) be loss function. Then, the influence function of a training data T. over test data IR is the loss of computed from the model trained without 2; minus that computed from the model trained with x4. If the difference is large, then 2; should be very influential for z. Formally, the influence function is defined below.
Definition (Influence functions [Koh and Liang, 2017)). Ler X-4 X {x.). Then, the influence of x; over z is defined as IF x. (I,, 2) = {22 A(X-i)) l(2; ((2 A(X)). IfIF x.A(x;; 2) >> 0, say I; is a proponent of 2; otherwise, we say 2; is an opponent of
For big models A such as deep neural networks, doing retraining and obtaining A(X can be expensive. The following TracIn score is fast approximation IF.
Definition 2 (TracIn scores [Pruthi al., 2020]). Suppose A(X) is obtained by minimizing L(X; A) via stochastic gradient descent. Let {f(4)e= be checkpoints during the training procedure. Then, the estimated influence of x;; over is defined TracIn (dr;; Ec V((x;; V (V((==))))
We are also interested in the influence of a training sample over itself. Formally, we define this quantity the self influence of or Self-IF (x) IF (x, In supervised learning, self influences provide rich information about memorization properties of training samples. Intuitively, high self influence samples are atypical, ambiguous or mislabeled, while low self influence samples are typical [Feldman and Zhang, 2020].
Influence Functions for Classical Unsupervised Learning
In this section, we analyze influence functions for unsupervised learning. The goal is to provide intuition on what influence functions should tell us in the unsupervised setting. Specifically, we look at three classical methods: the non non-parametric k-ncarest-neighhor (k-NN) density estimator, the non- non-parametric kernel density estimator (KDE), and the parametric Gaussian mixture models (GMM). We let the loss function & to be the negative likeliiel;oood: {(=) - log p(2).
The R--Nearest-Niighbor (k--NN density estimator. The k-NN density estimator defined as PaNN (2; X) = k/(NVa X)",, where R₂ (æ; X) is the distance between æ and its k-th nearest neighbor in X and Va is the volume of the unit ball in Rd. Then, we have the following influence function for the k--N density estimator:
N -1 +
d dog
(2:1(2:)) Re(x:X)
|| =11 S Rx(2;X)
IF (x,ANN(E;;; logg
(I)
N
0
otherwise
See Appendix A.1.1 for proof. Note, when 2 is fixed, there are only two possible values for training data influences: log N-1 and log N-1 "N= dlog (x=1(::)) As for Self-IF (x;) samples with RR(xxX)
3

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the largest self influences are those with the largest R ( Intuitively, these samples belong to cluster of size exactly k and the cluster is far away from other samples.
Kernel Density Estimators (KDE). The KDE is defined PkDe(a;;)) # K2(2 xz).
where Kg is the Gaussian (0, o² T)). Then, we have the following influence function for KDE:
1 Ko (2 - xz)
IF xx,KDE(Z,,2) = log N
log 1+
PKDE PKDB(2; (Z x4
See Appendix A.1.2 for proof. For a fixed 2, an z; with larger || - æ;| has higher influence over 2. Therefore, the strongest proponents of are those closest in the lg distance, and the strongest opponents are the farthest. As for Self-IFx X.KDE(x,))) (2;), samples with the largest self influences are those with the least likelihood (KDE( X). Intuitively, these samples locate very sparse regions and have few nearby samples. On the other hand, samples with the largest likelihood PKDE (x;; X), or those in the high density area, have the least self influences.
Gaussian Mixture Models (GMM). As there is no closed-form expression for general GMM, we make the following well well-separation assumption simplify the problem.
Assumption 1. X U-- X where each Xx is cluster. We assume these clusters are well- separated: min x x E XR,
x' E Xxx > max [|æ x'1l xx, E Xx)-
; let k = arg min; d(x,, X2). Then, we define the well-separated spherical GMM (WS-GMM) of K mixtures PWS-GMM N where the parameters are given by the maximum likelihood estimates
Let X = N₂ and N = EK- N₂- For x E IR.
Ny N(x; pk: rf.
Pk
NO
Σ I. xEXt
N
1
Σ zE
N&d
Σ xEXt
x x
1 Juc kn.
(3)
For conciseness, we let test sample from cluster zero: ccon((Xo). Then, we have the following influence function for WS-GMM. If xz & Xo. IF x, (I
O (N- Otherwise,
d+2 2
|| 1 1 ||= 1:11²
1 N +NON)
IF (X.WS.CMM(EI
(4)
2No
2No
o
See Appendix A.1.3 for proof. A surprising finding is that some x; E Xy may have very negative influences over (i.e. strong opponents of from the same class). This happens with high probability if || 2.,12 & (1 og)d + 2n for large dimension d. Next, we compute the self
influence of an 2; Xg. According to (4),
d+2
ue 2
Self-IF X.Ws))MM(.;)
N +
(5)
2Nx
2N
Within each cluster Xg. samples far away to the cluster center μa have large self influences and vice versa. Across the entire dataset, samples in cluster X₃ whose N₂ or is small tend to have large self influences, which is very different from k-NN or KDE.
3.1 Summary
We summarize the intuitions of influence functions classical unsupervised learning in Table Among these methods, the strong proponents are all nearest samples, but self influences and strong opponents are quite different. We then visualize an example of six clusters of 2D points in Figure in Appendix B.I. In Figure 6, We plot the self influences of these data points under different density estimators. For test data point z, we plot influences of all data points over 2 in Figure 7.
Instance-based Interpretations for Variational Auto-encoders
In this section, we show how to compute influence functions for class of deep generative models called variational auto-encoders (VAB). Specifically, we look 3-VAE [Higgins et al., 2016] defined below, which generalizes the original VAE by Kingma and Welling [2013].

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Definition 3 (3-VAE [Higgins et al., 2016]). Let distent be the latent dimension. Let Pg. R. 7 Ft be the decoder and Qs _² + R+ be the encoder, where 6 and vP are the parameters of the networks. Let 0 (Øb 1b). Let the latent distribution Piatent be (O, 1). For 3 > the B-VAE model minimizes the following loss:
Lg(X;0) E Es 8) B.E KL (Qo(+))) Piatent) BEw.xEeEEeeeeee log Pa(x) (6)
In practice, the encoder Q = Qu outputs two vectors, lQ and σQ. so that Q(-1z) = (x)((z), diag(o)(z))))). The decoder P Pa outputs vector pr so that log ((=(5)
constant times [(p((E) 11²² plus a constant.
Let A the 3-VAE that returns A(X) = arg ming (x( Let 0 = A(X) and 8 = A(x-;)) Then, the influence function of z; test point (0(2;02,) Ez(2; (*), which equals to
IF (A) 3 KT. (- (1=2)) - KT. vQ (-1=) |a Platent)
[e- (+2= log Poi, (x), B - (-1=) log Po- (21())
Challenge. The first challenge is that IF in (7) involves expectation over the encoder, so it cannot be precisely computed. To solve the problem, we compute the empirical average of the influence function over samples. In Theorem I, we theoretically prove that the empirical influence function close to the actual influence function with high probability when m is properly selected. The second challenge is that IF is hard compute. To solve this problem, in Section 4.1, we propose VAE-TracIn, a computationally efficient solution to VAE.
(m)) probabilistic bound on influence estimates. Let IF X.VAE be the empirical average of the influ- ence function over m i.dd samples. We have the following result.
Theorem (Error bounds on influence estimates (informal, see formal statement in Theorem 4)). Under mild conditions, for any small > 0 and 5 >> 0, there exists an m θ (2+8) such thut
Prob (IIF ([FEx.AA((E,))) - -1F X. (2,, z) 2 S.
(8)
Formal statements and proofs are in Appendix A.2.
4.1 VAE-TracIn
In this section, we introduce VAE-TracIn, a computationally efficient interpretation method for VAE. is built based on Tracin (Definition 2). According to (6), the gradient of the loss ég (z; 0)
can be written as (2la(=; 0) = |V (=; V wala(z; TT, where
Valo(=;0) Bc-Q,((1m) (- V log Px((E)) =: (i v)), and Vela(=;0) Bgrqu((1a) V log Qx
(E|z) log (a(()(
B Blog Pintent (C)
=: V(,
The derivations are based on the Stochastic Gradient Variational Bayes estimator [Kingma and Welling, 2013], which offers low variance [Rezende al., 2014], See Appendix A.3 for full details of the derivation. We estimate the expectation Ee by averaging over m i.idd samples. Then, for training data x and test data z, the VAE-TracIn score over is computed as
C
m U(r, 019,
m
VAE-Tracli 2)
Σ
Ul, Eg 014) V(o)
m
j=l
Var, E.te 01)) v(e)
V Stt Mic s/1e)
(10)
where the notations U, V are from (9), 0,4 [dt wheil is the e-th checkpoint, ( are i.i.d. 1m are i..d. samples from Q (-1z).
samples from Q (-|æ), and 1m
5

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Table 1: Sanity check on the frequency that training sample is the most influential one over itself. Results on MNIST, CIFAR, and the averaged result on CIFAR subclasses are reported.
MNIST
CIFAR
Averaged CIFAR subclass
diatent = 64 diatent- = 128
diatent = 64
diatent = 128
dlatent = 64
0.992
1.000
0.609
0.602
0.998
Connections between VAE-TracIn and influence functions [Koh and Liang, 2017]. Koh and Liang [2017] use the second assumption that the loss function is convex. The TracIn algorithm [Pruthi al., 2020] uses the first-order (gradient-based) approximation to the change of loss during the training process under the assumption that (stochastic) gradient descent the optimizer. We expect these methods to give similar results in the ideal situation. However, implemented the method by Koh and Liang [2017] and found it to be inaccurate for VAE. A possible reason is that the Hessian vector product used to approximate the second order term is unstable.
besssian-based approximation to the change of loss under the
Complexity of VAF-TracIn. The run-time complexity of VAE-TracIn linear in the number of samples (N), checkpoints (C), and network parameters ((0).
Experiments
In this section, we aim answer the following questions.
Does TAE-TracIn pass a sanity check for instance-based interpretations?
Which training samples have the highest and lowest self influences, respectively?
Which training samples have the highest influences over (i.e. are strong proponents of) test sample? Which have the lowest influences over it (i.e. are its strong opponents)?
These questions examined in experiments on the MNIST [LeCun et al., 2010] and CIFAR-10 (Krizhevsky et al., 2009] datasets.
5.1 Sanity Check
Question. Does VAE-TracIn find the most influential training samples? In perfect balancee-based interpretation for good model, training samples should have large influences over themselves. As a sanity check, we examine if training samples the strongest proponents over themselves. This is analogous to the identical subclass test by Hanawa et al. [2020].
Methodology. We train separate VAE models MNIST, CIFAR, and each CIFAR subclass (the set of five thousand CIFAR samples in each class). For each model, we examine the frequency that a training sample is the most influential one among all samples over itself. Due computational limits we examine the first 128 samples. The results for MNIST, CIFAR, and the averaged result for CIFAR subclasses reported in Table 1. Detailed results for CIFAR subclasses are in Appendix B.3.
Results. The results indicate that VAE-Trach can find the most influential training samples in MNIST and CIFAR subclasses. This is achieved even under the challenge that many training samples are very similar to each other. The results for CIFAR is possibly due to underfitting as it is challenging to train a good VAE on this dataset. Note, the same VAE architecture is trained on CIFAR subclasses.
Visualization. We visualize some correctly and incorrectly identified strongest proponents in Figure 1. On MNIST or CIFAR subclasses, even if a training sample is not exactly the strongest proponent of itself, it still ranks very high in the order of influences.
5.2 Self Influences
Question. Which training samples have the highest and lowest self influences, respectively? Self influences provide rich information about properties of training samples such as memorization. In supervised learning, high self influence samples can be atypical, ambiguous or mislabeled, while

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samples
stronges propenents
sumplees
strogges propenennt
samples
strongost properents
S
S 5
3
5
O
0
0
o 4
0
4
4
(a) MNIST
(b) CIFAR
(c) CIFAR subelass
Figure 1: Certain training samples and their strongest proponents in the training set (sorted from left to right). A sample т; is marked in green box if it is more influential than other samples over itself (i.e. it is the strongest proponent of itself) and otherwise in red box.
1.0
1.0
1.0
0.35)
Inner rog IR² 0.56)
linca
0.5
0.5
0.5
VIs:
0.0
0.0
-0.010
-0.00
005
-0. -0.025 -0 020 -0 01500.000 00.005
-0.012 010 -0.008 -0 .0.0000-000 -0.004
-12000
-t200)
(a) MNIST
(b) CIFAR
(c) CIFAR-Airplan
Figure 2: Scatter plots of self influences versus negative losses of all training samples in several datasets. The linear regressors show that high self influence samples have large losses.
low self influence samples are typical [Feldman and Zhang, 2020]. We examine what self influences reveal in VAE.
Methodology. We train separate VAE models on MNIST, CIFAR, and each CIFAR subclass. We then compute the self influences and losses of each training sample. We show the scatter plots of self influences versus negative losses in Figure 2. We fit linear regression models to these points and report the I?2 scores of the regressors. More comparisons including the marginal distributions and the joint distributions can be found in Appendix B.4 and Appendix B.5.
Results. We find the self influence of a training sample Hy tends to be large when its loss (o(x;) is large. This finding in VAE is consistent with KDE and GMM (see Figure 6). In supervised learning, Pruthi et al. [2020] find high self influence samples come from densely populated areas while low self influence samples come from sparsely populated areas. Our findings indicate significant difference between supervised and unsupervised learning in terms of self influences under certain scenarios.
Visualization. We visualize high and low self influence samples in Figure 3 (more visualizations in Appendix B.5). High self influence samples are either hard to recognize or visually high-contrast, while low self influence samples share similar shapes or background. These visualizations are consistent with the memorization analysis by Feldman and Zhang [2020] in the supervised setting. We also notice that there is a concurrent work connecting self influences on log- likeliodd and memorization propertics in VAE through cross validation and retraining [van den Burg and Williams, 2021]. Our quantitative and qualitative results are consistent with their results.
Application on unsupervised data cleaning. potential application on unsupervised data cleaning is to use self influences to detect unlikely samples and let human expert decide whether to discard them before training. The unlikely samples may include noisy samples, contaminated samples, or incorrectly collected samples due bugs in the data collection process. For example, they could be unrecognizable handwritten digits in MNIST or objects in CIFAR. Similar approaches in supervised learning use self influences to detect mislabeled data KKhh and Liang, 2017, Pruthi et al., 2020, Yeh et al., 2018] or memorized samples [Feldman and Zhang, 2020]. We extend the application self influences scenarios where there are no labels.
We use the negative loss because it relates the log- likelihood of when 3 = 1, (xx) S logp(z).
7

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4 D
6
7
4
5
60
4
65
0685008
(a) MNIST (highest self-inf)
(b) CIFAR (highest self-inf)
(c) CIFAR-Airplane (highest self-inf)
S
Saplen
(d) MNIST (lowest self-inf)
(c) CIFAR (lowest self-inf)
(f) CIFAR- -IFAl-Airp (lowest self-inf)
Figure 3: High and low self influence samples from several datasets. High self influence samples are hard to recognize or high-contrast. Low self influence samples share similar shapes or background.
To test this application, we design an experiment to see if self influences can find a small amount of extra samples added the original dataset. The extra samples are from other datasets: 1000 EMNIST [Cohen et al., 2017] samples for MNIST, and 1000 CelebA [Liu et al., 2015] samples for CIFAR, respectively. In Figure 15, we plot the detection curves to show fraction of extra samples found when all samples are sorted in the self influence order. The area under these detection curves (AUC) are 0.887 in the MNIST experiment and 0.760 in the CIFAR experiment. Full results and more comparisons can be found in Appendix B.6. The results indicate that extra samples generally have higher self influences than original samples, so has much potential to apply unsupervised data cleaning.
to
5.3 Influences over Test Data
Question. Which training samples strong proponents opponents of test sample, respectively? Influences over test sample = provide rich information about the relationship between training data and In supervised learning, strong proponents help the model correctly predict the label of while strong opponents harm it. Empirically, strong proponents are visually similar samples from the same class, while strong opponents tend confuse the model [Pruthi et al., 2020]. In unsupervised learning, we expect that strong proponents increase the likelihood of = and strong opponents reduce it. We examine which samples strong proponents opponents in VAE.
Methodology. We train separate VAE models on MNIST, CIFAR, and each CIFAR subclass. We then compute VAE-Trachn scores of all training samples over 128 test samples.
In MNIST experiments, we plot the distributions of influences according to whether training and test samples belong to the same class (See results on label zero in Figure 18 and full results in Figure 19). We then compare the influences of training over test samples their distances in the latent space in Figure 20f. Quantitatively, strongest proponents/opponents. Then, we report the fraction of the strongest proponentslopponents that belong to the same class as the test sample and the statistics of pairwise distances in Table 2. Additional comparisons can be found in Appendix B.7.
define samples that have the 0.1% highest/lowest influences as the
In CIFAR and CIFAR subclass experiments, we compare influences of training over test samples to the norms of training samples in the latent space in Figure 22 and Figure 23, Quantitatively, we report the statistics of the norms in Table 3. Additional comparisons can be found in Appendix B.8.
Results. In MNIST experiments,
find many strong proponents and opponents of test sample are its similar samples from the same class. In terms of class information, many (~ 80%) strongest proponents and many (~~ 40%) strongest opponents have the same label test samples. terms of distances in the latent space, it is shown that the strongest proponents and opponents are close (thus
AUC 1 means the detection perfect, and AUC z 0.5 means the detection is random.

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Table 2: Statistics of influences, class information, and dis- tances of train-test sample pairs in MNIST. "+" means top- 0.1% strong proponents. "-" means top-0.1% strong oppo- nents, and "all" means the train set. The first two rows are fractions of samples that belong the same class as the test sample. The bottom three rows are means + standard errors of latent space distances between train-test sample pairs.
Table 3: The means + standard errors of latent space norms of training samples in CIFAR and CIFAR-Airplane. Notations fol- low Table 2. It is shown that strong proponents tend to have very large norms.
diatent
64
96
128
(+)
7.42 1.10
same class rate (1)
81.9%
84.0%
82. 82.1%
CIFAR
(-)
3.89 + 1.26
same class rate ()
37.3%
43.3%
40.3%
(all)
5.06 + 1.18
distances (+)
0.94 + 0.53
0.94 + 0.55
0.76 = 0.51
(1)
4.73 1 0.78
CIFAR-
distances
1.78 + 0.75
1.84 0.78
1.29 + 0.67
Airplane
-
4.26 + 0.91
distances (all) 2.54 0.90
2.57 0.91
2.23 + 0.92
(all)
4.07 0.83
Table 4: High level summary of influence functions in classical unsupervised learning methods (k-NN, KDE and GMM) and VAE. In terms of self influences. VAE is similar to KDE, non-parametric method. In terms of proponents and opponents, VAE trained on MNIST is similar to GMM, a parametric method. In addition, VAE trained on CIFAR is similar to supervised methods [Hanawa et al., 2020, Barshan et al., 2020].
Method
high self influence samples in a cluster of size exactly k in low density (sparse) region
low self influence samples
k-NN
KDE
in high density region
GMM
far away to cluster centers
near cluster centers
large loss visually complicated or high-contrast
small loss
VAE
simple shapes or simple background
Method
strong proponents
strong opponents
k-NN
k nearest neighbours
other than i nearest neighbours
KDE
nearest neighbours
farthest samples
GMM
nearest neighbours
possibly far away samples in the same class Tar away samples in the same class different colors
(MNIST))
nearest neighbors in the same class"
VAE(CIFAR)
large norms and similar colors
similar) samples, while far away samples have small absolute influences. These findings are similar to GMM discussed in Section 3, where the strongest opponents may come from the same class (see Figure 7). The findings are also related the supervised setting in the sense that dissimilar samples from a different class have small influences.
Results in CIFAR and CIFAR subclass experiments indicate strong proponents have large norms in the latent space. This observation also happens to many instance-based interpretations in the supervised setting including classification methods [Hanawa et al., 2020] and logistic regression [Barshan et al., 2020], where large norm samples can impact a large region in the data space, so they influential to many test samples.
Visualization. We visualize the strongest proponents and opponents in Figure 4. More visualizations can be found in Appendix B.7 and Appendix B.8. In the MNIST experiment, the strongest proponents look very similar to test samples. The strongest opponents are often the same but visually different digits. For example, the opponents of the test "two" have very different thickness and styles. In CIFAR and CIFAR subelass experiments, we find strong proponents seem to match the color of the images - including the background and the object - and they tend to have the same but brighter colors. Nevertheless, many proponents are from the same class. Strong opponents, on the other hand, tend to have very different colors as the test samples.
5.4 Discussion
VAE-TracIn provides rich information about instance-level interpretability in VAE. In terms of self influences, there is correlation between self influences and VAE losses. Visually, high self influence
'Large norm samples can be outliers, high-contrast samples, very bright samples.

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samplee
steongest progonents
strongos opponiets
O
0
D
1
1
1
V
2
2
2
2
2
2
2
2
2
7
2
G
2
3
3
3
3
3
3
3
6
6
3
8
6
S
5 4
4
4
9
9
4
9
4
4 9
9
9
9
4
4
x
CIFAR-Airplane
Figure 4: Test samples from several datasets, their strongest proponents, and strongest opponents. In MNIST the strongest proponents visually similar while the strongest opponents often the same digit but are visually different. In CIFAR and CIFAR-Airplane the strongest proponents seem match the colors and are often very bright high-contrast.
samples are ambiguous or high-contrast while low self influence samples are similar in shape or background. In terms of influences over test samples, for VAE trained on MNIST, many proponents and opponents have large norms in the latent space. We summarize these high level intuitions of influence functions in VAE in Table 4. We observe there are strong connections between these findings and influence functions KDE, GMM, classification and simple regression models.
similar samples in the same class, and for VAE trained on CIFAR, proponents
Conclusion
Influence functions in unsupervised learning can reveal the responsible training samples that increase the likelihood (or reduce the loss) of a particular test sample. In this paper, we investigate influence functions for several classical unsupervised learning methods and deep generative model with extensive theoretical and empirical analysis. We present VAE-TracIn, a theoretical sound and computationally efficient algorithm that estimates influence functions for VAE, and evaluate on real world datasets.
One limitation of our work is that it is still challenging apply VAE-TracIn to modern, huge models trained on a large amount of data, which is an important future direction. There are several potential ways to scale up VAE-Trach for large networks and datasets. First, observe both positively and negatively influential samples (i.e. strong proponents and opponents) are similar to the test sample. Therefore, could train embedding space or tree structure (such as the kd-tree) and then only compute VAF-TracIn values for similar samples. Second, because training at earlier epochs may be more effective than later epochs (as optimization is convergence then), but optimal subset of checkpoints to compute VAE- AracIn... Finally, we could use gradients of certain layers (e.g. the last fully-connected layer of the network in Pruthi et al. [2020]).
could select smaller
Another important future direction is to investigate down-stream applications of VAE-TracIn such as detecting memorization or bias and performing data deletion debugging.
10

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Acknowledgements
We thank NSF under IS 1719133 and CNS 1804829 for research support. We thank Casey Mechan, Yao-Yuan Yang, and Mary Anne Smart for helpful feedback.
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Checklist
1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper's
contributions and scope? [Yes]
(b) Did you describe the limitations of your work? [Yes]
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(a) Did you state the full set of assumptions of all theoretical results? [Yes] See Section Section 4, and Appendix A.
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(c) Did you report error bars (c.g., with respect to the random seed after running experi- ments multiple times)? [Yes]
(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] We use one NVIDIA 2080Ti GPU,
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13


Reviews:
[{"review_id": "ukR80tCHG36", "reviewer": "Reviewer_yzrG", "summary": "The paper applies the method TracIn to VAEs (beta-VAE) in order to estimate influence functions of training samples with new ones.", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "The novelty of the paper is not clear. Looking at the technical part, section 3 presents at length classical influence functions, but there is no novel component. Section 4 begins by explaining beta-VAEs for the most part, which is also background. Section 4.1 is where the new approach is explained, but is less than half a page long and, as far as I can tell, it is just straight application of TracIn. \n\nThe authors do outline as a unique challenge the impossibility to directly compute the expectation over the encoder Q, and proposed as a novel insight/method using samples to estimate the IF over this expectation (theorem 1). However, the estimation of the expectation over Q through sampling is just a standard feature of VAEs and the fact that this remains a valid approximation in estimating the IF instead of the ELBO directly derives from plugging in the ELBO into TracIn, and therefore it is not a technical novelty.", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "No unstated limitations or societal concerns.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "2"}, "extra_text": {}}, {"review_id": "du542sHeT3-", "reviewer": "Reviewer_55DE", "summary": "The paper characterizes influence estimation techniques to understand instance based interpretations for deep generative models.  Specifically, the authors carry out a comprehensive empirical and theoretical study about influence functions for generative models. \n", "questions": "", "limitations": "", "rating": 7, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "Influence estimation techniques have not been studied well for unsupervised learning and this paper does a good job of filling that gap.  The authors primarily study the effectiveness of influence function based methods to understand example based explanations and data cleaning. In the paper, influence functions for classical unsupervised learning (e.g. KDE/GMM) is first studied where the authors provide closed form solutions for influence estimations. Then, to compute influence estimates for deep generative models,  instead of using the classical influence function (Koh et al. , Cook et al.), the authors use TracIn (Pruthi et al. ) and design it specifically for VAEs. With VAE-TracIn, the authors conduct multiple empirical studies to understand in details about self-influence and top/bottom influential samples. \n\nAlthough the general influence estimation technique used in the paper (TracIn) is not entirely novel and borrowed from Pruthi et al. , the authors adapt it nicely for generative models which is not trivial to do. The empirical studies on VAE models give relevant information about what influential samples constitutes for in generative models (e.g. samples having high self-influence scores are generally atypical examples are important takeaways specifically for generative models and could be used in data cleaning). The related works section is comprehensive and the authors have covered most of the recent works. Overall, the paper is solid and can open up more research directions on using influence function based methods for generative models. However I have the following concerns/ questions to the authors:\n\n(1) While the general influence function based method (Koh et al. 2017) is computationally expensive, it has a principled theory behind it. What is the relationship between VAE-TracIn and influence function? Would using both the methods give similar inferences? \n\n(2) The experiments on VAE are on small networks and from a practical standpoint how can VAE-TracIn be scaled up for large networks ?\n\n(3) How can VAE-TracIn be used for applications other than data cleaning?\n\n(4) How precise is VAE-TracIn to quantify the ground-truth influence (e.g change in loss for a test sample when a training sample is removed)? \n\n\nAlthough the experiments are relatively small scale, I feel this paper is a very good starting point for instance based interpretability for generative models and I lean towards an acceptance. I also hope the authors provide answers to (1)-(4) in the rebuttal. \n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Although the paper is a very good starting for instance based explanations for generative models, the paper has the following limitations:\n\n(a) Experiments are relatively small scale on small networks/datasets. More experiments on modern VAEs will improve the paper from a practical standpoint.\n\n(b) No solid discussion about how VAE-TracIn compares with other influence estimation technique (Koh et al. 2017). A small discussion would be useful to understand if the generally used influence technique used in supervised learning can be readily adapted for VAEs.\n\n\nNo negative societal impact.", "review_text": "", "extra_scores": {"ethical_concerns": "No ethical concerns", "needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "3.5"}, "extra_text": {}}, {"review_id": "JHWY1BDABwo", "reviewer": "Reviewer_n5Rx", "summary": "This paper provides the mathematical frame for influence functions (IFs) for several unsupervised machine learning approaches such as kNN, KDE, and GMM, and proposes a way to empirically compute VAE's IF, and performs several experiments to describe the rationale for using IF for unsupervised tasks. Their approach, VAE-TracIn, can answer which training samples are most responsible for increasing the likelihood of itself (i.e. self-influences) and/or a test sample. The authors claim that the self-influences can be used to discard undesired training samples before training and provide several experimental results showing that IFs for a test sample can identify most and least responsible training samples.\n", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "- Originality: This paper proposes a way to calculate IFs for VAE and performs several experiments showing the ability of IFs and what they reveal about the impact of training samples. There is no big concern about originality.\n\n- Quality and Significance: This paper provides experimental results describing how VAE-TracIn answers the three questions (L186-L189). However, an experiment should be performed in section 5.2 to evaluate how much can a VAE be improved by removing x% of samples with the highest self-influence from the training samples. This can explain why it is important to identify training samples with the highest and lowest self-influence training samples. Also, it is not clear what are the benefits of knowing which training samples have the highest/lowest influence over a test sample. What is the possible application of IFs over test data in unsupervised learning tasks?\n\n- Clarity: The technical challenge and rationale are well described in the introduction. However, the benefit of using VAE-TracIn interpretation from the user's perspective is not well described in the introduction and abstract. The authors describe these in the later sections but I recommend the authors clearly describe the benefits and potential applications of VAE-TracIn in the introduction and/or abstract. Also, here are minor comments for improving the clarity. \n  1. L49: Make italic or use different font styles for the terms \"proponents\" and \"opponents\" when they're first introduced.\n  2. L84: Complimentary - Is this a typo?\n  3. L363: Please update the journal/conference name. \n\n- In summary, the main concern is that there is no or weak experimental evidence for illustrating the benefits and potential applications of using VAE-TracIn in training and test sample. The reviewer's score can be modified upon the authors' response and additional experimental results.", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "The author briefly mentions in the conclusion that it is challenging to apply their approach to millions of samples. ", "review_text": "", "extra_scores": {"ethical_concerns": "N/A", "needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "5"}, "extra_text": {}}, {"review_id": "8ap9BWMfj2K", "reviewer": "Reviewer_FRdn", "summary": "The paper investigates influence functions for several classical unsupervised learning methods and VAE. Influence functions in unsupervised learning can reveal the most responsible training samples that increase the likelihood (or reduce the loss) of a particular test sample. A particular usage of the proposed approach is for data cleaning. The authors demonstrated their approach on the MNIST and CIFAR-10 datasets. ", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "I am not an expert on instance-based interpretation methods so I cannot evaluate the significance of this work. But the methods proposed here look useful to me. The extension of instance-based interpretation from supervised learning models to unsupervised learning models is non-trivial. A potential drawback of this work is that the authors do not validate their experimental results with other existing data cleaning methods. Therefore, I am not sure the approach proposed here provides meaningful practical usage in data cleaning. I, therefore, refer to other reviewers on the evaluation of the significance of this work. ", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "See my comments in \"Main Review\".", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "1"}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper investigates the instance-based interpretation method based on influence functions in the Variational Auto-Encoder framework. The paper is well-organized and rather easy to follow. The reviewers tend to agree on the positive aspects of the paper, such as:
- Even though some ideas were introduced elsewhere, the application to VAEs is novel and interesting.
- The paper is well-positioned in the literature review.

The main disadvantage of the paper is the experimental part:
- The paper does not compare their experimental results with other existing data cleaning methods.
- There is little experimental evidence for illustrating the benefits and potential applications of using VAE-TracIn in training and test samples.

Overall, all reviews have a tendency towards the acceptance with one reviewer being positive about the acceptance. Therefore, I believe that the paper could be accepted.

Decision: Accept (Poster)