Error Discovery By Clustering Influence Embeddings
Authors: Fulton Wang
Julius Adebayo
Sarah Tan
Diego Garcia-Olano
Narine Kokhlikyan
Venue: NeurIPS 2023 poster (2023)
Keywords: Debugging
interpretability
influence functions
TL;DR: use influence functions to find clusters where the model is wrong for the same reasons

Abstract: We present a method for identifying groups of test examples---slices---on which a model under-performs, a task now known as slice discovery. We formalize coherence---a requirement that erroneous predictions, within a slice, should be wrong for the same reason---as a key property that any slice discovery method should satisfy.  We then use influence functions to derive a new slice discovery method, InfEmbed, which satisfies coherence by returning slices whose examples are influenced similarly by the training data.  InfEmbed is simple, and consists of applying K-Means clustering to a novel representation we deem influence embeddings. We show InfEmbed outperforms current state-of-the-art methods on 2 benchmarks, and is effective for model debugging across several case studies.

Error Discovery By Clustering Influence Embeddings
Fulton Wang" Meta
Julius Adebayo' Prescient Design Genentech
Sarah Tan Cornell University
Diego Garcia-Olano Meta
Narine Kokhlikyan
Meta
Abstract
We present method for identifying groups of test examples-slices sli-sssiicssss which a model under-performs, a task now known as slice discovery. We formalize coherence-a requirement that erroneous predictions, within slice, should be wrong for the same reason-as key property that any slice discovery method should satisfy. We then influence functions derive a new slice discovery method, InfEmbed, which satisfies coherence by returning slices whose examples are influenced similarly by the training data. Infembed is simple, and consists of applying K-Means clustering to a novel representation we deem inffuence embeddings. We show Enbed outperforms current state-of-the-ar methods on 2 benchmarks, and is effective for model debugging across several case studies."
1 Introduction
Error analysis is longstanding challenge in machine learning [Pope, 1976, Amershi et al., 2015, Sculley et al., 2015, Chung al., 2019, Chakarov et al., 2016, Cadamuro et al., 2016, Kearns et al., 2018, Kim et al., 2019, Zinkevich Martin, 2020]. Recently, Eyuboglu et al. [2022b] formalized type of eгror analysis termed the slice discovery problem. In the slice discovery problem, given a multi-class classification model and test data, the goal is to partition the test data into a set of slices-groups of test examples-by by model performance. An effective slice discovery method (SDM) should satisfy two desiderata:
1. Error surfacing: Identify under-perfon performpr slices, i.e. slices with low accuracy, if they exist.
2. Coherence: Return slices that are coherent in the following sense: Erroneous predictions in a given slice should have the same root cause, i.c. be "wrong for the same reason".
To illustrate, suppose we have pre-trained model that detects hip fractures from x-ray, trained on data from one hospital, that is now deployed new hospital. The model might err frequently on two groups: 1) examples containing spurious signal like the frequency signature of the scanner from the first hospital [Badgeley et al., 2019]. and 2) examples from patients not represented in the new hospital. An effective SDM should uncover these two groups The slice discovery problem has taken on renewed importance given that models trained via empirical risk minimization exhibit degraded performance on certain data groups- slices due distribution shift [Koh al., 202 l]. noisy labels [Northcutt et al., 2021] I], and imbalanced data [Idrissi et al., 2022].
In developing SDM, there two key challenges. First, formalizing the coherence property, iee, defining the root cause an error, identifying whether two errors have similar root causes, and ultimately, defining whether slice is coherent. Second, identifying the representation use:
"The two authors contributed equally. Correspondence
"Code to replicate our findings is available at: https: // com http///dbbyooo//fbbbeee
37th Conference on Neural Information Processing Systems (NcurIPS 2023).

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slices similarly influenced test errors
most influential training points
similarly influenced test errors
training points
test points
influential training points
Compute
Cluster
influence embeddings
influence embeddings
Figure I: Schematic of the Emfed slice discovery method. In slice discovery, the goal is to partition the test points into groups slices by model performance. First, for each test point, we compute its influence embeddings a low dimensional representation of the vector of influence scores of the training set. In the second stage, we cluster this influence embedding set to group points by training set influence.
returning the x-ray example, there typically no additional metadata that indicates whether an x-ray contains the spuriously correlated scanner pattern. The input to SDM will typically be the x-ray or a representation thereof, without additional metadata that indicates which features are spurious. Thus, identifying representation that can help distinguish under- mnder-performing groups from high performing ones while maintaining coherence is a key challenge.
Influence Explanations: To formalize the coherence property. we appeal to influence functions [Koh and Liang, 2017]. which tractably estimate the "effect" of training example test example how much the loss of the test example would change if the model were re-trained without that training example. We define the influence explanation of given example be the vector of the 'influence' of each training example on the test example. Since a test example's influence explanation is the "cause" of its prediction from training dataset perspective, we say a SDM achieves coherence if the slices it returns have similar influence explanations.
Influence Embeddings: To create coherent slices, might naively cluster test examples' influence explanations. However, each influence explanation is high dimensional-equal to the size of the training dataset. Instead, we transform influence explanations into influence embeddings. We define influence embeddings to be vectors with the dot-product property: that the influence of a training example on a test example the dot-product of their respective embeddings.
Given the coherence-inducing properties of influence embed- dings, we propose Embed, an SDM that applies K-means clustering to the influence embeddings of all test examples. Despite its simplicity, InfEmbed satisfies the two aforementioned desiderata. First, it is error-surfacing, mis- mis-predicted examples that are "wrong for the same reasons", i.e. have similar influence explanations, tend to be grouped into the same slice. Second, InfEmbed satisfies several desirable properties. It produces coherent slices, and can accomodate user-defined choices regarding the subset of a model's representations to use, ranging from all to only the last layer [d'Fon al., 2022, Sohoni et al., 2020]. In addition, the slices produced by InfEmbed tend to have label homogeniety, i.c. examples are homogeneous in terms of the true and predicted label. Unlike other SDMs, these properties are implicit consequences the rigorous influence function-based derivation.
Clustering Influence Embeddings:
Contributions: We summarize our contributions as follows:
1. New slice discovery method:
We propose InfEmbed, a SDM to identify under-performing groups of data that applies K-Means to new representation: influence embeddings.
2. Coherence & Theoretical Justification. We formalize the coherence property, which requires that erroneous predictions within a slice should be "wrong for the same reason". We then define influence embeddings, and prove that procedure which clusters them achieves coherence: it returns slices whose examples are influenced similarly by the training data.
3. Empirical Performance.
We show that Inf Embed outperforms previous state-of-the-arr [Eyuboglu et al., 2022a]. and SpotCheck [Plumb et al., 2022]. InfEmbed scalable, able to recover known errors in variety of settings, and identifies coherent slices on realistic models.
SDMs on 2 Denchmarks-DCHBee

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2 Background
At a high level, the goal of the slice discovery problem is to partition a test dataset into slices groups of data points-such that some slices are uuder-performing i.c. have low accuracy, and slices are coherent, i.e. erroneous predictions in given slice are "wrong for the same reason". We first formally define the slice discovery problem, and then overview influence functions, which we use to define what means for slice to be coherent one of our key contributions, and which will be detailed in Section 3.
Given a trained multi-class classification model [[: () over C classes with model parameters 0 and an example x from some input space X, f.x;;; E TRC is the prediction for the example and [x;; f) is the pre-softmax prediction for class c. Given test dataset Z = [=t,-... 2x]
Slice Discovery Problem:
with N examples where 2 == (z 31) and z E X, E y = [0, _Jc, the goal of the slice discovery problem is to partition the test dataset into K slices: { [0 where Da S [MV] Dx 2x = 0 for + K, UK 8 = [N As we will make precise shortly, each slice 4x should be coherent, and some slices should be under-performing slices. The model is assumed to be trained on a training dataset with N' examples, Z. = 1z',;-.,,''''' where 2'1 == (4'i,'') with x'; E X, v' E V/ so that 0 = argming N' >E L(2';;0') We assume the loss function L is cross- loss. Thus, given test point 2 (z. with E X, y .y yc] E V, L(2; 0) exp(f(a:0)e) p= [p...... Pcl and Pe E. exp(/(2:0) i.e. softmax(f(x; 0)).
1
E. ye log leg Pe, with
Influence functions estimate the effect of given training example.c', on test example, 2, for a pre-trained model. Specifically, the influence function approximates the change in loss for given test example when a given training example z' is removed from the training data and the model is retrained. Koh and Liang [2017] derive the aforementioned influence to be 7(z²,=) == V (2'; H VoL(2 0), where H, is the loss Hessian for the pre-trained model: Hg == 1/n Et= V2L(2; 9), evaluated at the pre-trained model's 's final parameter checkpoint. The loss Hessian is typically estimated with random mini-batch of data.
Influence Functions:
The main challenge in computing influence that it is impractical to explicitly form Hg unless the model is small, or if one only considers parameters in a few layers. Schioppa et al. [2022] address this problem by forming a low-rank approximation of Hg via a procedure that does not explicitly form Hg. In brief,, they run the Amolli iteration [Trefethen and Bau III, 1997] for P iterations to get a P-dimensional Krylov subspace for Hes which requires only P Hessian-vectus products. Then, they find a rank- approximation of the restriction of Ho to the Krylov subspace, small P P matrix, via eigendecomposition. Their algorithm, FactorHessian (see Appendix B for details), which we use in our formulation, returns factors of a low-rank approximation of H
M, A = Factor SenetorHessiin(... (Z', O, P, D) where
H = MX-1M = Hg!
(2)
M E R X E DxD is a diagonal matrix. |0| is the parameter count, Hg approximates Hy' D is the rank of the approximation, P is the Arnoldi dimension, i.c. number of Arnoldi iterations, and here and everywhere, configuration θ = (L, f) denotes the loss function, model and parameters. ft is then used to define the practical influence of training example z' on test example 2:
i(z', = VoL 6) VoL
3 Error Discovery By Clustering Influence Embeddings
We propose slice discovery method (SDM) whose key contribution is to require returned slices have coherence, and unlike past works, rigorously define coherence, which we do using influence functions. In this section, we derive our SDM as follows: 1) We formally define that a slice has coherence if its examples have similar influence explanations wwich we define using influence functions, and leverage them to formalize a clustering problem. 2) Because influence explanations are high-dimensional, we derive influence embeddings, a low-dimensional similarity- -weerrtt--preeevingg approximation. 3) Using influence embeddings, we give a simple and efficient procedure to solve the clustering problem to give our SDM, Infen. 4) We then describe an extension, Inf Embed-Run Rule, which instead of returning partition with user-specified number of slices, returns the largest slices
3

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satisfying user-specified rules. 5) We then show that the proposed formulation implicitly encourages a property, label homogeneity, that previous methods needed to explicitly encourage. in a manner that requires specifying hard-to-tune hyperparameter. 6) Lastly, explain slices using training samples.
3.1 Influence Explanations and Problem Formulation
We seek to partition the test data into slices such that predictions for examples in the same slice have similar root causes as quantified by influence functions. Therefore, we propose to represent the root cause of the prediction for test example, 2, with its influence explanation Ex)):
E(2) {f(22 (,2]]==1,
(4)
the vector containing the influence of every training example on the test example, Thus, test examples with similar influence explanations are "influenced similarly by the training data". We then want each slice be coherent the following sense: the Euclidean distance between influence explanations of examples in the same slice should be low.
Algorithm 2 Our SDM, Infembed, applies K-Means to influence embeddings of test examples.
1: procedure GETEMBEDDINGSIZ,,2 Z', θ, P)
2: Outputs: D-dimensional embeddings for test data
3: M, À & FactorHessiann('' θ, P. D)
D. see Equation 2
4: p4 f 11 0) for i I,....N
Equation 6
5:
1 - -I pN]
6: Return: f
7: end procedure
8: procedure INFEMBED(K, Z, Z', P.
9: Inputs: number of slices K, training dataset Z', test dataset Z, configuration θ Arnoldi
dimension P. influence embedding dimension D
10:
Outputs: partition of test dataset into slices, +b
11: f1
Get Embeeddins (Z, 7 P, D)
12:
ri...., rN] - K-Means( (t. K)
D compute cluster assignments for all N examples D convert cluster assignments a partition
Return: {{i E [N] : r; = k} for k [K]}
14:
end procedure
Therefore, we solve the slice discovery problem described in Section 2 via the following formulation- return partition of the test dataset into slices, { that minimizes the total Euclidean distance between influence explanations of examples in the same slice. Formally, we seek:
argmin H Σx Erves. [[,),,) El
(5)
3.2 Influence Embeddings to approximate Influence Explanations
To form coherent slices, naive approach would be to form the influence explanation of each test example, and apply K-Means clustering with Euclidean distance to them. However, influence explanations are high-dimensional, with dimensionality equal to the size of the training data, N'. Instead, we will use influence embeddings: vectors such that the practical influence a training example a test example is the dot-product of their respective influence embeddings. Looking at Equations 2 and 3, we see the influence embedding p(z) of an example 2 must be defined as
p(z) = 1/2 V (zL(2;
(6)
and M, X are as defined in Section 2, because for any training example z' and test example i(=', =) ===')T Procedure GetEmbeddings of Algorithm 1 finds D D-dimensional influence embeddings for the test dataset Z,, which has runtime linear in the Arnoldi dimension P. The rank D of the factors from FactorHessian determines the dimension of the influence embeddings.
Influence embeddings satisfy critical property-that if two examples have similar influence em- beddings, they also tend to have similar influence explanations. This is formalized by the following lemma, whose proof follows from the Cauchy-Sch wartwrtzz inequality, and that influence the dot- product of influence embeddings (see Appendix Section B.
Lemma 1: There is a constant C > 0 such that for any test examples 2,, ||E(2,) E(, ClIn(()) (2

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3.3 InfEmbed: Discovering Problematic Slices by Clustering Influence Embeddings
We will solve the formulation of Equation 5 via a simple procedure which applies K-Means with Euclidean distance to the influence embeddings of the test dataset, fe == [f(21),.... ((x))) This
procedure is justified as follows: Using Lemma 1, given any partition { { ,, we know
Ex Eiveo [1((2) ((zz)( )12 S &E Eures, lll(zz)
m(z))11²
However, the quantity in the right of Equation 7 is the surrogate objective minimized by this procedure-it is the K-Means objective applied to influence embeddings, scaled by C' > 0 which does not matter. The quantity in the left of Equation 7 is the actual objective which the formulation of Equation 5 minimizes. Therefore, the procedure is minimizing a surrogate objective which upper bounds the actual objective we care about. Put another way. by applying K- K-Means to influence embeddings, examples within the same slice will not only have similar influence embeddings, but also similar influence explanations as desired, by Lemma 1. Algorithm describes Inf Embed. In addition, we find that normalizing the centroid centers in each iteration of K-Means is helpful, perhaps due to lessening the effect of outlier influence embeddings. Overall, instead of K-means, other clustering algorithms can used as part of formulation.
Computational Complexity: We now take a step back to examine the computational complexity of the proposed procedure. The main computational bottleneck is the implicit Hessian estimation required to compute influence embeddings. Schioppa et al. [2022]'s Is approach, which we rely on, has complexity O(P) where P is the Amoldi dimension. This is because each of the P steps in the Arnolii iteration requires computing a deessannvvecto product. Consequently, the complexity of computing influence embeddings exactly the same that of the influence function (IF) estimation. For the K-means portion, the complexity is Onumber of samples x number of k-means iterations x number of clusters). In practice, the clustering step is near instantaneous, so the method is dominated by implicit Hessian estimation step.
3.4 InfEmbed-Rule Finding Slices Satisfying a Rule
The key prpeer-prrmmttr of the InfEmbed method is K, the number of slices return. In practice, it may not be intuitive for user to choose K. Instead, the user may want to know if there exist any coherent slices that are problematic, where 'problematic' is pre-specified by a rule, e.g., slice with greater than T samples and that has accuracy less than A. Without specifying the rule, a practitioner has iterate through all the returned slices to figure out which one problematic. Therefore we propose a procedure that recursively clusters influence embeddings until slices satisfying a pre-specified rule are found, or until the slices are too small.
The proposed approach, InfEmbed-Ruil is analogous to building a tree to identify slices satisfying the rule, where the splits are determined by K-Means clustering of influence embeddings. In addition to letting the user specify more intuitive hyperparameters, this procedure also has the advantage that if a large slice with sufficiently low accuracy is found, will not be clustered further. Appendix Algorithm B.2 outlines this Embed RRule method. Its inputs are the same as Inf except instead of K, one specifies accuracy threshold A, size threshold S, and branching factor B, which sets how many clusters the K-means call in each step of the recursion should return. It outputs set of slices each with accuracy less than A and size greater than S.
3.5 Properties of Discovered Slices
We now show that the slices discovered by InfEmbed possess properties similar to past SDM's. To derive these properties, we consider what factors result in two examples = (x., y), 2' (a', y²) having influence embeddings with low Euclidean distance, so that K-Means would place them in the same slice. To simplify analysis, we note that two examples would be placed in the same slice if their gradients have high dot-product, because influence embeddings are linearly transformed gradients (see Equation 6) and Euclidean distance is equal to their negative dot-product plus a constant. For further simplicity, we consider the case when the model f is linear model, i.e. when only gradients in the last fully-comm cullecconeected layer are considered when computing influence embeddings. Then, a straightforward computation shows that two examples s, 2' will be placed in the same slice if
OzL(2;))VVoLl = (v - Ty) (0'''''''''"" p')x
(8)
5

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is high, where L is cross-entrope loss, are the parameters, and for example 2, p :== softmax(f=)) is the predicted probabilities for each class, y is the one-hot encoded label, is its last-layer representation, and p', y xx are defined analogously for example z'.
[pi..... PC]
that two examples will be placed in the same slice if, all else being equal, their last-layer representations are similar as measured by the dot-product. Past SDM's [d' Eon et al., 2022, Sohoni et al., 2020] also use heuristic similarity measures involving last-layer representations to decide if two examples should be in the same slice. On the other hand, our influence function- based derivation shows that their dot- dot-product the "canonical" similarity measure to use. Note that since influence embeddings consider gradients in layers beyond the last, they are able to consider information beyond the last-layer representations, generalizing past SDM's. s..
Looking the x'xx term of Equation 8,
Looking at the (y- - T - -p') term of Equation 8, we see that two examples will be placed in the same slice if, all else being equal, their margins are similar, as measured by dot-product. The margin of an example, eg- (y p) or (v' p²). is the vector containing the difference between the true label and predicted probability for each class. Thus, examples with similar labels and predictions will tend to be in the same slice, i.e. slices tend to have label homogeniety. Past SDM's [Eyuboglu et al., 2022b] also produce slices that tend to have label homogeniety, but do so through an "error-aware" mixture model that requires choosing a hard-to-tune tradeoff parameter. On the other hand, our influence function- to encourage label homogeniety, and does not require tuning a trade-off constant. Note that label homogeniety is not the only factor in consideration when forming slices.
derivation shows that the dot-product of margins is the "canonical" way
3.6 Slice Explanation via Slice Opponents
Given an under-performing slice, we are interested in identifying the root-cause of the erroncous predictions in it. We therefore compute the top-k slice opponents and consider them the root-cause. These are the k training examples whose influence on the slice are most harmful, i.e. whose influence on the total loss over all examples in the slice is the most negative.
Formally, given a slice h we first compute the influence embeddings of the training data, (=(=))...-,, (= Then, we compute the sum of the influence embeddings of the slice, v == Eieo (=zz)). Finally, due to the properties of influence embeddings, the top-k slice oppo- nents of the given slice, O, are the k training examples 2' for which eT =²2") is the most negative, i.e. eT (Eieo- p((z))). Inspecting slice opponents can provide insight into the key features responsible for low performance in a slice.
O = argminorgiv"
4 Results
Here, we perform quantitative evaluation of Infenbed the dcbench benchmark, and also perform several case studies showing Embed's usefulness. Please see Appendix Section H on a case study applying InfEmbed to small dataset where additional metadata can help explain slices. For scalability, to compute influence embeddings, at times we will only consider gradients in some layers of the model when calculating the gradient Lo 0) and mnverse-Hessian factors M, 1/2 in Equation 6. Furthermore, following Schioppa et al. [2022], we do not compute those factors using the entire training data, i.e. we pass in a subset of the training data to FactorHessian For all experiments, we use Arnoldi dimention P 500, and influence embedding dimension D 100, unless noted otherwise. In the experiments that use Infemdd-Ruull -Rle, we used branching factor B=3. The rationale is that B should not be too large, to avoid unnecessarily dividing large slices with sufficient low accuracy into smaller slices. In practice B=2 and B=3 did not give qualitatively different results.
4.1 InfEmbed on dcbench
Overview & Experimental Procedure: dcbench [Eyuboglu et al., 2022a] provides 1235 pre-trained models that are derived from real-world data, where the training data is manipulated in 3 ways to lead to test-time errors. This gives 3 tasks discovering slices whose errors are due to the following manipulations: 1) "rare" (examples in given class are down-sampled), 2) "correlation" (a spurious correlation is introduced), and "noisy label" (examples in given class have noisy labels). Since the error causing slice is known priori, dcbench can serve way to assess new SDM. The

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underlying datasets for those tasks include 3 input types: natural images, medical images, and medical time-series. Eyuboglu et [2022a] also introduced Domino, the SDM that currently the best-performing SDM on dcbench, which uses multi-modal CLIP embeddings as input to an error-aware mixture model. Following Eyuboglu et al. [2022a], we compare InfEmbed to Domino using the precision-at-k measure, which measures the proportion of the top k (k 10) elements in the discovered slice that are in the ground truth slice. We evaluate in the setting where errors are due to trained model.
Results: Table compares Emfed to Domino across the 3 tasks and 3 input types. InfEmbed always beats Domino except on the
Correlation
Noisy Label
SDM Approach
"noisy" task for the EEG Medical Time se-
Natural Images
ries input type. We can also compute coher- ence scores- the k-means objective within each cluster-for for each setting in the domino bench- mark,
Domino Inf- inf-Emeed
0.4
0.45
0.6
0.65
0.55
0.67
Medical Images
Domino Inf inf-Emeed
0.39
0.6
0.58
0.57
0.62
0.73
Medical Time Series
We also compute the coherence scores, the K- means objective that we cluster, across all slices for influence embedding clustering as well as for domino. We find similar correspondence to the previous results. In every setting where we outperform Domino (1219 out of 1235 trained settings), our coherence scores are better, in some cases, by almost 50 percent. Regarding label homogeneity: In high performing clusters, we find that influence embedding clustering has lower label homogeneity than domino. However, for error clusters we find the reverse. The goal of our scheme is to find clusters where the model is 'wrong for the same reason'. We conjecture that influence embeddings are most effective for these settings.
Domino Inf-Emnedd
0.6
0.55
0.9
0.64
0.65
0.81
Table I: InfEmbed generally outperforms Domino across 3 input types and 3 tasks on the dcbench benchmark in terms of precision (higher is better).
4.2 Infed on the SpotCheck Benchmark
Overview & Experimental Procedure. The SpotCheck benchmark Plumb et al. [2022] is based on synthetic task consist- ing of 3 semantic features that can be eas- ily controlled to determine the number of
DR
FDR
SDM Approach
Barlow
0,43(0.04)
0.03(0.01)
Spotlight
0.79(0.03)
0.09 (0.55)
"blindspots' in a dataset. A blindspot is a feature responsible for model's mistake. Similar to dcbench, a model trained on data generated from SpotCheck is induced to make mistakes on an input that has a set of blindspots. The task is to predict the pres- ence of square in the image. Attributes of the input image such the background, object color, and co-occurrence can be varied to induce mistakes in models obtained from the manipulated data generation process. Plumb et al. [2022] find that current SDM approaches struggle in the presence of several blindspots. Consequently. they propose PlaneSpot, an SDM that uses the representation of the model's last layer, projected to 2 dimensions, along with the model's 'confidence" as part of mixture model to partition the dataset. They show that PlaneSpot outperforms current approaches as measured by discovery and false discovery rates. We replicate the SpotCheck benchmark, for subset of features, and compare InfEmbed to PponeSpot. We refer to the Appendix Section D. for additional details.
Domino
0.64 (0.04)
0.07(0.01)
PlaneSpot
0.85 (0.03)
0.07 (0.01)
Inf-Embed
0.91 (0.09)
0.15 (0.62)
Table 2: Spotcheck Benchmark results.
Results. We report the performance of the influence embedding approach compared to PlaneSpot on the Spotcheck benchmark in Table 2. We find that Inf-Embed approaches improves upon PlaneSpot's performance on the benchmark, which indicates that the proposed approach is an effective SDM.
Alternative Clustering Approaches. To check that the proposed scheme robust to the choice of clustering algorithm, we replicate the spotcheck experiments with 2 additional clustering algorithms: 1) DBSCAN, and 2) Spectral Clustering. Similar to Table 2, we find that the discovery rate for these algorithms are: 0.92, and 0.89 respectively. In both both cases, the performance of the InfEmbed
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Distribution Labels Predictions
Examples from Sioee
Opponents" Slice
Poat furies
Held incl
meette Phont
Sloc1
1000 003P
Accuracy
39%
F-bood oomees
araag 2uch 4.0-1g
Pead Takan Park
Slioe 2
Accursty
K100g
Figure 2: Overview of Influence embedding Clustering for a Resnet-18 on ImageNet. We show two slices from applying the influence embedding approach to the ImageNet data. We apply the RuleFind algorithm and identify slices with accuracy under 40 percent and atleast 25 samples. Left: On the left, we show the distribution of labels and model predictions in that slice (Labels are in red, and Prediction are blue). Middle: Representative examples from the slice. Right: We show four opponent examples for the identified slice.
procedure does not degrade, which indicates that the influence embedding representations are robust to the clustering algorithm used.
4.3 Imagenet & High Dimensional Settings
Overview & Experimental Procedure: Here, we identify problematic slices in a "natural" dataset- the test split of Imagenet [Deng et al., 2009], containing 5000 examples. This image-classification problem is a 1000-class problem, and the analyzed model is a pre-trained Resnet-18 18 model achieving 69% test accuracy. To compute influence embeddings, we consider gradients in the "fc" and "layer4" layers, which contain 8.9M parameters. We use Embed Ruule to find slices with at most 40% accuracy, and at least 25 examples. To diagnose the root-cause of errors for a slice, we examine the slice's 's opponents, following Section 3.6.
We find 25 slices, comprising total of 954 examples, whose overall accuracy is 34.9%. Figure displays 2 of these slices (See Section E for additional examples). For each slice, we show the distribution of predicted and true labels, the 4 examples nearest the slice center in influence embedding space, and the 4 strongest opponents of the slice. First, we see that the slices have the label homogeneity property explained in Section 3h5-the predicted and true labels typically come from a small number of classes. In the first slice, the true label is mostly "sidewinder", and often predicted to be "horned viper". The slice's strongest opponents are horned vipers, hard examples of another class, which the model considers to be similar to sidewinders. In the second slice, we observe two dominant labels. Here, vine snakes and analog clocks are often mis- pridspeed..ed. A priori, one would not have guess the predictions, for these seemingly, unrelated classes would be wrong for the same reasons. However, looking at the opponents, which turns out to be similar to a snake. Also, because the opponents are of a variety of classes, the mis- predictions are also of a variety of classes. Overall, we demonstrate here that procedure can identify problematic slices with low performance and then trace the cause of the errors to subset of the training data.
Results:
all of them have clock hand,
4.4 AGNews & Mis-labelled Data
Overview & Experimental Procedure: We identify problematic slices in another "natural" dataset- the test split of AGNews [Zhang et al., 2015], comprising 7600 examples. This text-classffication problem is a 4-class problem, and the pre-trained model is a BERT base model fine-tuned on the training dataset? 9, achieving 93% test accuracy, making 475 errors. We use Imbed with K 25, and examine slices with at most 10% accuracy, and at least 10 examples.
Results: We find 9 such slices, comprising total of 452 examples, whose overall accuracy is only 3%. Table 3 shows 2 slices, both of which have label homogeniety. In the first slice, the model achieves 0% accuracy, systematically mis mis-predic "business" examples to be "world", perhaps predicting any text with a country name in it to be "world". In the second slice, most labels are
8

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Slice with 45 examples, 0% accuracy. Predictions: 100% "world". labels: 100% "business"
British grocer Tesco sees group sales rise 12. 12.0-percent. Britain's 's biggest supermarket chain ...
2. "Nigerian Senate approves $1.5 bin claim on Shell LAGOS- Nigeria's is Senate has passed...
3. "Cocoa farmers issue strike threat. Unions are threatening a general strike in the Ivory Coast ...
Slice with 139 examples, 3% accuracy. Predictions: 93% "business", labels: 94% "sciltech"
1. "Googl Unveils Desktop Search, Takes on Microsoft. Google Inc. rolled out a version of its
2. PallmOne play with Windows Mobile? Rumors of Treo's 's using Microsoft operating system....
3. "ntel Posts Higher Profit, Salex. Computer-chip maker Intel Corp. said yesterday that earnings...
Table 3: Representative examples for slices obtained using Imbed on AGnews. In the 1st slice, the model mis-predicts business text containing country names to be "world". The 2nd slice appears to be mis-labelled, showing Infed can detect mis-labeled data.
"science/technology". and most predictions are "business". However looking at the examples, they may in fact be mis-labeled, as they are about businesses and their technology. This shows that InfEmbed can also identify mis-labeled data.
4.5 Detecting Spurious Signals in Boneage Classification
Overview & Experimental Procedure: We now artificially inject signal that spuriously correlates with the label in training data, and see if InfEmbed-rule can detect the resulting test errors as well as the spurious correlation. Here, the classification problem is bone-age classification 4- to predict which of 5 age groups a person is in given their x-ray. Following the experimental protocol of Zhou al. [2022] and Adebayo al. [2021], we consider possible spurious signals: tag, stripe, or blur (see Appendix for additional details). For a given signal, we manipulate the bone-age training dataset by injecting the spurious signal into all training examples from the "mid-pub" class (short for mid- mid-puberty). Importantly, we leave the test dataset untouched, so that we expect model trained on the manipulated training data to err on "mid-pub" test examples.
This setup mimics a realistic scenario where x-ray training data comes from a hospital where one class is over-represented, and also contains spurious signal (i.e. x-rays from the hospital might have the hospital's 's tag). We then train a resnet50 on the manipulated training data, and apply Emfed-RuRll on the untouched test data discovered slices with at least 25 examples and at most 40% accuracy (same as for the Imagenet analysis). We repeat this manipulate-train-siic discovery procedure for runs: once for each possible signal.
Results: For all runs / signals, InfEmbed-Rie Rule returned 2 slices satisfying the rule. From the distribution of labels and predictions, we observe that for each signal, the labels in the most problematic slice are of the "mid-pub" class, showing that InfEmbed-Rui RRule is able to detect test errors due to the injected spurious training signal. Furthermore, for each run, around 80% of the slice's ' opponents are training examples with the spurious training signal, i.e. examples with the "mid-pub" label. Together, this shows Inf Embed-RoRoe Rulle can not only detect slices that the model errs on because of a spurious training signal, but also identify training examples with the spurious training signal by using slice opponents.
5 Related Work
Error analysis, more generally, has been studied extensively even within the deep learning literature; however, here we focus our discussions the recently formalized slice discovery problem- as termed by Eyuboglu et al. [2022b].
To the best of our knowledge, Infedbed is the first SDM based on influence functions. More broadly, it represents the first use of influence functions for clustering and global explainability; they have been used for local explainability, i.e. returning the influence explanation for a single test example, but not for clustering, as we do. Pruthi et al. [2020] also define influence embeddings, but use a different definition and also use it for different purpose- accelerating the retrieval of an example's 's most influential training examples. Crucially, their embeddings were high- nigh-dimensional (exceeding the number of model parameters) and thus ill-suited
Error Analysis, Slice Discovery, Related Problems:
'https:/ peittps://ww.. http://wwwww.ggl..oo com haaaa//ll.ooo//ttttstt/tmeeeeeee bwwp://rwwaaaabaaaaaeeeeeeeeeeee age

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for clustering. and lacked any dimension reduction procedure preserving the dot-product property, aside from random projections.
Previous SDMs use a variety of representations as input into a partitioning procedure. Some rely on external feature extractors, using PCA applied to pre-trained CLIP embeddings [Eyuboglu et al., 2022b] along with an "error-aware" mixture model, or features from a robustly pre-trained image model [Singla et al., 2021] along with decision tree. This external reliance limits the generalizability of these SDMs-what if one wanted to perform slice discovery on a domain very different from the one CLIP was trained on, or wanted consider new modality, like audio or graphs? Other SDM's 's use the pre-trained model's ' last-layer representations Ed'Eon et al., 2022], or transform them via SCVIS [Plumb et al., 2022] or UMAP [Sohoni et al., 2020] before applying K-Means clustering. While they are performant, it is less clear why this is. In contrast, InfEmbed does not rely on external feature extractors and is theoretically framed by our influence function- baseed definition of coherency.
Other methods solve problems related to, but not the same as slice discovery. Rajani et al. [2022], Wiles et al. [2022], Hua et al. [2022] leverages the aforementioned SDM's to create interactive systems for discovering slices, focusing on improving the user interface and how to explain slices. In contrast, we focus on solving the core slice discovery problem. All these methods use last-layer representations along with K-Means or an "error-aware" mixture model [Hua et al., 2022]. Jain et al. [2022] globally ranks examples to identify examples in single failure mode, but being unable to output multiple slices, is not a SDM. Numerous works also rank examples using scores capturing various properties: the degree to which a sample out-of-distribution [Liu et al., 2020]. reliability
[Schulam and Saria, 2019], or difficulty Simsek et al. [2022]. Complementary to SDMs are methods for identifying low-performance subgroups when tabular metadata is available [Lakkaraju et al., 2017].
Influence Functions: Influence functions [Koh and Liang, 2017] have been used to calculate influences, which can be used for ranking training examples for local explainability [Barshan et al., 2020] for both supervised and unsupervised models [Kong and Chaudhuri, 2021], identifying mis- labeled data interactively [Tees et al.. 2021] 11. data augmentation [Lee et al., 2020], active learning [Liu et al., 2021], quantifying reliability [Schulam and Saria, 2019], and identifying mabela data [Pruthi et al., 2020]. However, influence functions have not been used for slice discovery or global explainability.
Conclusion
We present a method, Embed, that addresses the slice discovery problem. Our proposed solution departs from previous work in that it both identifies slices-group of data points- on which a model under-performs, and that satisfy a certain coherence property. We formalize being wrong for the same reasons within a slice-as as a key property that all slice discovery methods should satisfy. In particular, we leverage influence functions [Koh and Liang, 2017] to define coherent slices as ones whose examples are influenced similarly by the training data, and then develop a procedure to return coherent slices based on clustering a representation we call influence embeddings. We demonstrate on several benchmarks that Infed out-performs current approaches on a variety of data modalities. In addition, we show through case studies that InfEmber is able to recover known errors in a variety of settings, and identifies coherent slices on realistic models.
interence--rediettinn
Limitations. One major limitation of the proposed approach is that the validation set needs to include the kind of error that one is seeking to detect. In addition, the proposed procedure suggests to inspect the slice opponents as the root cause of the wrong predictions in a slice. However, it is still up to the user of the proposed Algorithm to infer which feature(s) of the slice opponents is responsible for causing the errors. Despite these limitations, error discovery by clustering influence embeddings can be seen as additional tool in the ML model debugging toolbox.
Acknowledgments and Disclosure of Funding
Fulton Wang, Sarah Tan, Diego Garcia- and Narine Kokhlikyan were employed by Meta, while Julius Adebayo was employed by Genentech over the course of work on this project. No external funding was received or used by the authors over the course of the project.
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Reviews:
[{"review_id": "UgHJKcp24T", "reviewer": "Reviewer_2vT2", "summary": "This paper proposes a method (InfEmbed) for discovering coherent slices of data such that the model fails on samples within a slice due to similar reasons. InfEmbed uses k-means to cluster a representation proposed in the work called influence embeddings, where samples with similar influence embeddings have similar influence explanations. The proposed slicing method outperforms prior work on standard slice discovery benchmarks.", "questions": "My main concerns are around hyperparameter selection, missing details and analysis.\n1. It would be great if the paper can provide error bars in Table 1 for both methods, as Figure 3 of Domino shows high standard deviations across the different settings. Also, it would be useful to report the total number of settings (out of 1235 settings), where InfEmbed outperforms Domino.\n2. Can the authors report coherence scores (Equation 4) for different methods in Table 1?\n3. Can authors measure the label homogeneity (entropy of predicted distributions) of InfEmbed and compare it with Domino?\n4. Can authors provide guidelines on how a practitioner could select the branching factor $B$ and desired accuracy $K$ to get slices with “at most K% accuracy with at least $m$ samples”, which can surface all error modes ranked by accuracy? Currently it seems like the user may have to try a bunch of values to arrive at a reasonable set of slices. \n5. What is the value of $K$ used for K-means for the results in Table 1 and Table 2. Does it match the values used by Domino?\n6. Section 5: why are the influence embeddings expected to be real valued? In other words, why is  $\\lambda$ guaranteed to have positive values? \n7. What do the practical values for $C$ in Lemma 1 [L582] look like? Can a theoretical bound be established on the values of $C$?\n8. On L188: Why are norm terms treated as a constant?\n9. Any reason for marking the reproducibility as N/A?\n10. Can the performance be further improved if k-means is replaced with Domino’s error-aware mixture modeling that more directly promotes label homogeneity?\n\n\n\nTypos:\n* L42: “influence explanation, L\n* L215: “whose influence on the slice is most harmful”\n* L297: “Apriori one would not have guessed”\n* L560: $H_\\theta^{P - 1} b$\n", "limitations": "Yes.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 4, "strengths": "1. The work attempts to solve an important problem of surfacing failure modes of a model automatically.\n2. The work has sufficient technical novelty: it proposes influence embeddings, relating them to influence explanations and using them in the context of slice discovery.\n3. The work does a thorough evaluation by evaluating the proposed method against multiple slice discovery benchmarks in addition to multiple case studies.\n", "weaknesses": "My main concerns are around hyperparameter selection and missing analysis.\n1. The work formalizes coherence (Equation 4), but does not compare to prior works (mainly Domino) in terms of coherence.\n2. It is not clear how the method promotes label homogeneity when all gradients of the neural network are considered. The analysis in 3.5 works only when the gradients in the fully connected layer are considered.\n3. For a practical use case, the approach still requires the users to determine and specify the branching factor $B$ and the maximum accuracy $A$ of a slice. \n4. The work doesn’t provide discussion on the selection of hyperparameter choices: e.g. L231 - the values of P and D, which are used in FactorHessian and the number of clusters ($K$) for the results in Table 1.\n5. It is unclear how the authors choose the number [L286] of layers for which gradients are considered. Is it based on the total number of parameters?\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your detailed feedback. We address your concerns below.\n\n**Weaknesses**\n- **Coherence scores and Experiments**: See discussion in the first bullet point of the next section.\n- **How does InfEmbed promote label homogeneity?**: While the analysis in Section 3.5 only considered gradients in the last layer, it does also explain why label homogeneity is encouraged, even when considering gradients in all layers.  This is because for two examples, the dot-product between their gradients in all layers is equal to the dot-product between their gradients in the last-layer, plus the dot-product between their gradients in the remaining layers, and the presence of the former encourages label homogeneity. We will add this clarification to the paper.\n- **How to select B (branching factor) and A (minimum accuracy of a slice) when running InfEmbed-Rule**:\n1) **Regarding B**: In all experiments, we used branching factor B=3.  The rationale is that B should not be too large, to avoid unnecessarily dividing large slices with sufficient low accuracy into smaller slices.  In practice, B=2 and B=3 did not give qualitatively different results for the case studies where InfEmbed-Rule was used.\n2) **Regarding A**: The choice of A (seek slices with at most A% accuracy) is application specific, depending on the overall accuracy over the entire test data, and the level of accuracy tolerable for the application. We suggest starting with a low value of A, see if slices are discovered, and increase A if not.  Fortunately, the clustering step is fast, so that this process can be done interactively.\n- **Hyper-parameters (P, D, K)**: Thank you for raising this point. For the case studies, we used a consistent hyper-parameter scheme across the board, and for the baselines, we match the underlying approaches. We will update the draft with all of these hyper-parameters. \n- **Choosing layers**: As is customary in the influence function literature, we choose the last layer. Note that we did not tune this choice for the case study, and it is possible to scale the approach to all parameters for the models we consider.\n\n**Questions**\n- **Error bars, Coherence Scores, and Label Homogeneity**: Thank you for these suggestions. Starting with the domino benchmark, we compute the coherence scores, which corresponds exactly to the K-means objective that we cluster, across all slices for influence embedding clustering as well as for domino. We find a correspondence between our current results (Table 1 in the main paper), and the coherence scores. In every setting where we outperform Domino (1219 out of 1235 trained settings), our coherence scores are better, in some cases, by almost 50 percent. Regarding label homogeneity: In high performing clusters, we find that influence embedding clustering has lower label homogeneity than domino. However, for error clusters we find the reverse. The goal of our scheme is to find clusters where the model is ‘wrong for the same reason’. We conjecture that influence embeddings are most effective for these settings. We will update Table 1 and 2 with error bars, label homogeneity scores, and coherence scores.\n- **Choosing hyper-parameters B, A for InfEmbed-Rule**: Please see response to Weaknesses #3.\n- **Choosing hyperparameter K**: For K (number of slices), we follow past work to ensure comparability of quantitative results: when comparing to Domino in Table 1, we follow Domino and use K=25, and when comparing to various methods on the SpotCheck benchmark in Table 2, we set the number of clusters K using the Bayesian information criterion (BIC), following how the SpotCheck benchmark authors chose K for their slice discovery method, PlaneSpot.\n- **Why are the influence embeddings real-valued?** Thank you for pointing this out. In finding a low-rank approximation of the Hessian, we actually only use the top eigenvectors / eigenvalues for which the eigenvalues are positive. Effectively, this means we first project the Hessian to the closest (in terms of L2 norm) symmetric matrix that is positive definite, before finding its low-rank approximation.  This means that in the algorithm FactorHessian, V, $\\lambda** should actually be the top-D eigenvectors / eigenvalues for which the eigenvalues are positive.  We will add this clarification to the paper.\n- **Values of C from the Lemma**: C is the sum of the L2 norms of the influence explanations of the test data, and is thus large.  However, given the strong quantitative results, and that examples within discovered slices tend to have the same opponents, the high value of this theoretical constant does not affect practical results.\n- **Norms**: We apologize for the confusion - what that line is trying to say is that clustering to maximize intra-cluster dot-product similarity is equivalent to clustering to minimize intra-cluster Euclidean distance, because the objective function of the former is equal to the objective function of the latter plus a constant that does not depend on the specific clustering. Therefore, when considering when 2 examples would be placed in the same slice, we just need to consider their dot-product, which is easier to analyze than their Euclidean distance.\n- **Why N/A for reproducibility**: This was a mistake. We will release all our code, datasets, and artifacts to replicate our analyses.\n- **Error aware mixture modeling**: One of the advantages of our method is that by simply applying K-Means to the right representation, influence embeddings, it achieves the same goal as Domino’s error aware mixture model, i.e., the influence embeddings themselves are ‘error-aware’. Having said this, a mixture model can be applied on top of the influence embeddings, and it might be possible to endow the influence embeddings with other qualities. Studying the effects of using influence embeddings with an error aware-mixture model is an interesting future direction.\n\nWe would be happy to address additional questions."}, {"review_id": "4F0tRc0P0d", "reviewer": "Reviewer_uzoE", "summary": "In this paper, the authors propose InfEmbed on the slice discovery problem. The method is derived from the influence function and surrogate embedding representations are proposed for reducing complexity. Overall, the paper is well written and the derivation of the method is reasonable. Some pros and cons are discussed as follows:", "questions": "Do the authors validate the performance of the method with different clustering algorithms?\nHow is the runtime complexity using influence embedding compared with directly using influence functions?\n", "limitations": "Some results are not well visualized and shown. It is recommened that the authors reorganize the result section to make it more readable. ", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper proposes a new slice discovery method, which is based on K-Means and the influence embedding the authors proposed. The method is well-designed and I believe is relatively easy for deployment. The overall derivation of the method is clear and theoretical analysis is supported.\n\nThe question that the authors are addressing is of importance and the experiment results seem to validate the performance of this method.\n\nThe proposed method, if it is effective like the paper states, can provide insights in other domains that require identifying groups of data.\n", "weaknesses": "The K-means is sometimes unstable. As the authors claim the clustering algorithms can be others, they should also validate the results on more cluster algorithms such as the spherical k-means. \n\nThe result section is not organized well. It is recommended that the authors discuss datasets, baselines, and results in different subsections and provide more details of data description and baseline methods.\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your detailed feedback. We address your concerns below.\n\n**Weaknesses**\n- **Trying other clustering algorithms**: We actually did try 3 other clustering algorithms (DBSCAN, spectral clustering, gaussian mixture model) on the Spotcheck benchmark, and found that no matter what clustering algorithm was used on influence embeddings, the performance beat the Domino and PlaneSpot baselines.  These results were unfortunately hidden in Appendix J, and will be moved to the main body.\n- **Organization of results section**: Thank you for the suggestion. We will update the submission to reorganize the results section, and also provide additional overview data and baseline method details.\n\n**Questions**\n- **Trying other clustering algorithms**: Please see the response to weaknesses bullet point 1.\n- **Runtime complexity using influence embedding compared with directly using influence functions**: Directly using influence functions to generate the influence explanation of every test example, and then clustering them would be computationally impractical because for each test example, the length of its influence explanation is the same as the training data size (i.e. if the training data had 1M examples, each test example’s influence explanation would be 1M-dimensional and computing the influence of every training examples on the test example).  In that sense, clustering influence embeddings (instead of influence explanations) is far more efficient, as the dimension of what is being clustered is much smaller. \n\nWe would be happy to answer any additional questions that you may have."}, {"review_id": "UTkg7n74cc", "reviewer": "Reviewer_MYaZ", "summary": "This paper proposes a heuristic clustering-based method for identifying errorneous groups of test examples. I'm erring on the side of caution here and go with a weak reject, but I have limited familiarity with the subfield.\n\nAC note: score increased 4 -> 6 after rebuttal.", "questions": "* What is the computational complexity of the method? Is the main computational bottleneck in the implicit Hessian estimation?\n* Is there any intuition or an ablation study on the number of elements needed for the Hessian estimation?", "limitations": "N\\A", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "* Error analysis tools are useful for many different ML systems; better algorithms for discovering slices might have have significant practical impact.\n* The paper is reasonably clearly written and easy to follow.", "weaknesses": "* The main algorithm for estimating the seems to be taken from Schioppa et al.; without it, the methodological contribution of the paper seems quite limited in its nature.\n* The paper seems rather simplistic in its approach and not very exciting in its results – cf. Table 2, which is the main evaluation where the technique from the paper is compared to the related work.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your detailed feedback. We address your concerns below.\n\n**Weaknesses**\n- **Methodological Contribution**: We agree that the actual computation of the influence embeddings being clustered in the main algorithm is a straightforward application of Schioppa et al. [2022]. However, we believe we make 3 key contributions, which we clarify here, and will add to the paper:\n1. Our work is the first to connect one popular problem in model debugging (slice discovery), with a popular tool in model debugging (influence functions).\n2. A main contribution is to theoretically derive what the representation to cluster in a slice discovery algorithm should be (influence embeddings), in terms of an influence function-based definition of coherency. This is significant, given that past slice discovery algorithms choose to cluster various representations (CLIP embeddings, last-layer representations, last-layer representations projected by SCVIS) without theoretically explaining why clustering those representations should enable effective slice discovery.  In doing so, we have improved over previously published slice discovery algorithms (e.g. methods that extracted CLIP embeddings or last-layer representations), where the actual computation of those representations did not require any methodological contribution.\n3. We are the first to use influence functions (of any kind, including but not limited to Schioppa et al) for global explainability (simultaneously explaining predictions for the entire test dataset), unlike past work, which solely focused on local explainability (explaining one prediction at a time, independent of other predictions).\n\n- **Simplicity and Results**: We agree with your take, the algorithm itself is simple; a benefit in practice. However, its development required theoretical insight that past slice discovery approaches missed. In addition, our results (**Table 1 and Table 2**) show that we outperform competing approaches. Lastly, across a series of case studies, we are able to use the scheme to identify underperforming slices in real-world tasks, which is an important challenge in practice. Slice discovery is a critical and challenging task, in practice; the proposed approach is a step towards developing reliable tools for solving that problem.\n\n**Questions**\n- **Computational Complexity**:  Indeed, the main computational bottleneck is the implicit Hessian estimation. Although Schioppa et al. do not explicitly state their complexity, it is O(P), where P is the Arnoldi dimension. This is because each of the P steps in the Arnoldi iteration requires computing a Hessian-vector product, where the Hessian is typically computed on a subset of the training data for tractability. The complexity of computing influence embeddings is then exactly the same as that of the influence function (IF) implementation of Schioppa et al. We also wish to point out the following note in Schioppa et al.’s paper: “Empirically, IFs with Arnoldi iteration achieve speedups of 3-4 orders of magnitude over the LISSA-powered IFs (Koh and Liang 2017) and of 10x over TracIn (Pruthi et al. 2020), a heuristic gradient-only alternative to IFs, with better or similar accuracy.” For the K-means portion, the complexity is O(n_samples$\\times$n_iterations$\\times$k_clusters). In practice, the clustering step is near instantaneous, so the method is dominated by Hessian estimation.\n\n- **Number of elements needed for hessian estimation**: We estimate the hessian with a random sampling (representative mini-batch) of the training set. In our experiments on Spotcheck and Domino benchmark, we performed ablations to estimate the number of critical samples and found that less than 5 percent of the training samples is needed for such settings. We will update the paper with detailed discussion of these experiments.\n\nWe thank you for the feedback, and would be happy to address any other questions that you have. We encourage you to reconsider your score in light of our response."}, {"review_id": "PILT9PPaHT", "reviewer": "Reviewer_6oL7", "summary": "The paper presents a method to discover groups of test examples on which the model performs badly, and the misclassification of the examples is caused by the same reason (defined as coherence).\nThis problem is known as slice discovery. The method leverages influence functions to compute the influence explanation for each test example. This generates a vector of the influence of each training example on the test example. Given that this vector is high-dimensional, they derive influence embeddings that are used to cluster them by applying K-Means. The authors propose a variant in which there is no need to specify the number of clusters but only the minimum size of the slice and the maximum accuracy. The method is evaluated on several datasets.", "questions": "- how has been selected the number of cluster K for each experiment with InfEmbed?\n- I couldn't find the value of the hyperparameter P, D and K on dcbench and SpotCheck", "limitations": "A very brief discussion of the limitations is addressed in the supplementary material. It would be useful for the reader to have the limitation section in the main text, even if there is limited space for this addition.", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- the paper is well written and structured clearly. The contributions are clearly reported, and the related works are described. \n- the problem is relevant because finding the training examples that influence a group of test examples on which the model underperforms is the first step to debug a model and to fix sporious correlations in the data.\n- using influence functions to address this problem is novel according to the provided references and interesting\n- the author formalizes the coherence desideratum\n- the metho is evaluated extensively on multiple datasets (text and images) and considering multiple types of causes that induce the model to make a classification error", "weaknesses": "_Reproducibility_: it is not clear if the source code of the experiments will be made available upon acceptance. The datasets are publicly available.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your detailed feedback. We address your concerns below.\n\n**Weaknesses**\n- **Reproducibility**: Regarding reproducibility, we plan to release all our code, datasets, and additional artifacts to replicate our analyses.\n\n**Questions**\n- **Choosing number of clusters K for InfEmbed experiments**: For K (number of clusters), we follow past work to ensure comparability of quantitative results: when comparing to Domino in Table 1, we follow Domino and use K=25, and when comparing to various methods on the SpotCheck benchmark in Table 2, we set the number of clusters K using the Bayesian information criterion (BIC), following how the SpotCheck benchmark authors chose K for their slice discovery method, PlaneSpot\n- **Hyperparameters**: For K (number of slices), we follow past work to ensure comparability of quantitative results: when comparing to Domino in Table 1, we follow Domino and use K=25, and when comparing to various methods on the SpotCheck benchmark in Table 2, we set the number of clusters K using the Bayesian information criterion (BIC), following how the SpotCheck benchmark authors chose K for their slice discovery method, PlaneSpot. We will update the draft to include all these details.\n\nWe thank you for the feedback, and would be happy to address any other questions that you have."}, {"review_id": "7RQOOP7Vmp", "reviewer": "Reviewer_yzdi", "summary": "This work presents a method for discovering subsets of the test set of a multi-class classification task on which a trained model incorrectly classifies a large portion due to the same root cause. The method uses a factorized low-rank approximation of a bilinear influence function, parameterized using the Hessian of loss function with respect to the parameters of the classifier, to project high dimensional influence explanations into lower dimensional representations of test examples, naturally called influence embeddings. The influence embeddings preserve structure of high dimensional influence explanation space and are used to cluster the test examples into subsets using k-means. The influence embeddings are also shown to have purpose in finding slices of the test set that satisfy certain size and failure properties and can be used to find the most problematic training examples causing the failures. Empirical results show that the proposed method outperforms other slice discovery methods on benchmarks and is able to find known errors in several settings.", "questions": "* Why is k-means the clustering algorithm of choice? It seems like specifying the number of slices could hinder the methods effectiveness? DBSCAN was used in an ablation and seems like a better (maybe not the best) choice since it is density based.\n* Why is a hierarchical clustering (tree or DAG shaped clustering) not used for InfEmbed-Rule? Why do we need to re-cluster using k-means to find slices with user defined properties?\n* How is the subset of the training set passed to FactorHessian chosen? (Section 4, lines 229-230)\n* How would this slice discovery method be used to improve the model?\n* How does the quality of the low-rank approximation of the influence function affect the quality of slices discovered?\n", "limitations": "The choice of influence function seems to be chosen by fiat and not really justified. The entire method hinges on the specified influence function being the best. It does seem to be used in the sub-community, but is not justified by the authors. I think the limitations of the method with respect to the issues with the choice of influence function should be addressed.\n\nIs it possible that this method could be used to cause harm in some way. Could the authors address this with respect to the method presented?", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "* The problem is important and well-motivated\n* The general method is easy to understand and seems scalable\n* Empirical evidence is strong and shows the method is very suitable for the specified problem\n* The paper as a whole is fairly well-written", "weaknesses": "* Given the work of Schioppa et al. [2022], the methodological contribution seems quite small and straight-forward\n* The clustering algorithms and procedures using clustering as a subroutine (InfEmbed-Rule) seem to treat the clustering algorithm as an after-thought. There is no justification for why k-means is used.\n* The theoretical results are slightly over-stated in the introduction. It seems like more can be done on the theoretical side to motivate the method. Lemma 1 and Section 3.5 contain interesting results that could be strengthened and elaborated to improve the justification for the presented method.\n* The writing in the background section could be improved a bit. It seems like notation is introduced in an unnatural order and in a slightly imprecise way. Also the definition of the Hessian (line 98) is unclear: which examples are used to define the Hessian? Training examples? Test examples? Or maybe both?\n* Figure 1 really is not that helpful in understanding the method and takes up quite a bit of space.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your detailed feedback. We address your concerns below.\n\n**Weaknesses**\n- **Methodological Contribution**: We agree that the actual computation of the influence embeddings being clustered in the main algorithm is builds on the work of Schioppa et al. [2022]. However, we make 3 key contributions:\n1. Our work is the first to connect one popular problem in model debugging (slice discovery), with a popular tool in model debugging (influence functions).\n2. We theoretically derive what the representation to cluster in a slice discovery algorithm should be in terms of an influence function-based definition of coherency. This is significant, given that past slice discovery algorithms choose to cluster various representations (CLIP embeddings, last-layer representations etc) without theoretically explaining those representations should enable effective slice discovery.\n3. We are the first to use influence functions (of any kind, including but not limited to Schioppa et al) for global explainability (simultaneously explaining predictions for the entire test dataset), unlike past work, which solely focused on local explainability (explaining one prediction at a time, independent of other predictions).\n\n- **Why K-Means**: Our core contribution is identifying what representation should be clustered for slice discovery, and not how to cluster those representations. Furthermore, we tested 3 clustering algorithms (DBSCAN, spectral clustering, gaussian mixture model) on the Spotcheck benchmark, and found that 1) performance was not sensitive to the choice of clustering algorithm and 2) no matter what clustering algorithm was used on influence embeddings, the performance beat the Domino and PlaneSpot baselines.  These results were unfortunately hidden in Appendix J, and will be moved to the main body.\n\n- **Theoretical Results**:  Thank you for the suggestion. We will rewrite the presentation of the theoretical results. The slice discovery field has to-date been without much theoretical underpinning, and this submission hopes to be a first step towards addressing that gap. In this work, we focus on showing intuition for why simply using K-Means to cluster influence embeddings implicitly leads to desirable properties that other slice discovery methods needed to explicitly encourage, suggesting that influence embeddings are the right representation to cluster.\n- **Background Section**: Thank you for the writing feedback; we will rewrite to clarify that training examples are used to define the Hessian.\n- **Figure 1**: Thank you for the feedback, we will update the figure to make the steps more clear.\n\n**Questions**\n- **Why K-Means (and not DBSCAN)?**: We definitely agree using DBSCAN and not having to choose the K, the number of slices, beforehand is a good idea that would make InfEmbed easier to use, especially since the various clustering algorithms performed similarly (Appendix J results).  We will highlight this possibility in the main body.\n- **Hierarchical Clustering**:  InfEmbed-Rule can actually be viewed as a form of top-down hierarchical clustering (using K-means to subdivide existing clusters) with 1 important modification - we do not further cluster a cluster if it already has sufficiently low accuracy and large size. The reason for doing so is that for error analysis, we want to find the largest cluster with sufficiently low accuracy.\n- **Factor Hessian**: We chose an uniformly random subset. We will add the detailed numbers in the Appendix.\n- **How would this slice discovery method be used to improve the model**: This is a great question. Once we identify a high-error slice, we can identify the slice opponents - the examples whose inclusion in the training data increases the loss on the slice (see Section 3.6), remove them, and either retrain or fine-tune the model.  If the examples in a high-error slice turn out to be mis-labeled (as was the case for some slices in the AGNews case study of Section 4.4), we can identify their proponents (training examples which supported the prediction for the wrong label) and remove them. Model improvement is a particularly important challenge, and interesting future work.\n- **Quality of low-rank approximation**: The quality of the low-rank approximation can indeed affect the quality of the discovered slices.  We found that a small Arnoldi dimension, rank, or size of training data subset used for the Hessian could lead to worse quantitative and qualitative results. However, we also found that following the recommendations by Schioppa et al. [2022] led to satisfactory results.\n\n**Limitations**\n- **Choice of influence function**: We chose to use influence functions because 1) they have been shown to be powerful model debugging tools (Koh and Liang, 2017, Han et al 2020, Kong et al 2020), 2) they provide actionable next steps for model improvement (remove opponents of examples in high-error slices and retrain), and 3) using them leads us to derive influence embeddings, which possess appealing properties when clustered. We will provide additional justification in the draft.\n- **Harms**: As with every model diagnosis or understanding method, there is a risk of the method giving the practitioner a false sense of safety. If the diagnosis is wrong, and an action is taken based on the wrong diagnosis, this may have unintended consequences.\n\nWe thank you for the feedback, and would be happy to address any other questions that you have. We encourage you to reconsider your score in light of our response.\n\n**References**\\\n[Koh and Liang, 2017]: Understanding black-box predictions using influence functions.  ICML 2017.\\\n[Han et al, 2020]: Explaining black-box prediction and unveiling data artifacts through influence functions.  ACL 2020.\\\n[Kong et al, 2020]: Understanding instance-based interpretability of variational autoencoders.  Neurips 2020."}]

Author rebuttal:
**General  Response**\
We thank all the reviewers for their generous feedback, and for noting that the work addresses an important problem (**Reviewers 2vT2, uzoE, yzdi, MYaZ**), is novel (**Reviewers 2vT2, 6oL7**), provides a thorough empirical evaluation (**Reviewers 2vT2, 6oL7, yzdi**), and well-written (**Reviewers MYaZ, 6oL7**).

In addition to the point-by-point response to each reviewer, here, we address some general concerns:

- **Methodological contribution:** A few reviewers asked us to contrast with the closely related work of Schioppa et. al [2022]. We make the following key contributions: 
1. We theoretically derive what the representation to cluster in a slice discovery algorithm should be—influence embeddings—in terms of an influence function-based definition of coherency. This is important because past slice discovery algorithms choose to cluster various representations (CLIP embeddings, last-layer representations, last-layer representations projected by SCVIS) without theoretically explaining why clustering those representations should enable effective slice discovery.  In doing so, we have improved over previously published slice discovery algorithms (e.g. methods that extracted CLIP embeddings or last-layer representations).

2. Our work is the first to use influence functions for the slice discovery task. In addition, we are the first to use influence functions (of any kind, including but not limited to Schioppa et. al. 2022) for global explainability (simultaneously explaining predictions for the entire test dataset), unlike past work, which solely focused on local explainability (explaining one prediction at a time, independent of other predictions). 

- **Experiments relating to coherence & label homogeneity**: We have now conducted additional experiments to compare the label homogeneity and coherence scores of InfEmbed compared to baselines in several settings. We find that Infembed returns clusters that have higher coherence scores and identifies low-performing clusters that are more label homogenous than baselines.

- **K-means & the importance of the clustering component of Inf-Embed**: In slice discovery, the goal is to partition a set of examples in groups on which a model has high performance versus low performance. So far, this task has been formulated as a search or clustering one. However, the key challenge in this area is to identify a suitable representation that would allow search or clustering algorithms to reliably find low-performing clusters. Consequently, our key contribution is to propose influence embeddings as such a representation for any clustering or search algorithm for slice discovery.

Meta-review: This paper proposes to use k-means clustering influence encodings (InfEmbed) to identify coherent groups of test examples, where the model is wrong (or right) for the same reason. They evaluate on a range of slice discovery benchmarks, including large and diverse datasets such as ImageNet.

Reviewers agree that this is an elegant approach to an important problem, and found the experiments convincing. Initial reviews expressed some concerns about choice of clustering methods, that the methodological contribution was fairly simple, and that the work may be only incremental over Scioppa et al. 2022, but these concerns were largely addressed by the authors' rebuttal, leading three reviewers to raise their scores significantly.
Decision: Accept (poster)