Title: Error Discovery By Clustering Influence Embeddings
Venue: NeurIPS 2023
Accepted: yes
Keywords: Debugging

interpretability

influence functions
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

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. 2

1 Introduction

Error analysis is a longstanding challenge in machine learning [Pope, 1976, Amershi et al., 2015, Sculley et al., 2015, Chung et 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 a type of error 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 model performance. An effective slice discovery method (SDM) should satisfy two desiderata:

Error surfacing : Identify under-performing slices , i.e. slices with low accuracy, if they exist.

Coherence : Return slices that are coherent in the following sense: Erroneous predictions in a given slice should have the same root cause , i.e. be 'wrong for the same reason'.

To illustrate, suppose we have a pre-trained model that detects hip fractures from an x-ray, trained on data from one hospital, that is now deployed in a new hospital. The model might err frequently on two groups: 1) examples containing a 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 to distribution shift [Koh et al., 2021], noisy labels [Northcutt et al., 2021], and imbalanced data [Idrissi et al., 2022].

In developing a SDM, there are two key challenges. First, formalizing the coherence property , i.e., defining the root cause of an error, identifying whether two errors have similar root causes, and ultimately, defining whether a slice is coherent . Second, identifying the representation to use :

training points

test points

1

most

influential

training points

similarly

influenced

test errors

2

most

influential

training points

slices

of similarly

influenced

test errors

3

Compute

influence embeddings

Cluster

influence embeddings

returning to the x-ray example, there is typically no additional metadata that indicates whether an x-ray contains the spuriously correlated scanner pattern. The input to a SDM will typically be the x-ray or a representation thereof, without additional metadata that indicates which features are spurious. Thus, identifying a representation that can help distinguish under-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 a training example on a 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 a given test example to 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 a training dataset perspective, we say a SDM achieves coherence if the slices it returns have similar influence explanations.

Influence Embeddings : To create coherent slices, one 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 is the dot-product of their respective embeddings.

Clustering Influence Embeddings : Given the coherence-inducing properties of influence embeddings, we propose InfEmbed , 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, as 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'Eon et al., 2022, Sohoni et al., 2020]. In addition, the slices produced by InfEmbed tend to have label homogeniety , i.e. 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.

Contributions : We summarize our contributions as follows:

New slice discovery method : We propose InfEmbed , a SDM to identify under-performing groups of data that applies K-Means to a new representation: influence embeddings .

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 a procedure which clusters them achieves coherence: it returns slices whose examples are influenced similarly by the training data.

Empirical Performance. We show that InfEmbed outperforms previous state-of-the-art SDMs on 2 benchmarks-DCBench [Eyuboglu et al., 2022a], and SpotCheck [Plumb et al., 2022]. InfEmbed is scalable, able to recover known errors in variety of settings, and identifies coherent slices on realistic models.

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 under-performing , i.e. have low accuracy, and slices are coherent , i.e. erroneous predictions in a 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 it means for a slice to be coherent - one of our key contributions, and which will be detailed in Section 3.

̸

Slice Discovery Problem: Given a trained multi-class classification model f ( · ; θ ) over C classes with model parameters θ and an example x from some input space X , f ( x ; θ ) ∈ R C is the prediction for the example and f ( x ; θ ) c is the pre-softmax prediction for class c . Given a test dataset Z := [ z 1 , . . . , z N ] with N examples where z i := ( x i , y i ) and x i ∈ X , y i ∈ Y := [0 , 1] C , the goal of the slice discovery problem is to partition the test dataset into K slices: { Φ k } K k =1 , where Φ k ⊆ [ N ] , Φ k ∩ Φ k ′ = ∅ for k = k ′ , ∪ K k =1 Φ k = [ N ] . As we will make precise shortly, each slice Φ k 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 ′ := [ z ′ 1 , . . . , z ′ N ′ ] where z ′ i := ( x ′ i , y ′ i ) with x ′ i ∈ X , y ′ i ∈ Y , so that θ = argmin θ ′ 1 N ′ ∑ N ′ i = 1 L ( z ′ i ; θ ′ ) . We assume the loss function L is cross-entropy loss. Thus, given test point z = ( x, y ) with x ∈ X , y = [ y 1 , . . . , y C ] ∈ Y , L ( z ; θ ) = -∑ c y c log p c , with p = [ p 1 , . . . , p C ] and p c := exp( f ( x ; θ ) c ) ∑ c ′ exp( f ( x ; θ ) c ′ )) , i.e. p = softmax( f ( x ; θ )) .

Influence Functions : Influence functions estimate the effect of a given training example, z ′ , on a test example, z , for a pre-trained model. Specifically, the influence function approximates the change in loss for a given test example z 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 I ( z ′ , z ) := ∇ θ L ( z ′ ; θ ) ⊺ H -1 θ ∇ θ L ( z ; θ ) , where H θ is the loss Hessian for the pre-trained model: H θ := 1 /n ∑ n i =1 ∇ 2 θ L ( z ; θ ) , evaluated at the pre-trained model's final parameter checkpoint. The loss Hessian is typically estimated with a random mini-batch of data.

The main challenge in computing influence is that it is impractical to explicitly form H θ 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 H -1 θ via a procedure that does not explicitly form H θ . In brief, they run the Arnoldi iteration [Trefethen and Bau III, 1997] for P iterations to get a P -dimensional Krylov subspace for H θ , which requires only P Hessian-vector products. Then, they find a rankD approximation of the restriction of H θ to the Krylov subspace, a 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 -1 θ

M ∈ R | θ |× D , λ ∈ R D × D is a diagonal matrix, | θ | is the parameter count, ˆ H -1 θ approximates H -1 θ , D is the rank of the approximation, P is the Arnoldi dimension, i.e. number of Arnoldi iterations, and here and everywhere, configuration Θ := ( L, f, θ ) denotes the loss function, model and parameters. ˆ H -1 θ is then used to define the practical influence of training example z ′ on test example z :

3 Error Discovery By Clustering Influence Embeddings

We propose a 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 -which 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-preserving approximation. 3) Using influence embeddings, we give a simple and efficient procedure to solve the clustering problem to give our SDM, InfEmbed . 4) We then describe an extension, InfEmbed-Rule , which instead of returning a partition with a user-specified number of slices, returns the largest slices

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 a 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 a test example, z , with its influence explanation E ( z ) :

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 in the following sense: the Euclidean distance between influence explanations of examples in the same slice should be low.

Therefore, we solve the slice discovery problem described in Section 2 via the following formulationreturn a partition of the test dataset into slices, { Φ k } K k =1 , that minimizes the total Euclidean distance between influence explanations of examples in the same slice. Formally, we seek:

3.2 Influence Embeddings to approximate Influence Explanations

To form coherent slices, a 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 of a training example on a test example is the dot-product of their respective influence embeddings. Looking at Equations 2 and 3, we see the influence embedding µ ( z ) of an example z must be defined as

and M , λ are as defined in Section 2, because for any training example z ′ and test example z , ˆ I ( z ′ , z ) = µ ( z ′ ) ⊺ µ ( z ) . Procedure GetEmbeddings of Algorithm 1 finds 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 a critical property-that if two examples have similar influence embeddings, they also tend to have similar influence explanations. This is formalized by the following lemma, whose proof follows from the Cauchy-Schwartz inequality, and that influence is the dotproduct of influence embeddings (see Appendix Section B.1)

Lemma 1: There is a constant C > 0 such that for any test examples z i , z j , || E ( z i ) -E ( z j ) || 2 ≤ C || µ ( z i ) -µ ( z j )) || 2 .

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, µ := [ µ ( z 1 ) , . . . , µ ( z N )] . This procedure is justified as follows: Using Lemma 1, given any partition { Φ k } , we know

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-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 1 describes InfEmbed . 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 approach, which we rely on, has complexity 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. Consequently, the complexity of computing influence embeddings is exactly the same as that of the influence function (IF) estimation. For the K-means portion, the complexity is O (number of samples × number of k-means iterations × 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 hyper-parameter of the InfEmbed method is K , the number of slices to return. In practice, it may not be intuitive for a 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., a slice with greater than T samples and that has accuracy less than A . Without specifying the rule, a practitioner has to iterate through all the returned slices to figure out which one is 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-Rule , 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, it will not be clustered further. Appendix Algorithm B.2 outlines this InfEmbed-Rule method. Its inputs are the same as InfEmbed , 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 a 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 z = ( x, y ) , z ′ = ( x ′ , y ′ ) 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 a linear model, i.e. when only gradients in the last fully-connected layer are considered when computing influence embeddings. Then, a straightforward computation shows that two examples z, z ′ will be placed in the same slice if

is high, where L is cross-entropy loss, θ are the parameters, and for example z , p := [ p 1 , . . . , p C ] = softmax( f ( x )) is the predicted probabilities for each class, y is the one-hot encoded label, x is its last-layer representation, and p ′ , y ′ , x ′ are defined analogously for example z ′ .

Looking at the x ⊺ x ′ term of Equation 8, we see 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-product is 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.

Looking at the ( y -p ) ⊺ ( y ′ -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 ( y ′ -p ′ ) , 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-based derivation shows that the dot-product of margins is the 'canonical' way 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.

3.6 Slice Explanation via Slice Opponents

Given an under-performing slice, we are interested in identifying the root-cause of the erroneous predictions in it. We therefore compute the topk slice opponents and consider them as 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 Φ , we first compute the influence embeddings of the training data, [ µ ( z ′ 1 ) , . . . , µ ( z ′ N ′ )] . Then, we compute the sum of the influence embeddings of the slice, v := ∑ i ∈ Φ µ ( z i ) . Finally, due to the properties of influence embeddings, the topk slice opponents of the given slice, O , are the k training examples z ′ for which v ⊺ µ ( z ′ ) is the most negative, i.e. O = argmin O ′ ⊆ [ N ′ ] , | O ′ | = k v ⊺ ( ∑ i ∈ O ′ µ ( z ′ i ′ )) . Inspecting slice opponents can provide insight into the key features responsible for low performance in a slice.

4 Results

Here, we perform a quantitative evaluation of InfEmbed on the dcbench benchmark, and also perform several case studies showing InfEmbed 's usefulness. Please see Appendix Section H on a case study applying InfEmbed to a 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 ∇ θ L ( z ; θ ) and inverse-Hessian factors M,λ 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 InfEmbed-Rule , 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 a given class are down-sampled), 2) 'correlation' (a spurious correlation is introduced), and 3) 'noisy label' (examples in a given class have noisy labels). Since the error causing slice is known a priori, dcbench can serve as a way to assess a new SDM. The

underlying datasets for those tasks include 3 input types: natural images, medical images, and medical time-series. Eyuboglu et al. [2022a] also introduced Domino, the SDM that is 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 a trained model.

Results : Table 1 compares InfEmbed to Domino across the 3 tasks and 3 input types. InfEmbed always beats Domino except on the 'noisy' task for the EEG Medical Time series input type. We can also compute coherence scores -the k-means objective within each cluster-for each setting in the domino benchmark,

We also compute the coherence scores, the Kmeans objective that we cluster, across all slices for influence embedding clustering as well as for domino. We find a 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.

4.2 InfEmbed on the SpotCheck Benchmark

Overview & Experimental Procedure.

The SpotCheck benchmark Plumb et al. [2022] is based on a synthetic task consisting of 3 semantic features that can be easily controlled to determine the number of 'blindspots' in a dataset. A blindspot is a feature responsible for a 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 presence of a square in the image. Attributes of

the input image such as 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 a 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 a subset of features, and compare InfEmbed to PlaneSpot. We refer to the Appendix Section D.1 for additional details.

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 is 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

Distribution of Labels & Predictions

Label

: sidewinder

Pred

: horned viper

Label

: sidewinder

Pred

: sidewinder

Label

: sidewinder

Pred

: horned viper

Label

: sidewinder

Pred

: horned viper

Label

: horned viper

Pred

: horned viper

Label

: horned viper

Pred

: sidewinder

Label

: horned viper

Pred

: horned viper

Label

: horned viper

Pred

: horned viper

Examples from Slice

'

Opponents'

of Slice

Slice 1

35%

Slice 2

36%

Label

: vine snake

Pred

: green snake

Label

: vine snake

Pred

: green lizard

Label

: analog clock

Pred

: odometer

Label

: analog clock

Pred

: magnetic compass

Label

: barometer

Pred

: analog clock

Label

: odometer

Pred

: analog clock

Label

: gong

Pred

: analog clock

Label

: odometer

Pred

: odometer

sidewinder

horned viper

fiddler crab

envelope

diamondback

cliff dwelling

bubble

buckle

black swan

Polaroid camera

vine snake

triceratops

0

25

scoreboard

green snake

hatchet

iPod

nail

0

SyM

干

espresso maker

dishwasher

analog clock

0

6

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 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 ImfEmbed-Rule 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 opponents, following Section 3.6.

Results: We find 25 slices, comprising a total of 954 examples, whose overall accuracy is 34.9%. Figure 2 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 as explained in Section 3.5-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-predicted. 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, we see all of them have a clock hand, 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 our procedure can identify problematic slices with low performance and then trace the cause of the errors to a subset of the training data.

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-classification problem is a 4-class problem, and the pre-trained model is a BERT base model fine-tuned on the training dataset 3 , achieving 93% test accuracy, making 475 errors. We use ImfEmbed with K = 25 , and examine slices with at most 10% accuracy, and at least 10 examples.

Results : We find 9 such slices, comprising a 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-predicting '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

Slice with 45 examples, 0% accuracy. Predictions: 100% 'world', labels: 100% 'business'

Slice with 139 examples, 3% accuracy. Predictions: 93% 'business', labels: 94% 'sci/tech'

Table 3: Representative examples for 2 slices obtained using InfEmbed 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 InfEmbed can detect mis-labeled data.

4.5 Detecting Spurious Signals in Boneage Classification

Overview & Experimental Procedure : We now artificially inject a 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 et al. [2022] and Adebayo et al. [2021], we consider 3 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-puberty). Importantly, we leave the test dataset untouched, so that we expect a 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 a spurious signal (i.e. x-rays from the hospital might have the hospital's tag). We then train a resnet50 on the manipulated training data, and apply InfEmbed-Rule on the untouched test data to discovered slices with at least 25 examples and at most 40% accuracy (same as for the Imagenet analysis). We repeat this manipulate-train-slice discovery procedure for 3 runs: once for each possible signal.

Results : For all 3 runs / signals, InfEmbed-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-Rule 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 InfEmbed-Rule 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 on the recently formalized slice discovery problem -as termed by Eyuboglu et al. [2022b].

Error Analysis, Slice Discovery, Related Problems : To the best of our knowledge, InfEmbed 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 a different purpose - accelerating the retrieval of an example's most influential training examples. Crucially, their embeddings were high-dimensional (exceeding the number of model parameters) and thus ill-suited

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 a 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 to consider a new modality, like audio or graphs? Other SDM's use the pre-trained model's last-layer representations [d'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-based 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 a 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 is 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 mislabeled data interactively [Teso et al., 2021], data augmentation [Lee et al., 2020], active learning [Liu et al., 2021], quantifying reliability [Schulam and Saria, 2019], and identifying mis-labeled data [Pruthi et al., 2020]. However, influence functions have not been used for slice discovery or global explainability.

6 Conclusion

We present a method, InfEmbed , 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 coherence-predictions being wrong for the same reasons within a slice-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 InfEmbed out-performs current approaches on a variety of data modalities. In addition, we show through case studies that InfEmbed is able to recover known errors in a variety of settings, and identifies coherent slices on realistic models.

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-Olano 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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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS:\n* 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\n\nWEAKNESSES:\n* 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.\n\nQUESTIONS:\n* 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\nLIMITATIONS:\nThe 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?", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-06-27"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe 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.\n\nSTRENGTHS:\n- 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\n\nWEAKNESSES:\n_Reproducibility_: it is not clear if the source code of the experiments will be made available upon acceptance. The datasets are publicly available.\n\nQUESTIONS:\n- 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\n\nLIMITATIONS:\nA 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.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-03"}

comments: 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.

AC note: score increased 4 -> 6 after rebuttal.

STRENGTHS:
* Error analysis tools are useful for many different ML systems; better algorithms for discovering slices might have have significant practical impact.
* 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.
* 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.

QUESTIONS:
* What is the computational complexity of the method? Is the main computational bottleneck in the implicit Hessian estimation?
* Is there any intuition or an ablation study on the number of elements needed for the Hessian estimation?

LIMITATIONS:
N\A
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2023-07-05

comments: 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:

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.

The question that the authors are addressing is of importance and the experiment results seem to validate the performance of this method.

The proposed method, if it is effective like the paper states, can provide insights in other domains that require identifying groups of data.

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. 

The 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.

QUESTIONS:
Do the authors validate the performance of the method with different clustering algorithms?
How is the runtime complexity using influence embedding compared with directly using influence functions?

LIMITATIONS:
Some results are not well visualized and shown. It is recommened that the authors reorganize the result section to make it more readable.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2023-07-06

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS:\n1. 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\nWEAKNESSES:\nMy 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\nQUESTIONS:\nMy 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\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-09"}