1 Introduction
What is encoded in vector representations of textual data, and can we control it? Word embeddings, pretrained language models, and more generally deep learning methods emerge as very effective techniques for text classification. Accordingly, they are increasingly being used for predictions in realworld situations. A large part of the success is due to the models’ ability to perform
representation learning, coming up with effective feature representations for the prediction task at hand. However, these learned representations, while effective, are also notoriously opaque: we do not know what is encoded in them. Indeed, there is an emerging line of work on probing deeplearning derived representations for syntactic Linzen et al. (2016); Hewitt and Manning (2019); Goldberg (2019), semantic Tenney et al. (2019) and factual knowledge Petroni et al. (2019). There is also evidence that they capture a lot of information regarding the demographics of the author of the text Blodgett et al. (2016); Elazar and Goldberg (2018).What can we do in situations where we do not want our representations to encode certain kinds of information? For example, we may want a word representation that does not take tense into account, or that does not encode partofspeech distinctions. We may want a classifier that judges the formality of the text, but which is also oblivious to the topic the text was taken from. Finally, and also our empirical focus in this work, this situation often arises when considering fairness and bias of languagebased classification. We may not want our wordembeddings to encode gender stereotypes, and we do not want sensitive decisions on hiring or loan approvals to condition on the race, gender or age of the applicant.
We present a novel method for actively removing certain kinds of information from a representation. Previous methods are either based on projection on a prespecified, userprovided direction Bolukbasi et al. (2016), or on adding an adversarial objective to an endtoend training process Xie et al. (2017). Both of these have benefits and limitations, as we discuss in the related work section (§2). Our proposed method, Iterative Nullspace Projection (INLP), presented in section 4, can be seen as a combination of these approaches, capitalizing on the benefits of both. Like the projection methods, it is also based on the mathematical notion of linear projection, a commonly used deterministic operator. Like the adversarial methods, it is datadriven in the directions it removes: we do not presuppose specific directions in the latent space that correspond to the protected attribute, but rather learn those directions, and remove them. Empirically, we find it to work well. We evaluate the method on the challenging task of removing gender signals from word embeddings Bolukbasi et al. (2016); Zhao et al. (2018). Recently, Gonen and Goldberg (2019) showed several limitations of current methods for this task. We show that our method is effective in reducing many, but not all, of these (§4).
We also consider the context of fair classification, where we want to ensure that a classifier’s decision is oblivious to a protected attribute such as race, gender or age. There, we need to integrate the projectionbased method within a pretrained classifier. We propose a method to do so in section 5, and demonstrate its effectiveness in a controlled setup (§6.2) as well as in a realworld one (§6.3).
Finally, while we propose a general purpose informationremoval method, our main evaluation is in the realm of bias and fairness applications. We stress that this calls for some stricter scrutiny, as the effects of blindly trusting strong claims can have severe realworld consequences on individuals. We discuss the limitations of our model in the context of such applications in section 7.
2 Related Work
The objective of controlled removal of specific types of information from neural representation is tightly related to the task of disentanglement of the representations (Bengio et al., 2013; Mathieu et al., 2016)
, that is, controlling and separating the different kinds of information encoded in them. In the context of transfer learning, previous methods have pursued representations which are
invariant to some properties of the input, such as genre or topic, in order to ease domain transfer (Ganin and Lempitsky, 2015). Those methods mostly rely on adding an adversarial component (Goodfellow et al., 2014; Ganin and Lempitsky, 2015; Xie et al., 2017; Zhang et al., 2018) to the main task objective: the representation is regularized by an adversary network, that competes against the encoder, trying to extract the protected information from its representation.While adverserial methods showed impressive performance in various machine learning tasks, and were applied for the goal of removal of sensitive information
Elazar and Goldberg (2018); Coavoux et al. (2018); Resheff et al. (2019); Barrett et al. (2019), they are notoriously hard to train. Elazar and Goldberg (2018) have evaluated adverserial methods for the removal of demographic information from representations. They showed that the complete removal of the protected information is not trivial: even when the attribute seems protected, different classifiers of the same architecture can often still succeed in extracting it. Another drawback of these methods is their reliance on a maintask loss in addition to the adverserial loss, making them less suitable for tasks such as debiasing pretrained word embeddings.Xu et al. (2017) utilized a ”nullspace cleaning” operator for increasing privacypreserving in classifiers. Given a pretrained maintask model, they remove from the input a subspace that contains the nullspace (but is not limited to it). By doing so, they aim to remove from the representations information that is not used for the main task (and can be protected), while minimally impairing the maintask performance. While similar in spirit to our method, several key differences exist. As the complementary setting – removing the nullsapce of the maintask classifier vs. projection onto the nullspace of protected attribute classifiers – aims to achieve a distinct goal (privacy preseving), there is no notion of exhaustive, iterative cleaning. Furthermore, since debiasing is not a goal, they do not remove protected attributes that are used by the pretrained maintask classifier (for example, where the main task classifier conditions on gender).
A recent line of work focused on projecting the representation to a subspace which does not encode the protected attributes. Under this methodology, one identifies specific directions in the latent space that correspond to the protected attribute, and removes them. In a seminal work, Bolukbasi et al. (2016) aimed to identify a “gender subspace” in wordembedding space by calculating the main directions in a subspace spanned by the differences between gendered word pairs, such as the direction. They suggested to zero out the components of neutral words in the direction of the “gender subspace” first principle components, and actively pushed neutral words to be equally distant from male and femalegendered words. However, Gonen and Goldberg (2019) have recently shown that these methods only cover up these biases and that in fact, the information is deeply ingrained in the representations. A key drawback of this approach is that it relies on an intuitive selection of a few (or a single) gender directions, while, as we reveal in our experiments, the gender subspace is actually spanned by dozens to hundreds of orthogonal directions in the latent space, which are not necessarily as interpretable as the direction. This observation aligns with the analysis of Ethayarajh et al. (2019) who demonstrated that debiasing by projection is theoretically effective provided that one removes all directions in the latent space, and not only the first principle component.
3 Objective and Definitions
Our main goal is to “guard” sensitive information, so that it will not be encoded in a representation. Given a set of vectors , and corresponding discrete attributes , (e.g. race or gender), we aim to learn a transformation , such that cannot be predicted from . In this work we are concerned with “linear guarding”: we seek a guard such that no linear classifier can predict from with an accuracy greater than that of a decision rule that considers only the proportion of labels in . We also wish for to stay informative: when the vectors are used for some end task, we want to have as minimal influence as possible on the end task performance, provided that remains guarded. We use the following definitions:
Guarded w.r.t. a hypothesis class
Let be a set of vectors, with corresponding discrete attributes , . We say the set is guarded for Z with respect to hypothesis class (conversely Z is guarded in X) if there is no classifier that can predict from at better than guessing the majority class.
Guarding function
A function is said to be guarding X for Z (w.r.t. to class ) if the set is guarded for w.r.t. to .
We use the term linearly guarded to indicate guarding w.r.t. to the class of all linear classifiers.
4 Iterative Nullspace Projection
Given a set of vectors and a set of corresponding discrete^{1}^{1}1
While this work focuses on the discrete case, the extension to a linear regression setting is straightforward: A projection to the nullspace of a linear regressor
enforces for every , i.e., each input is regressed to the noninformative value of zero. protected attributes , we seek a linear guarding function that remove the linear dependence between and .We begin with a highlevel description of our approach. Let be a trained linear classifier, parameterized by a matrix , that predicts a property with some accuracy. We can construct a projection matrix such that for all , rendering useless on dataset . We then iteratively train additional classifiers and perform the same procedure, until no more linear information regarding remains in . Constructing is achieved via nullspace projection, as described below. This method is the core of the INLP algorithm (Algorithm 1).
Nullspace Projection
The linear interaction between and a new test point has a simple geometric interpretation: is projected on the subspace spanned by ’s rows, and is classified according to the dot product between and ’s rows, which is proportional to the components of in the direction of ’s rowpsace. Therefore, if we zeroed all components of in the direction of ’s rowspace, we removed all information used by for prediction: the decision boundary found by the classifier is no longer useful. As the orthogonal component of the rowspace is the nullspace, zeroing those components of is equivalent to projecting on ’s nullspace. Figure 2 illustrates the idea for the 2 dimensional binaryclassification setting, in which is just a 2dimensional vector.
For an algebraic interpretation, recall that the nullspace of a matrix is defined as the space . Given the basis vectors of we can construct a projection matrix into , yielding .
This suggests a simple method for rendering linearly guarded for a set of vectors : training a linear classifier that is parameterized by to predict from , calculating its nullspace, finding the orthogonal projection matrix onto the nullspace, and using it to remove from those components that were used by the classifier for predicting .
Note that the orthogonal projection is the least harming linear operation to remove the linear information captured by from , in the sense that among all maximum rank (which is not full, as such transformations are invertible—hence not linearly guarding) projections onto the nullspace of , it carries the least impact on distances. This is so since the image under an orthogonal projection into a subspace is by definition the closest vector in that subspace.
Iterative Projection
Projecting the inputs on the nullspace of a single linear classifier does not suffice for making linearly guarded: classifiers can often still be trained to recover from the projected
with above chance accuracy, as there are often multiple linear directions (hyperplanes) that can partially capture a relation in multidimensional space. This can be remedied with an iterative process: After obtaining
, we train classifier on , obtain a projection matrix , train a classifier on and so on, until no classifier can be trained. We return the guarding projection matrix , with the guarding function . Crucially, the th classifier is trained on the data after the projection on the nullspaces of classifiers , …, and is therefore trained to find separating planes that are independent of the separating planes found by previous classifiers.In Appendix §A.1 we prove three desired proprieties of INLP: (1) any two protectedattribute classifiers found in INLP are orthogonal (Lemma A.1); (2) while in general the product of projection matrices is not a projection, the product calculated in INLP is a valid projection (Corollary A.1.2); and (3) it projects any vector to the intersection of the nullspaces of each of the classifiers found in INLP, that is, after INLP iterations, is a projection to (Corollary A.1.3). We further bound the damage causes to the structure of the space (Lemma A.2). INLP can thus be seen as a linear dimensionalityreduction method, which keeps only those directions in the latent space which are not indicative of the protected attribute.
During iterative nullspace projection, the property becomes increasingly linearlyguarded in . For binary protected attributes, each intermediate is a vector, and the nullspace rank is . Therefore, after iterations, if the original rank of was , the rank of the projected input is at least .
The entire process is formalized in Algorithm 1.
Implementation Details
A naive implementation of Algorithm 1 is prone to accumulating numerical errors. Those stem mainly from the accumulative projectionmatrices multiplication . To mitigate this problem, we use the formula of BenIsrael (2015), which connects the intersection of nullspaces with the projection matrices to the corresponding rowspaces:
(1) 
Where is the orthogonal projection matrix to the rowspace of a classifier . Accordingly, in practice, we do not multiply but rather collect rowspace projection matrices for each classifier . In place of each input projection , we recalculate according to 1, and perform a projection . Upon termination, we once again apply 1 to get the final nullspace projection matrix , and return it. We make our code publicly available.^{2}^{2}2https://github.com/Shaul1321/nullspace_projection
5 Application to Fair Classification
The previous section described the INLP method for producing a linearly guarding function
for a set of vectors. We now turn to describe its usage in the context of providing fair classification by a (possibly deep) neural network classifier.
In this setup, we are given, in addition to and also labels , and wish to construct a classifier , while being fair with respect to . Fairness in classification can be defined in many ways Hardt et al. (2016); Madras et al. (2019); Zhang et al. (2018). We focus on a notion of fairness by which the predictor is oblivious to when making predictions about .
To use linear guardedness in the context of a deep network, recall that a classification network can be decomposed into an encoder followed by a linear layer : , where is the last layer of the network and is the rest of the network. If we can make sure that is linearly guarded in the inputs to , then will have no knowledge of when making its prediction about , making the decision process oblivious to . Adversarial training methods attempt to achieve such obliviousness by adding an adversarial objective to make itself guarding. We take a different approach and add a guarding function on top of an already trained .
We propose the following procedure. Given a training set , and protected attribute , we first train a neural network to best predict . This results in an encoder that extracts effective features from for predicting .
We then consider the vectors , and use the INLP method to produce a linear guarding function that guards in .
At this point, we can use the classifier to produce oblivious decisions, however by introducing (which is lower rank than ) we may have harmed s performance. We therefore freeze the network and finetune only to predict from , producing the final fair classifier . Notice that only sees vectors which are linearly guarded for during its training, and therefore cannot take into account when making its predictions, ensuring fair classification.
We note that our notion of fairness by obliviousness does not, in the general case, correspond to other fairness metrics, such as equality of odds or of opportunity. It does, however,
correlate with fairness metrics, as we demonstrate empirically.Further refinement. Guardedness is a property that holds in expectation over an entire dataset. For example, when considering a dataset of individuals from certain professions (as we do in §6.3), it is possible that the entire dataset is guarded for gender, yet if we consider only a subset of individuals (say, only those who work as nurses), we may still be able to recover gender with above majority accuracy within that subpopulation. As fairness metrics are often concerned with classification behavior also within groups, we propose the following refinement to the algorithm, which we use in the experiments in §6.2 and §6.3: in each iteration, we train a classifier to predict the protected attribute not on the entire training set, but only on the training examples belonging to a single (randomly chosen) maintask class (e.g. profession). By doing so, we push the protected attribute to be linearly guarded in the examples belonging to each of the maintask labels.
6 Experiments and Analysis
6.1 “Debiasing” Word Embeddings
In the first set of experiments, we evaluate the INLP method in its ability to debias word embeddings Bolukbasi et al. (2016). After “debiasing” the embeddings, we repeat the set of diagnostic experiments of Gonen and Goldberg (2019).
Data.
Our debiasing targets are the uncased version of GloVe word embeddings (Zhao et al., 2018), after limiting the vocabulary to the 150,000 most common words. To obtain labeled data , for this classifier, we use the 7,500 most malebiased and 7,500 most femalebiased words (as measured by the projection on the direction), as well as 7,500 neutral vectors, with a small component (smaller than 0.03) in the gender direction. The data is randomly divided into a test set (30%), and training and development sets (70%, further divided into 70% training and 30% development examples).
Procedure
6.1.1 Results
Classification. Initially, a linear SVM classifier perfectly discriminates between the two genders (100% accuracy). The accuracy drops to 49.3% following INLP. To measure to what extent gender is still encoded in a nonlinear
way, we train a 1layer ReLUactivation MLP. The MLP recovers gender with accuracy of 85.0%. This is expected, as the INLP method is only meant to achieve
linear guarding^{3}^{3}3Interestingly, nonlinear SVMs with different kernels, such as RBF, all achieve random accuracy..Humanselected vs. Learned Directions. Our method differs from the common projectionbased approach by two main factors: the numbers of directions we remove, and the fact that those directions are learned iteratively from data. Perhaps the benefit is purely due to removing more directions? We compare the ability to linearly classify words by gender bias after removing 10 directions by our method (i.e., running Algorithm 1 for 10 iterations) with the ability to do so after removing (i.e., projecting to the intersection of nullspaces) 10 manuallychosen directions defined by the difference vectors between gendered pairs ^{4}^{4}4We use the following pairs, taken from Bolukbasi et al. (2016): (“woman”, “man”), (“girl”, “boy”), (“she”, “he”), (“mother”, “father”), (“daughter”, “son”), (“gal”, “guy”), (“female”, “male”), (“her”, “his”), (“herself”, “himself”), (“mary”, “john”).. INLPbased debiasing results in a very substantial drop in classification accuracy (54.4%), while the removal of the predefined directions only moderately decreases classification accuracy (80.7%). This shows that datadriven identification of genderdirections outperforms manually selected directions: there are many subtle ways in which gender is encoded, which are hard for people to imagine.
Discussion.
Both the previous method and our method start with the main genderdirection of . However, while previous attempts take this direction as the information that needs to be neutralized, our method instead considers the labeling induced by this gender direction, and then iteratively find and neutralize directions that correlate with this labeling. It is likely that the direction is one of the first to be removed, but we then go on and learn a set of other directions that correlate with the same labeling and which are predictive of it to some degree, neutralizing each of them in turn. Compared to the 10 manually identified genderdirections from Bolukbasi et al. (2016), it is likely that our learned directions capture a much more diverse and subtle set of gender clues in the embedding space.
Effect of debiasing on the embedding space. In appendix §A.2 we provide a list of 40 random words and their closest neighbors, before and after INLP, showing that INLP doesn’t significantly damage the representation space that encodes lexical semantics. We also include a short analysis of the influence on a specific subset of inherently gendered words: gendered surnames (Appendix §A.4).
Additionally, we perform a semantic evaluation on the debiased embeddings by evaluating on multiple word similarities datasets (e.g. SimLex999 Hill et al. (2015)). We find large improvements in the quality of the embeddings after the projection (e.g. on SimLex999 the correlation improves from 0.373 to 0.489) and we elaborate more on these findings in Appendix A.3.
Clustering. Figure 1 shows tSNE (Maaten and Hinton, 2008) projections of the 2,000 most femalebiased and 2,000 most malebiased words, before the projection and after , and
projection steps. The results clearly demonstrate that the classes are no longer linearly separable: this behavior is qualitatively different from previous word vector debiasing methods, which were shown to maintain much of the proximity between female and malebiased vectors
(Gonen and Goldberg, 2019). To quantify the difference, we perform Kmeans clustering to
clusters on the vectors, and calculate the Vmeasure (Rosenberg and Hirschberg, 2007) which assesses the degree of overlap between the two clusters found in Kmeans and the binary gender bias of the words. For the tSNE projected vectors, the measure drops from 83.88% overlap before the debiasingprojection, to 0.44% following the projection; and for the original space, the measure drops from 100% to 0.31%.WEAT. While our method does not guarantee attenuating the biasbyneighbors phenomena that is discussed in Gonen and Goldberg (2019), it is still valuable to quantify to what extent it does mitigate this phenomenon. We repeat the Word Embedding Association Test (WEAT) from Caliskan et al. (2017) which aims to measure the association in vector space between male and female concepts and stereotypically male or female professions. Following Gonen and Goldberg (2019), we represent the male and female groups with common names of males and females, rather than with explicitly gendered words (e.g. pronouns). Three tests evaluate the association between a group of male names and a groups of female names to (1) career and familyrelated words; (2) art and mathematics words; and (3) artistic and scientific fields. In all three tests, we find that the strong association between the groups no longer exists after the projection (nonsignificant pvalues of 0.855, 0.302 and 0.761, respectively).
BiasbyNeighbors. To measure biasbyneighbors as discussed in (Gonen and Goldberg, 2019), we consider the list of professions provided in (Bolukbasi et al., 2016) and measure the correlation between biasby projection and bias by neighbors, quantified as the percentage of the top 100 neighbors of each profession which were originally biasedbyprojection towards either of the genders. We find strong correlation of 0.734 (compared with 0.852 before), indicating that much of the biasbyneighbors remains.^{5}^{5}5Note that if, for example, STEMrelated words are originally biased towards men, the word “chemist” after the projection may still be regarded as malebiased by neighbors, not because an inherent bias but due to its proximity to other originally biased words (e.g. other STEM professions).
6.2 Fair Classification: Controlled Setup
We now evaluate using INLP with a deeper classifier, with the goal of achieving fair classification.
Classifier bias measure: TPRGAP.
To measure the bias in a classifier, we follow DeArteaga et al. (2019) and use the TPRGAP measure. This measure quantifies the bias in a classifier by considering the difference (GAP) in the True Positive Rate (TPR) between individuals with different protected attributes (e.g. gender/race).
The TPRGAP is tightly related to the notion of fairness by equal opportunity (Hardt et al., 2016): a fair classifier is expected to show similar success in predicting the task label for the two populations, when conditioned on the true class. Formally, for a binary protected attribute and a true class , define:
(2)  
(3) 
where
is a random variable denoting binary protected attribute,
and denote its two values, and , are random variables denoting the correct class and the predicted class, respectively.Experiment setup. We begin by experimenting with a controlled setup, where we control for the proportion of the protected attributes within each maintask class. We follow the setup of Elazar and Goldberg (2018) which used a twitter dataset, collected by Blodgett et al. (2016), where each tweet is associated with “race” information and a sentiment which was determined by their belonging to some emoji group.
Naturally, the correlation between the protected class labels and the mainclass labels may influence the fairness of the model, as high correlation can encourage the model to condition on the protected attributes. We measure the TPRGAP on predicting sentiment for the different race groups (African American English (AAE) speakers and Standard American English (SAE) speakers), with different imbalanced conditions, with and without application of our “classifier debiasing” procedure.
In all experiments, the dataset is overly balanced with respect to both sentiment and race (50k instances for each). We change only the proportion of each race within each sentiment class (e.g., in the 0.7 condition, the “happy” sentiment class is composed of 70% AAE / 30% SAE, while the “sad” class is composed of 30% AAE / 70% SAE).
Sentiment  TPRGap  

Ratio  Original  INLP  Original  INLP 
0.5  0.76  0.75  0.19  0.16 
0.6  0.78  0.74  0.29  0.22 
0.7  0.81  0.66  0.38  0.24 
0.8  0.84  0.67  0.45  0.15 
Our classifier is based on the DeepMoji encoder Felbo et al. (2017), followed by a 1hideenlayer MLP. The DeepMoji model was trained on millions of tweets in order to predict their emojis; a model which was proven to perform well on different classification tasks Felbo et al. (2017), but also encodes demographic information Elazar and Goldberg (2018). We train this classifier to predict sentiment. We then follow the procedure in §5: training a guarding function on the hidden layer of the MLP, and retraining the final linear layer on the guarded vectors. Table 1 presents the results.
As expected the TPRGAP grows as we increase the correlation between class labels and protected attributes. The accuracy grows as well. Applying our debiasing technique significantly reduced the TPR gap in all settings, although hurting more the main task accuracy in the highlyimbalanced setting. In Appendix A.5, we give some more analysis on the balance between performance and TPRGap and show that one can control for this ratio, by using more iterations of INLP.
6.3 Fair Classification: In the Wild
We now evaluate the fair classification approach in a less artificial setting, measuring gender bias in biography classification, following the setup of DeArteaga et al. (2019).
BoW  FastText  BERT  

Accuracy (profession)  Original  78.2  78.1  80.9 
+INLP  80.1  73.0  75.2  
Original  0.203  0.184  0.184  
+INLP  0.124  0.089  0.095 
They scraped the web and collected a dataset of short biographies, annotated by gender and profession. They trained logistic regression classifiers to predict the profession of the biography’s subject based on three different input representation: bagofwords (BOW), bag of wordvectors (BWV), and RNN based representation. We repeat their experiments, using INLP for rendering the classifier oblivious of gender.
Setup. Our data contains 393,423 biographies.^{6}^{6}6The original dataset had 399,000 examples, but 5,557 biographies were no longer available on the web. We follow the train:dev:test split of DeArteaga et al. (2019), resulting in 255,710 training examples (65%), 39,369 development examples (10%) and 98,344 (25%) test examples. The dataset has 28 classes (professions), which we predict using a multiclass logistic classifier (in a onevsall setting). We consider three input representations: BOW, BWV and BERT (Devlin et al., 2019) based classification. In BOW, we represent each biography as the sum of onehot vectors, each representing one word in the vocabulary. In the BWV representation, we sum the FastText token representations (Joulin et al., 2016) of the words in the biography. In BERT representation, we represent each biography as the last hidden state of BERT over the token. Each of these representations is then fed into the logistic classifier to get final prediction. We do not finetune FastText or BERT.
We run INLP with scikitlearn Pedregosa et al. (2011) linear classifiers. We use 100 logistic classifiers for BOW, 150 linear SVM classifiers for BWV, and 300 linear SVM classifiers for BERT.
Bias measure. We use the TPRGAP measure for each profession.
Following Romanov et al. (2019), we also calculate the rootmean square of over all professions , to get a single pergender bias score:
(4) 
where is the set of all labels (professions).
DeArteaga et al. (2019) have shown that strongly correlates with the percentage of women in profession , indicating that the true positive rate of the model is influenced by gender.
6.3.1 Results
Main results
The results are summarized in Table 2. INLP moderately changes maintask accuracy, with a 1.9% increase in BOW, a 5.1% decrease in performance in BWV and a 5.51% decrease in BERT. is significantly decreased, indicating that on average, the true positive rate of the classifiers for male and female become closer: in BOW representation, from 0.203 to 0.124 (a 38.91% decrease); in BWV, from 0.184 to 0.089 (a 51.6% decrease); and in BERT, from 0.184 to 0.095 (a 48.36% decrease). We measure the correlation between for each profession , and the percentage of biographies of women in that profession. In BOW representation, the correlation decreases from 0.894 prior to INLP to  0.670 after it (a 33.4% decrease). In BWV representation, the correlation decreases from 0.896 prior to INLP to 0.425 after it (a 52.5% decrease). In BERT representation, the correlation decreases from 0.883 prior to INLP to 0.470 following it (a 46.7% decreases; Figure (b)b). DeArteaga et al. (2019) report a correlation of 0.71 for BWV representations when using a “scrubbed” version of the biographies, with all pronouns and names removed. INLP significantly outperforms this baseline, while maintaining all explicit gender markers in the input.
Analysis. How does imposing fairness influence the importance the logistic classifier attribute to different words in the biography? We take advantage of the BOW representation and visualize which features (words) influence each prediction (profession), before and after the projection.
According to Algorithm 1, to debias an input , we multiply . Equivalently, we can first multiply by to get a “debiased” weight matrix .
We begin by testing how much the debiased weights of words that are considered to be biased were changed during the debiasing, compared to random vocabulary words. We compare the relative change before and after the projection of these words, for every occupation.
Biased words undergo an average relative change of x1.23 compared to the average change of the entire vocabulary, demonstrating that biased words indeed change more.
The perprofession breakout is available in Figure 2 in Appendix §A.6.1.
Next, we test the words that were changed the most during the INLP process. We compare the weight difference before and after the projection. We sort each profession words by weight, and average their location index for each professions. Many words indeed seem gender specific (e.g. ms., mr., his, her, which appears in locations 1, 2, 3 and 4 respectively), but some seem unrelated, perhaps due to spurious correlations in the data.
The complete list is available in Table 4 in the Appendix §A.6.1; an analogous analysis for the FastText representation is available at Appendix §A.6.2.
7 Limitations
When dealing with bias and fairness, it is important to also disclose the limitations. The main limitation of our method when used in the context of fairness is that, like other learning approaches, it depends on the data , that is fed to it, and works under the assumption that the training data is sufficiently large and is sampled i.i.d from the same distribution as the test data. This condition is hard to achieve in practice, and failure to provide sufficiently representative training data may lead to biased classifications even after its application. Like other methods, there are no magic guarantees, and the burden of verification remains on the user. It is also important to remember that the method is designed to achieve a very specific sense of protection: removal of linear information regarding a protected attribute. While it may correlate with fairness measures such as demographic parity, it is not designed to ensure them. Finally, it is designed to be fed to a linear decoder, and the attributes are not protected under nonlinear classifiers.
8 Conclusion
We present a novel method for removing linearlyrepresented information from neural representations. We focus on bias and fairness as case studies, and demonstrate that our method is capable of attenuating societal biases that are expressed in representations learned from data. Our method is also shown applicable for increasing fairness in multiclass classification setting: predicting a profession from a biography of a person. We demonstrate that across three increasingly complicated architectures—bag of words, word embeddings and BERT representations—INLP is robust in decreasing bias. We also perform a controlled experiment on a DeepMoji model, to assess the influence of uneven division of protected attributes among maintask labels—as is commonly the case in many realworld application.
While this work focuses on societal bias and fairness, Iterative Nullspace Projection has broader possible usecases, and can be utilized to remove specific components from a representation, in a controlled and deterministic manner. This method can be applicable for other end goals, such as styletransfer, disentanglement of neural representations and increasing their interpretability. We aim to explore those directions in a future work.
Acknowledgements
We thank Jacob Goldberger and Jonathan Berant for fruitful discussions. This project has received funding from the Europoean Research Council (ERC) under the Europoean Union’s Horizon 2020 research and innovation programme, grant agreement No. 802774 (iEXTRACT).
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Appendix A Appendix
a.1 INLP Guarantees
In this section, we prove, for the binary case, an orthogonality property for INLP classifiers: each two classifiers and from two iterations steps and are orthogonal (Lemma A.1). Several useful properties of the matrix that is returned from INLP emerge as a direct result of orthogonality: the product of the projection matrices calculated in the different INLP steps is commutative (Corollary A.1.1); P is a valid projection (Corollary A.1.2); and P projects to a subspace which is the intersection of the nullspaces of all INLP classifiers (Corollary A.1.3). Furthermore, we bound the influence of on the structure of the representation space, demonstrating that its impact is limited only to those parts of the vectors that encode the protected attribute (Lemma A.2).
We prove those properties for two consecutive projection matrices and from two consecutive iterations of Algorithm 1, presented below in 5. The general property follows by induction.


= GetProjectionMatrix())



= GetProjectionMatrix())
INLP Projects to the Intersection of Nullspaces.
Lemma A.1.
if is initialized as the zero vector and trained with SGD, and the loss is convex, then is orthogonal to , that is, .
Proof.
In line 4 of the algorithm, we calculate . For a convex and a linear model , it holds that the gradient with respect to is a linear function of : for some scalar . It follows that after stochastic SGD steps, is a linear combination of input vectors . Since we constrain the optimization to , and considering that fact the nullspace is closed under addition, at each step in the optimization it holds that . In particular, this also holds for the optimal ^{7}^{7}7If we performed proper dimensionality reduction at stage 3 – i.e., not only zeroing some directions, but completely removing them – the optimization in 4 would have a unique solution, as the input would not be rankdeficient. Then, we could use an alternative construction that relies on the Representer theorem, which allows expressing as a weighted sum of the inputs: , for some scalars . As each is inside the nullspace, so is any linear combinations of them, and in particular .. ∎
We proceed to prove commutativity based on this property.
Corollary A.1.1.
Proof.
By Lemma A.1, , so , where is the projection matrix on the rowspace of . We rely on the relation and write:
.
Similarly,
, which completes the proof. ∎
Corollary A.1.2.
is a projection, that is, .
Proof.
, where follows from Corollary A.1.1 and follows from and being projections. ∎
Corollary A.1.3.
is a projection onto .
Proof.
Let . , as is the projection matrix to . Similarly, , so . Conversely, let . Then , so , so is mapped by to .
∎
Note that in practice, we enforce Corollary A.1.3 by using the projection Equation 1 (section 4). As such, the matrix that is returned from Algorithm 1 is a valid projection matrix to the intersection of the nullspaces even if the the conditions in Lemma A.1 do not hold, e.g. when is nonconvex or is not initialized as the zero vector.
INLP Approximately Preserves Distances.
While the projection operations removes the protected information from the representations, ostensibly it could have had a detrimental impact on the structure of the representations space: as a trivial example, the zero matrix
is another operator that removes the protected information, but at a price of collapsing the entire space into the zero vector. The following lemma demonstrate this is not the case. The projection minimally damages the structure of the representation space, as measured by distances between arbitrary vectors: the change in (squared) distance between is bounded by the difference between the “gender components” of and .Lemma A.2.
Let be a unit gender direction found in one INLP iteration, and let be arbitrary input vectors. Let be the nullspace projection matrix corresponding to . Let and be the distances between before and after the projection, respectively. Then the following holds:
Proof.
notation: we denote the th entry of a vector by .
Since is the parameter vector of a gender classifier, a point can be classified to a gender according to the sign of the dot product . Note that in the binary case, the nullspace projection matrix is given by
(5) 
Where is the outer product. By definition, if is in the direction of one of the axes, say without loss of generality the first axis, such that , then the following holds:
(6) 
Such that is the zero matrix except its entry, and then is simplified to
(7) 
I.e, the unit matrix, except of a zero in the position. Hence, the projection operator keeps intact, apart from zeroing the first coordinate . We will take advantage of this property, and rotate the axes such that is the direction of the first axis. We will show that the results we derive this way still apply to the original axes system.
Let be a rotation matrix, such that after the rotation, the first coordinate of is aligned with :
(8) 
One can always find such rotation of the axes. Let be another point in the same space. Given the original squared distance between and :
(9) 
Our goal is to bound the squared distance between the projected points in the new coordinate system:
(10) 
Where denotes the projection matrix in the rotated coordinate system, which takes the form 7.
Note that , being a rotation matrix, is orthogonal. By a known result in linear algebra, multiplication by orthogonal matrices preserves dot product and distances. That means that the distance is the same before and after the rotation: , so we can safely bound and the same bound would hold in the original coordinate system.
By 7,
(11)  
Note that in general it holds that for any
(12)  
(13) 
From 11 one can also trivially get
(14)  
(15)  
Or, equivalently, after subtracting from all elements and multiplying by 1:
So
Note that this result has a clear interpretation: the difference between the distance of the projected and the distance of the original is bounded by the difference of and in the gender direction . In particular, if and are equally malebiased, their distance would not change at all; if is very malebiased and is very femalebiased, the projection would significantly alter the distance between them.
∎
a.2 Influence on Local Neighbors in Glove Space
Word  Neighbors before  Neighbors after 

order  orders, ordering, purchase  orders, ordering, ordered 
crack  keygen, cracks, torrent  keygen, cracks, warez 
craigslist  ebay, craiglist, ads  ebay, craiglist, freecycle 
populations  population, species, communities  population, species, habitats 
epub  ebook, mobi, pdf  mobi, ebook, kindle 
finals  semifinals, playoffs, championship  semifinals, semifinal, quarterfinals 
installed  install, installing, installation  install, installing, installs 
identifiable  disclose, identify, identifying  disclose, pii, distinguishable 
photographs  photograph, photos, images  photograph, images, photos 
ta  si, tu, ti  que, bien, ele 
couch  sofa, sitting, bed  sofa, couches, loveseat 
cooler  coolers, cooling, warmer  coolers, cooling, warmer 
becky  debbie, kathy, julie  debbie, steph, jen 
appreciated  appreciate, greatly, thanks  appreciate, muchly, thanks 
negotiation  negotiating, negotiations, mediation  negotiating, negotiations, mediation 
initial  subsequent, prior, following  intial, inital, subsequent 
chloe  chanel, emma, lauren  chloé, chanel, handbags 
filipino  pinoy, filipinos, philippine  filipinos, pinoy, tagalog 
relying  rely, relied, relies  rely, relied, relies 
perpetual  eternal, continual, irrevocable  irrevocable, datejust, perpetuity 
himself  him, herself, his  herself, oneself, he 
seaside  beach, beachside, picturesque  beachside, idyllic, seafront 
measure  measures, measuring, measured  measures, measuring, measured 
yorkshire  staffordshire, leeds, lancashire  staffordshire, dales, lancashire 
merchandise  goods, items, apparel  goods, items, merchandize 
sub  subs, k, def  subs, subbed, svs 
tones  tone, hues, muted  tone, polyphonic, muted 
therapist  therapists, psychologist, therapy  therapists, physiotherapist, psychologist 
leaned  sighed, smiled, glanced  leant, leaning, sighed 
tho  nnd, cuz, tlie  nnd, tlio, tlie 
lawyers  attorneys, lawyer, attorney  attorneys, lawyer, attorney 
compile  compiling, compiler, compiles  compiling, compiler, compiles 
chord  chords, progressions, guitar  chords, progressions, voicings 
aims  aim, aimed, aiming  aim, aimed, aiming 
ensure  ensuring, assure, ensures  ensuring, ensures, assure 
aerospace  aviation, engineering, automotive  aeronautics, aviation, aeronautical 
clubhouse  pool, playground, amenities  clubhouses, pool, playground 
locking  lock, locks, latch  lock, locks, latch 
reign  reigns, emperor, throne  reigns, reigned, emperor 
vulnerable  susceptible, fragile, affected  susceptible, vunerable, fragile 
a.3 Quantitative Influence of Gender Debiasing on Glove Embeddings
In Appendix A.2 we provide a sample of words to qualitatively evaluate the influence of INLP on semantic similarity in Glove word embeddings (Section 6.1
). We observe minimal change to the nearest neighbors. To complement this measure, we use a quantitative measure: measuring performance on established wordsimilarity tests, for the original Glove embeddings, and for the debiased ones. Those tests measure correlation between cosine similarity in embedding space and human judgements of similarity. Concretely, we test the embeddings similarities using three dataset, which contain four similarity tests that measure similarity or relatedness between words. We use the following datasets: SimLex999
Hill et al. (2015), WordSim353 Agirre et al. (2009) which contain two evaluations, on words similarity and relatedness and finally on Mturk771 Halawi et al. (2012).The test sets are composed of word pairs, where each pair was annotated by humans to give a similarity or relatedness score. To evaluate a model against such data, each pair is given a score (in the case of word embedding, cosine similarity) and then we calculate Spearman correlation between all the score pairs. The results on the regular Glove embeddings before and after the gender debiasing are presented in Table 3. We observe a major improvements across all evaluation sets after the projection: between 0.044 to 0.116 points.
This major difference in performance is rather surprising. It is not clear how to interpret the positive influence on correlation with human judgements. This puzzle is further compounded by the fact the projection reduces the rank of the embedding spaces, and by definition induces loss of information. We hypothesize that many of the words in the embedding space contain a significant gender component, which is not correlated with humans judgements of similarity. While intriguing, testing this hypothesis is beyond the scope of this work, and we leave the more rigorous answer to a future work.
a.4 Influence on Local Neighbors of Surnames Representations in Glove Space
Word  Neighbors before  Neighbors after 

ruth  helen, esther, margaret  etting, esther, gehrig 
charlotte  raleigh, nc, atlanta  raleigh, greensboro, nc 
abigail  hannah, lydia, eliza  hannah, phebe, josiah 
sophie  julia, marie, lucy  moone, bextor, marceau 
nichole  nicole, kimberly, kayla  nicole, mya, heiress 
emma  emily, lucy, sarah  grint, frain, watson 
olivia  emma, rachel, kate  munn, thirlby, wilde 
ava  devine, zoe, isabella  viticultural, devine, appellation 
isabella  sophia, josephine, isabel  rossellini, beeton, ferdinand 
sophia  anna, lydia, julia  hagia, antipolis, topkapi 
mia  bella, mamma, mama  bangg, mamma, culpa 
amelia  earhart, louisa, caroline  earhart, fernandina, bedelia 
james  john, william, thomas  jassie, nightfire, perse 
john  james, william, paul  deere, scatman, betjeman 
robert  richard, william, james  pattinson, mccammon, blacksportsonline 
michael  david, mike, brian  micheal, franti, moorcock 
william  henry, edward, james  edward, henry, sir 
david  stephen, richard, michael  bisbal, magen, sylvian 
richard  robert, william, david  clayderman, brautigan, rorty 
joseph  francis, charles, thomas  joesph, dreamcoat, abboud 
thomas  james, william, john  szasz, deshaun, tomy 
ariel  sharon, alexis, hanna  peterpan, mermaid, cinderella 
mike  brian, chris, dave  mignola, birbiglia, dave 
The results in Table 1 suggest that, as expected, the projection has little influence on the lexical semantics of unbiased words, as measured by their closest neighbors in embedding space. But how does the projection influence inherently gendered words? Table 2 contains the closestneighbors to the Glove representations of gendered surnames, before and after the projection. We observe an interesting tendency to move from neighbors which are other gendered surnames, towards family names, which are by definition genderneutral (for instance, the closest neighbor of “Robert” changes from “Richard” to “Pattinson”). Another interesting tendency is to move towards place names bearing a connection to that surnames (For instance, the closest neighbor of “Sophia” changs to “Hagia”). At the same time, some gendered surnames remain close neighbors even after the projection.
a.5 Performance and “Fair Classification” as a Function of INLP Iterations
Eval  Before  After 

SimLex999  0.373  0.489 
WordSim353  Sim  0.695  0.799 
WordSim353  Rel  0.599  0.698 
Mturk771  0.684  0.728 
In Section 6.2 where we compare the accuracy and TPRGap before and after using INLP for a certain amount of iterations. The number of iterations chosen is somehow arbitrary, but we emphasize that this can be controlled for as the number of iterations used with INLP. By sacrificing the main task performance, one can improve the TPRGap of their model. In Figure 1 we detail these tradeoffs for the ratio, where the original TPRGap originally is the highest.
We note that the performance is minimally damaged for the first 180 iterations, while the TPRGap improves greatly, afterwhich, both metric account for larger drops. Using this tradeoff, one can decide how much performance they are willing to sacrifice in order to get a less biased model.
a.6 Biographies dataset: Words MostAssociated with Gender
a.6.1 BagofWords Model
In this section, we present the raw results of the experiment aimed to assess the influence of INLP on specific words, under the bagofwords model, for the biographies experiments (Section 6.3.1).
Table 4 lists the words most influenced by INLP projection (on average over all professions) after the debiasing procedure explained in Section 6.3.
Figure 2 presents the relative change of biased word for each profession, compared to a random sample.
Most Changed Words 

ms., mr., his, her, he, she, mrs., specializes, 
english, practices, ’,’, him, spanish, 
speaks, with, affiliated, and, medicine, ms, 
state, #, the, medical, michael, in, 
residency, at, of, psychology, dr., ’s, 
law, research, practice, about, where, 
business, education, 5, , is, first, 
women, america, insurance, more, john, 
university, location, ph.d., surgery, (, 
mental, ), that, engineering, graduated, 
language, bs, litigation, collection, 
united, 1, graduate, humana, cpas, 
cancer, npi, completed, 10, book, hospital, c, 
out, family, or, when, oklahoma, certified, 
ohio, number, training, for, like, a, 
than, be, nursing, ], _, can, writing, 
patients, no, orthopaedic, attorney, 
over, ny, mr, “, 
a.6.2 BagofWordVectors Model
Gender direction  Malebiased  Femalebiased 

0  his, he, His, himself  herself, she, She, her 
1  himself, him, Gavin, His  Hatha, midwifery, Midwifery, feminist 
2  Mark, Jon, Darren, Luke  Actress, Zumba, Diana, woman 
3  Gordon, he, wind, charge  hers, recipe, Challenge, cookbooks 
4  1935, 1955, namely, 1958  Roots, Issue, FHM, yoga 
5  M, Mickey, KS, Bethesda  Vietnam, Subject, Elle, Ecuador 
6  Keys, correct, address, fuel  leap, Embedded, textile, femininity 
7  Papers, Categories, wherein, Newark  Botox, LASIK, periodontal, UnityPoint 
8  binding, closely, MT, command  Aventura, brunette, HTML, Disclosure 
9  82, 92, 91, 86  ASP.NET, committer, Twilight, Seth 
10  t, Cisco, Philips, Sharp  preschool, caregivers, homeowners, Preschool 
11  Toulouse, Aviv, scored, commended  intersectional, Equality, equality, ASME 
12  addressing, segment, inequalities, segments  Wire, loose, anything, Vincents 
13  comparison, Hart, 480, refereed  Matthew, independence, couples, LGBTQ 
14  manufacturer, organizers, scope, specifications  homeschooling, ligament, loyalty, graduating 
In this section, we present an analysis for the influence of INLP projection on the FastText representation of individual words, under the bagofwordvectors model, for the biographies experiments (Section 6.3.1). We begin by ordering the vocabulary items by their cosine similarity to each of the top 15 gender directions found in INLP (i.e., their similarity to the weight vector of each classifier). For each gender direction , we focus on the 20,000 most common vocabulary items, and calculate the closest words to (to get malebiased words) as well as the closest words to (to get femalebiased words). The result are presented in Table 5.
The first gender direction seems to capture pronouns. Other gender directions capture socially biased terms, such as “preschool” (direction 10), “cookbooks” (direction 3) or other genderrelated terms, such as “LGBTQ” (direction 15) or “femininity” (direction 6). Interestingly, those are mostly femalebiased terms. As for the malebiased words, some directions capture surnames, such as “Gordon” and “Aviv”. Other words which were found to be malebiased are less interpretable, such as words specifying years (direction 4), organizational terms such as “Organizers”, “specifications” (direction 14), or the words “Papers”, “Categories” (direction 7). It is not clear if those are the result of spurious correlations/noise, or whether they reflect actual subtle differences in the way the biographies of men and women are written.
Gender rowspace
The above analysis focuses on what information do individual gender directions convey. Next, we aim to demonstrate the influence of the final INLP projection on the representation of words. To this end, we rely on the rowspace of the INLP matrix . Recall that the rowspace is the orthogonal complement of the nullspace. As the INLP matrix projects to the intersection of nullspaces of the gender directions, the complement projects to the union of rowspaces of individual gender directions. This is a subspace which is spanned by all gender directions, and thus can be thought of as an empirical gender subspace within the representation space.
For a given word vector , the “gender norm” – the norm of its projection on the rowspace, – is a scalar quantity which can serve as a measure for the genderbias of the word. We sort the vocabulary by the ratio between the gender norm and the original norm, and present the 200 most gendered words (Table 6).
Top words by component on the gender subspace 
motherhood, SSHRC, 
microfinance, preschool, genocide, IFP, 
CSE, intersectional, student, 
homeschooling, photoshoot, 
intersectionality, 920, breastfeeding, STEM, 
photojournalistic, haiku, kindergarten, 
FreeOnes, UNESCO, menstrual, 
turbulence, NTR, ASME, HFN, ECE, IEEE, 
feminism, noir, Jadavpur, Motherhood, 
reportage, Contra, TU, WebSphere, 
counsellor, photovoltaic, J2EE, 
contraception, university, PEN, 
masculinities, parenting, EAP, 
Politecnico, Feminism, trauma, 
Universiti, counselling, curriculum, 
Kanpur, women, edits, Pune, Nanjing, 
ethnographic, Pinterest, surrealist, 
taught, Hindustan, students, CNRS, 
Bangalore, Mumbai, consortium, tooth, 
Vitae, Kindergarten, nanoscale, 
school, ACL, scholarships, cloud, 
Goa, NIJC, Montessori, JSPS, 
scholarship, Neha, DAAD, endometriosis, 
carrier, UCI, activism, Ambedkar, 
EECS, semiconductor, scholar, 
microfluidic, bikini, Raising, teacher, 
Feminist, vinyasa, NBER, ethnography, 
Twilight, Sunil, Shankar, viral, 
earthquake, semiconductors, 
historiography, vampire, HMO, PSU, bioenergy, 
historian, Ravi, Breastfeeding, Raman, 
resettlement, Shweta, ICTs, UNDP, NVIDIA, 
HIV, Counselling, HEC, KDD, 
Hyderabad, contraceptive, macro, 
Ghaziabad, sexuality, CAS, 
documentary, mic, biography, postdoc, 
transnationalism, AMD, CFD, B.Tech, physicist, 
LGBT, parenthood, HKU, HIP, 
internationalization, M.Tech, BDS, acne, theorist, 
HPV, Meerut, ageing, smile, 
Rajesh, psychoeducational, PUNE, 
grief, AHA, Essays, discourses, 
secrets, Swati, EPFL, coaching, IIE, 
Manoj, BIDMC, infertility, 
fashion, Chicana, Vaishali, 
Graduation, sociologist, Gender, EA, MIT, 
teach, gift, IETF, NPPA, counselor, 
JPL, gender, menopause, LGBTQ, 
Waseda, perceptions, praxis, 
birthday, Jawaharlal, fertility, 
gendered, coverage, stills, PIH, 
Balaji, Tagged, baking, USM, 
postpartum, Goenka, Pooja, forgiveness 

As before, we see a combination of inherentlygendered words (“motherhood”, “women”, “gender”, “masculinities”), sociallybiased terms (“teacher”, “raising”, “semiconductors”, “B.Tech”, “IEEE”, “STEM”, “fashion”) and other words whose connection to gender is less interpretable, and potentially represent spurious correlations (“trauma”, “Vitae”, “smile”, “920”, “forgiveness”).
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