Rationales that select the most relevant parts of an input text can help explain model decisions for a range of language understanding tasks Lei et al. (2016); DeYoung et al. (2019). Models can be faithful to a rationale by only using the selected text as input for end-task prediction. However, there is almost always a trade-off between interpretable models that extract sparse rationales and more accurate models that are able to use the full context but provide little explanation for their predictions Lei et al. (2016); Weld and Bansal (2019). In this paper, we show that it is possible to better manage this trade-off by optimizing a novel bound on the Information Bottleneck Tishby et al. (1999) objective (Figure 1).
We follow recent work in representing rationales as binary masks over the input text Lei et al. (2016); Bastings et al. (2019). During learning, it is common to encourage sparsity by minimizing a norm on the rationale masks (e.g. or ) Lei et al. (2016); Bastings et al. (2019). As we will see in Section 5, it is challenging to control the sparsity-accuracy trade-off in norm-minimization methods; we show that these methods seem to push too directly for sparsity at the expense of accuracy. Our approach, in contrast, allows more control through a prior that specifies task-specific target sparsity levels that should be met in expectation across the training set.
More specifically, we formalize the problem of inducing controlled sparsity in the mask using the Information Bottleneck (IB) principle. Our approach seeks to extract a rationale as an optimal compressed intermediate representation (the bottleneck) that is both (1) minimally informative about the original input, and (2) maximally informative about the output class. We derive a novel variational bound on the IB objective for our case where we constrain the intermediate representation to be a concise subsequence of the input, thus ensuring its interpretablity.
Our model consists of an explainer that extracts a rationale from the input, and an end-task predictor
that predicts the output based only on the extracted rationale. Our IB-based training objective guarantees sparsity by minimizing the Kullback–Leibler (KL) divergence between the explainer mask probability distribution and a prior distribution with controllable sparsity levels. This prior probability affords us tunable fine-grained control over sparsity, and allows us to bias the proportion of the input to be used as rationale. We show that, unlike norm-minimization methods, our KL-divergence objective is able to consistently extract rationales with the specified sparsity levels.
Across different tasks from the ERASER interpretability benchmark DeYoung et al. (2019) and the BeerAdvocate dataset McAuley et al. (2012), our IB-based sparse prior objective has significant gains over previous norm-minimization techniques—0.5% to 5% relative improvement in task performance metrics and 6% to 80% relative improvement in agreement with human rationale annotations. Our interpretable model achieves task performance within of a model of comparable size that uses the entire input. Furthermore, we find that in the semi-supervised setting, adding a small proportion of gold rationale annotations (approximately of the training examples) bridges this gap—we are able to build an interpretable model without compromising performance.
2.1 Task and Method Overview
We assume supervised text classification or regression data that contains tuples of the form . The input document can be decomposed into a sequence of sentences . is the category, answer choice, or target value to predict. Our goal is to learn a model that not only predicts , but also extracts a rationale or explanation —a latent subsequence of sentences in with the following properties:
Model prediction should rely entirely on and not on its complement DeYoung et al. (2019).
must be compact yet sufficient, i.e., it should contain as few sentences as possible without sacrificing the ability to correctly predict .
2.2 Formalizing Interpretability Using Information Bottleneck
The Information Bottleneck (IB) method is used to learn an optimal compression model that transmits information from a random variableto another random variable through a compressed representation . The IB objective is to minimize the following:
where is mutual information. This objective encourages to only retain as much information about as is needed to predict
. The hyperparametercontrols the trade-off between retaining information about either or in . Alemi et al. (2016) derive the following variational bound on Equation 1:222For brevity and clarity, objectives are shown for a single data point. More details of this bound can be found in Appendix A.1 and Alemi et al. (2016).
where is a parametric approximation to the true likelihood ; , the prior probability of , is an approximation to ; and is the parametric posterior distribution over .
The information loss term in Equation 2 reduces by decreasing the KL divergence333To analytically compute the KL-divergence term, the posterior and prior distributions over are typically K-dimensional multivariate normal distributions. Compression is achieved by setting
are typically K-dimensional multivariate normal distributions. Compression is achieved by setting, the input dimension of . between the posterior distribution that depends on and a prior distribution that is independent of . The task loss encourages predicting the correct label from to increase .
Our Variational Bound for Interpretability
The learned bottleneck representation , found via Equation 2, is often not human-interpretable33footnotemark: 3. We consider an interpretable latent representation , where is a boolean mask on the input sentences in . We assume that the mask variables over individual sentences are conditionally independent given the input , i.e. , where and indexes sentences in the input text. Because , the posterior distribution over is a mixture of dirac-delta distributions:
where is the dirac-delta probability distribution that is zero everywhere except at .
Our prior is that the rationale needed for prediction is sparse; we encode this prior as a distribution over masks for some constant , which also induces a distribution on via the relationship . In contrast to Alemi et al. (2016), our prior has no trainable parameters; instead of using an expressive to approximate , we use a fixed prior to force the marginal of the learned distribution over to approximate the prior . Our parameterization of the prior and the posterior achieves compression of the input via sparsity in the latent representation, in contrast to compression via dimensionality reduction Alemi et al. (2016).
For the masked representation , we can decompose we can decompose as:
Since is a constant with respect to , it can be dropped. Hence, we obtain the following variational bound on IB with interpretability constraints, described in more detail in Appendix A.2:
The first term is the expected cross-entropy term for the task which can be computed by drawing samples . The information-loss term encourages the mask to be independent of by reducing the KL divergence of its posterior from a prior that is independent of . However, this does not necessarily remove information about in . For instance, a mask consisting of all ones is independent of , but and the rationale is no longer concise. In the following section, we present a simple way to avoid this degenerate case in practice.
2.3 The Sparse Prior Objective
The key to ensuring that is strictly a subsequence of lies in the fact that is our prior belief about the probability of a sentence being important for prediction. For instance, if humans annotate of the input text as a rationale, we can fix our prior belief that a sentence should be a part of the mask is , setting . We refer to this prior probability hyperparameter as the sparsity threshold .
can be estimated as the expected sparsity of the mask from expert annotations. If such a statistic is not available, it can be explicitly tuned using, for the desired trade-off between end task performance and rationale length.
Consequently, we can control the amount of sparsity in the mask that is eventually sampled from the learned posterior distribution by appropriately setting the prior belief in Equation LABEL:ivib. Since is restricted to be small, the sampled mask is generally sparse. As a result, the intermediate representation , which is our human-interpretable rationale, is guaranteed to be a subsequence of and reduce . We refer to this training objective as the sparse prior (Sparse IB) method in our experiments.
Our model (Figure 2) consists of an explainer which extracts rationales from the input, and an end-task predictor which predicts the output based on the explainer rationales. In our experiments, both the explainer and the predictor are Transformers with BERT pretraining Devlin et al. (2019).
Given an input consisting of sentences, the explainer produces a binary mask over the input sentences which is used to derive a rationale . It maps every sentence to its probability, of being selected as part of where is a binary distribution.
The explainer contextualizes the input sequence at the token level, and produces sentence representations where is obtained by concatenating the contextualized representations of the first and last tokens in sentence . When an optional query sequence is available,444For question answering tasks in ERASER. and are encoded together in the sequence while assuming that is fully unmasked i.e.
. A linear layer is used to transform these representations into logits (log probabilities) of a Bernoulli distribution. We choose the Bernoulli distribution since its sample can be reparameterized as described in Section3.2, and we can analytically compute the KL-divergence term between two Bernoulli distributions. The mask is constructed by independently sampling each from . We define as the rationale representation , an element-wise dot product between and the corresponding sentence representation .
End-task Predictor :
The end-task predictor applies the mask from the explainer to the input sequence555Once again, the sequence is used if query is available, i.e., we assume no masking over as it is assumed to be essential to predict . to predict the output variable . The same attention mask is applied to all end-task transformer layers at every head to ensure prediction relies only on
. The predictor further consists of a log-linear classifier layer over the[CLS] token, similar to Devlin et al. (2019).
3.2 Reparameterization and Training
to overcome non-differentiability. We found REINFORCE to be quite unstable with high variance in results. Instead, we employ reparameterizationKingma et al. (2015) to facilitate end-to-end differentiability of our approach. We use the Gumbel-Softmax reparameterization trick Jang et al. (2017) for categorical (here, binary) distributions to reparameterize the Bernoulli variables . A random perturbation is added to the log probability (logit), . The reparameterized binary variable is generated as follows:
is the Sigmoid function,is a hyperparameter for the temperature of the Gumbel-Softmax function, and is a sample from the Gumbel(0,1) distribution Gumbel (1948). is a continuous and differentiable approximation to with low variance.
During inference, we extract the top sentences, where corresponds to the threshold hyperparameter described in Section 2.3. Previous work Lei et al. (2016); Bastings et al. (2019) samples from during inference. Such an inference strategy is non-deterministic, making comparison of different masking strategies difficult. Moreover, it is possible to appropriately scale values to obtain better inference results, thereby not reflecting if are correctly ordered. By allowing a fixed budget of per example, we are able to fairly judge how well the model fills the budget with the best rationales.
3.3 Semi-Supervised Setting
As we will show in Section 5, despite better control over the sparsity-accuracy trade-off, there is still a gap in task performance between our unsupervised approach and a model that uses full context. To bridge this gap and better manage the trade-off at minimal annotation cost, we experiment with a semi-supervised setting where we have annotated rationales for part of the training data.
For input example and a gold mask , we use the following semi-supervised objective:
We set to simplify experiments. For examples where the rationale supervision is not available, we only consider the task loss.
4 Experimental Setup
4.1 End Tasks
We evaluate performance on five text classification tasks from the ERASER benchmark DeYoung et al. (2019) and one regression task used in previous work Lei et al. (2016). All these datasets have sentence-level rationale annotations for validation and test sets. Additionally, the ERASER tasks contain rationale annotations for the training set, which we only use for our semi-supervised experiments.
Movies Pang and Lee (2004): Sentiment classification of movie reviews from IMDb.
FEVER Thorne et al. (2018): A fact extraction and verification task adapted in ERASER as a binary classification of the given evidence supporting or refuting a given claim.
MultiRC Khashabi et al. (2018): A reading comprehension task with multiple correct answers modified into a binary classification task for ERASER, where each (rationale, question, answer) triplet has a true/false label.
BoolQ Clark et al. (2019): A Boolean (yes/no) question answering dataset over Wikipedia articles. Since most documents are considerably longer than BERT’s maximum context window length of 512 tokens, we use a sliding window to select a single document span that has the maximum TF-IDF score against the question.
Evidence Inference Lehman et al. (2019): A three-way classification task over full-text scientific articles for inferring whether a given medical intervention is reported to either significantly increase, significantly decrease, or have no significant effect
on a specified outcome compared to a comparator of interest. We again apply the TF-IDF heuristic.
BeerAdvocate McAuley et al. (2012): The BeerAdvocate regression task for predicting 0-5 star ratings for multiple aspects like appearance, smell, and taste based on reviews. We report on the appearance aspect.
4.2 Evaluation Metrics
We adopt the metrics proposed for the ERASER benchmark to evaluate both agreement with comprehensive human rationales as well as end task performance. To evaluate quality of rationales, we report the token-level Intersection-Over-Union F1 (IOU F1), which is a relaxed measure for comparing two sets of text spans. For task accuracy, we report weighted F1 for classification tasks, and the mean square error for the BeerAdvocate regression task.
|3. Task Only||82.8||35.5||60.1||20.8||78.2||37.9||62.5||10.9||43.0||09.0||.018||47.3|
|4. Sparse Norm||83.1||44.0||59.7||20.4||78.6||34.7||62.5||7.1||38.9||6.3||.017||35.5|
|5. Sparse Norm-C||83.3||44.9||61.7||21.7||81.9||34.4||63.7||9.1||44.7||8.0||.018||49.0|
|6. Sparse IB (Us)||84.7||45.5||62.1||24.3||84.0||39.6||65.2||16.5||46.3||10.0||.017||52.3|
|8. data (Us)||88.8||66.6||66.4||54.4||85.4||43.4||63.4||32.3||46.7||13.3||-||-|
Norm Minimization (Sparse Norm):
Controlled Norm Minimization (Sparse Norm-C):
For fair comparison against our approach for controlled sparsity, we can modify Equation 4 to ensure that the norm of is not penalized when it drops below the threshold .
Explicit control over sparsity in the mask through the tunable prior probability naturally emerges from IB theory in our Sparse IB approach, as opposed to the modification adopted in norm-based regularization (Equation 5).
No Sparsity (Task Only):
This method only optimizes for the end-task performance without any sparsity-inducing loss term, and serves as a common baseline for evaluating the effect of sparsity inducing objectives in Sparse IB, Sparse Norm, and Sparse Norm-C.
In addition to the above baselines, we also compare to models that don’t predict rationales:
Full Context (Full):
This method uses the entire context to make prediction, and allows us to estimate the loss in performance as a result of our interpretable hard attention model that only usesof the input.
Gold Rationale (Gold):
For datasets with human-annotated rationales available at training time, we train a model that uses these rationale annotations for training and inference to estimate an upper-bound that can be achieved on task and rationale performance metrics with transformers.
4.4 Implementation Details
We use BERT-base with a maximum context-length of 512 to instantiate the explainer and end-task predictor. Models are tuned based on their performance on the rationale IOU F1 as it is available for all the datasets considered in this work. When rationale IOU F1 is not available, the sparsity-accuracy trade-off (Figure 4) can be used to determine an operation point. We tune the prior probability/threshold to ensure strictly concise rationales. For our Sparse IB approach, we observe less sensitivity to hyperparameter and set it to 1 to simplify experimental design. For baselines, we tune the values of the Lagrangian multipliers, as norm-based techniques are more sensitive to . More details about the hyperparameters and model selection are presented in Appendix B.
Table 1 compares our Sparse IB approach against baselines described in Section 4.3. Our Sparse IB approach outperforms norm-minimization approaches (rows 4-6) in both agreement with human rationales and task performance across all tasks. We perform particularly well on rationale extraction with relative improvements ranging from 5 to 80% over the better performing norm-minimization variant Sparse Norm-C. Sparse IB also attains task performance within of the full-context model (row 1), despite using of the input sentences on average. All unsupervised approaches still obtain a lower IOU F1 compared to the full context model for the Movies and MultiRC datasets, primarily due to their considerably lower precision on these benchmarks.
Our results also highlight the importance of explicit controlled sparsity inducing terms as essential inductive biases for improved task performance and rationale agreement. Specifically, sparsity-inducing methods consistently outperform the Task Only-baseline (row 3). One way to interpret this result is that sparsity objectives add input-dimension regularization during training, which results in better generalization during inference. Moreover, Sparse Norm-C, which adds the element of control to norm-minimization, performs considerably better than Sparse Norm. Finally, we observe a positive correlation between task performance and agreement with human rationales. This is important since accurate models that also better emulate human rationalization likely engender more trust.
In order to close the performance gap with the full-context model, we also experiment with a setup where we minimize the task and the rationale prediction loss using rationale annotations available for a part of the training data (Section 3.3).
Figure 3 shows the effect of incorporating an increasing proportion of rationale annotation supervision for the FEVER and MultiRC datasets. Our semi-supervised model is even able to match the performance of the full-context models for both FEVER and MultiRC with only 25% of rationale annotation supervision. Furthermore, Figure 3 also shows that these gains can be achieved with relatively modest annotation costs since adding more rationale supervision to the training data seems to have diminishing returns.
Table 1 compares our interpretable model (row 8), which uses rationale supervision for 25% of the training data, with the full-context model and Lehman et al. (2019)’s fully-supervised pipeline approach (row 7). Lehman et al. (2019) learn an explainer and a task predictor independently in sequence, using the output of the explainer for inference in the predictor. On three (FEVER, MultiRC, and BoolQ) out of five datasets for which rationale supervision is available, our interpretable models match the end task performance of the full-context models while recording large gains in IOU (17-30 F1 absolute). Our approach outperforms the pipeline-based approach in task performance (for FEVER, MultiRC, Movies, and BoolQ) and IOU (for MultiRC and Movies). These gains may result from better exploration due to sampling and inference based on a fixed budget of sentences. Our weakest results are on Evidence Inference where the TF-IDF preprocessing often fails to select relevant rationale spans.666Selected document spans have gold rationales 51.8% of the time Our overall results suggest that the end-task model attention can be supervised with a small proportion of human annotations to be more interpretable.
|Dataset||Sparse Norm-C||Sparse IB|
|Examples from Error Analysis|
|The original Babe gets my vote as the best family film since the princess bride, and it’s sequel has been getting rave reviews|
|from most internet critics, both Siskel and Ebert sighting it more than a month ago as one of the year’s finest films. So,|
|naturally, when I entered the screening room that was to be showing the movie and there was nary another viewer to be found,|
|this notion left me puzzled. It is a rare thing for a children’s movie to be praised this highly, so wouldn’t you think|
|that every parent in the entire city would be flocking with their kids to see this supposedly “magical” piece of work?|
|Looking back, I should have taken the hint and left right when I entered the theater. Believe me; I wanted to like Babe: Pig|
|in the City. The plot seemed interesting enough; It is here that we meet an array of eccentric characters, the most|
|memorable being the family of chimps led by Steven Wright. Here is where the film took a wrong turn unfortunately, the|
|story wears thin as we are introduced to a new set of animals that the main topic of discussion it just didn’t feel right|
|and was more painful to watch than it was funny or entertaining, and the same goes for the rest of the movie.|
|Statement : Unforced labor is a reason for human trafficking.|
|Ground Truth: REFUTES|
|DOC: Human trafficking is the trade of humans, most commonly for the purpose of forced labour, sexual slavery, or comm-|
|-ercial sexual exploitation for the trafficker or others. This may encompass providing a spouse in the context of forced|
|marriage, or the extraction of organs or tissues, including for surrogacy and ova removal. Human trafficking can occur|
|within a country or transnationally. Human trafficking is a crime against the person because of the violation of the victim’s|
|rights of movement through coercion and because of their commercial exploitation In 2012, the I.L.O. estimated that|
|21 million victims are trapped in modern-day slavery|
|Statement: Atlanta metropolitan area is located in south Georgia.|
|DOC: Metro Atlanta , designated by the United States Office of Management and Budget as the Atlanta-Sandy Springs-|
|Roswell, GA Metropolitan Statistical Area, is the most populous metro area in the US state of Georgia and the ninth-largest|
|metropolitan statistical area (MSA) in the United States. Its economic, cultural and demographic center is Atlanta, and|
|it had a 2015 estimated population of 5.7 million people according to the U.S. Census Bureau. The metro area forms the core|
|of a broader trading area, the Atlanta – Athens-Clarke – Sandy Springs Combined Statistical Area.|
|The Combined Statistical Area spans up to 39 counties in north Georgia and had an estimated 2015 population of 6.3 million|
|people. Atlanta is considered an “ alpha world city ”. It is the third largest metropolitan region in the Census Bureau’s|
|Southeast region behind Greater Washington and South Florida.|
Accurate Sparsity Control
Table 2 compares average sparsity rates in rationales extracted by Sparse IB with those extracted by norm-minimization methods. We measure the sparsity achieved by the explainer during inference by computing the average number of one entries in the input mask over sentences (the hamming weight) for 100 runs. Our Sparse IB-approach consistently achieves the sparsity level used in the prior while the norm-minimization approach (Sparse Norm-C) converges to a lower average sparsity for the mask.
Figure 4 shows the variation in task and rationale agreement performance as a function of the sparsity rate for Sparse IB and Sparse Norm-C on the FEVER dataset. Both methods extract longer rationales with increasing value of that results in a decrease in agreement with sparse human rationales, while model accuracy improves. However, Sparse IB consistently outperforms Sparse Norm-C in terms of task performance.
In summary, our analysis indicates that unlike norm-minimization methods, our KL-divergence objective is able to consistently extract rationales with the specified sparsity rates, and achieves a better trade-off with accuracy. We hypothesize that optimizing the KL-divergence of the posterior may be able to model input salience better than an implicit regularization (through ). The sparse prior term can learn adaptive to different examples, while encourages uniform sparsity across examples.777Unlike the norm , the derivative of KL-divergence term is proportional to This can be seen explicitly in Table 2, where the variance in sampled mask across examples is higher for our objective.
A qualitative analysis of the rationales extracted by the Sparse IB approach indicates that such methods struggle when the context offers spurious—or in some cases even genuine but limited—evidence for both output labels (Figure 3). For instance, the model makes an incorrect positive prediction for the first example from the Movies sentiment dataset based on sentences that: (a) praise the prequel of the movie, (b) still acknowledge some critical acclaim, and (c) sarcastically describe the movie as magical. We also observed incorrect predictions based on shallow lexical matching (likely equating forced and unforced in the second example) and world knowledge (likely equating south Georgia, southeastern United States, and South Florida in the third). Overall, there is scope for improvement through better incorporation of exact lexical match, coreference propagation, and representation of pragmatics in our sentence representations.
6 Related Work
Previous work on explaining model predictions can be broadly categorized into post hoc explanation methods and methods that integrate explanations into the model architecture. Post hoc explanation techniques Ribeiro et al. (2016); Krause et al. (2017); Alvarez-Melis and Jaakkola (2017) typically approximate complex decision boundaries with locally linear or low complexity models. While post hoc explanations often have the advantage of being simpler, they are not faithful by construction.
On the other hand, methods that condition predictions on their explanations can be more trustworthy. Extractive rationalization Lei et al. (2016) is one of the most well-studied of such methods in NLP, and has received increased attention with the recently released ERASER benchmark DeYoung et al. (2019). Building on Lei et al. (2016), Chang et al. (2019) and Yu et al. (2019) consider benefits like class-wise explanation extraction while Chang et al. (2020) explore invariance to domain shift. Bastings et al. (2019)
employ a reparameterizable version of the bi-modal beta distribution (instead of Bernoulli) for the binary mask. This more expressive distribution may be able to complement our approach, as KL-divergence for it can be analytically computedNalisnick and Smyth (2017).
While many methods for extractive rationalization, including ours, have focused on unsupervised settings due to the considerable cost of obtaining reliable annotations, recent work Lehman et al. (2019) has also attempted to use direct supervision from rationale annotations for critical medical domain tasks. Finally, Latcinnik and Berant (2020) and Rajani et al. (2019) focus on generating explanations (rather than extracting them from the input), since the extractive paradigm could be unfavourable for certain tasks like common sense question answering where the given input provides limited context for the task.
The Information Bottleneck (IB) principle Tishby et al. (1999) has recently been adapted in a number of downstream applications like parsing Li and Eisner (2019), summarization West et al. (2019), and image classification Alemi et al. (2016); Zhmoginov et al. (2019). Alemi et al. (2016) and Li and Eisner (2019)
use IB for optimal compression of hidden representations of images and words respectively. We are interested in compressing the number of cognitive units (like sentences) to ensure interpretability of the bottleneck representation. Our work is more similar toWest et al. (2019) in that the input (words) is compressed rather than the embedding dimension. However, while West et al. (2019) use brute-force search to optimize IB for summarization, we directly optimize a parametric variational bound on IB for rationales.
IB has also been previously used for interpretability—Zhmoginov et al. (2019) use a VAE to estimate the prior and posterior distributions over the intermediate representation for image classification. Bang et al. (2019) use IB for post-hoc explanation for sentiment classification. They do not enforce a sparse prior, and as a result, cannot guarantee that the rationale is strictly smaller than the input. This also means controlled sparsity, which we have shown to be crucial for task performance and rationale extraction, is harder to achieve in their model.
We propose a new sparsity objective derived from the Information Bottleneck principle to extract rationales of desired conciseness. Our approach outperforms existing norm-minimization techniques in task performance and agreement with human annotations for rationales for tasks in the ERASER benchmark. The sparse prior objective also allows for a straight-forward and accurate control of the amount of sparsity desired in the rationales. We also obtain better a trade off of accuracy vs. sparsity using our objective. We are able to close the gap with models that use the full input with rationale annotations for a majority of the tasks. In future work, we would like to explore the application of our approach on longer contexts and tasks such as document-level QA.
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Appendix A Information Bottleneck Theory
We first present an overview of the variational bound on IB introduced by Alemi et al. (2016) and then derive a modified version amenable to interpretability.
a.1 Variational Information Bottleneck (Alemi et al. (2016))
The objective is to parameterize the information bottleneck objective
using neural models and use SGD to optimize. Consider the joint distribution:
under the Markov chain. As mutual information is hard to compute, the following bounds are derived on both MI terms:
This marginal is intractable. Let be a variational approximation to this marginal. Since ,
If and are of a form that KL divergence can be analytically computed, we get:
Typically, the distributions and are instantiated as multivariate Normal distributions to analytically compute the KL-divergence term.
is a neural network which outputs the K-dimensional mean of z andoutputs the covariance matrix . This also allows us to reparameterize samples drawn from .
Again, as this is intractable, is used as a variational approximation to and is instantiated as a transformer model with its own set of parameters
. As Kullback Leibler divergence is always positive:
The term can be dropped as it is constant with respect to parameters . Thus, we minimize
Thus the IB objective is bounded by the loss function:
|(Sparsity threshold (%))||40||10||25||20||20||20|
|(weight on SR)||0.5||0.05||1.00E-04||0.01||0.001||0.01|
a.2 Deriving the sparse prior objective
The latent space learned in Appendix A.1 is not easy to interpret. Instead we consider a masked representation of the form , where is a binary mask sampled from a distribution . This is an adaptive masking strategy, defined by data-driven relevance estimators . The distributions over and induce a distribution on defined by the conditionals
Our prior, based on human annotations, is that rationale needed for a prediction is sparse; we encode this prior as a distribution over masks . The prior also induces a distribution on given by
We want to enforce a constraint ; i.e. that the marginal distribution matches our prior . This is difficult to do directly, but as in Appendix A.1, we can construct an upper bound the mutual information between and :
The inequality is tight if . By optimizing to minimize mutual information , we will implicitly learn parameters that approximate the desired constraint on the marginal.
In contrast to Alemi et al. (2016), our prior has no parameters; rather than using an expressive model to approximate the , we instead use the fixed prior to force the learned conditionals to assume a form such that the marginal approximately matches the marginal of the prior. Average mask sparsity values in Table 2 corroborate this.
By a limiting argument, we can compute the divergence between and :
The term is a divergence between two Bernoulli distributions and has a simple closed form. If and are uncorrelated then
The term is constant with respect to the parameters and can be dropped.
We use the same, standard cross-entropy bound discussed in Appendix A.1 to estimate , leading us to our variational bound on IB with interpretability constraints
Appendix B Experimental Details
We use a sequence length of 512, batch size 16 and Adam optimizer with a learning rate of 5e-5. We do not use warmup or weight decay. Hyper-parameter tuning is done on the validation set for the rationale performance metric (IOU F1) for ERASER tasks and on the test set for BEER (only test set contains rationale annotations). Instead of explicitely tuning or annealing the Gumbel softmax parameter, we fix it to 0.7 across all our experiments. We found that Sparse IB approach is not as sensitive to the parameter and fix it to 1 to simplify experimental design. Hyperparameters for each dataset used for the final results are presented in Table 4.
For ERASER tasks, we use the preprocessed training and validation sets from DeYoung et al. (2019) and for BEER, we use data pre-processsed for the appearance aspect in Bastings et al. (2019). For BoolQ and Evidence Inference, we use a sliding window of 20 sentences (with step 5) over the document to find the span that has maximum TF-IDF overlap with the query.