I NTERPRETABILITY WITH FULL COMPLEXITY BY CONSTRAINING FEATURE INFORMATION
Anonymous authors
Paper under double-blind review
A BSTRACT
Interpretability is a pressing issue for machine learning.
Common approaches to interpretable machine learning constrain interactions between features of the input, rendering comprehensible the effects of features on a model’s output at the expense of model complexity.
We approach interpretability from a different angle: constrain the information about the features utilized by the model, without any restrictions on the complexity of the model.
Borrowing from information theory, we use the
Distributed Information Bottleneck to find optimal compressions of each feature that maximally preserve information about the output.
The learned information allocation, by feature and by feature value, is a rich source of interpretability well-suited to problems with many features and complex feature interactions.
The central object of analysis is not a single trained model, but rather a spectrum of models serving as approximations that leverage variable amounts of information about the inputs.
Information is allocated to features by their relevance to the output, tackling the problem of feature selection with inclusion/exclusion existing on a learned continuum.
The optimal compression of each feature—at every stage of approximation—allows fine-grained inspection of how feature values are similar or distinct with regards to the prediction.
We develop a framework for extracting insight from the spectrum of approximate models and demonstrate its utility on a range of tabular datasets.
I NTRODUCTION
Interpretability is a pressing issue for machine learning (ML) (Doshi-Velez & Kim, 2017; Fan et al., 2021; Rudin et al., 2022).
Models are growing more complex, and machine learning is increasingly integrated into fields where flawed decisions can have serious ramifications (Caruana et al., 2015;
Rudin et al., 2018; Rudin, 2019).
Interpretability is not a binary property that machine learning methods have or do not: it is the degree to which a learning system can be probed and comprehended (Doshi-Velez & Kim, 2017), with many distinct routes to interpretability (Lipton, 2018).
Various constraints on the learning system are incorporated that bring about a degree of comprehensibility (Molnar et al., 2020; Molnar, 2022).
As opposed to explainable AI that happens post-hoc after a black-box model is trained, interpretable methods engineer the constraints into the model from the outset (Rudin et al., 2022).
A common route to interpretability enforces that the effects of features combine in a simple (e.g. linear) manner, greatly restricting the space of possible models in exchange for comprehensibility.
In this work, we introduce a novel route to interpretability that places no restrictions on model complexity, and instead tracks how much and what information is most important for predictive power.
By identifying optimal information from features of the input, the method grants a measure of salience of each feature, a spectrum of models utilizing different amounts of optimal information about the input, and a learned compression scheme for each feature that highlights in fine-grained detail from where the information is coming.
The central object of interpretation is no longer a single optimized model, but rather a family of models revealing how predictive information is distributed across features at all levels of fidelity.
The information constraint we incorporate is a variant of the Information Bottleneck (IB) (Tishby et al., 2000; Asoodeh & Calmon, 2020).
The IB is a framework for extracting the relevant information in a relationship, though it lacks interpretability because the extracted information is free be any
[Figure]
black-box function of the input.
Instead, by siloing the information from each feature and using a sum-rate constraint in the form of the Distributed IB (Estella Aguerri & Zaidi, 2018; Murphy &
Bassett, 2022), we introduce a constraint at an appropriate level in the model’s processing: after every feature is processed and before any interaction effects between the features can be included.
Before and after the bottlenecks, arbitrarily complex processing can occur, allowing the Distributed IB to be used in scenarios with many features and complex feature interactions.
By restricting the flow of information into a model, we acquire a novel signal about the relationship in the data and the inner workings of the model.
We develop a framework for extracting insight from the signal of interpretability acquired by the Distributed IB, and demonstrate the following strengths:
Capacity to find the most important features and the most important feature values.
Interpretability comes from the fine-grained decomposition of the most relevant information in the data, found over the full spectrum of approximations and allowing a comprehensible entry point to analyzing the relationship.
Full complexity of feature processing and interactions.
Compatible with arbitrary neural network architectures before and after the bottlenecks, the method gains interpretability without sacrificing predictive accuracy.
Intuitive to implement, minimal additional overhead.
The method is a probabilistic encoder for each feature, employing the same loss penalty as β -VAE (
Higgins et al., 2017).
A single model is trained once while sweeping the magnitude of the information bottleneck.
We compare the Distributed IB to approaches in interpretable ML and feature selection on a variety of tabular datasets.
R ELATED WORK
We propose a method that analyzes the information content of features of an input with respect to an output, and its generality bears relevance to several broad areas of research.
Feature selection and variable importance.
Feature selection methods isolate the most important features in a relationship (Fig. 1a), both for interpretability and to reduce computational burden (Chandrashekar & Sahin, 2014; Cai et al., 2018).
When higher order feature interactions are present, the combinatorially large search space presents an NP-hard optimization problem (Chandrashekar &
Sahin, 2014).
Modern feature selection methods navigate the search space with feature inclusion weights, or masks, that can be trained as part of the full stack with the machine learning model (Fong & Vedaldi, 2017; Balın et al., 2019; Lemhadri et al., 2021; Kolek et al., 2022).
Many classic methods of statistical analysis work to identify the most important features in data.
Canonical correlation analysis (CCA) (Hotelling, 1936) relates sets of random variables by finding transformations that maximize linear correlation.
Modern extensions broaden the class of transformations to include kernel methods (Hardoon et al., 2004) and deep learning (Andrew et al., 2013; Wang et al., 2016).
Another line of research generalizes the linear correlation to use mutual information instead (Painsky et al., 2018).
Analysis of variance (ANOVA) finds a decomposition of variance by features (Fisher, 1936; Stone, 1994); extensions incorporate ML-based transforms (Märtens & Yau, 2020).
Variable importance (VI) methods quantify the effect of different features on an outcome (Breiman, 2001; Fisher et al., 2019).
Interpretable machine learning.
Interpretability is often broadly categorized as intrinsic — the route to comprehensibility was engineered into the learning model from the outset, as in this work—or post hoc —constructed around an already-trained model (Rudin et al., 2022).
Another common distinction is local —the comprehensibility built up piecemeal by inspecting individual examples (e.g., LIME (Ribeiro et al., 2016))—or global —the comprehensibility applies to the whole space of inputs at once, as in this work.
Intrinsic interpretabilty is generally achieved through constraints incorporated to reduce the complexity of the system.
Generalized Additive Models (GAMs) require feature effects to combine linearly (Fig. 1b), though each effect can be an arbitrary function of a feature (Chang et al., 2021;
Molnar, 2022).
By using neural networks to learn the transformations from raw feature values to effects, Neural Additive Models (NAMs) maintain the interpretability of GAMs with more expressive effects (Agarwal et al., 2021).
Pairwise interactions between effects can be included for more complex models (Yang et al., 2021; Chang et al., 2022), though complications arise involving degeneracy and the number of effect terms to inspect grows rapidly.
Regularization can be used to encourage sparsity in the ML model so that fewer components—weights in the network (Ng, 2004) or even features (Lemhadri et al., 2021)—conspire to produce the final prediction.
Information Bottleneck.
The proposed method can also be seen as a regularization to encourage sparsity of information.
It is based upon the Information Bottleneck, a broad framework that extracts the most relevant information in a relationship (Tishby et al., 2000) and has proven useful in representation learning (Alemi et al., 2016; Wang et al., 2019; Asoodeh & Calmon, 2020).
However, the standard IB has one bottleneck for the entire input, preventing an understanding of where the relevant information originated and hindering interpretability.
Bang et al.
(2021) and Schulz et al.
(2020) employ the IB to explain a trained model’s predictions as a form of post-hoc interpretability, whereas the current method is built into the training of a model and thus serves as intrinsic interpretability.
Distributing bottlenecks to multiple sources of data has been of interest in information theory as a solution to multiterminal source coding, or the “CEO problem” (Berger et al., 1996; Estella Aguerri & Zaidi, 2018; Aguerri & Zaidi, 2021; Steiner & Kuehn, 2021).
Only recently has the Distributed IB been utilized for interpretability (Murphy & Bassett, 2022), whereby features of data serve as the distributed sources.
With this formulation, the extracted information from each feature can be used to understand the role of the features in the relationship of interest.
M ETHODS
At a high level, we introduce a penalty on the amount of information used by a black-box model about each feature of the input.
Independently of all others, features are compressed and communicated through a noisy channel; the messages are then combined for input into a prediction network.
The optimized information allocation and the learned compression schemes—for every feature and every magnitude of the penalty—serve as our windows into feature importance and into feature effects.
3. D ISTRIBUTED V ARIATIONAL I NFORMATION B OTTLENECK
Let X, Y ∼ p ( x, y ) be random variables representing the input and output of a relationship to be modeled from data.
The mutual information is a general measure of the statistical dependence of two random variables, and is the reduction of entropy of one variable after learning the value of the other:
[Equation]
with H ( Z ) =
E z ∼ p
( z ) [ − log p ( z )]
Shannon’s entropy (Shannon, 1948).
The Information Bottleneck (IB) (Tishby et al., 2000) is a learning objective to extract the most relevant information from X about Y .
It optimizes a representation U that conveys information about
X , realized as a stochastic channel of communication U ∼ p ( u | x ) .
The representation is found by minimizing a loss comprising competing mutual information terms balanced by a scalar parameter β ,
[Equation] [Equation]
[Figure]
The first term is the bottleneck—acting as a penalty on information passing into U —with β determining the strength of the bottleneck.
Optimization with different values of β produces different models of the relationship between X and Y , and a sweep over β produces a continuum of models that utilize different amounts of information about the input X .
In the limit where β → 0 , the bottleneck is fully open and all information from X may be freely conveyed into U in order for it to be maximally informative about Y .
As β increases, only the most relevant information in X about Y becomes worth conveying into U , until eventually it is optimal for U to be uninformative random noise.
Because mutual information is generally difficult to estimate from data (Saxe et al., 2019; McAllester & Stratos, 2020), variants of IB replace the mutual information terms of Eqn.
2 with bounds amenable to deep learning (Alemi et al., 2016; Achille & Soatto, 2018; Kolchinsky et al., 2019).
We follow the
Variational Information Bottleneck (VIB) (Alemi et al., 2016), which learns representations U in a framework resembling that of β -VAEs (Higgins et al., 2017).
A sample x is encoded as a distribution in representation space p ( u | x ) =
f ( x, ϕ, ϵ ) with a neural network parameterized by weights ϕ and a source of noise ϵ ∼ N (0 , 1) to facilitate the “reparameterization trick” (Kingma & Welling, 2014).
The bottleneck restricting the information from X takes the form of the Kullback-Leibler (KL) divergence— D KL ( p ( z ) ||
q ( z )) =
E z ∼ p
( z ) [ − log ( q ( z ) /p
( z ))] — between the encoded distribution p ( u | x ) and a prior distribution r ( u ) =
N (0 , 1) .
As the KL divergence tends to zero, all representations are indistinguishable from the prior and from each other, and therefore uninformative.
A representation u is sampled from p ( u | x ) and then decoded to a distribution over the output Y with a second neural network parameterized by weights ψ , q ( y | u )
=
g ( u, ψ ) .
In place of Eqn. 2, the following may be minimized with standard gradient descent methods:
[Equation]
The Distributed IB (Estella Aguerri & Zaidi, 2018) concerns the optimal scheme of compressing multiple sources of information as they relate to a single relevance variable Y .
Let { X i } be the features of X ; the Distributed IB encodes each feature X
i independently of the rest, and then integrates the information from all features to predict Y .
A bottleneck is installed after each feature in the form of a compressed representation U i (Fig. 1c), and the set of representations U X
= { U i } is
[Figure]
used as input to a model to predict Y .
The same mutual information bounds of the Variational IB (Alemi et al., 2016) can be applied in the Distributed IB setting (Aguerri & Zaidi, 2021),
[Equation]
Arbitrary error metrics with feature information regularization.
Instead of quantifying predictive power with cross entropy, another error metric can be used in combination with the constraint on the information about the features.
For instance, mean squared error (MSE) may be used instead,
[Equation]
with y and ˆ y the prediction and ground truth values, respectively.
The connection to the IB is diminished, as β is no longer a unitless ratio of information about Y to the sum total information about the features of X , and the optimal range must be found anew for each problem.
Optimization will nevertheless yield a spectrum of approximations prioritizing the most relevant information about the input features; the heart of the proposed framework is regularization on the information about input features, inspired by the Distributed Information Bottleneck rather than a strict application of it.
Boosting expressivity for low-dimensional features.
Similarly with Agarwal et al.
(2021), we found the feature encoders struggled with expressivity due to the low dimensionality of the features.
We positionally encode feature values (Tancik et al., 2020) before they pass to the neural network, taking each value z to [ sin ( ω 1 z ) , sin ( ω 2 z ) , ... ] where ω k are the frequencies for the encoding.
V ISUALIZING THE OPTIMALLY PREDICTIVE INFORMATION
Distributed information plane.
Tishby & Zaslavsky (2015) popularized the visualization of learning dynamics in the “information plane”, a means to display how successfully information conveyed about the input of a relationship is converted to information about the output.
We employ the distributed information plane (Fig. 2a) to compare the total information utilized about the features (cid:80) I
( U i ; X i ) to information about the output I ( { U i } ; Y ) .
Considering the space of possible allocations of information to the input features, the Distributed IB finds the set of allocations that minimize (cid:80)
I
( U i ; X i ) while maximizing I ( { U i } ; Y ) .
All possible allocations converge on two universal endpoints: the origin (no information in, no information out; red point in Fig. 2a) and at ( (cid:80) H ( X i ) , I ( X, Y ))
(all information in, mutual information between X and Y out; green point in Fig.
2a).
A pragmatic alternative to the distributed information plane replaces (cid:80) I
( U i ; X i ) with its upper bound used during optimization, the sum of KL divergences across the features.
Additionally, we replace the vertical axis
I ( { U i } ; Y ) with the error used for training.
Switching the vertical axis from mutual information to an error has the effect of inverting the trajectory found by the Distributed IB: the uninformed point is now the maximum error with zero (cid:80) D KL (red point in Fig. 2b).
[Table]
The entire curve of optimal information allocations in Fig.
2b may be found in a single training run
[Table]
by increasing β logarithmically over the course of training as Eqn. 4 is optimized.
Training begins with negligible information restrictions ( β ∼ 0 ) so that the relationship may be found without obstruction (Wu et al., 2020; Lemhadri et al., 2021), and ends once all information is removed.
Each value of β corresponds to a different approximation to the relationship between X and Y , and we can use the KL divergence for each feature as a measure of its overall importance in the approximation.
Confusion matrices to visualize feature compression schemes.
Transmitting information about a feature is equivalent to conveying distinctions between the feature’s possible values.
In the course of optimizing the Distributed IB, the spectrum of models learned range from all input values perfectly distinguishable ( β → 0, unconstrained machine learning) to all inputs perfectly indistinguishable (the model utilizes zero information about the input).
Along the way, the distinctions most important for predicting Y are selected, and can be inspected via the encoding of each feature value.
Each feature value x i is encoded to a probability distribution p ( u i | x i ) in channel space, allowing the compression scheme to be visualized with a confusion matrix showing the similarity of feature values (Fig. 3a).
The Bhattacharyya coefficient (Kailath, 1967) is a symmetric measure of the similarity between probability distributions p ( z ) and q ( z ) , and is straightforward to compute for normal distributions.
The coefficient is 0 when the distributions are perfectly disjoint (i.e., zero probability is assigned by p ( z ) everywhere that q ( z ) has nonzero probability, and vice versa), and 1 when the distributions are identical.
In other words, if the Bhattacharyya coefficient between encoded distributions p ( u | x = a ) and p ( u | x = b ) is 1, the feature values x = a and x = b are indistinguishable to the prediction network and the output is independent of whether the actual feature value was a or b .
On the other extreme where the Bhattacharyya coefficient is 0, the feature values are perfectly distinguishable and the prediction network can yield distinct outputs for x = a and x = b .
We display in Fig.
3b,c possible compression schemes containing one bit of information about a hypothetical x with four equally probable values.
The compression scheme can be a hard clustering of the values into two sets where x = a and x = b are indistinguishable, and similarly for x = c and x = d (Fig. 3b).
Another possibility with one bit of information is the soft clustering shown in Fig.
3c, where each feature value is partially ‘confused’ with other feature values.
In the case of soft clustering, a learned compression cannot be represented by a decision tree; rather, the confusion matrices are the fundamental distillations of the information preserved about each feature.
An example in Fig.
3d shows the learned compression scheme of the weekday feature in the Bikeshare dataset (dataset details below).
The soft clustering distinguishes between the days of the week to different degrees, with the primary distinction between weekend and weekdays.
E XPERIMENTS
We analyzed a variety of tabular datasets with the Distributed IB and compared the interpretability signal to what is obtainable through other methods.
Our emphasis was on the unique capabilities of the Distributed IB to produce insight about the relationship in the data modelled by the neural network.
Performance results are included in Tab. 1 for completeness.
Note that the Distributed IB places no constraints on the machine learning models used and in the β → 0 regime could hypothetically be as good as any finely tuned black-box model.
The benefit of freedom from complexity constraints
[Figure]
was particularly clear in the smaller regression datasets, whereas performance was only middling on several other datasets (refer to App.
B for extended results and App for implementation
4. T HE FEATURES WITH THE MOST PREDICTIVE INFORMATION
We used the Distributed IB to identify informative features for three datasets in Fig. 4.
Bikeshare (Dua & Graff, 2017) is a regression dataset to predict the number of bikes rented given 12 descriptors of the date, time, and weather; MiceProtein (Higuera et al., 2015) is a classification dataset relating the expression of 77 genes to normal and trisomic mice exposed to various experimental conditions;
MIMIC-II (Johnson et al., 2016) is a binary classification dataset to predict hospital ICU mortality given 17 patient measurements.
We compared to LassoNet (Lemhadri et al., 2021), a feature selection method that attaches an L1 regularization penalty to each feature and introduces a proximal algorithm into the training loop to enforce a constraint on weights.
By gradually increasing the regularization,
LassoNet finds a series of models that utilize fewer features while maximizing predictive accuracy, providing a signal about what features are necessary for different levels of accuracy.
[Figure]
display the learned compression schemes for four of the top features found in Figure 4.
Following the scheme in Fig.
3, the pairwise distinguishability of feature values visualizes the information preserved about each the features
, workingday , season corresponds to a unique value, and the are aligned were
Like LassoNet, the Distributed IB probes the tradeoff between information about the features and predictive performance, though with important differences.
LassoNet can only vary the information used for prediction through a discrete-valued proxy: the number of features.
Binary inclusion/exclusion of features is a common limitation for feature selection methods (Bang et al., 2021; Kolek et al., 2022).
By contrast, the Distributed IB compresses each feature along a continuum, selecting the most informative information from across all input features.
At every point along the continuum is an allocation of information to each of the features.
For example, in Fig.
4a we found that only a few bits of information ( (cid:80) D KL ), mostly coming from the hour feature, was necessary to achieve most of the performance of the full-information model, and to outperform the linear models (Tab. 1).
The capacity to extract partial information from features shines in comparison to the LassoNet performance (Fig. 4b), which improves only after multiple whole features have been incorporated.
In contrast to Bikeshare, where decent approximations required only a handful of features, the
MiceProtein (Fig.
4c,d) and MIMIC-II (Fig.
4e,f) datasets were found to require information from many features simultaneously.
The information allocation is dynamic, with some features saturating quickly, and others growing in information usage at different rates.
The signal obtained by the
Distributed IB can help decompose datasets with a large number of features and complex interaction effects, all within a single training run.
4. F INE - GRAINED INSPECTION OF FEATURE EFFECTS
In addition to a measure of feature importance in the form of average information allocation, the
Distributed IB offers a second source of interpretability: the relative importance of particular feature values, revealed through inspection of the learned compression schemes of each feature.
One of the primary strengths of a GAM is that the effects of the feature values are straightforwardly obtained, though at the expense of interactions between the features.
By contrast, the Distributed IB permits arbitrarily complex feature interactions, and still maintains a window into the role specific feature values play toward the ultimate prediction.
In Fig.
5 we compared the compression schemes found by the Distributed IB to the feature effects obtained with NAM (Agarwal et al., 2021).
The Distributed IB allows one to build up to the maximally predictive model by approximations that gradually incorporate information from the input features.
As an example, Fig. 5 displays approximations to the relationship between weather and bike rentals in the Bikeshare dataset.
To a first approximation (Fig. 5a), with only two bits of information about all features, the scant distinguishing power was allocated to discerning certain times of day—commuting periods, the middle of the night, and everything else—in the hour feature.
Some information discerned hot from cold from the apparent temperature ( atemp ), and no information about the season or whether it was a working day was used for prediction.
We are unable to say how this information was used for prediction—the price of allowing the model to be more complex—but we can infer much from where the discerning power was allocated.
At four bits, and performing about as well as the linear models in
Tab. 1, the Boolean workingday was incorporated for prediction, and the evening commuting time was distinguished from the morning commuting time.
With eight and sixteen bits of information about the inputs, nearly all of the information was used from the hour feature, the apparent temperature became better resolved, and in season , winter was distinguished from the warmer seasons.
The feature effects retrieved with NAM (Fig. 5e) corroborated some of the distinguishability found by the Distributed IB, particularly in the hour feature (e.g., the commuting times, the middle of the night, and everything else form three characteristic divisions of the day in terms of effect).
However, we found the interpretability of the NAM-effects to vary significantly across datasets (see App.
B for more examples).
For example, the apparent temperature ( atemp ) and temperature ( temp ) features were found to have contradictory effects (Fig. 10) that were difficult to comprehend, presumably arising from feature interactions that NAM could not model.
Thus while the Distributed IB sacrifices a direct window to feature effects, tracking the information on the input side of a black-box model may serve as one of the only sources of interpretability in cases with complex feature interactions.
D ISCUSSION
Interpretability can come in many forms (Lipton, 2018); the Distributed IB provides a novel sources of interpretability by illuminating where information resides in a relationship with regards to features of the input.
Without having to sacrifice complexity, the Distributed IB is particularly useful in scenarios where obtaining a foothold is difficult due to large numbers of features and complex interaction effects.
The capacity for successive approximation, allowing detail to be incorporated gradually so that humans’ limited cognitive capacity (Cowan, 2010; Rudin et al., 2022) does not prevent comprehension of the full relationship, plays a central role in the Distributed IB’s interpretability.
The signal from the Distributed IB—particularly the compression schemes of feature values (e.g.
Fig.
5)—is vulnerable to over-interpretation.
This is partly due to a difficulty surrounding intuition about compression schemes (i.e., it is perhaps most natural to imagine 1 bit of information as a single yes/no question, or a hard clustering (Fig. 3b), rather than as a confusion matrix (Fig. 3c)), and partly due to the inability to make claims like “changing input A to input B changes output C to output D ”.
Rather than supplanting other interpretability methods, the Distributed IB should be seen as a new tool for interpretable machine learning, supplementing other methods.
A thorough analysis could incorporate methods where the effects are comprehensible, possibly using a reduced feature set found by the Distributed IB.
A S YNTHETIC EXAMPLE
We work through a synthetic example with the Distributed IB in Fig.
There are two binary-valued interventions A, B which we take to be the features X 1 , X 2 of the input X , affecting a binary outcome Y with a joint probability distribution shown in Fig. 6a. Using 10,000 samples from the joint distribution, we train the Distributed IB and recover the distributed information plane (Fig. 6b) revealing that intervention A is more important for predicting the outcome Y . Each feature is embedded to a two-dimensional Gaussian so that they may be directly visualized (Fig. 6c), alongside the computed confusion matrix.
Note that the feature values of the interventions
[Figure]
split along the cardinal axes because the embeddings are diagonal Gaussians.
That intervention A split horizontally while B split vertically is due to chance.
B E XTENDED VISUALIZATIONS
We show in Fig.
7 Distributed IB optimizations for the remaining three datasets mentioned in the manuscript, alongside their LassoNet counterparts.
In Figs.
8 and 9 we show the compression schemes for some of the most important features in the Wine and MIMIC-II datasets.
Finally, in Fig.
10, we display the compression schemes for all 12 features of the Bikeshare dataset over a dense sampling of information rates.
C I MPLEMENTATION DETAILS
All Distributed IB experiments were run on a single computer with a 12 GB GeForce RTX
GPU; 100k steps on the Bikeshare dataset takes a couple minutes, and on Microsoft about 2 hours.
Tensorflow code is attached to this submission and will be posted (in extended form) to Github upon acceptance.
The data loading and set up is heavily based on the publically released code from
NODE-GAM
(Chang et al., 2022).
[Figure]
We subclassed tensorflow.keras.
Model so that training can leverage the keras high level
API, and is only one step more difficult than the model.compile(); model.fit() flow.
The additional step is the creation of a tensorflow.keras.callbacks.
Callback to control the β ramp over the course of training.
All of the information allocations can be recovered from the tensorflow.keras.callbacks.
History object returned at the end of training.
Training hyperparameters and architecture details are shown in Tab.
(common to all datasets) and
Tab. 3 (dataset-specific).
In the first stage of hyperparameter tuning, we explored parameter space in an unstructured manner and found that most parameters affected all datasets similarly.
After some tuning, we settled on the values listed in Tab. 2.
We found the effects of dropout rate and the number of annealing steps to differ by dataset, and selected optimal values out of { 0 , 0 . 1 , 0 . 3 , 0 .
5 } and { 2 . 5 × 10 4 , 5 × 10 4 , 10 5 } , respectively (Tab. 3).
[Table]
[Table]
[Figure]
The features ( PFratio , Age , Renal , and Bilirubin ) were the first four to contribute to the approximations in the low (cid:80) D KL regime of Fig.
For the categorical variable Renal the barchart displays the frequency of the particular values.
As (cid:80) D KL increases the entropy C. D ATASETS
We used the dataset preprocessing code released with NODE-GAM (Chang et al., 2022) on Githubwith a minor modification to use one-hot encoding 2 for categorical variables instead of leave-one-out encoding 3 when input to the Distributed IB models, unless there were more than 100 categories. encoders/onehot.htmlhttps:// scikit .org/ _encoders/leaveoneout.html
C. C OMPARISON METHODS LassoNet
(Lemhadri et al., 2021)
We used the publicly released Pytorch code 4 with only minor modifications: (i) fixed a typo in interfaces.py 5 (delete ‘=’), (ii) changed the objective outputs during training 6 to be solely the error metric (excluding the regularization terms) for the plots of
Fig.
We used the LassoNetClassifier and LassoNetRegressor classes, instead of the crossvalidation versions that were released after the original publication, and performed our own crossvalidation with the same splits for the other methods.
For each dataset we tuned the following hyperparameters with a grid search: L2-regularization weight γ ∈ { 0 , 0 . 01 , 0 . 1 , 1 } and the dropout fraction in { 0 , 0 . 1 , 0 . 3 , 0 . 5 } .
We selected the parameters with the best integrated performance on the validation set (metric × number of features), and used the same hidden dimensions as the joint encoder for the Distributed IB method (2 layers of 256 units).
Neural additive models
(Agarwal et al., 2021)
We used the publicly released Tensorflow code 7 .
We modified the size of the feature encoders: we encountered memory issues with the default feature encoder sizes of [ M , 64, 32], with M a function of the number of unique values in the training set for that feature 8 .
Instead we made each feature encoder size [256, 256, 256] for the datasets with under 20 features, and [128, 128, 128] for the rest.
For each dataset we tuned the following hyperparameters with a grid search: hidden activation in { relu , exu } , dropout in { 0 , 0 . 1 , 0 . 3 } , weight decay (L2 regularization) in { 0 , 10 − 5 , 10 − 4 } .