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\noindentKeywords: Admissible machine learning; InfoGram; L-Features; Information-theory; ALFA-testing, Algorithmic risk management; Fairness; Interpretability; COREml; FINEml.
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First-generation “prediction-only” machine learning technology has served the tech and eCommerce industry pretty well. However, ML is now rapidly expanding beyond its traditional domains into highly regulated or safety-critical areas---such as healthcare, criminal justice systems, transportation, financial markets, and national security---where achieving high predictive-accuracy is often as important as ensuring regulatory compliance and transparency in order to ensure the trustworthiness. We thus focus on developing admissible machine learning technology that can balance fairness, interpretability, and accuracy in the best manner possible. How to systematically go about building such algorithms in a fast and scalable manner? This article introduces some new statistical learning theory and information-theoretic graphical exploratory tools to address this question. \vskip.35em {\bf Going Beyond “Pure” Prediction Algorithms}. Predictive accuracy is not the be-all and end-all for judging the `quality' of a machine learning model. Here is a dazzling example: Researchers at the Icahn School of Medicine at Mount Sinai in New York City found that zech2018variable,reardon2019rise a deep-learning algorithm, which showed more than 90% accuracy on the x-rays produced at Mount Sinai, performed poorly when tested on data from other institutions. Later it was found that “the algorithm was also factoring in the odds of a positive finding based on how common pneumonia was at each institution—not something they expected or wanted.” This sort of unreliable and inconsistent performance can be clearly dangerous. As a result of these safety concerns, despite lots of hype and hysteria around AI in imaging, only about 30% of radiologists are currently using machine learning (ML) for their everyday clinical practices allen2021. To apply machine learning appropriately and safely-- especially when human life is at stake--we have to think beyond predictive accuracy. The deployed algorithm needs to be comprehensible (by end-users like doctors, judges, regulators, researchers, etc.) in order to make sure it has learned relevant and admissible features from the data, which is meaningful in light of investigators’ domain knowledge. The fact of the matter is, an algorithm that is solely focused on what is learned, without reasoning how it learned what it has learned, is not intelligent enough. We next expand on this issue using two real data applications. \vskip.35em {\bf Admissible ML for Industry}. Consider the UCI Credit Card data (discussed in more details in Sec (ref)), collected in October 2005, from an important Taiwan-based bank. We have records of $n=30,000$ cardholders. The data composed of a response variable $Y$ denoting: default payment status (Yes = 1, No = 0), along with $p=23$ predictor variables (e.g., gender, education, age, history of past payment, etc.). The goal is to accurately predict the probability of default given the profile of a particular customer.
On the surface, this seems to be a straightforward classification problem for which we have a large inventory of powerful algorithms. yeh2009comparisons performed an exhaustive comparison between six machine learning methods (logistic regression, K-nearest neighbor, neural net, etc.) and finally selected the neural network model, which attained $83\%$ accuracy on a 80-20 train-test split of the data. However, traditionally build ML models are not deployable, unless it is admissible under the financial regulatory constraints\footnote{The Equal Credit Opportunity Act (ECOA) is a major federal financial regulation law enacted in 1974.} wall2018some, which demand that (i) the method should not discriminate people on the basis of protective features\footnote{https://en.wikipedia.org/wiki/Protected_group}, here based on gender and age; and (ii) The method should be simpler to interpret and transparent (compared to those big neural-nets or ensemble models like random forest and gradient boosting).
To improve fairness, one may remove the sensitive variables and go back to business as usual by fitting the model on the rest of the features--known as `fairness through unawareness.' Obviously this is not going to work because there will be some proxy attributes (e.g, zip code or profession) that share some degree of correlation (information-sharing) with race, gender, or age. These proxy variables can then lead to the same unfair results. It is not clear how to define and detect those proxy variables to mitigate hidden biases in the data. In fact, on a recent review by roth2020 on algorithmic fairness, the authors forthrightly stated
Currently, there exists no systematic method to directly construct an admissible algorithm that can mitigate bias. To quote a real practitioner of a reputed AI-industry: “I ran 40,000 different random forest models with different features and hyper-parameters to search a fair model.” This ad-hoc and inefficient strategy could be a significant barrier for an efficient large-scale implementation of admissible AI technologies. Fig. (ref) shows a fair and shallow tree classifier with four decision nodes, which attains 82.65% accuracy; this was built in a completely automated manner without any hand-crafted manual tuning. Section (ref) will introduce the required theory and methods behind our procedure. Nevertheless, this simple and transparent anatomy of the final model makes it easy to convey which are the key drivers of the model: variables Pay_0 and Pay_2\footnote{Pay_0 and Pay_2 denote the repayment status of the last two months (-1=pay duly, 1=payment delay for one month, 2=payment delay for two months, and so on).} are the most important indicators to default. These variables have two key characteristics: they are highly predictive and at the same time safe to use in the sense that they share very little predictive information with the sensitive attributes age and gender, and for that reason, we call them admissible features. The model also convey \textit{how} the key variables impacting credit risk: the simple decision tree shown in Fig. (ref) is fairly self-explanatory, and its clarity facilitates an easy explanation of the predictions. \vskip.35em {\bf Admissible ML for Science}. Legal requirement is not the only reason why we want to build admissible ML. In scientific investigations, it is important to know whether the deployed algorithm helps researchers to better understand the phenomena by refining their “mental model.” Consider, for example, the prostate cancer data where we have $p=6033$ gene expression measurements from 52 tumor and 50 normal specimens. Fig. (ref) shows a 95% accurate classification model for prostate data with only two “core” driver genes! This compact model is admissible in the sense that it confers the following benefits: (i) it identifies a two-gene signature (composed of gene-1627 and gene-2327) as the top factor associated with prostate cancer. They are \textit{jointly} overexpressed in the tumor samples but interestingly they have very little marginal information (not individually differentially expressed, as shown in Fig. (ref)). Accordingly, traditional linear-model-based analysis will fail to detect this gene-pair as a key biomarker. (ii) The simple decision tree model in Fig. (ref) provides a mechanistic understanding and justification as to why the algorithm thinks a patient has prostate cancer or not. (iii) Finally, it provides the needed guidance on what to do next by having a control over the system. In particular, a cancer biologist can choose between different diagnosis and treatment plans with the goal to regulate those two oncogenes.
\vskip.35em {\bf Goals and Organization}. The primary goal of this paper is to introduce some new fundamental concepts and tools to lay the foundation of admissible machine learning that are efficient (enjoy good predictive accuracy), fair (prevent discrimination against minority groups), and interpretable (provide mechanistic understanding) to the best possible extent.
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Our statistical learning framework is grounded in the foundational concepts of information theory. The required statistical formalism (nonparametric estimation and inference methods) and information-theoretic principles (entropy, conditional entropy, relative entropy, and conditional mutual information) are introduced in Section (ref). A new nonparametric estimation technique for conditional mutual information (CMI) is proposed that scales to large datasets by leveraging the power of machine learning. For statistical inference, we have devised a new model-based bootstrap strategy. The method was applied to the problem of conditional independence testing and integrative genomics (breast cancer multi-omics data from Cancer Genome Atlas). Based on this theoretical foundation, in Section (ref), we laid out the basic elements of admissible machine learning. Section (ref) focuses on algorithmic interpretability: how can we efficiently search and design self-explanatory algorithmic models by balancing accuracy and robustness to the best possible extent? Can we do it in a completely model-agnostic manner? Key concepts and tools introduced in this section are: Core features, infogram, L-features, net-predictive information, and COREml. The procedure was applied to several real datasets, including high-dimensional microarray gene expression datasets (prostate cancer and SRBCT data), MONK's problems, and Wisconsin breast cancer data. Section (ref) focuses on algorithmic fairness, which tackles the challenging problem of designing admissible ML algorithms that are simultaneously efficient, interpretable, and equitable. There are several key techniques introduced in this section: admissible feature selection, ALFA-testing, graphical risk assessment tool, and FINEml. We illustrate the proposed methods using examples from criminal justice system (ProPublica's COMPAS recidivism data), financial service industry (Adult income data, Taiwan credit card data), and marketing ad campaign. We conclude the paper in Section (ref) by reviewing the challenges and opportunities of next-generation admissible ML technologies.
The foundation of admissible machine learning relies on information-theoretic principles and nonparametric methods. The key theoretical ideas and results are presented in this section to develop a deeper understanding of the conceptual basis of our new framework.
Let $Y$ be the response variable taking values $\{1,\ldots,k\}$, $\X=(X_1,\ldots,X_p)$ denotes a $p$-dimensional feature matrix, and $\Sb=(S_1,\ldots,S_q)$ is additional set of $q$ covariates (e.g., collection of sensitive attributes like race, gender, age, etc.). A variable is called mixed when it can take either discrete, continuous, or even categorical values, i.e., completely unrestricted data-types. Throughout, we will allow both $\X$ and $\Sb$ to be mixed. We write $Y \perp\!\!\!\perp \X$ to denote the independence of $Y$ and $\X$. While, the conditional independence of $Y$ and $\X$ given $\Sb$ is denoted by $Y \perp\!\!\!\perp \X \mid \Sb$. For a continuous random variable, $f$ and $F$ denote the probability density and distribution function, respectively. For a discrete random variable the probability mass function will be denoted by $p$ with proper subscript.
Our theory starts with an information-theoretic view of conditional dependence. Under conditional independence: $$Y \perp\!\!\!\perp \Xb \mid \Sb$$ the following decomposition holds for all $y,\xb,\sb$ \[ f_{Y,\X|\Sb}(y,\xb|\sb)\,=\, f_{Y|\Sb}(y|\sb) f_{\X|\Sb}(\xb|\sb).\] More than testing independence, often the real interest lies in quantifying the conditional dependence: the average deviation of the ratio \beq \dfrac{f_{Y,\X|\Sb}(y,\xb|\sb)}{f_{Y|\Sb}(y|\sb) f_{\X|\Sb}(\xb|\sb)}, \eeq which can be measured by conditional mutual information wyner1978.
{\bf Two Important Properties}. (P1) One of the striking features of CMI is that it captures multivariate non-linear conditional dependencies between the variables in a completely nonparametric manner. (P2) CMI possesses the necessary and sufficient condition as a measure of conditional independence, in the sense that \beq \MI(Y,\X|\Sb)=0 if and only if Y \perp\!\!\!\perp \Xb \mid \Sb.\eeq
Conditional independence relation can be described using graphical model (also known as Markov network), as shown the figure below:
One of the major significances of CMI as a measure of conditional dependence comes from its interpretation in terms of additional `information gain' on $Y$ learned through $\Xb$ when we already know $\Sb$. In other words, CMI measures the Net-Predictive Information (NPI) of $\boldmath{\Xb}$---the exclusive information content of $\X$ for $Y$ beyond what is already subsumed by $\Sb$. To formally arrive at this interpretation, we have to look at CMI from a different angle, by expressing it in terms of conditional entropy. Entropy is a fundamental information-theoretic uncertainty measure. For a random variable $Z$, entropy $H(Z)$ is defined as $-\Ex_Z[\log f_Z]$.
The proof involves some standard algebraic manipulations, and is given in Appendix (ref).
The basic formula (ref) of conditional mutual information (CMI) that we have presented in the earlier section, is, unfortunately, not readily applicable for two reasons. First, the practical side: in the current form, (ref) requires estimation of $f_{Y,\X|\Sb}$ and $f_{\X|\Sb}$, which could be a herculean task, especially when $\X=(X_1,\ldots,X_p)$ and $\Sb=(S_1,\ldots,S_q)$ are large-dimensional. Second, the theoretical side: since the triplet $(Y,\X,\Sb)$ is mixed (not all discrete or continuous random vectors) the expression (ref) is not even a valid representation. The necessary reformulation is given in the next theorem.
To prove it, first rewrite the dependence-ratio (ref) solely in terms of conditional distribution of $Y$ as follows: \[\dfrac{\Pr(Y=y\mid \X=\xb,\Sb=\sb)}{\Pr(Y=y\mid \Sb=\sb)}~=~\dfrac{p_{Y\mid \Xb,\Sb}(y|\xb,\sb) }{p_{Y|\Sb}(y|\sb)}\] Next, substitute this into (ref) and express it as \[ \MI(Y,\Xb \mid \Sb)~=~ \iint_{\xb,\sb} \Bigg[ \sum_y\, p_{Y\mid \X,\Sb}(y|\xb,\sb) \,\log\left(\frac{p_{Y| \X,\Sb}(y|\xb,\sb) }{p_{Y|\Sb}(y|\sb)}\right)\Bigg] \dd F_{\X,\Sb}\] Replace the part inside the square brackets by (ref) to finish the proof. \qed
{\bf Estimator}. Goal is to develop a practical nonparametric algorithm for estimating CMI from $n$ i.i.d samples $\{\xb_i,y_i,\sb_i\}_{i=1}^n$ that works for large($n,p,q$) settings. Theorem (ref) immediately leads to the following estimator of (ref): \beq \widehat{\MI}(Y, \Xb \mid \Sb) = \dfrac{1}{n}\sum_{i=1}^n \log \dfrac{ \widehat{\Pr}(Y=y_i|\xb_i,\sb_i) }{ \widehat{\Pr}(Y=y_i|\sb_i)}.\eeq \vskip.25em
{\bf Algorithm 1}. Conditional mutual information estimation: the proposed ML-powered nonparametric estimation method consists of three simple steps:
Step 1. Choose a machine learning classifier (e.g., support vector machines, random forest, gradient boosted trees, deep neural network, etc.), and call it ML$_0$.
Step 2. Train the following two models: \beas ML.train_{y|\xb,\sb}&\leftarrow &ML_0\big( Y \sim [\X, \Sb] \big)\\ ML.train_{y|\sb}&\leftarrow & ML_0\big( Y \sim \Sb \big) \eeas
Step 3. Extract the conditional probability estimates $\widehat{\Pr}(Y=y_i|\xb_i,\sb_i)$ from $\texttt{ML.train}_{y|\xb,\sb}$, and $ \widehat{\Pr}(Y=y_i|\sb_i)$ from $\texttt{ML}_0\big( Y \sim \Sb \big)$, for $i=1,\ldots, n$.
Step 4. Return $\widehat{\MI}(Y, \Xb \mid \Sb)$ by applying formula (ref).
The proposed style of nonparametric estimation provides some important practical benefits:
$\bullet$ Flexibility: Unlike traditional conditional independence testing procedures candes2018panning,berrett2019conditional, our approach requires neither the knowledge of the exact parametric form of high-dimensional $F_{X_1,\ldots,X_p}$ nor the knowledge of the conditional distribution of $\Xb \mid \Sb$, which are generally unknown in practice.
$\bullet$ Applicability: (i) Data-type: The method can be safely used for mixed $\X$ and $\Sb$ (any combination of discrete, continuous, or even categorical variables). (ii) Data-dimension: The method is applicable to high-dimensional $\X=(X_1,\ldots, X_p)$ and $\Sb=(S_1,\ldots, S_q)$.
$\bullet$ Scalability: Unlike traditional nonparametric methods (such as kernel density or $k$-nearest neighbor-based methods), our procedure is scalable for big datasets with large($n,p,q$).
One can even perform statistical inference for our ML-powered conditional-mutual-information statistic. In order to test $H_0: Y \perp\!\!\!\perp \X \mid \Sb$, obtain bootstrap-based p-value by noting that under the null $\Pr(Y=y|\X=\xb,\Sb=\sb)$ reduces to $\Pr(Y=y|\Sb=\sb)$.
{\bf Algorithm 2}. Model-based Bootstrap: The inference scheme proceeds as follows:
Step 1. Let $$\widehat{p}_{i|\sb}~=~ \widehat{\Pr}(Y_i=1|\Sb=\sb_i),~~\text{for}~ i=1,\ldots,n$$ as extracted from (already estimated) the model $\texttt{ML.train}_{y|\sb}$ (step 2 of Algorithm 1).
Step 2. Generate the null $Y_{n \times 1}^*=(Y^*_1,\ldots,Y^*_n)$ by $$Y_i^* ~\leftarrow~{\rm Bernoulli}(\widehat{p}_{i|\sb}),~~\text{for}~ i=1,\ldots,n$$
Step 3. Compute $\widehat{\MI}(Y^*,\X \mid \Sb)$ using the Algorithm 1.
Step 4. Repeat the process $B$ times (say, $B=500$); compute the bootstrap null distribution, and return the p-value.
How to design admissible machine learning algorithms with enhanced efficiency, interpretability, and equity?\footnote{However, the general premise of admissible ML is extremely broad and flexible, and will continue to evolve with the regulatory requirements to ensure rapid development of trustworthy algorithmic methods.} A systematic pipeline for developing such admissible ML models is laid out in this section, which is grounded in the earlier information-theoretic concepts and nonparametric modeling ideas.
One of the first tasks of any predictive modeling is to identify the key drivers that are affecting the response $Y$. Here we will discuss a new information-theoretic graphical tool to quickly spot the “core” decision-making variables, which are going to be vital in building interpretable models. One of the advantages of this method is that it works even in the presence of correlated features, as the following example illustrates; also see Appendix (ref). \vskip1em
Note that the imitator $X_p$ is highly predictive for $Y$ due to its association with the core variables. We have simulated $n=500$ samples with $p=50$. For each feature we compute, \beq R_j = overall relevance score of $j$th predictor, j=1,\ldots,p.\eeq The bottom-left corner of Fig. (ref) shows the relative importance scores (scaled between 0 and 1) for the top seven features using gbm algorithm\footnote{based on whether a particular variable was selected to split on during learning a tree, and how much it improves the Gini impurity or information gain.}, which correctly finds $\{X_1,X_2,X_{50}\}$ as the important predictors. However, it is important to recognise that this modus operandi---irrespective of the ML algorithm---can not distinguish the `fake imitator' $X_{50}$ from the real ones $X_1$ and $X_2$. To enable refined characterization of the variables, we have to `add more dimension' to the classical machine learning feature importance tools.
We introduce a tool for identification of core admissible features based on the concept of net-predictive information (NPI) of a feature $X_j$.
{\bf L-Features}. The highlighted L-shaped area contains features that are either irrelevant or redundant. For example, notice the position of $X_{50}$ in the plot, indicating that it is highly predictive but contains no new complementary information for the response. Clearly, there could be an opposite scenario: a variable carries valuable net individual information for $Y$, despite being moderately relevant (not ranked among the top few); see Sec. (ref).
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How does one distill a compact (parsimonious) ML model by balancing accuracy, robustness, and interpretability to the best extent? To answer that, we introduce COREtree, whose construction is guided by infogram. The methodology is illustrated using two real datasets, namely Prostate cancer and SRBCT tumor data. The main findings are striking: it shows how one can systematically search and construct robust and interpretable shallow decision tree models (often with just two or three genes) for noisy high-dimensional microarray datasets that are as powerful as the most elaborate and complex machine learning methods.
Step 1. Identifying CoreGenes. GBM-selected top 50 genes are shown in Fig. (ref). We generate the infogram\footnote{To reduce unnecessary clutter, we have displayed the infogram using top 50 features, since the rest of the genes will be cramped inside the nonessential L-zone anyway.} of these $50$ variables (displayed on the top-right corner), which identifies five core-genes $\{1627, 2327, 77, 1511, 1322\}$.
\vskip.4em Step 2. Rank-transform: Robustness and Interpretability. Instead of directly operating on the gene expression values, we transform them into their ranks. Let $\{x_{j1},\ldots, x_{jn}\}$ be the measurements on $j$th gene with empirical cdf $\wtF_j$. Convert the raw $x_{ji}$ to $u_{ji}$ by \beq u_{ji}=\wtF_j(x_{ji}), i=1,\ldots,n\eeq and work on the resulting $\mathbf{U}_{n\times p}$ matrix instead of the original $\X_{n\times p}$. We do this transformation for two reasons: first, to robustify, since it is known that gene expressions are inherently noisy. Second, to make it unit-free, since the raw gene expression values depend on the type of preprocessing, thus carries much less scientific meaning. On the other hand, percentiles are much more easily interpretable to convey “how overexpressed a gene is.”
Step 3. Shallow Robust Tree. We build a single decision tree using the infogram-selected coregenes. This is displayed in the bottom-right panel of Fig. (ref). Interestingly, the CoreTree retained only two genes $\{1627, 2327\}$ whose scatter plot (in the rank-transform domain) is shown in the bottom-left corner of Fig. (ref). A simple eyeball estimate of the discrimination surfaces are shown in bold (black and red) lines, which closely matches with the decision tree rule. It is quite remarkable that we have reduced the original $6033$-dimensional problem to a simple bivariate two-sample one, just by wisely selecting the features based on the infogram.
\vskip.34em Step 4. Stability. Note the tree that we build is based only on the infogram-selected core features. These features have less redundancy and high relevance, which provide an extraordinary stability (over different runs on the same dataset) to the decision-tree--a highly desirable characteristic.
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Step 5. Accuracy. The accuracy of our single decision tree (on a randomly selected 20% test set, averaged over 100 times) is more than $95\%$. On the other hand, the full-data gbm (with $p=6033$ genes) is only $75\%$ accurate. Huge simplification of the model-architecture with significant gain in the predictive performance!
\vskip.34em Step 6. Gene Hunting: Beyond Marginal Screening. We compute two-sample $t$-test statistic for all $p=6033$ genes and rank them according to their absolute values (the gene with the largest absolute $t$-statistic gets ranked 1--the most differentially expressed gene). The $t$-scores for the coregenes along with their p-values and ranks are: \beas \big | t_{1627} \big|= 0.15; & $p$-value= 0.88; & rank= 5383.\\ \big | t_{2327} \big|= 1.40; & $p$-value= 0.17; & \text{rank}= 1228. \eeas Thus, it is hopeless to find \texttt{coregenes} by any marginal-screening method--they are \textit{too weak marginally (in isolation), but jointly an extremely strong predictor}. The good news is that our approach can find those multivariate hidden gems in a completely nonparametric fashion.
\vskip.34em Step 7. Lasso Analysis and Results. We have used the glmnet R-package. Lasso with $\lambda_{{\rm min}}$ (minimum cross-validation error) selects $70$ genes, where as $\lambda_{1{\rm se}}$ (the largest lambda such that error is within 1 standard error of the minimum) selects $60$ genes. Main findings are:
\hskip2em(i) The coregenes $\{1627,2327\}$ were never selected, probably because they are marginally very weak; and the significant interaction is not detectable by standard-lasso.
\hskip2em(ii) Accuracy of Lasso with $\lambda_{{\rm min}}$ is around $78\%$ (each time we have randomly selected 85% data for training; computed the $\lambda_{{\rm cv}}$ for making prediction; averaged over 100 runs).
\vskip.4em Step 8. Explainability. The final “two-gene model” is so simple and elegant that it can be easily communicated to doctors and medical practitioners: a patient with overexpressed gene $1627$ and gene $2327$ has a higher risk of getting prostate cancer. Biologists can use these two genes as robust prognostic markers for decision-making (or for recommending the proper drug). It is hard to imagine there could be a more accurate algorithm, one that is at least as compact as the “two-gene model.” We should not forget that the success behind this dramatic model-reduction hinges on discovering multivariate coregenes, which: (i) help us to gain insights into biological mechanisms [clarifying `who' and `how'], and (ii) provide a simple explanation of the predictions [justifying `why'].
Step 1. Infogram Construction: Fig. (ref) displays the infogram, which provides a quick understanding of the phenomena by revealing its `core.' Noteworthy points: (i) there are three highly predictive inadmissible features (green bubbles in the plot: perimeter_worst, area_worst, and concave_points_worst), which have large overall predictive importance but almost zero net individual contributions. We have called these variables `Imitators' in Sec. (ref). (ii) Three among the four `core' admissible features (texture_worst, concave_points_mean, and texture_mean) are not among the top features based on usual predictive information, yet they contain a considerable amount of new exclusive information (net-predictive information) that is useful for separating malignant and benign tumor cells. In simple terms, infogram help us to track down where the `core' discriminatory information is hidden.
\vskip.35em Step 2. Core-Scatter plot. The right panel of Fig. (ref) shows the scatter plot of the top two core features and how they separate the malignant and benign tumor cells.
\vskip.5em Step 3. Infogram-assisted CoreGLM model: The simplest possible model that one could build is a logistic regression based on those four admissible features. Interestingly, the Akaike information criterion (AIC) based model selection further drops the variable texture_mean, which is hardly surprising considering that it has the least net and total information among the four admissible core features. The final logistic regression model with three core variables is displayed below (output of glm R-function):
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This simple parametric model achieves a competitive accuracy of $96.50\%$ (on a 15% test set; averaged over $50$ trials). Compare this with full-fledged big ML models (like gbm, random forest, etc.) which attain accuracy in the range of $95-97\%$. This example again shows how infogram can guide the design of a highly transparent and interpretable CoreGLM model with a few handful of variables---which is as powerful as complex black-box ML methods.
ML-systems are increasingly used for automated decision-making in various high-stakes domains such as credit scoring, employment screening, insurance eligibility, medical diagnosis, criminal justice sentencing, and other regulated areas. To ensure that we are making responsible decisions using such algorithms, we have to deploy admissible models that can balance Fairness, INterpretability, and Efficiency (FINE) to the best possible extent. This section discusses principles and tools for designing such FINE-algorithms.
Imagine that a machine learning algorithm is used by a bank to accurately predict whether to approve or deny a loan application based on the probability of default. This ML-based risk-assessing tool has access to the following historical data:
\hskip1.5em\,
\ $Y$: \{$0, 1$\} Loan status variable--$1$ whether the loan was approved and $0$ if denied.
\hskip1.5em\,
\ $\X$: Feature matrix \{income, loan amount, education, credit history, zip code\}
\hskip1.5em\,
\ $\Sb$: Collection of protected attributes \{gender, marital status, age, race\}.
To automate the loan-eligibility decision-making process, the bank wants to develop an accurate classifier that will not discriminate among applicants on the basis of their protected features. Naturally, the question is: how to go about designing such ML-systems that are accurate and at the same time provide safeguards against algorithmic discrimination?
{\bf Approach 1} Non-constructive: We can construct a myriad of ML models by changing and tuning different hyper-parameters, base learners, etc. One can keep building different models until one finds a perfect one that avoids adverse legal and regulatory issues. There are at least two problems with this `try until you get it right' approach: first, it is non-constructive. The whole process gives zero guidance on how to rectify the algorithm to make it less-biased. Second, there is no single definition of fairness---more than twenty different definitions have been proposed over the last few years narayanan21def. And the troubling part is that these different fairness measures are mutually incompatible to each other and cannot be satisfied simultaneously kleinberg2018; see Appendix (ref). Hence this laborious process could end up being a wild-goose chase, resulting in a huge waste of computation. \vskip.5em {\bf Approach 2} Constructive: Here we seek to construct ML models that---by design---mitigate bias and discrimination. To execute this task successfully, we must first identify and remove proxy variables (e.g., zip code) from the learning set, which prevent a classification algorithm from achieving desired fairness. But how to define a proper mathematical criterion to detect those surrogate variables? Can we develop some easily interpretable graphical exploratory tools to systematically uncover those problematic variables? If we succeed in doing this, then ML developers can use it as a data filtration tool to quickly spot and remove the potential sources of biases in the pre-modeling (data-curation) stage, in order to mitigate fairness issues in the downstream analysis.
We offer a diagnostic tool for identification of admissible features that are predictive and safe. Before going any further, it is instructive to formally define what we mean by `safe.'
Construction. In the context of fairness, we construct the infogram by plotting $\{(R_j, F_j)\}_{j=1}^p$, where recall $R_j$ denotes the relevance score (ref) for $X_j$. The goal of this graphical tool is to assist identification of admissible features which have little or no information-overlap with sensitive attributes $\Sb$, yet are reasonably predictive for $Y$. Interpretation. Fig. (ref) displays an infogram with six covariates. The L-shaped highlighted region contains variables that are either inadmissible (the horizontal slice of L) or inadequate (the vertical slice of L) for prediction. The complementary set ${\rm L}^c$ comprises of the desired admissible features. Focus on variables A and B: both have the same predictive power, but are gained through a completely different manner. The variable B gathered information for $Y$ entirely through the protected features (verify it from the graphical representation of B), and is thus inadmissible. On the other hand, the variable \texttt{A} carries direct informational value, having no connection with the prohibitive $\Sb$, and is thus totally admissible. Unfortunately, though, reality is usually more complex than this clear-cut black and white \texttt{A}-\texttt{B} situation. The fact of the matter is: admissibility (or fairness, per se) is not a yes/no concept, but a matter of \textit{degree}\footnote{“Zero bias” is an illusion. All models are biased (to a different degree), but some are admissible. The real question is how to methodically construct those admissible ones from possibly biased data.}, which is explained at the bottom two rows of Fig. (ref) utilizing variables C to F.
Legal doctrine. Note that in our framework the protected variables are used only in the pre-deployment phase to determine what other (admissible) attributes to include in the algorithm to mitigate unforeseen downstream bias, which is completely legal hellman2020. It is also advisable that once inadmissible variables are identified using an infogram, not to throw them (especially the highly predictive ones such as the feature B in Fig. (ref)) blindly from the analysis without consulting domain experts---including some of them may not necessarily imply violation of the law; ultimately, it is up to the policymakers and judiciary to determine their appropriateness (legal permissibility) based on the given context. Our job as statisticians is to discover those hidden inadmissible L-features (preferably in a fully data-driven and automated manner) and raise a red flag for further investigation.
Step 1. Trust in data. Is there any evidence of built-in bias in the data? That is to say, whether a `significant' portion of the decision-making ($Y$ is greater or less than 50k per year) was influenced by the sensitive attributes $\Sb$ beyond what is already captured by other covariates $\Xb$? One may be tempted to use $\MI(Y,\Sb \mid \Xb)$ as a measure for assessing fairness. But we need to be careful while interpreting the value of $\MI(Y,\Sb \mid \Xb)$. It can take a `small' value for two reasons: First, a genuine case of fair decision-making where individuals with similar $\xb$ received a similar outcome irrespective of their age, gender, and other protected characteristics; see Appendix (ref) for one such example. Second, there is a collusion between $\Xb$ and $\Sb$ in the sense that $\Xb$ contains some proxies of $\Sb$ which reduce its effect-size---leading one to falsely declare a decision-rule fair when it is not.
Step 2. Infogram to identify inadmissible proxy features. The infogram, shown in the left panel of Fig. (ref), finds four admissible features \[~~~~~~\Xb_A~=~\big\{ \texttt{Capital\_gain}, ~\texttt{Capital\_loss}, ~\texttt{Occupation}, ~\texttt{Education} \big\}.~~\] They share very little information with $\Sb$ yet are highly predictive. In other words, they enjoy high relevance and high safety-index. Next, we also see that there is a feature that appears at the lower-right corner
$\hskip12em \Xb_R~=~\big\{ \texttt{Relationship}\big\}$ \vskip.2em which is the prime source of bias; the subscript `R' stands for risky. The variable relationship represents the respondent's role in the family---i.e., whether the breadwinner is husband, wife, child, or other relative.
\vskip.5em Step 3. ALFA-test and encoded bias. We can construct an honest fairness assessment metric by conditioning CMI with $\X_{A}$ (instead of $\Xb)$: \beq \widehat{\MI}(Y,\Sb \mid \Xb_A)=0.13, with pvalue almost $0$. \eeq This strongly suggests historical bias or discrimination is encoded in the data. Our approach not only quantifies but also allows ways to mitigate bias to create an admissible prediction rule; this will be discussed in Step 4. The preceding discussions necessitate the following, new general class of fairness metrics.
{\bf Three Different Interpretations}. The ALFA-statistic (ref) can be interpreted from three different angles.
$\bullet$ It quantifies the trade-off between fairness and model performance: how much net-predictive value is contained within $\Sb$ (and its close associates)? This is the price we pay in terms of accuracy to ensure a higher degree of fairness.
$\bullet$ A small $\mathlarger{\alpha}$-inadmissibility value ensures that individuals with similar `admissible characteristics' receive a similar outcome. Note that our strategy of comparing individuals with respect to only (infogram-learned) “admissible” features allows us to avoid the (direct and indirect) influences of sensitive attributes on the decision making.
$\bullet$ Lastly, the $\mathlarger{\alpha}$-statistic can also be interpreted as “bias in response $Y$.” For a given problem, if we have access to several “comparable” outcome variables\footnote{e.g, obermeyer2019 showed that healthcare cost can be a poor proxy of health, especially for Black patients; similarly, blattner2021costly showed that credit scores could be a poor proxy for creditworthiness especially for low-income and minority groups.} then we choose the one which minimizes the $\mathlarger{\alpha}$-inadmissibility measure. In this way, we can minimize the loss of predictive accuracy while mitigating the bias as best as we can.
Step 4. FINEtree. The inherent historical footprints of bias (as noted in eq. (ref)) need to be deconstructed to build a less-discriminatory classification model for the income data. Fig. (ref) shows FINEtree---a simple decision tree based on the four admissible features, which attains 83.5% accuracy.
Step 5. Trust in algorithm through risk assessment and ALFA-ranking: The current standard for evaluating ML models is primarily based on predictive accuracy on a test set, which is narrow and inadequate. For an algorithm to be deployable it has to be admissible; an unguided ML carries the danger of inheriting bias from data. To see that, consider the following two models: \vskip.2em \hskip7em Model$_{\rm A}$: Decision tree based on $\Xb_A$ (FINEtree)
\hskip7em Model$_{\rm R}$: Decision tree based on $\Xb_A \cup \{ \text{relationship}\}$. \vskip.2em Both models have comparable accuracy around $83.5\%$. Let $\whY_{\rm A}$ and $\whY_{\rm R}$ be the predicted labels based on these two models, respectively. Our goal is to compare and rank different models based on their risk of discrimination using ALFA-statistic: \bea \mathlarger{\widehat\alpha}_{\rm A} &=&\widehat{\MI}(\whY_A,\Sb \mid \Xb_A) = 0.00042, with pvalue $0.95$ \\ \mathlarger{\widehat\alpha}_{\rm R} &=&\,\widehat{\MI}(\whY_{\rm R},\Sb \mid \Xb_A) = 0.195, with pvalue almost $0$. \eea $\mathlarger{\alpha}$-inadmissibility statistic measures how much the final decision (prediction) was impacted by the protective features. A smaller value is better in the sense that it indicates improved fairness of the algorithm's decision. Eqs (ref)-(ref) immediately imply that Model$_{\rm A}$ is better (less discriminatory without being inefficient) than Model$_{\rm R}$, and can be safely put into production. \vskip.2em
The infogram, shown in the left panel of Fig. (ref), clearly selects the variable Pay_0 and Pay_2 as the key admissible factors that determine the likelihood of default. Once we know the admissible features, the next question is: `how' Pay_0 and Pay_2 are impacting the credit risk? Can we extract an admissible decision rule? For that we construct the FINEtree: a decision tree model based on the infogram-selected admissible features; see Fig. (ref). The resulting predictive model is extremely transparent (with shallow yet accurate decision trees\footnote{One can slightly improve accuracy by combining hundreds or thousands of trees (based on only the admissible features) using random forest or boosting. But the opacity of such models renders them unfit for deployment in financial and bank sectors fahner2018fico.}) and also mitigates unwanted bias by avoiding inadmissible variables. Lenders, regulators, and bank managers can use this model for automating credit decisions.
We are interested in the following question: how does one systematically build fairness-enhancing parametric statistical algorithms, such as a generalized linear model (GLM)?
Customer targeting using admissible machine learning. Our approach is summarized below:
Step 1. Graphical tool for algorithmic risk management. Fig. (ref) shows the infogram, which identifies two admissible features for loan decision: Income (annual income in \$000), and CCAvg (Avg. spending on credit cards per month). However, the two highly predictive variables education (education level: undergraduate, graduate, or advanced) and family (family size of the customer) turn out to be inadmissible, even though they look completely “reasonable” on the surface. Consequently, including these variables in a model can do more harm than good by discriminating against minority applicants.
Step 2. FINE-Logistic model: We train a logistic regression model based on the two admissible features, leading to the following model: \beq \logit \left\{\mu(x)\right\} \,=\,-6.13\,+\,.04 \,Income\,+\,.06 \,CCAvg, \eeq where $\mu(x)=\Pr(Y=1|\Xb=\xb)$. This simple model achieves 91% accuracy. It provides a clear understanding of the `core' factors that are driving the model's recommendation.
Step 3. FINElasso. One natural question would be, How can we extend this idea to high-dimensional glm models? In particular, we are interested in the following question: Is there any way we can directly embed `admissibility' into the lasso regression model? The key idea is as follows: use adaptive regularization by choosing the weights to be the inverse of safety-indices, as computed in formula (ref) of definition (ref). Estimate FINElasso model by solving the following adaptive version: \beq \widehat{\bb}_{{\rm FINE}}\,=\, \argm_{\bb} \sum_{i=1}^n \Big[ -y_i(\xb_i^T\bb)+ \log\big(1+e^{\xb_i^T\bb}\big) \Big] - \lambda \sum_{j=1}^p w_j \big| \beta_j \big|, \eeq where the weights are defined as \beq w_j^{-1}\,=\MI\big(Y,X_j \mid \{ S_1,\ldots,S_q\}\big).\eeq The adaptive penalization in (ref) acts as a bias-mitigation mechanism by dropping (that is, heavily penalizing) the variables with very low safety-indices. This whole procedure can be easily implemented using the penalty.factor argument of glmnet R-package friedman2010regularization. No doubt a similar strategy can be adopted for other regularized methods such as ridge or elastic-net. For an excellent review on different kinds of regularization procedures, see hastie2020ridge.
Over the last $60$ years or so---since the early 1960s---there's been an explosion of powerful ML algorithms with increasing predictive performance. However, the challenge for the next few decades will be to develop sound theoretical principles and computational mechanisms that transform those conventional ML methods into more safe, reliable, and trustworthy ones.
The fact of the matter is that doing machine learning in a `responsible' way is much harder than developing another complex ML technique. A highly accurate algorithm that does not comply with regulations is (or will soon be) unfit for deployment, especially in safety-critical areas that directly affect human lives. For example, the Algorithmic Accountability Act\footnote{Also see, EU's “Artificial Intelligence Act” released on April 21, 2021, whose key points are summarized in Appendix (ref).} (see Appx. (ref)) introduced in April 2019 requires large corporations (including tech companies, as well as banks, insurance, retailers, and many other consumer businesses) to be cognizant of the potential for biased decision-making due to algorithmic methods; otherwise, civil lawsuits can be filed against those firms. As a result, it is becoming necessary to develop tools and methods that can provide ways to enhance interpretability and efficiency of classical ML models while guarding against bias. With this goal in mind, this paper introduces a new kind of statistical learning technology and information-theoretic automated monitoring tools that can guide a modeler to quickly build “better” algorithms that are less-biased, more-interpretable, and sufficiently accurate.
\vskip.3em One thing is clear: rather than being passive recipients of complex automated ML technologies, we need more general-purpose statistical risk management tools for algorithmic accountability and oversight. This is critical to the responsible adoption of regulatory-compliant AI-systems. This paper has taken some important steps towards this goal by introducing the concepts and principles of `Admissible Machine Learning.'
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The author thanks the editor, associate editor, and four anonymous reviewers for their helpful suggestions. I would like to specially thank Erin LeDell for bringing this problem to my attention. The author was benefited from many useful discussions with Michael Guerzhoy, Hany Farid, Julia Dressel, Beau Coker, and Hanchen Wang on demystifying some aspects of COMPASS data; Daniel Osei on the data pre-processing steps of Lending Club loan data. This research was supported by H2O.ai.
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