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$\texttt{findr}$: Transparent and Fair Credit Risk Decisions through Semi-Structured Regressions

Victor Medina-Olivares, Stefan Lessmann, Jonathan Crook

arXiv:2608.24582v1stat.MLcs.AIcs.LGq-fin.RM

TL;DR

Credit-risk models must balance predictive accuracy with transparent explanations and auditable fairness, but linear models can miss nonlinear structure and flexible models often rely on post-hoc explanations. findr combines an interpretable structured logit with an orthogonal neural residual, applies an in-processing Wasserstein penalty to score distributions, and adds attribution diagnostics. Across simulations and eight credit datasets, it stays close to logistic regression for approximately linear signals while recovering much of neural models’ predictive gain when nonlinear structure matters, making the trade-offs explicit.

  • Problem

    Credit-risk modelling needs transparent, auditable fairness alongside predictive flexibility, while logistic regression can miss nonlinear structure and flexible models may not explain their decision rule.

  • Method

    findr decomposes the logit into an interpretable structured component and orthogonal neural residual, penalises Wasserstein score-distribution disparities during training, and evaluates attribution with diagnostics.

  • Results

    Across simulations and eight credit datasets, findr behaves close to logistic regression for linear signals and recovers much of neural models’ predictive gain when nonlinear structure is relevant.

  • Takeaways & Limitations

    Semi-structured modelling makes performance, fairness, and interpretability trade-offs explicit while indicating when residual variation must also be examined.

  • Takeaways & Limitations

    The score-level frontier is a numerical validation-sample benchmark, not the attainable boundary for any specified family of fitted models.

Abstract

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Credit risk models increasingly need to combine predictive accuracy with transparent explanations and auditable fairness constraints. Logistic regression remains attractive because its coefficients are easy to interpret, but it can miss nonlinear structure. Flexible models can improve prediction, but their explanations are often post-hoc and may not describe the decision rule itself. We introduce $\texttt{findr}$, short for flexible, interpretable deep regression, a semi-structured framework for binary credit risk modelling that decomposes the logit into an interpretable structured component and an orthogonal neural residual. The orthogonalisation separates coefficient-based effects from residual nonlinear variation, while an in-processing Wasserstein penalty mitigates group disparities by comparing score distributions during training. The framework also includes diagnostics that measure the structured component's contribution to logit variation, decision agreement, and local directional consistency. We evaluate $\texttt{findr}$ in a simulation study and on eight public credit datasets using score-level accuracy-fairness frontiers. The results show that $\texttt{findr}$ behaves close to logistic regression when the signal is approximately linear, while recovering much of the predictive gain of neural models when nonlinear structure is relevant. The diagnostics identify when coefficient-based explanations remain close to the full fitted model and when residual variation must also be examined. These findings support semi-structured modelling as a practical way to make performance, fairness, and interpretability trade-offs explicit in credit risk decisions.

1 Introduction

Credit-risk modelling must reconcile predictive flexibility with transparent, auditable explanations and formal fairness assessment. findr addresses this by combining structured coefficient effects, neural flexibility, orthogonal attribution, and in-processing disparity mitigation.

  • Motivation: Logistic regression is interpretable and auditable but can underfit nonlinearities and feature interactions in credit portfolios.Flexible models may improve prediction, but their explanations are often post-hoc rather than the decision rule itself.
  • Motivation: Credit-risk governance increasingly emphasizes explainability, auditability, model-risk control, and monitoring discriminatory or disparate impacts.These frameworks do not prescribe one mathematical fairness criterion, but they reinforce the need to reconcile predictive gains with accountable disparity mitigation.
  • Fairness: Fairness criteria compare outcomes, errors, or score distributions across protected groups, because accuracy alone does not ensure fair outcomes.Mitigation methods are commonly organized as pre-processing, in-processing, or post-processing interventions.
  • Fairness: Group fairness is more directly auditable at portfolio level, whereas individual fairness requires a justified similarity metric that can be costly and domain-dependent.The two notions protect against different forms of harm and encode different operational requirements.
  • Fairness: Independence, separation, and sufficiency define distinct group-fairness requirements concerning predictions, errors conditional on outcomes, and predictive information.Equalised odds and equal opportunity are special cases of separation.
  • Proposed framework: findr decomposes the log-odds into an interpretable structured component and an orthogonal neural residual, while penalising Wasserstein distances between sensitive-group score distributions.It also introduces diagnostics for structured contribution, decision agreement, and reliability of coefficient-based interpretations.

2 Related Work

Prior work exposes a joint gap: flexible predictors complicate intrinsic attribution, while fairness methods typically address selected disparities without establishing that explanations describe the deployed decision rule. findr combines identifiable attribution with fairness imposed on the complete score.

  • Research gap: The literature spans interpretable credit-risk models and methods for explaining flexible predictors, alongside statistical fairness criteria and disparity-reducing penalties.These two strands motivate a framework that treats explanation and fairness together.
  • Attribution: Orthogonalisation removes from the neural representation the part explained by structured variables, creating a defined allocation between coefficient effects and residual patterns.Identifiability does not establish that the structured specification captures relevant data structure or that the residual is small.
  • Fairness criteria: Fairness criteria encode different requirements and generally cannot all hold when group base rates differ and prediction is imperfect.Credit-scoring conclusions about fairness, performance, and profitability depend on the selected criterion and mitigation method.
  • Fairness penalties: Mean-based or threshold-based penalties may leave differences in score dispersion, tails, or threshold-crossing shares uncontrolled.Wasserstein penalties instead compare fuller score distributions, distinguishing them from distributionally robust fairness methods.
  • Research gap: Post-hoc explanations can describe a black box without being its decision rule, while intrinsic additive models restrict remaining interaction structure.Semi-structured models restore flexibility but require diagnostics beyond identifiability to show whether the structured term explains decisions.
  • findr: findr applies an in-processing Wasserstein penalty to the complete deployed score and separately diagnoses whether its fair score remains attributable to coefficient effects.Fairness is therefore not imposed only on the structured component or at one decision threshold.

3 Methodology

The methodology represents binary default risk with a structured linear logit plus a neural residual, then enforces orthogonality so shared linear effects are attributed to the structured component. This separation makes coefficient-based and residual contributions distinguishable.

  • Semi-structured logistic model: For binary default modelling, findr decomposes each observation’s total log-odds into structured and unstructured contributions.The structured inputs and neural inputs may coincide, include additional variables, or come from separate information sources.
  • Semi-structured logistic model: The structured contribution is a linear logit with explicit coefficients, while the unstructured contribution is a neural-network residual on the same logit scale.The fitted default probability is obtained by applying the logistic function to the total logit.
  • Identifiability via orthogonalisation: Using the same covariates in both components makes the decomposition non-identifiable because linear effects can be represented by either component.Orthogonalisation addresses this overlap by projecting the neural representation away from the structured design’s linear span.
  • Identifiability via orthogonalisation: The structured design matrix includes an intercept and covariates, and its QR decomposition defines the column space used for projection.The construction assumes full column rank and n ≥ p + 1.
  • Identifiability via orthogonalisation: After projection, encoder features contain no intercept or linear combination of the structured covariates, so linear effects are attributed to coefficients and residual variation lies outside that subspace.The residual-layer intercept can be absorbed into the structured intercept.

3.3 In-processing fairness via Wasserstein distances

findr mitigates fairness during training by comparing Wasserstein distances between predicted-probability distributions across protected groups, optionally conditional on outcomes. This distribution-level penalty targets disparities beyond a single threshold or summary statistic.

  • Motivation and penalty target: Post-hoc threshold corrections may satisfy fairness at one deployment threshold but fail when credit-scoring thresholds change.findr therefore applies its penalty to score distributions themselves during training.
  • Wasserstein fairness: The predicted probability of default is compared across protected groups using a p-Wasserstein distance, optionally conditioning on the true outcome.The empirical analysis uses p = 1.
  • Wasserstein fairness: The discrete Wasserstein distance is the minimum average cost of transporting one empirical score distribution into another.For one-dimensional scores, the ground metric is absolute distance between scores.
  • Wasserstein computation: Because credit scores are one-dimensional, the Wasserstein distance has an exact quantile-matching form without entropic regularisation.This form supports efficient evaluation of the training penalty.
  • Wasserstein computation: The one-dimensional penalty is evaluated exactly by sorting scores and is differentiable almost everywhere, allowing gradients to propagate through training.The formulation is written for a chosen Wasserstein order, with p = 1 used empirically.
  • Fairness criteria: Equalised odds compares score distributions within each outcome group, equal opportunity compares them for Y = 0, and independence compares overall group distributions.The selected fairness criterion determines which conditional score distributions enter the penalty.

3.4 Training objective

The training objective combines predictive fit with a Wasserstein fairness penalty, producing models indexed by a fairness multiplier. A score-level frontier then benchmarks empirical accuracy–fairness trade-offs using validation scores and reference risks.

  • Training objective: The total loss is a weighted combination of predictive fit and a fairness penalty, with λ governing the accuracy–fairness trade-off.Gradients update both structured and neural parameters through orthogonalisation and the Wasserstein distance.
  • Score-level accuracy–fairness frontier: The frontier evaluates how much predictive accuracy is attainable at each value of the Wasserstein unfairness criterion used during training.This creates a common score-level reference for comparing fitted models rather than using only thresholded error-rate frontiers.
  • Score-level accuracy–fairness frontier: Candidate predictors assign one probability score to each validation observation, while realised labels and groups determine the fairness and prediction-loss calculations.The reference risk is the true conditional risk in simulations or an unconstrained training-data model score otherwise.
  • Score-level accuracy–fairness frontier: For equalised odds, unfairness compares empirical score distributions conditional on realised outcome and protected group; the same construction extends to other criteria.Other fairness criteria are handled by replacing the score-distribution penalty with the corresponding empirical functional.
  • Score-level accuracy–fairness frontier: Expected Bernoulli log loss is decomposed into reference-risk entropy plus average KL divergence from the reference risks to candidate scores.The unconstrained optimum is s = r, so the frontier measures expected loss relative to the reference risk surface rather than realised-label sampling noise.
  • Score-level accuracy–fairness frontier: The frontier is a numerical validation-sample benchmark: increasing ξ generally lowers empirical unfairness while increasing expected log loss.It is useful for common-scale model comparison, not a population-optimal boundary for a specified fitted-model family.

3.6 Diagnostics for interpretable-component contribution

The diagnostics assess how closely the structured component represents the full fitted model across score variation, binary decisions, and feature-level directions. Together, they distinguish transparent explanations that remain reliable from cases where neural residuals materially alter the model.

  • Diagnostic overview: The diagnostics measure the structured component’s role in score variation, binary decisions, and feature-level directional interpretations.They compare the structured logit with the full logit and neural residual.
  • Explained Variance Ratio (EVR): EVR measures the share of total logit-scale variation attributable to the structured component.Orthogonalisation makes total logit variance decompose into structured and residual variation.
  • Explained Variance Ratio (EVR): A high EVR indicates that most score variation is explained by the transparent linear component, whereas a low EVR indicates material neural-residual variation.EVR is therefore a continuous-scale measure of how much fitted score variation the structured component captures.
  • Decision Disagreement Rate (DDR): DDR measures whether the structured component and full model produce the same binary decision at the default threshold.DDR equals zero for identical classifications; larger values indicate more disagreements when using the structured component alone.
  • Counterfactual Sign Error Rate (CSER): CSER measures the share of observations where a perturbation deemed risk-increasing by a structured coefficient lowers the full-model logit.It evaluates directional contradictions after adding the neural residual, with reported minimum and maximum values summarising feature-level reliability.

4 Simulation Study

The simulation tests whether findr preserves logistic-regression behaviour for linear signals while using its orthogonal neural residual for nonlinear structure. Varying the Wasserstein penalty traces explicit accuracy–fairness and interpretability trade-offs through score distributions and component diagnostics.

  • Design: The simulation compares logistic regression, a neural model, and findr across simple linear and complex nonlinear data-generating processes.The simple mode contains linear effects, whereas the complex mode adds nonlinear terms and interactions; all settings include two sources of score-distribution unfairness.
  • Design: The models vary the fairness multiplier λ from 0 to 4, repeat configurations over 10 random seeds, and report equalised-odds results.Performance uses out-of-sample AUC and Brier score; fairness uses the equalised-odds Wasserstein penalty, alongside coefficient paths and interpretability diagnostics.
  • Out-of-sample performance: In simple mode, findr behaves like the linear model, while in complex mode it follows the neural model more closely as the residual captures nonlinear signal.As λ increases, performance can decline when fitted scores move toward smaller score-distribution gaps.
  • Fairness: Increasing λ moves model paths toward lower equalised-odds Wasserstein distance, with a mild or stronger predictive trade-off depending on whether the correction aligns with model structure.Figure 3 compares paths with predictor frontiers using fairness distance on the horizontal axis and excess log loss on the vertical axis.
  • Fairness: The penalty changes group-conditional score distributions within outcome-by-group cells rather than adjusting a single decision threshold.Different model classes can reach smaller Wasserstein gaps while retaining distinct final score-distribution shapes.
  • Interpretability: In simple mode, findr’s structured coefficients follow logistic regression, while EVR stays near one, DDR is essentially zero, and CSER is negligible.In complex mode, EVR ranges from about 0.44 to 0.56, DDR becomes nonzero, and the residual carries substantial logit-scale variation while coefficients remain coherent with LR.
  • Interpretability: Increasing λ shrinks the fairness-affected coefficient β1 toward zero while leaving other covariate effects relatively unchanged.The simulation encodes feature-level disparities in X1, linking the coefficient-path change to the fairness intervention.

5 Application

The application evaluates findr against logistic and neural benchmarks on eight public credit datasets, revealing heterogeneous accuracy–fairness and interpretability trade-offs. Findr follows flexible models where nonlinear signal matters and remains close to logistic regression in approximately linear regimes, while diagnostics show when residual information must be examined.

  • Evaluation design: Eight public credit datasets are evaluated using 70/10/20 splits, a λ grid from 0 to 2, and 10 random seeds per configuration.Age is the sensitive attribute, binarised at 25; metrics are evaluated out of sample on the test split.
  • Out-of-sample performance: At λ = 0, findr reaches AUC 0.774 in Taiwan and 0.850 in GMSC, recovering most neural-model gains over logistic regression.The neural model's AUC gains are about 5.5 and 18.7 percentage points in Taiwan and GMSC, respectively.
  • Out-of-sample performance: At λ = 1, findr AUC falls from 0.774 to 0.665 in Taiwan and from 0.641 to 0.513 in Vehicle Loan.Taiwan and Vehicle Loan show the largest predictive cost at high fairness penalties.
  • Accuracy–fairness frontiers: In nonlinear regimes, findr stays close to the flexible model and nearer the accuracy–fairness frontier, while in approximately linear regimes it is usually comparable to simpler alternatives.South German is the main exception, with logistic regression moving more directly toward the low-gap region.
  • Interpretability diagnostics: At λ = 0, EVR is 0.957 in South German, 0.935 in MyHome, 0.981 in Thomas, and 0.946 in PAKDD 2010, with small DDR values.Vehicle Loan and UK Credit have lower EVR values of 0.805 and 0.749, but DDR remains close to zero.
  • Interpretability diagnostics: Taiwan and GMSC have EVR values of 0.527 and 0.399 at λ = 0, showing that neural residuals carry substantial logit-scale variation.Taiwan also shows nonzero DDR and high maximum CSER, whereas GMSC's residual changes continuous scores more than thresholded decisions.

6 Conclusion

The conclusion presents findr as a semi-structured response to the tension among credit-risk performance, interpretability, and fairness. Across simulation and eight datasets, it combines coefficient-based explanations with flexible residual modelling and makes trade-offs and explanation sufficiency more explicit.

  • Motivation: Credit scoring faces a tension between auditable linear coefficients, nonlinear predictive structure, and fairness requirements.Flexible models may not identify which fitted-score component is interpretable or whether structured explanations suffice for decisions.
  • Framework: findr combines a readable linear logit, an orthogonal neural residual, and an in-processing Wasserstein fairness penalty.The structured component could be replaced by richer models such as GAMs while retaining a residual for difficult-to-represent information.
  • Diagnostics: EVR, DDR, and CSER separately assess structured logit variation, binary decision agreement, and local directional consistency.Score-level frontiers benchmark predictive loss associated with lower fairness gaps on the same validation sample.
  • Findings: In simulation, findr behaves like logistic regression for linear signal and uses more residual variation when nonlinearities or interactions are present.The Wasserstein penalty moves scores toward improved fairness, with associated predictive trade-offs described in the study.
  • Findings: Across eight credit datasets, findr recovers much of the neural gain in Taiwan and GMSC while remaining near the linear baseline elsewhere.Fairness costs vary across datasets and evaluation criteria, and diagnostics indicate when coefficients are insufficient for full decision-level explanation.
  • Implications: Semi-structured modelling makes performance, fairness, and interpretability trade-offs more explicit without removing the need to choose fairness criteria or evaluate context.The approach reports how much decision-relevant information remains outside the coefficient component when nonlinear structure matters.
  • Future work: Future work could extend fairness penalties to multi-group, intersectional, and individual-based notions, and support variable selection under explicit interpretability criteria.These are proposed extensions rather than evaluated capabilities of the present framework.

A Computational construction of the predictor frontier

The predictor frontier is constructed by evaluating fairness and log-loss objectives over candidate score vectors, then solving across a penalty grid. The implementation uses empirical CDF gaps, soft-cell fairness directions, candidate retention, and optional refinement.

  • Objective construction: For a score vector s, the code evaluates the fairness objective by exact step-function integration of empirical CDF gaps within realised validation cells.The log-loss objective uses expected Bernoulli log loss under the reference risk rather than realised validation log loss.
  • Solver setup: The default penalty grid is {0} ∪ logspace(−2, 2, 40), with ξ = 0 returned in closed form as s = r.For ξ > 0, the solver starts from the best candidate among reference, fitted-model, and previous-multiplier scores.
  • Fairness direction: The fairness-direction update uses weighted empirical CDFs for outcome–group cells and a clipped effective sign.A positive Θ_i increases observation i's next score, and larger scores represent higher predicted default risk.
  • Selection and refinement: The accepted solution is the best objective value among the initial score, periodic iterates, and the second-half Polyak–Ruppert average.Candidate scores are added before envelope extraction, and an optional refinement pass inserts up to fifteen multipliers at large normalised chord lengths.

B Datasets summary statistics

Table 7 summarizes dataset statistics, including overall default rates and rates by sensitive group A = 0 versus A = 1.

  • Table 7 reports overall default rates and default rates separately for sensitive groups A = 0 and A = 1.

C Hyperparameter search

The empirical experiments use separate discrete random searches for each fairness-penalty value, with validation-based selection and early-stopped training under a fixed optimization budget.

  • Configurations are ranked by their mean validation objective across two optimization restarts.
  • Training is capped at 6,000 epochs, with early stopping after 20 epochs without validation improvement during tuning.
  • All models use AdamW optimization and binary cross-entropy with logits.
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