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

Victor Medina-Olivares; Stefan Lessmann; Jonathan Crook

stat.MLAug 25, 2026

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

Authors: Victor Medina-Olivares, Stefan Lessmann, Jonathan Crook

Organizations: University of Edinburgh, Edinburgh, UK · Humboldt-Universität zu Berlin, Berlin, Germany

Abstract

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.

Figures & tables

Figure 2: Out-of-sample performance in the simple and complex simulation settings. Curves show the mean across

10

random seeds as the fairness multiplier

\lambda

varies under equalised odds.

Figure 3: Predictor frontiers and model paths under equalised odds. The panels compare the simple and complex simulation settings. Black curves show the true and estimated predictor frontiers in the fairness–log-loss plane, where the horizontal axis is the equalised-odds Wasserstein distance $\widehat{\Phi}(s;y,a)$ and the vertical axis is the excess log loss relative to the reference-risk entropy $\widehat{H}(r)$ . Coloured lines trace the solutions obtained by each model as the regularisation parameter $\lambda$ varies, and the point colour gradient indicates the value of $\lambda$ . Points closer to the lower-left region correspond to lower fairness violation and lower predictive loss.

Figure 4: Score distributions in the simple simulation setting. The panels show fitted score distributions across groups and outcome strata as

\lambda

varies.

Figure 5: Score distributions in the complex simulation setting. The panels show fitted score distributions across groups and outcome strata as

\lambda

varies.

Figure 6: Estimated structured coefficients in the simple simulation setting. The true signal is linear, and the structured coefficients from findr follow the same coefficient pattern as LR .

$\lambda$	EVR	DDR	CSER (min)	CSER (max)
0.0	1.000 (0.000)	0.000 (0.000)	0.000 (0.000)	0.000 (0.000)
0.1	1.000 (0.000)	0.001 (0.001)	0.000 (0.000)	0.000 (0.000)
0.5	0.997 (0.009)	0.002 (0.004)	0.000 (0.000)	0.002 (0.006)
1.0	0.984 (0.019)	0.001 (0.001)	0.000 (0.000)	0.005 (0.014)

Table 1: Simple mode: Equalised Odds. EVR, DDR, CSER (min) and CSER (max) reported as mean (SD) for the findr model by

\lambda

Figure 7: Estimated structured coefficients in the complex simulation setting. The structured coefficients from findr remain coherent with LR , while the neural residual accounts for nonlinear terms and interactions.

$\lambda$	EVR	DDR	CSER (min)	CSER (max)
0.0	0.556 (0.013)	0.123 (0.004)	0.000 (0.000)	0.038 (0.007)
0.1	0.525 (0.006)	0.120 (0.001)	0.000 (0.000)	0.055 (0.013)
0.5	0.436 (0.016)	0.112 (0.004)	0.000 (0.000)	0.289 (0.032)
1.0	0.469 (0.017)	0.068 (0.003)	0.011 (0.006)	0.424 (0.020)

Table 2: Complex mode: Equalised Odds. EVR, DDR, CSER (min) and CSER (max) reported as mean (SD) for the findr model by

\lambda

Figure 8: Out-of-sample discrimination and calibration across datasets under the equalised-odds penalty. Curves show AUC and Brier score as the fairness multiplier

\lambda

varies. Dashed horizontal lines show the unpenalised CatBoost reference.

	$\lambda=0$
	AUC ( $\uparrow$ )			Brier ( $\downarrow$ )
Dataset	LR	NN	findr	LR	NN	findr
Nonlinear regime
Taiwan	0.722	0.777 (0.001)	0.774 (0.002)	0.144	0.134 (0.000)	0.135 (0.000)
GMSC	0.669	0.856 (0.002)	0.850 (0.003)	0.059	0.050 (0.000)	0.052 (0.000)
Linear regime

Table 3: Out-of-sample AUC and Brier score for the unpenalised models. Bold marks the best result per metric within each row. Datasets are grouped by predictive regime.

	$\lambda=0.1$
	AUC ( $\uparrow$ )			Brier ( $\downarrow$ )
Dataset	LR	NN	findr	LR	NN	findr
Nonlinear regime
Taiwan	0.718	0.772 (0.002)	0.774 (0.001)	0.147	0.136 (0.001)	0.135 (0.000)
GMSC	0.669	0.833 (0.019)	0.849 (0.003)	0.059	0.051 (0.001)	0.052 (0.000)
Linear regime

Table 4: Out-of-sample AUC and Brier score for the three models at

\lambda=0.1

. Bold marks the best result per metric within each row. Datasets are grouped by predictive regime.

Figure 9: Score-level predictor frontiers and fitted model paths for the application datasets under equalised odds. The horizontal axis is the empirical equalised-odds Wasserstein gap, and the vertical axis is excess expected log loss relative to the reference risk. The colour of each point indicates the value of $\lambda$ .

Figure 10: Taiwan Credit Default structured coefficients across seeds under equalised odds for selected values of

\lambda

Figure 11: MyHome Loan structured coefficients across seeds under equalised odds for selected values of

\lambda

Nonlinear regime
	$\lambda=0$
Dataset	EVR	DDR	Min CSER	Max CSER
Taiwan	0.527 (0.008)	0.084 (0.002)	0.088 (0.050)	0.470 (0.046)
GMSC	0.399 (0.034)	0.034 (0.004)	0.018 (0.010)	0.215 (0.195)
Linear regime
South German	0.957 (0.017)	0.030 (0.013)	0.000 (0.000)	0.158 (0.126)

Table 5: EVR, DDR, and CSER for findr at

\lambda=0

under equalised odds. Entries are mean and standard deviation across seeds. Datasets are grouped by predictive regime.

Nonlinear regime
	$\lambda=0.1$
Dataset	EVR	DDR	Min CSER	Max CSER
Taiwan	0.505 (0.009)	0.095 (0.002)	0.039 (0.028)	0.462 (0.034)
GMSC	0.392 (0.019)	0.021 (0.002)	0.011 (0.012)	0.145 (0.040)
Linear regime
South German	0.965 (0.011)	0.033 (0.013)	0.000 (0.000)	0.147 (0.085)

Table 6: EVR, DDR, and CSER for findr at

\lambda=0.1

under equalised odds. Entries are mean and standard deviation across seeds. Datasets are grouped by predictive regime.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	$n$	Default rate (%)	Sensitive share (%)	Default rate $A{=}0$ (%)	Default rate $A{=}1$ (%)
South German	1,000	30.00	19.00	27.16	42.11
Taiwan	30,000	22.12	12.90	21.45	26.66
GMSC	150,000	6.68	2.02	6.59	11.16
Vehicle loan	233,154	21.71	22.84	21.03	24.00
MyHome	7,000	40.00	7.03	39.67	44.31
Thomas	1,225	26.37	7.67	24.49	48.94

Appendix

Table 7: Dataset summary statistics. Default rates are reported overall and by sensitive group (

A=0

vs.

A=1

Hyperparameter	Candidate values
Learning rate	$\{3\!\times\!10^{-4},10^{-3},3\!\times\!10^{-3},10^{-2}\}$
Weight decay	$\{0,10^{-5},10^{-4},10^{-3}\}$
Batch size	$\{32,64,128,256\}$
Hidden-layer widths	$\{[4],[16,8],[64,32],[128,64]\}$
Activation	{ReLU, GELU, tanh}
Dropout rate	$\{0,0.1,0.25\}$

Appendix

Table 8: Hyperparameter search space.

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