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Learning Monotonic Probabilities with a Generative Cost Model
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Reformulates strict and implicit monotonic probability modeling as learning a latent cost variable via a generative model (GCM, and IGCM with a latent kernel variable), trained with ELBO-style losses and evaluated on simulated quantile regression and six public datasets.
Problem
Monotonic input-output relationships, both strict and implicit, are hard to enforce in neural networks; construction and regularization methods either restrict architectures or do not guarantee strict monotonicity.
Approach
Treat monotonic probability P(y=1|x,r) as P(c < r) for a latent cost variable c independent of r given x, and model c with a variational generative model (GCM); IGCM adds a latent kernel variable k with a positive projection so that r and y are monotone in k, giving implicit monotonicity.
Key results
- On simulated quantile regression, GCM achieves the lowest MAE at every quantile (e.g., 0.0742 at r=0.5 vs 0.0799 for the next best CMNN).
- IGCM achieves the best AUC on Adult (0.7891) and Diabetes (0.8285) and the best RMSE on Blog Feedback (0.1569) and Auto MPG (3.1000); the paper says IGCM beats GCM on five of six datasets.
- GCM reports up to 72% inference time reduction versus the quickest baseline at 1,024 revenue inputs on COMPAS.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use GCM instead of the strongest listed baseline | Quantile regression · Simulated quantile regression (r=0.1, 0.3, 0.5, 0.7, 0.9) | MAE: 0.1536 -> 0.0961; 0.1096 -> 0.0817; 0.0799 -> 0.0742; 0.1140 -> 0.0792; 0.1379 -> 0.0899 | Helped |
| Use GCM instead of MM | Monotonic classification · Adult | AUC: 0.7791 -> 0.7844 | Helped |
| Use IGCM instead of MM | Implicit monotonic classification · Adult | AUC: 0.7791 -> 0.7891 | Helped |
| Use GCM instead of MM | Monotonic classification · COMPAS | ACC: 0.6976 -> 0.6945 | Hurt |
| Use IGCM instead of MM | Implicit monotonic classification · COMPAS | ACC: 0.6976 -> 0.6934 | Hurt |
| Use IGCM instead of MM | Implicit monotonic classification · Diabetes | AUC: 0.8263 -> 0.8285 | Helped |
| Use IGCM instead of SMM | Implicit monotonic regression · Blog Feedback | RMSE: 0.1579 -> 0.1569 | Helped |
| Use IGCM instead of SMNN | Implicit monotonic classification · Loan Defaulter | ACC: 0.6527 -> 0.6528 | No clear effect |
| Use IGCM instead of PosNN | Implicit monotonic regression · Auto MPG | RMSE: 3.1473 -> 3.1000 | Helped |
| Vary latent dimension D and sample number N (ablation) | GCM · Adult | AUC: 0.7822–0.7857 | Mixed |
| Vary latent dimension D and sample number N (ablation) | IGCM · Adult | AUC: 0.7889–0.7924 | Mixed |
Surprising findings
- The non-strict IGCM outperforms the strictly monotone GCM on five of six public datasets, which the authors attribute to real-world relationships being correlational rather than causal.
Limitations
- GCM and IGCM are most efficient only when many r values share a single x; with few r values, GCM is slower than MM and PWL (about 11.7 ms vs 1.5 ms at r=1).
- IGCM does not guarantee strict monotonicity, and the paper's own results show its non-strict gains on the public datasets.
- Unbounded revenue variables can cause out-of-distribution issues, which the paper handles only with a heuristic bounded-probability hypothesis or a sigmoid transform.
Experimental setup
Datasets: Adult COMPAS Diabetes Blog Feedback Loan Defaulter Auto MPG synthetic quantile regression simulation
Benchmarks: Adult COMPAS Diabetes Blog Feedback Loan Defaulter Auto MPG synthetic quantile regression
Metrics: MAE AUC ACC RMSE inference time
Baselines: PosNN MM SMM CMNN SMNN Hint PWL
Builds on: IWAE variational inference reparameterization trick Min-Max monotonic network monotonicity hints
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