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Decision Making under the Exponential Family: Distributionally Robust Optimisation with Bayesian Ambiguity Sets
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Proposes DRO with Bayesian ambiguity sets (DRO-BAS), two KL-ball ambiguity sets built from the posterior predictive (PP) or posterior expectation (PE), giving single-stage dual programs solved by sample average approximation, and compares against Bayesian DRO on Newsvendor and Portfolio problems.
Problem
Decision making under uncertainty when the data-generating process is unknown and Bayesian posterior risk minimisation inherits estimation error (optimiser's curse), so decisions can be overly optimistic under limited or noisy data.
Approach
Builds worst-case (not expected worst-case) KL-divergence ambiguity sets centred on the posterior predictive (DRO-BAS-PP) or posterior-expected KL (DRO-BAS-PE); for exponential family models with conjugate priors the PE set reduces to a KL ball around the posterior-mean model, yielding a closed-form or single-stage dual.
Key results
- On the Newsvendor problem, DRO-BAS-PE and DRO-BAS-PP Pareto dominate BDRO in out-of-sample mean-variance for well-specified multivariate normal DGP, and DRO-BAS-PE dominates in the misspecified truncated normal case for M=25 and 100 samples.
- On the Portfolio problem, average solve time is 0.012 s for DRO-BAS-PE and 0.213 s for DRO-BAS-PP versus 3.381 s for BDRO (the table reports 1.34 s and 4.47 s, inconsistently with the text).
- DRO-BAS-PE has a closed-form dual for linear objectives with Gaussian likelihood, requiring no sampling.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use DRO-BAS-PE instead of BDRO | Portfolio problem · Portfolio problem | out-of-sample mean-variance tradeoff | No clear effect |
| Use DRO-BAS-PE instead of BDRO | Portfolio problem · Portfolio problem | solve time: 0.01 (0.01) vs 4.47 (0.62) | Helped |
| Use DRO-BAS-PP instead of BDRO | Portfolio problem · Portfolio problem | solve time: 1.34 (0.22) vs 4.47 (0.62) | Helped |
| Use DRO-BAS-PE instead of BDRO | Portfolio problem · Portfolio problem | sample time: N.A. vs 44.45 (3.11) | Helped |
| Use DRO-BAS-PP instead of BDRO | Portfolio problem · Portfolio problem | sample time: 3.48 (0.91) vs 44.45 (3.11) | Helped |
| Use DRO-BAS-PE instead of DRO-BAS-PP | Portfolio problem · Portfolio problem | out-of-sample mean-variance tradeoff | Helped |
| Use DRO-BAS-PE instead of BDRO | Portfolio problem · Portfolio problem | cumulative return | No clear effect |
| Use DRO-BAS instead of BDRO with M=25 or M=100 | Newsvendor problem, well-specified multivariate normal DGP · Newsvendor problem | out-of-sample mean-variance tradeoff | Helped |
| Use DRO-BAS instead of BDRO with M=25 or M=100 | Newsvendor problem, well-specified exponential DGP · Newsvendor problem | out-of-sample mean-variance tradeoff: ε_BDRO ≥ 0.5 | Mixed |
| Use DRO-BAS-PE instead of BDRO and DRO-BAS-PP with M=25 or M=100 | Newsvendor problem, truncated normal DGP with normal likelihood · Newsvendor problem | out-of-sample mean-variance tradeoff | Helped |
| Increase the sample size to M=900 | Newsvendor problem, truncated normal DGP with normal likelihood · Newsvendor problem | out-of-sample mean-variance tradeoff: M=900 | No clear effect |
Surprising findings
- DRO-BAS-PP is Pareto dominated by BDRO and DRO-BAS-PE on the Portfolio problem, despite needing 3600 posterior predictive samples.
Limitations
- DRO-BAS-PP requires a finite MGF of the objective under the Student-t posterior predictive, which fails for some exponential family cases, so no exact dual exists.
- Relies on the posterior, so it is vulnerable to severe model misspecification.
- DRO-BAS-PE's dual is derived only for exponential family models with conjugate priors.
Experimental setup
Datasets: DowJones weekly stock returns (28 stocks, 1363 weeks) synthetic exponential, multivariate normal and truncated normal DGPs
Benchmarks: Newsvendor problem Portfolio problem (DowJones)
Metrics: out-of-sample mean-variance tradeoff solve time cumulative return
Baselines: BDRO BRO PDRO
Builds on: Bayesian distributionally robust optimisation (BDRO) Bayesian risk optimisation KL-based distributionally robust optimisation
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