Conformal-DRO: Distributionally Robust Optimization with Conformalized Ambiguity Set
Luhao Zhang, Shixiang Zhu
math.OC, stat.ML
2026-09-10
Nested conformal sets cover the next latent law in finite samples. Newsvendor law coverage is 0.95 vs 0.345 for mixture WDRO; K=8 regret is 1.319 vs 0.848 for Mixture SP.
A decision maker sees context \(x\), picks an action \(z\), and only then observes the outcome \(y\). Data-driven distributionally robust optimization (DRO) hedges by taking the worst expected cost over a set of plausible laws, rather than trusting a single fitted model. The usual construction treats the relevant conditional law as fixed. The ambiguity set is there to cover sampling error, and it is supposed to shrink as more data arrives.
That story fails when each instance has its own latent law. Two patients with the same chart can have different response distributions because a subtype was never recorded. Each instance contributes one outcome, so pooling estimates the mixture law, not the law of the next case. Even a perfectly known mixture leaves the next latent law random. An ambiguity set that tightens around the mixture gets better at describing the average patient and can still miss the patient in front of you.
Conformal-DRO starts from split conformal prediction. Under exchangeability, a set \(\mathcal{C}\gamma(x)\) contains the next outcome with probability at least \(1-\gamma\). For a candidate latent law \(Q\), the mass it places outside \(\mathcal{C}\gamma\) is its leakage. The expected leakage equals the observable miscoverage rate, hence is at most \(\gamma\). Dividing by \(\gamma\) yields a conformal incompatibility score whose expectation is at most 1: an e-statistic.
One level only splits "inside the set" from "outside". Nested conformal regions cut the outcome space into shells, like onion layers. The innermost shell sits in every set and pays penalty 0; outer shells look less like the calibration sample and pay more. The aggregation rule \(\nu\) puts weights on the \(\gamma\) levels and sets the relative cost of moving mass between shells, which is the transport geometry. The latent-law miscoverage level \(\alpha\) sets the radius \(1/\alpha\). Markov's inequality turns the moment bound into containment: laws whose incompatibility is at most \(1/\alpha\) cover the future latent law with probability at least \(1-\alpha\). Historical latent laws and the mixing kernel are never estimated.
The inner worst-case problem then collapses. The adversary only allocates probability across shells and, inside each shell, piles mass on the outcome with the largest loss. The constraints are a simplex plus one linear incompatibility budget. An extreme point has at most two positive coordinates, so an optimal adversarial law is supported on at most two shells. The dual is a one-dimensional convex minimization over a multiplier \(\lambda\). \(\nu\) can be chosen jointly with \(z\) on a held-out tuning sample, with budget \(K\) on how many conformal levels are used. Final calibration must stay independent of that tuning step, or the finite-sample cover fails.
In a covariate-free synthetic study with \(\alpha=0.2\) and 500 tuning plus 500 calibration points, the learned shells recover the shape of a population-optimal geometry. Raising \(K\) cuts the robust value quickly, then flattens. At \(K=100\) the gap to the population oracle is about 0.054 for a Gaussian and 0.176 for Student-\(t3\); heavy tails make extreme quantiles noisier.
The sharper test is a two-product newsvendor with a hidden regime. Context \(X\) is uniform on \([-1,1]^2\). An unobserved event \(W\sim\mathrm{Bernoulli}(p(x))\) with marginal probability about 19% switches the demand law. Baselines are handed the exact mixture \(Qx\), so the comparison isolates target mismatch from estimation error. At \(\alpha=0.2\), heterogeneity \(\delta=2\), ten repeats, discrete demand:
| Method | Regret | Law coverage | Certificate coverage |
| Mixture SP | 0.848 | n/a (no law-level set) | 0.804 |
| Mixture WDRO | 0.853 | 0.345 | 0.849 |
| Conformal-DRO, \(K=1\) | 1.616 | 0.927 | 1.000 |
| Conformal-DRO, \(K=4\) | 1.338 | 0.946 | 1.000 |
| Conformal-DRO, \(K=8\) | 1.319 | 0.945 | 1.000 |
Mixture methods win on mean regret and lose on protection: the Wasserstein ball covers 34.5% of realized laws. Conformal-DRO covers about 0.95 of them, above the nominal 0.8, with certificate coverage 1.0. Raising \(K\) from 1 to 8 drops regret from 1.616 to 1.319 (18.4%) with coverage essentially unchanged. In the same discrete design, mean cost at \(K=8\) is 4.727 versus 4.256 for Mixture SP; P90 cost is 8.040 versus 10.072. The certificate is loose: 19.665 on average, far above realized cost. When \(\delta=0\) and the hierarchy collapses, costs of all methods sit together, so the gap is the mismatched target, not a built-in conformal advantage. A continuous truncated-Gaussian design points the same way.
For contextual decisions where the conditional law may still vary across instances, this is an ambiguity set that never estimates a mixing prior or any individual latent law. Geometry comes from the conformal path in the data; the radius is the chosen miscoverage level. The infinite-dimensional DRO problem becomes a linear program over shells, and the adversary is sparse.
This is protection bought with average efficiency. Mixture methods still spend less on the mean. Conformal-DRO buys a finite-sample certificate for the next random law. The certificate is large, so \(\alpha\) and \(K\) are operating knobs.
The coverage guarantee is marginal: it averages over calibration data, context, and the future latent law. It is not conditional on \(x\). Exchangeability is a hard assumption; drift or policy interventions break calibration. Markov's step is loose. Empirical law coverage near 0.95 against a nominal 0.8, and a mean certificate about four times realized cost, make the robust value a usable upper bound and a slack one.
Regional losses must be finite; an unbounded outer shell can send the robust risk to infinity. Those regional suprema have to be computed, which is unpleasant in continuous outcome spaces. The aggregation rule \(\nu\) must be chosen independently of final calibration.
All experiments are synthetic newsvendors, and the baselines receive the exact \(Qx\). That isolates the target mismatch cleanly. It does not travel to real clinical or scheduling data where estimation error and score error show up together. The appendix also notes that the geometry-only synthetic code path does not compute decision regret or aggregate coverage across trials.