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The Bias-Variance Tradeoff in Data-Driven Optimization: A Local Misspecification Perspective
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Analyzes SAA, ETO and IEO for data-driven stochastic optimization under local misspecification (contiguity theory), deriving bias-variance tradeoffs that depend on misspecification amount and direction, validated on a newsvendor simulation.
Problem
Relative statistical performance (bias, variance, regret) of SAA versus model-based estimate-then-optimize (ETO) and integrated estimation-optimization (IEO) when the parametric model is only nearly well-specified, a regime previously unexplored beyond well-/mis-specification dichotomy.
Approach
Models misspecification as local perturbations Q_t with t = Theta(1/n^alpha), so the regime (mild, balanced, severe) sets whether bias, variance, or both dominate; uses contiguity/Le Cam's third lemma and influence functions to derive explicit bias and variance expressions for each method and identifies misspecification directions (score-function span) with zero bias.
Key results
- Under balanced misspecification (alpha=1/2), SAA has zero bias while ETO has the smallest variance and the largest bias, with IEO in between on both, establishing a bias-variance tradeoff.
- Under severe misspecification, bias ordering is SAA (zero) ≤ IEO ≤ ETO, with variance asymptotically zero for all methods, so regret is governed by bias.
- Under mild misspecification, bias vanishes for all methods and regret ordering reverses to ETO best, then IEO, then SAA, generalizing the well-specified result.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use ETO instead of IEO or SAA under mild misspecification (α>1/2) | ETO, IEO, and SAA · Mild misspecification | regret | Helped |
| Use SAA instead of IEO or ETO under severe misspecification (0<α<1/2) | SAA, IEO, and ETO · Severe misspecification | regret | Helped |
| Use SAA instead of IEO or ETO under severe misspecification | SAA, IEO, and ETO · Severe misspecification | bias | Helped |
| Use ETO instead of IEO or SAA under mild misspecification | ETO, IEO, and SAA · Mild misspecification | variance | Helped |
| Use SAA instead of IEO or ETO under balanced misspecification (α=1/2) | SAA, IEO, and ETO · Balanced misspecification | bias | Helped |
| Use ETO instead of IEO or SAA under balanced misspecification | ETO, IEO, and SAA · Balanced misspecification | variance | Helped |
| Use IEO instead of ETO or SAA | IEO, ETO, and SAA · Balanced misspecification; newsvendor experiments | regret | Helped |
| Use ETO instead of IEO or SAA | ETO, IEO, and SAA · Mild misspecification; newsvendor experiments | regret | Helped |
| Use SAA instead of IEO or ETO | SAA, IEO, and ETO · Severe misspecification; newsvendor experiments | regret | Helped |
| Choose a misspecification direction in the linear span of the model score function | ETO, IEO, and SAA · Local misspecification | bias | Helped |
Surprising findings
- Misspecification along directions in the span of the parametric score function yields zero decision bias for all three methods, even though the model is misspecified.
- In the balanced regime, IEO attained the lowest regret in the newsvendor experiments, despite having no uniform theoretical regret ordering.
Limitations
- Results are asymptotic in the local regime and rely on classical smoothness, interchangeability, and M-estimation assumptions.
- Analysis limited to unconstrained, non-contextual problems; extension to contextual or constrained optimization left as future work.
- Severe-regime analysis requires a novel technique because contiguity tools are unavailable there; the experiments are limited to a single newsvendor setting.
Experimental setup
Metrics: regret bias variance
Baselines: SAA ETO IEO
Builds on: Le Cam's third lemma integrated estimation-optimization (IEO) estimate-then-optimize (ETO)
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