Sample Robust Optimization: Methods and Applications

• Sample Robust Optimization is a data-driven framework that constructs uncertainty sets from observed samples to enable robust decision-making under stochastic variations.
• It encompasses variants like learning-based, two-stage, and unknown-objective SRO that address different challenges in robust optimization using sample information.
• Practical applications in areas such as energy systems, nurse redeployment, and autonomous driving demonstrate SRO’s ability to improve feasibility and performance compared to traditional methods.

Searching arXiv for recent and foundational papers on Sample Robust Optimization and related usages of the term. Sample Robust Optimization (SRO) denotes a family of data-driven robust optimization methodologies in which robustness is constructed from samples rather than from a fully specified objective or a fixed ex ante uncertainty set. The term does not have a single canonical meaning across the literature. In one strand, SRO learns uncertainty or prediction sets from i.i.d. data and uses them in robust counterparts of chance-constrained or parameterized programs; in another, it replaces a stochastic objective by the average of per-sample worst-case recourse costs over neighborhoods centered at observed scenarios; in a third, it addresses robust optimization when the objective itself is unknown and must be learned from noisy evaluations; and, in recent generative-model work, it has been extended to robustness against perturbations of the sampler induced by a learned generator (Hong et al., 2017, Bertsimas et al., 2019, Sessa et al., 2020, Zhang et al., 30 Apr 2026).

1. Conceptual scope and terminological usage

A concise way to organize the literature is by the object that is made robust from samples.

Usage	Robust object	Representative papers
Learning-based RO	Uncertainty or prediction set learned from data	(Tulabandhula et al., 2014, Hong et al., 2017)
Two-stage SRO	Sample-centered local uncertainty sets around observed scenarios	(Bertsimas et al., 2019, Liu et al., 2023, Liu et al., 9 Sep 2025)
Unknown-objective SRO	Unknown reward function learned from noisy samples	(Sessa et al., 2020)
Sampler-first SRO	Learned generator or sampler, perturbed in parameter space	(Zhang et al., 30 Apr 2026)

In learning-based robust optimization, the central task is to construct a set $U$ or $\mathcal{U}(D)$ from finite data so that a robust solution is feasible for the unknown future parameter with prescribed confidence. Tulabandhula and Rudin formulate the robust counterpart as

$$\min_\pi \max_{u \in U} f(\pi,u)\;\;\; \textrm{s.t.}\;\; F(\pi,u) \in K \textrm{ for all } u \in U,$$

and make the uncertainty set itself the statistical object to be learned from data (Tulabandhula et al., 2014). Bertsimas, Gupta, and Kallus place the same idea in a chance-constrained framework by constructing a $(1-\epsilon)$-content prediction set $\mathcal{U}(D)$ with confidence $1-\delta$, then solving the robust approximation over that set (Hong et al., 2017).

In two-stage SRO, the operative object is a neighborhood around each observed sample rather than a single global set. Bertsimas, Shtern, and Sturt define

$$\mathcal{U}_N^i \triangleq \{ \xi \in \Xi : \|\xi - \hat \xi^i\| \le \varepsilon_N \},$$

and optimize the average of sample-wise worst-case recourse costs. They further show that this formulation is equivalent to a two-stage DRO model with a type-$\infty$ Wasserstein ambiguity set centered at the empirical distribution (Bertsimas et al., 2019).

A different use of the term appears when the objective function is unknown. In "Mixed Strategies for Robust Optimization of Unknown Objectives" (Sessa et al., 2020), SRO refers to robust decision making when $f(x,\theta)$ must be learned from noisy point evaluations. The algorithm GP-MRO then seeks a robust mixed strategy over actions $x$.

A further extension appears in "Sampler-Robust Optimization under Generative Models" (Zhang et al., 30 Apr 2026). There the operational object of uncertainty is no longer an explicit probability law but the sampler induced by a learned generator $\mathcal{U}(D)$0. Robustness is imposed directly at the sampler level through perturbations of generator parameters. This suggests that, in current usage, SRO is best treated as an umbrella label for sample-driven robustification rather than a single formalism.

2. Learning uncertainty sets from data

A foundational formulation starts from the chance-constrained program

$\mathcal{U}(D)$1

and replaces it by the robust counterpart

$\mathcal{U}(D)$2

If $\mathcal{U}(D)$3 is a $\mathcal{U}(D)$4-content set, any feasible solution of the robust problem is feasible for the chance-constrained problem. The learning-based SRO question is therefore how to infer such a set from data with finite-sample guarantees (Hong et al., 2017).

The split-sample procedure of Bertsimas, Gupta, and Kallus makes this explicit. Data $\mathcal{U}(D)$5 are partitioned into $\mathcal{U}(D)$6 and $\mathcal{U}(D)$7. Phase 1 learns a tractable geometric shape $\mathcal{U}(D)$8 from $\mathcal{U}(D)$9, where $$\min_\pi \max_{u \in U} f(\pi,u)\;\;\; \textrm{s.t.}\;\; F(\pi,u) \in K \textrm{ for all } u \in U,$$0 is a scalar transformation associated with an ellipsoid, polytope, union, or intersection. Phase 2 calibrates the level $$\min_\pi \max_{u \in U} f(\pi,u)\;\;\; \textrm{s.t.}\;\; F(\pi,u) \in K \textrm{ for all } u \in U,$$1 from the order statistics of $$\min_\pi \max_{u \in U} f(\pi,u)\;\;\; \textrm{s.t.}\;\; F(\pi,u) \in K \textrm{ for all } u \in U,$$2 on $$\min_\pi \max_{u \in U} f(\pi,u)\;\;\; \textrm{s.t.}\;\; F(\pi,u) \in K \textrm{ for all } u \in U,$$3. If

$$\min_\pi \max_{u \in U} f(\pi,u)\;\;\; \textrm{s.t.}\;\; F(\pi,u) \in K \textrm{ for all } u \in U,$$4

and $$\min_\pi \max_{u \in U} f(\pi,u)\;\;\; \textrm{s.t.}\;\; F(\pi,u) \in K \textrm{ for all } u \in U,$$5, then

$$\min_\pi \max_{u \in U} f(\pi,u)\;\;\; \textrm{s.t.}\;\; F(\pi,u) \in K \textrm{ for all } u \in U,$$6

The sample-size requirement $$\min_\pi \max_{u \in U} f(\pi,u)\;\;\; \textrm{s.t.}\;\; F(\pi,u) \in K \textrm{ for all } u \in U,$$7 is independent of the dimensions of both the decision space and the probability space governing the stochasticity (Hong et al., 2017).

The geometry is chosen to preserve tractability. For ellipsoidal sets, the robust linear constraint admits the SOCP reformulation

$$\min_\pi \max_{u \in U} f(\pi,u)\;\;\; \textrm{s.t.}\;\; F(\pi,u) \in K \textrm{ for all } u \in U,$$8

For polyhedral uncertainty $$\min_\pi \max_{u \in U} f(\pi,u)\;\;\; \textrm{s.t.}\;\; F(\pi,u) \in K \textrm{ for all } u \in U,$$9, the robust counterpart becomes an LP through dual variables $(1-\epsilon)$0 satisfying

$(1-\epsilon)$1

Unions and intersections can also be handled without leaving the robust-optimization framework (Hong et al., 2017).

Tulabandhula and Rudin extend the same philosophy to supervised-learning-derived sets. One route learns a class of set functions $(1-\epsilon)$2 by minimizing the empirical miscoverage

$(1-\epsilon)$3

then defines

$(1-\epsilon)$4

Other routes construct $(1-\epsilon)$5 from conditional quantile regression, from a single set of “good” predictors $(1-\epsilon)$6 enlarged by a residual set $(1-\epsilon)$7, or from two sets of “good” quantile models $(1-\epsilon)$8 and $(1-\epsilon)$9 with quantile-deviation sets $\mathcal{U}(D)$0. The probabilistic guarantees are driven by Rademacher averages, McDiarmid’s inequality, Hoeffding’s inequality, and Ledoux–Talagrand contraction, yielding out-of-sample feasibility bounds that depend on hypothesis-class complexity rather than on a parametric model of the data-generating process (Tulabandhula et al., 2014).

A more specialized calibration problem arises when the uncertainty set is ellipsoidal and the aim is to select its scale as tightly as possible. In "Tightly Robust Optimization via Empirical Domain Reduction" (Yabe et al., 2020), the nominal parameterized problem has constraints $\mathcal{U}(D)$1 that are linear in $\mathcal{U}(D)$2, and the robust counterpart uses

$\mathcal{U}(D)$3

The standard SRO calibration chooses $\mathcal{U}(D)$4 so that $\mathcal{U}(D)$5 with probability at least $\mathcal{U}(D)$6, which asymptotically yields

$\mathcal{U}(D)$7

and therefore $\mathcal{U}(D)$8. The paper replaces this by a reduced-domain calibration based on the set

$\mathcal{U}(D)$9

and proves an asymptotic rate $1-\delta$0 with

$1-\delta$1

The associated guarantee is that

$1-\delta$2

The contrast with global ellipsoidal calibration is one of the clearest examples of SRO reducing conservatism by exploiting the effective active subspace of the optimization problem (Yabe et al., 2020).

3. Sample-centered two-stage and multistage formulations

The two-stage SRO formulation of Bertsimas, Shtern, and Sturt starts from the stochastic linear program

$1-\delta$3

with recourse function

$1-\delta$4

and replaces the expectation by the average of local worst cases: $1-\delta$5 This formulation interpolates between SAA and classical RO: SAA is recovered when $1-\delta$6, while a single uncertainty set corresponds to the robust-optimization limit. The same paper shows the equivalence to type-$1-\delta$7 Wasserstein DRO and introduces overlapping linear decision rules, assigning a local affine recourse policy $1-\delta$8 to each $1-\delta$9. The value hierarchy

$$\mathcal{U}_N^i \triangleq \{ \xi \in \Xi : \|\xi - \hat \xi^i\| \le \varepsilon_N \},$$0

formalizes the advantage of multi-policy over single-policy approximation, and under assumptions including $$\mathcal{U}_N^i \triangleq \{ \xi \in \Xi : \|\xi - \hat \xi^i\| \le \varepsilon_N \},$$1 the paper proves

$$\mathcal{U}_N^i \triangleq \{ \xi \in \Xi : \|\xi - \hat \xi^i\| \le \varepsilon_N \},$$2

with accumulation points of optimal first-stage decisions being almost surely optimal for the underlying stochastic problem (Bertsimas et al., 2019).

This sample-centered viewpoint has been adopted in application-driven models. In the integrated electricity–gas system scheduling problem, a two-stage SRO model is defined over historical wind samples $$\mathcal{U}_N^i \triangleq \{ \xi \in \Xi : \|\xi - \hat \xi^i\| \le \varepsilon_N \},$$3 by constructing sample-centered polyhedral sets

$$\mathcal{U}_N^i \triangleq \{ \xi \in \Xi : \|\xi - \hat \xi^i\| \le \varepsilon_N \},$$4

The original tri-level min–max–min UC–OEF model is simplified by linear decision rules for generator outputs, gas well outputs, and free-node pressures, then transformed through extreme-point reduction, inactive thermal limit elimination, and duality-based reformulation into a single-level MILP. The paper reports that on IEGS-6-7, SAA with 100 samples yields $$\mathcal{U}_N^i \triangleq \{ \xi \in \Xi : \|\xi - \hat \xi^i\| \le \varepsilon_N \},$$5 out-of-sample feasibility and $$\mathcal{U}_N^i \triangleq \{ \xi \in \Xi : \|\xi - \hat \xi^i\| \le \varepsilon_N \},$$6 with 1000 samples, while the SRO model achieves $$\mathcal{U}_N^i \triangleq \{ \xi \in \Xi : \|\xi - \hat \xi^i\| \le \varepsilon_N \},$$7–$$\mathcal{U}_N^i \triangleq \{ \xi \in \Xi : \|\xi - \hat \xi^i\| \le \varepsilon_N \},$$8 feasibility with just 100 samples by choosing $$\mathcal{U}_N^i \triangleq \{ \xi \in \Xi : \|\xi - \hat \xi^i\| \le \varepsilon_N \},$$9–$\infty$0. On IEGS-118-20, SRO reaches $\infty$1, $\infty$2, and $\infty$3 feasibility for $\infty$4, respectively, whereas SAA achieves $\infty$5 and RO is more conservative (Liu et al., 2023).

A multistage extension appears in dynamic nurse redeployment across hospitals. There the uncertainty is a demand trajectory $\infty$6, and for each observed sample path $\infty$7 the ambiguity set is an entrywise infinity ball,

$\infty$8

The SRO objective combines planned redeployment costs with the average of per-sample worst-case deployment costs, while adaptive decisions are approximated by affine decision rules in the observed demand history. Under $\infty$9-ball uncertainty, the positive-part terms admit exact linearization, and the semi-infinite robust LP becomes a finite LP by dualization. The implemented policy uses a rolling horizon, so only the first daily decision is executed after observing demand. Empirically, the paper reports that full connectivity reduces weekly average cost, redeployments, and travel distance relative to hub-and-spoke under baseline secondment settings, with FC SAA $f(x,\theta)$0 versus HS SAA $f(x,\theta)$1, and FC SRO $f(x,\theta)$2 versus HS SRO $f(x,\theta)$3. It also reports that SRO outperforms the traditional sample-average method in the presence of demand surges or under-forecasts by better anticipating emergency redeployments (Liu et al., 9 Sep 2025).

4. Unknown-objective robust learning and mixed strategies

In "Mixed Strategies for Robust Optimization of Unknown Objectives" (Sessa et al., 2020), SRO addresses a different problem class: the objective $f(x,\theta)$4 is unknown and must be learned from samples. The setting is a compact decision set $f(x,\theta)$5, a finite uncertainty set $f(x,\theta)$6, and a bounded reward function $f(x,\theta)$7 that can be queried through noisy point evaluations

$f(x,\theta)$8

At deployment, $f(x,\theta)$9 is uncontrollable and may be chosen adversarially. The goal is therefore not to optimize a nominal average, but to learn a mixed strategy $x$0 that maximizes the worst-case expected reward

$x$1

This contrasts with deterministic robust optimization,

$x$2

and the paper emphasizes that $x$3 and can be strictly larger, sometimes arbitrarily so.

The algorithm GP-MRO places a Gaussian process prior on $x$4, assumes $x$5 belongs to the RKHS $x$6 with $x$7, and uses posterior mean and variance

$x$8

$x$9

together with UCB/LCB confidence bounds. For bounded rewards, the bounds are truncated to $\mathcal{U}(D)$00. A kernel-dependent complexity term,

$\mathcal{U}(D)$01

governs the learning rate.

Algorithmically, GP-MRO simulates a zero-sum game between a learner choosing $\mathcal{U}(D)$02 and an adversary choosing $\mathcal{U}(D)$03. The adversary distribution $\mathcal{U}(D)$04 over $\mathcal{U}(D)$05 is updated by multiplicative weights using optimistic losses derived from GP-UCBs; the learner then chooses

$\mathcal{U}(D)$06

and queries the most uncertain adversarial state,

$\mathcal{U}(D)$07

After $\mathcal{U}(D)$08 rounds, the algorithm returns the uniform distribution over the queried actions,

$\mathcal{U}(D)$09

The main guarantee is a finite-sample, high-probability bound relative to the robust mixed-strategy optimum. If

$\mathcal{U}(D)$10

then, with probability at least $\mathcal{U}(D)$11,

$\mathcal{U}(D)$12

Equivalently, the optimality gap decreases as

$\mathcal{U}(D)$13

rather than as a cumulative-regret quantity. For squared exponential kernels on $\mathcal{U}(D)$14, the paper gives $\mathcal{U}(D)$15, which yields a polylogarithmic kernel-complexity contribution.

The empirical results reinforce the theoretical distinction between deterministic and randomized robustness. On synthetic functions and a robust polynomial task, GP-MRO concentrates probability on extremal decisions that hedge the worst-case $\mathcal{U}(D)$16 and outperforms deterministic robust baselines such as StableOpt. In an autonomous-vehicle overtaking scenario, the mixed policies randomize over left and right maneuvers, whereas deterministic max–min strategies brake behind the human-driven vehicle; the paper concludes that deterministic robust strategies can be overly conservative, while the mixed strategies found by GP-MRO significantly improve the overall performance (Sessa et al., 2020).

5. Sampler-first robustness under generative models

Recent work generalizes SRO from sample-centered neighborhoods in observation space to neighborhoods in generator-parameter space. In "Sampler-Robust Optimization under Generative Models" (Zhang et al., 30 Apr 2026), the context space is $\mathcal{U}(D)$17, the outcome space is $\mathcal{U}(D)$18, the decision set is $\mathcal{U}(D)$19, and uncertainty is represented by a conditional generator

$\mathcal{U}(D)$20

A nominal parameter $\mathcal{U}(D)$21 is learned from historical data, but downstream decisions are evaluated through Monte Carlo scenarios rather than through a tractable closed-form law. The paper therefore shifts the operational object of uncertainty from an explicit distribution $\mathcal{U}(D)$22 to the sampler $\mathcal{U}(D)$23 induced by $\mathcal{U}(D)$24.

The robust objective is defined directly over a parameter ball

$\mathcal{U}(D)$25

The nominal population objective is

$\mathcal{U}(D)$26

while the robust population and empirical objectives are

$\mathcal{U}(D)$27

$\mathcal{U}(D)$28

The SRO decision is

$\mathcal{U}(D)$29

A key design choice is shared-seed coupling: the same latent batch $\mathcal{U}(D)$30 is used in both the inner maximization over $\mathcal{U}(D)$31 and the outer minimization over $\mathcal{U}(D)$32. The resulting fixed-batch objective,

$\mathcal{U}(D)$33

reduces variance and provides a fair comparison against the nominal sample-average baseline.

The paper also gives a sharpness-aware decomposition. Defining the empirical sharpness

$\mathcal{U}(D)$34

SRO can be written as

$\mathcal{U}(D)$35

This formulation selects decisions whose empirical performance is stable under nearby generator perturbations.

Under a coverage condition stating that there exists $\mathcal{U}(D)$36 such that $\mathcal{U}(D)$37, together with Lipschitz assumptions on $\mathcal{U}(D)$38 and $\mathcal{U}(D)$39, the paper proves a one-sided reliability certificate. With probability at least $\mathcal{U}(D)$40, simultaneously for all $\mathcal{U}(D)$41,

$\mathcal{U}(D)$42

The correction term depends on the decision-class complexity through $\mathcal{U}(D)$43, not on the generator-parameter dimension or the complexity of $\mathcal{U}(D)$44. The paper interprets $\mathcal{U}(D)$45 as a slack induced by robustification that can partially absorb finite-simulation error.

Two minimax solvers are proposed. For small $\mathcal{U}(D)$46, a first-order worst-case sampler refinement linearizes the inner maximization at $\mathcal{U}(D)$47 and uses the dual-norm optimizer

$\mathcal{U}(D)$48

with

$\mathcal{U}(D)$49

For moderate or large $\mathcal{U}(D)$50, a two-timescale alternating minimax method performs slow projected ascent in $\mathcal{U}(D)$51 and fast projected descent in $\mathcal{U}(D)$52, with $\mathcal{U}(D)$53.

The portfolio experiments use a conditional LSTM-GAN with latent noise dimension $\mathcal{U}(D)$54 and hidden dimension $\mathcal{U}(D)$55, a long-only simplex decision set, and quadratic utility with $\mathcal{U}(D)$56. In a controlled generator-to-generator experiment, the paper reports empirical utility $\mathcal{U}(D)$57 for SRO versus $\mathcal{U}(D)$58 for nominal optimization, but oracle utility $\mathcal{U}(D)$59 for SRO versus $\mathcal{U}(D)$60 for nominal, with empirical-to-oracle gap $\mathcal{U}(D)$61 for SRO and $\mathcal{U}(D)$62 for nominal. On the 100-day out-of-sample test, SRO improves Sharpe from $\mathcal{U}(D)$63 to $\mathcal{U}(D)$64, improves $\mathcal{U}(D)$65 from $\mathcal{U}(D)$66 to $\mathcal{U}(D)$67, and reduces maximum drawdown from $\mathcal{U}(D)$68 to $\mathcal{U}(D)$69. In the real-data experiment on random 10-stock pools, SRO again improves mean return, standard deviation, Sharpe, $\mathcal{U}(D)$70, and maximum drawdown relative to nominal ERM (Zhang et al., 30 Apr 2026).

6. Relations to SAA, RO, DRO, and principal limitations

Across its variants, SRO is consistently defined in opposition to two simpler baselines. Sample Average Approximation replaces unknown expectations by empirical averages and optimizes nominally. Classical robust optimization fixes a global uncertainty set ex ante and enforces worst-case feasibility or performance over that set. SRO instead learns or constructs the robust object from data: a prediction set, a sample-centered neighborhood, an empirical reduced domain, a set of good predictors, a local Wasserstein ball around each observation, a mixed strategy against an adversarial latent parameter, or a perturbation set around a learned generator (Hong et al., 2017, Bertsimas et al., 2019, Sessa et al., 2020, Zhang et al., 30 Apr 2026).

The relation to DRO is more nuanced. In the two-stage linear setting, SRO with local uncertainty sets $\mathcal{U}(D)$71 is explicitly equivalent to DRO with a type-$\mathcal{U}(D)$72 Wasserstein ambiguity set (Bertsimas et al., 2019). In the electricity–gas scheduling paper, the adopted two-stage SRO model is described as equivalent to two-stage DRO under a type-c Wasserstein ambiguity set, while remaining computationally simpler because it operates on sample-centered polyhedral sets (Liu et al., 2023). By contrast, the nurse redeployment model is explicit that its ambiguity lies over trajectory realizations rather than over probability distributions, and the generative-model paper emphasizes that sampler-first robustness works even without explicit densities, unlike many divergence-based DRO constructions (Liu et al., 9 Sep 2025, Zhang et al., 30 Apr 2026).

Several recurring limitations also appear. Learning-based set construction can be conservative if the chosen shape family is crude, if hypothesis classes have large Rademacher complexity, or if the residual sets $\mathcal{U}(D)$73 and $\mathcal{U}(D)$74 must be large to satisfy Assumptions A or B (Tulabandhula et al., 2014). Split-sample calibration is dimension-free in its sample-size requirement, but small $\mathcal{U}(D)$75 still requires $\mathcal{U}(D)$76 on the order of $\mathcal{U}(D)$77, and continuity or i.i.d. assumptions are essential for the order-statistics argument (Hong et al., 2017). Domain-reduction methods rely on linearity in $\mathcal{U}(D)$78, convexity, moment bounds, and normal approximation; heavy-tailed or dependent samples may degrade finite-sample accuracy (Yabe et al., 2020).

In two-stage and multistage models, affine or linear decision rules are the primary tractability device. They collapse otherwise intractable adjustable problems into LP, SOCP, or MILP reformulations, but they introduce approximation error and can be sensitive to solver scale and the chosen norm. The nurse redeployment paper accordingly enforces feasibility through rolling-horizon re-optimization rather than through global implementability of the full LDR policy (Bertsimas et al., 2019, Liu et al., 9 Sep 2025). The electricity–gas formulation depends on DC power flow, radial gas networks with fixed flow directions, and accurate Weymouth linearization points (Liu et al., 2023).

The unknown-objective GP formulation has its own restrictions. The guarantees require $\mathcal{U}(D)$79 to lie in an RKHS with bounded norm, $\mathcal{U}(D)$80 to be finite, and $\mathcal{U}(D)$81 to grow sublinearly; GP inference also scales as $\mathcal{U}(D)$82 in vanilla form, and the dependence on $\mathcal{U}(D)$83 worsens with dimension (Sessa et al., 2020). The sampler-first generative formulation requires a nontrivial coverage assumption, Lipschitz continuity of both $\mathcal{U}(D)$84 and $\mathcal{U}(D)$85, and generally heuristic optimization of a nonconvex–nonconcave minimax problem; generator training instability can undermine the method, and excessively large $\mathcal{U}(D)$86 may over-conservatively flatten performance (Zhang et al., 30 Apr 2026).

A common misconception is that SRO denotes a single model class with a single notion of conservatism. The literature instead supports a more specific conclusion: SRO is a methodological pattern in which finite data define the robustification mechanism, but the underlying uncertainty object may be a future realization, a recourse scenario, an unknown objective, or a learned sampler. What unifies these formulations is the replacement of nominal empirical optimization by a worst-case construction tied explicitly to the available samples.