Pith. sign in

REVIEW 2 major objections 5 minor 282 references

Oracle-Based Distributionally Robust Optimization under Optimal Transport Ambiguity Sets

T0 review · 2 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A scalar budget allocation problem exactly captures the worst-case expectation over an optimal-transport ambiguity set.

desk verdict Genuinely new budget-allocation attack on OT-DRO that makes large instances tractable; the tightness proof for the dual support bound has a fixable n=1 gap and should not block a serious referee. read the letter →

arxiv 2608.03075 v1 pith:M2ZMWGBS submitted 2026-08-04 math.OC

Guixian Chen , Salar Fattahi , Soroosh Shafiee This is my paper · ORCID
classification math.OC MSC 90C4790C2549Q22
keywords distributionallyrobustoptimizationoptimaltransportambiguitysetworst-caseexpectationbudgetallocationleast-favorabledistributionconvex-piecewiseconcavelossoracle-basedalgorithmsupportsizebound
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Distributionally robust optimization with optimal-transport ambiguity sets is usually solved by lifting the minimax problem into one large conic program; this paper claims that the inner worst-case expectation can instead be solved exactly by allocating one scalar transport budget to each empirical sample. The central identity is that the worst-case expectation equals the optimum of a budget allocation problem whose local utility per sample is evaluated using at most two pieces of a convex-piecewise concave loss. This equivalence leads to an oracle-based algorithm that returns an epsilon-optimal worst-case distribution supported on at most $N+1$ points, with a running time that is near-linear in the sample size and poly-logarithmic in $1/\epsilon$, provided a local worst-case oracle exists for single loss pieces. The same machinery is embedded in a distributional best-response method for the primal DRO problem and extended to the dual DRO problem, where the paper proves that a least-favorable distribution needs at most $\min\{N+n+1, KN\}$ atoms and shows the bound is tight.

What carries the argument

The carrying object is the budget-allocation reformulation: each sample receives a local transport budget $b_i$, the total budget is constrained by $\rho$, and the local utility $V_i(b_i)$ is the maximum over pairs of loss pieces of a two-piece perspective program. This converts an infinite-dimensional adversarial distribution into $N$ scalar allocation decisions. Around this identity the paper hangs the local worst-case oracle (maximize a single loss piece over a transport ball), nested golden-section searches over mass splits and budget splits, bisection over the Lagrange multiplier of the aggregate budget, and a fractional-knapsack LP that selects the final $N+1$ atoms; for the dual problem, the same LP-counting argument with $n$ equilibrium constraints yields the $N+n+1$ atom bound.

What would settle it

Run Algorithm 5 on a piecewise-quadratic instance with $c(z,\hat z)=\|z-\hat z\|_2$ and compare its $N+1$-point output against a high-precision conic solution of the perspective reformulation; if the gap exceeds the stated $O(\epsilon + \eta^{\min\{1/p,1-1/p\}})$ bound, the budget-allocation equivalence in Theorem 1 is wrong. Alternatively, exhibit a loss piece whose maximization over $\{v : c(\hat z+v,\hat z)\le u\}$ is NP-hard, in which case no polynomial-time oracle exists and the $\tilde{O}(N K^2 \mathrm{Cost}_{\epsilon} \log^4(1/\eta))$ claim collapses to the oracle's true cost.

Watch

Extended reading notes

Core claim

Under the paper's standing assumptions, the worst-case expectation problem for a fixed decision is equivalent to maximizing $(1/N)\sum_{i=1}^N V_i(b_i)$ over local budgets $b_i$ whose average is at most $\rho$, where $V_i(b_i)$ is defined through a two-piece subproblem per empirical sample. The proof disintegrates any adversarial distribution into per-sample conditional measures, replaces each conditional measure by a Dirac at its mean via Jensen's inequality, and then observes that the remaining LP in the probability weights has exactly $N+1$ constraints, so at most $N+1$ atoms can be active. The paper turns this structural result into algorithms: a nested golden-section and bisection scheme driven by a local worst-case oracle for each loss piece, a fractional-knapsack post-processing step that compresses a $2N$-point construction down to $N+1$ points, a distributional best-response algorithm for the full primal DRO problem, and a convex-programming compression that extracts a least-favorable distribution on at most $N+n+1$ atoms for the dual problem.

Load-bearing premise

The entire complexity and correctness analysis rests on Assumption 2: for every loss piece, sample, and radius, a local worst-case oracle must return an $\epsilon$-optimal perturbation inside the transport ball in $\mathrm{Cost}_{k,\epsilon}$ time, and if maximizing a single loss piece over the transport ball is itself hard, the poly-logarithmic guarantees reduce to whatever that oracle costs.

Editorial extensions

If this is right

  • Inner worst-case expectations over optimal-transport balls can be computed without forming the lifted conic programs, so large-$N$ and large-$m$ instances that exhaust memory in lifting solvers remain solvable.
  • An $\epsilon$-optimal worst-case distribution with at most $N+1$ support points is produced in $\tilde{O}(N K^2 \mathrm{Cost}_{\epsilon} \log^4(1/\eta))$ time whenever each single-piece worst-case subproblem has a local oracle of cost $\mathrm{Cost}_{\epsilon}$.
  • The primal DRO problem admits a direct saddle-point algorithm: with projected online subgradient descent as the primal oracle, an $\epsilon$-saddle point is reached in $O(1/\epsilon^2)$ iterations, returning both a robust decision and a worst-case distribution.
  • The dual DRO problem has a least-favorable distribution supported on at most $\min\{N+n+1, KN\}$ atoms, a strict improvement over the $KN$ bound whenever $n < (K-1)N - 1$.
  • The tightness construction shows that least-favorable distributions at the optimal primal decision can require more atoms than worst-case distributions at a fixed decision.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the per-sample separability extends to other discrepancy sets whose unit balls decompose across samples, the same budget-allocation reduction could apply beyond optimal-transport balls.
  • The poly-logarithmic accuracy scaling suggests that with prox-friendly costs, optimal-transport DRO could run at nearly the cost of ordinary empirical risk minimization, making it a practical training objective rather than a post-hoc verification tool.
  • A streaming variant of the fractional-knapsack compression, applied online as the best-response iterations run, would avoid storing the full $T(N+1)$ historical atoms before compression.
  • The dual support bound implies that downstream consumers of least-favorable distributions, such as estimator or controller design, should expect support to grow with the decision dimension $n$ as well as the sample size $N$.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. This paper studies distributionally robust optimization (DRO) over optimal-transport ambiguity sets centered at an empirical distribution. For losses that are convex in the decision variable and piecewise concave in the uncertainty, the authors prove that the inner worst-case expectation problem is equivalent to a scalar budget-allocation problem and develop a nested golden-section algorithm that, given a local worst-case oracle, returns an epsilon-optimal worst-case distribution supported on at most 2N points in time O~(N K^2 Cost_epsilon log^4(1/eta)), with a fractional-knapsack post-processing step reducing the support to N+1. They embed this oracle in a distributional best-response method for the primal DRO problem and obtain an O(1/epsilon^2)-iteration saddle-point guarantee when projected subgradient is used. For the dual DRO problem, they prove an upper bound of min{N+n+1, KN} atoms and provide two-stage and tangent-based compression schemes in Theorems 5 and 6. Numerical experiments compare the proposed algorithms favorably with conic solver baselines.

Significance. If the results hold, the paper is significant: it gives one of the first tailored algorithms for the inner worst-case problem with explicit oracle-call complexity, it establishes a structural support bound for the dual DRO problem that improves over the generic KN bound in moderate dimension, and it ships public code together with detailed appendices. The main structural theorems are supported by internally coherent arguments, and the support bounds follow from basic feasible solution counting rather than parameter fitting. The two caveats are that Assumption 2 delegates the actual per-component optimization to an oracle, so the poly-logarithmic complexity is conditional, and the tightness claim in Lemma 5 currently has a proof gap for n=1.

major comments (2)
  1. [Appendix A.5, Lemma 5] The tightness construction is not valid for n=1. The proof requires N-1 distinct boundary points of the Euclidean unit ball in R^n that sum to zero and satisfy M:=max_{i,k} z_hat_i^T v_k < 1, where {v_k} are the vertices of a regular n-simplex. For n=1 the boundary has only two points; for N>=4 no such set exists, and for N=3 the only choice is {-1,1}, which coincides with the 1-simplex vertices and gives M=1, violating M<1. Hence the statement "for any n>=1" is not proved, and the claimed tightness of min{N+n+1, KN} and the accompanying remark that least-favorable distributions are provably denser than fixed-x worst-case distributions are not established in full generality. The lemma should be restricted to n>=2 or supplemented with a separate one-dimensional construction.
  2. [Appendix A.4, Proof of Theorem 2] The proof applies Lemma 4's guarantee to an unrestricted maximization. Lemma 4 bounds the error relative to max_{0<=b<=Nrho}{V_i(b)-lambda b}, but the proof then asserts V_i(hat b_i(lambda))-lambda hat b_i(lambda) >= max_{b>=0}{V_i(b)-lambda b} - E_inner and defines g(lambda) with max_{b>=0}. For lambda below the optimal multiplier, the unrestricted maximizer may lie above Nrho, and for lambda=0 with p>1 the supremum is infinite, so the displayed inequality is not implied. The argument can be repaired by working with the restricted dual g_R(lambda)=lambda rho + (1/N) sum_i max_{0<=b<=Nrho}{V_i(b)-lambda b}, which still weakly dominates the optimal value V* because every feasible allocation lies in the box [0,Nrho]. This is a local fix, but as written the proof of the central complexity theorem is incomplete.
minor comments (5)
  1. [Abstract and Section 1.1] The claim that the algorithm uses O(poly log(1/epsilon)) oracle calls should be clarified: the poly-logarithmic dependence is in the number of oracle calls, while the total runtime in Theorem 2 also carries the factor Cost_epsilon, which can depend on 1/epsilon and on problem data. Please state this distinction so readers do not infer an unconditional poly-logarithmic arithmetic complexity.
  2. [Algorithm 6 and Section 6.3] The experimental implementation uses tail-averaging with a 20% burn-in, whereas Algorithm 6's pseudocode averages all iterations uniformly. Please state that the experimental variant is a heuristic and note whether the theoretical regret analysis is intended to cover it.
  3. [Appendix B, Proposition 3] The proof of Proposition 3 is omitted with a reference to [SAKME19, Theorem 9]. If this proposition is claimed as a contribution, provide the proof or clearly mark it as an imported known result.
  4. [Algorithm 4] There is a mismatched parenthesis in the output line of Algorithm 4: the tuple (hat V_i(hat b_i(lambda)), hat s, hat k_1, hat k_2) should be closed before the assignment arrow.
  5. [Section 3.1, Assumption 2] Because Assumption 2 is load-bearing for all complexity statements, consider adding a sentence in the introduction or abstract emphasizing that the poly-logarithmic guarantee is relative to the per-component worst-case oracle; otherwise the abstract's claim of bypassing large lifted reformulations may be over-read as unconditional.

Circularity Check

1 steps flagged · score 2.0 of 10

No significant circularity; the central reductions are self-contained or oracle-conditional, with only a minor author-overlap citation in Lemma 1 and a separate n=1 correctness gap in Lemma 5's tightness construction.

  1. other [Lemma 1 and Appendix A.1 (saddle point / strong duality)]
    "The proof follows by verifying the conditions of [SADK26, Lemmas 3 & 4], which can be checked straightforwardly. For completeness, we provide a self-contained proof in Appendix A.1."

    The paper imports the minimax/strong-duality result from [SADK26], which is co-authored by S. Shafiee, an author of the present work, and then relies on it to assert that Problems (1) and (2) have equal finite values and admit a saddle point. Appendix A.1 does not provide an independent derivation; it verifies the hypotheses of the same cited lemmas. This is a genuine author-overlap citation, but it is not the source of the paper's main results: Proposition 1, Theorem 1, the support bounds, and the oracle-based algorithms are self-contained reductions based on Jensen's inequality, LP basic feasible solutions, and the stated oracle assumption.

full rationale

The main derivation chain is not circular. Proposition 1 reduces the worst-case expectation (4) to the finite nonconvex program (5) using disintegration and Jensen's inequality, and the N+1 support bound follows from counting the constraints of an LP; this is independent of the cited literature. Theorem 1 is an explicit value-function reformulation: the local utility V_i(b_i) is defined as the per-sample worst-case value under a local budget, so the budget allocation problem (7) is a transparent decomposition rather than a fitted parameter renamed as a prediction. The complexity guarantees in Theorem 2 are conditional on Assumption 2, a local worst-case oracle; this is a computational primitive, not a hidden input fitted to the target output, and the poly-logarithmic overhead is derived from concavity and Hölder-continuity arguments. Theorem 3's compression to N+1 atoms is a fractional knapsack problem, and Proposition 2's min{N+n+1,KN} support bound comes from LP BFS counting, not from self-citation. The only author-overlap dependence is Lemma 1, which imports a minimax theorem from [SADK26] co-authored by Shafiee; this is a minor, published, externally checkable result and is not the basis of the core reductions. Separately, the tightness proof in Appendix A.5 (Lemma 14) appears to have a correctness gap for n=1, because the unit sphere in R^1 has only two boundary points, so the required distinct boundary samples with zero sum and M<1 cannot exist for N>=3; this is a mathematical-support issue, not a circularity issue, and it does not affect Theorem 2 or the upper-bound Proposition 2. Against external benchmarks (MOSEK, Gurobi), the numerical claims are self-contained, so a score in the 0-2 range is appropriate.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no fitted free parameters; algorithmic tolerances (epsilon, eta, delta) are user-specified accuracy parameters. Invented entities: none. The main load-bearing premises are Assumptions 1-4 and standard convex-analysis theorems. The reliance on [SADK26] for Lemma 1 is a self-citation but not a fitted input.

assumptions (4)
  • domain assumption Assumption 1 (Regularity): convex-piecewise concave loss, c(z,z_hat) >= ||z-z_hat||^p, inf-compactness, sublinear growth if p=1
    Invoked throughout Sections 3-6; guarantees the budget allocation equivalence, attainment, and saddle point existence. If violated, the core reductions need not hold.
  • domain assumption Assumption 2 (Local Worst-Case Oracle WCO_epsilon^k)
    Computational primitive whose cost bounds every complexity claim; all algorithms (1-6) are oracle-based.
  • domain assumption Assumption 3 (Worst-case expectation oracle) and Assumption 4 (no-regret online algorithm)
    Used in the best-response framework (Theorem 4) for the O(R_T/T + delta) duality gap.
  • standard math Standard convex analysis facts: perspective functions preserve concavity, Fenchel duality, Sion's minimax theorem, LP BFS theorem, Everett's epsilon theorem
    Used in the proofs of Theorem 1, Proposition 2, and the complexity bounds; these are external standard results.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Oracle-Based Distributionally Robust Optimization under Optimal Transport Ambiguity Sets." pith.science (2026). https://pith.science/paper/M2ZMWGBS

@misc{pith2026260803075,
  author       = {Pith},
  title        = {Pith review of: Oracle-Based Distributionally Robust Optimization under Optimal Transport Ambiguity Sets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M2ZMWGBS}},
  note         = {Machine review of arXiv:2608.03075}
}
abstract

Distributionally robust optimization (DRO) with optimal transport ambiguity sets is traditionally solved by reformulating the minimax problem into a single-level convex program. While theoretically tractable, these reformulations introduce numerous auxiliary variables and demanding conic constraints that scale poorly in practice. In this paper, we address this challenge by reducing the inner worst-case expectation problem exactly to a scalar budget allocation task. This structural insight yields an efficient algorithm that bypasses large lifted reformulations, alongside a fast post-processing scheme to recover an optimal worst-case distribution supported on at most $N+1$ points, where $N$ denotes the sample size. We embed this procedure within an oracle-based distributional best-response framework to directly compute an approximate primal-dual solution to the overall DRO problem. Furthermore, we extend our analysis to the dual DRO formulation, proving the existence of a least-favorable distribution supported on at most $\min\{N+n+1, KN\}$ atoms, where $n$ and $K$ denote the decision dimension and number of loss components, respectively, and provide an efficient convex programming reduction to extract it from the solution of the primal DRO. Numerical experiments demonstrate that the proposed approach significantly outperforms state-of-the-art reformulation-based solvers.

Figures

Figures reproduced from arXiv: 2608.03075 by the authors.

Figure 1
Figure 1. Computational runtime and accuracy of the inner worst-case expectation solvers across [PITH_FULL_IMAGE:figures/full_fig_p029_1.png] view at source ↗
Figure 2
Figure 2. Relative duality gap versus support size across four post-processing methods: (1) primal [PITH_FULL_IMAGE:figures/full_fig_p031_2.png] view at source ↗

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

282 extracted references · 53 canonical work pages

  1. [1]

    2006 , publisher=

    Infinite dimensional analysis: a hitchhiker’s guide , author=. 2006 , publisher=

    2006

  2. [2]

    Glasgow Mathematical Journal , volume=

    Remarks on weak compactness in L1 ( , X) , author=. Glasgow Mathematical Journal , volume=. 1977 , publisher=

    1977

  3. [3]

    SIAM journal on Optimization , volume=

    Interior point methods in semidefinite programming with applications to combinatorial optimization , author=. SIAM journal on Optimization , volume=. 1995 , publisher=

    1995

  4. [4]

    SIAM review , volume=

    Semidefinite programming , author=. SIAM review , volume=. 1996 , publisher=

    1996

  5. [5]

    Mathematical Programming , volume=

    Distributionally robust shortfall risk optimization model and its approximation , author=. Mathematical Programming , volume=. 2019 , publisher=

    2019

  6. [6]

    2022 , publisher=

    Introduction to Algorithms , author=. 2022 , publisher=

    2022

  7. [7]

    SIAM Journal on Optimization , volume=

    Approximate primal solutions and rate analysis for dual subgradient methods , author=. SIAM Journal on Optimization , volume=. 2009 , publisher=

    2009

  8. [8]

    Operations Research Letters , volume=

    Mirror descent and nonlinear projected subgradient methods for convex optimization , author=. Operations Research Letters , volume=. 2003 , publisher=

    2003

Show all 282 references
  1. [9]

    arXiv:1206.4657 , year=

    Projection-free online learning , author=. arXiv:1206.4657 , year=

    arXiv

  2. [10]

    Conference on Learning Theory , pages=

    Faster projection-free online learning , author=. Conference on Learning Theory , pages=

  3. [11]

    Machine Learning , volume=

    Logarithmic regret algorithms for online convex optimization , author=. Machine Learning , volume=. 2007 , publisher=

    2007

  4. [12]

    International Conference on Machine Learning , pages=

    Online convex programming and generalized infinitesimal gradient ascent , author=. International Conference on Machine Learning , pages=

  5. [13]

    Foundations and Trends in Machine Learning , volume=

    Online learning and online convex optimization , author=. Foundations and Trends in Machine Learning , volume=

  6. [14]

    2004 , publisher=

    Convex Optimization , author=. 2004 , publisher=

    2004

  7. [15]

    On linear optimization over

    Yue, Man-Chung and Kuhn, Daniel and Wiesemann, Wolfram , journal=. On linear optimization over. 2022 , publisher=

    2022

  8. [16]

    Mathematical Finance: An International Journal of Mathematics, Statistics and Financial Economics , volume=

    Tractable robust expected utility and risk models for portfolio optimization , author=. Mathematical Finance: An International Journal of Mathematics, Statistics and Financial Economics , volume=. 2010 , publisher=

    2010

  9. [17]

    Operations Research , volume=

    On the relation between option and stock prices: a convex optimization approach , author=. Operations Research , volume=. 2002 , publisher=

    2002

  10. [18]

    1983 , publisher=

    Introduction to Sensitivity and Stability Analysis in Nonlinear Programming , author=. 1983 , publisher=

    1983

  11. [19]

    2000 , publisher=

    Perturbation Analysis of Optimization Problems , author=. 2000 , publisher=

    2000

  12. [20]

    Proceedings of the National Academy of Sciences , volume=

    Minimax theorems , author=. Proceedings of the National Academy of Sciences , volume=

  13. [21]

    The Annals of Statistics , volume=

    Learning models with uniform performance via distributionally robust optimization , author=. The Annals of Statistics , volume=. 2021 , publisher=

    2021

  14. [22]

    Faster rates for the

    Garber, Dan and Hazan, Elad , booktitle=. Faster rates for the

  15. [23]

    Foundations and Trends in Machine Learning , volume=

    Convex optimization: Algorithms and complexity , author=. Foundations and Trends in Machine Learning , volume=. 2015 , publisher=

    2015

  16. [24]

    2013 , publisher=

    Introductory lectures on convex optimization: A basic course , author=. 2013 , publisher=

    2013

  17. [25]

    International Conference on Learning Representations , year=

    Adversarial machine learning at scale , author=. International Conference on Learning Representations , year=

  18. [26]

    Proceedings of the third annual workshop on Computational learning theory , pages=

    Aggregating strategies , author=. Proceedings of the third annual workshop on Computational learning theory , pages=

  19. [27]

    European Conference on Computational Learning Theory , pages=

    Averaging expert predictions , author=. European Conference on Computational Learning Theory , pages=. 1999 , organization=

    1999

  20. [28]

    International Journal of Computer Vision , volume=

    Generalized out-of-distribution detection: A survey , author=. International Journal of Computer Vision , volume=. 2024 , publisher=

    2024

  21. [29]

    Artificial intelligence review , volume=

    A study of the effect of different types of noise on the precision of supervised learning techniques , author=. Artificial intelligence review , volume=. 2010 , publisher=

    2010

  22. [30]

    Mathematics of Operations Research , volume=

    Models for minimax stochastic linear optimization problems with risk aversion , author=. Mathematics of Operations Research , volume=. 2010 , publisher=

    2010

  23. [31]

    Foundations and Trends

    Regret analysis of stochastic and nonstochastic multi-armed bandit problems , author=. Foundations and Trends. 2012 , publisher=

    2012

  24. [32]

    SIAM journal on computing , volume=

    The nonstochastic multiarmed bandit problem , author=. SIAM journal on computing , volume=. 2002 , publisher=

    2002

  25. [33]

    Advances in Neural Information Processing Systems , volume=

    Adaptive sampling for stochastic risk-averse learning , author=. Advances in Neural Information Processing Systems , volume=

  26. [34]

    International conference on machine learning , pages=

    Risk-averse no-regret learning in online convex games , author=. International conference on machine learning , pages=

  27. [35]

    Proceedings of the ACM Web Conference 2022 , pages=

    Distributionally-robust recommendations for improving worst-case user experience , author=. Proceedings of the ACM Web Conference 2022 , pages=

    2022

  28. [36]

    Advances in Neural Information Processing Systems , volume=

    A model-based reinforcement learning with adversarial training for online recommendation , author=. Advances in Neural Information Processing Systems , volume=

  29. [37]

    Quantitative Finance , volume=

    Distributionally robust end-to-end portfolio construction , author=. Quantitative Finance , volume=. 2023 , publisher=

    2023

  30. [38]

    Management science , volume=

    Distributionally robust mean-variance portfolio selection with Wasserstein distances , author=. Management science , volume=. 2022 , publisher=

    2022

  31. [39]

    Advances in Neural Information Processing Systems , volume=

    Stochastic convex optimization with bandit feedback , author=. Advances in Neural Information Processing Systems , volume=

  32. [40]

    Wasserstein distributionally robust optimization:

    Kuhn, Daniel and Esfahani, Peyman Mohajerin and Nguyen, Viet Anh and Shafieezadeh-Abadeh, Soroosh , booktitle=. Wasserstein distributionally robust optimization:. 2019 , publisher=

    2019

  33. [41]

    The 22nd International Conference on Artificial Intelligence and Statistics , pages=

    Risk-averse stochastic convex bandit , author=. The 22nd International Conference on Artificial Intelligence and Statistics , pages=

  34. [42]

    2015 , publisher=

    Optimal Transport for Applied Mathematicians , author=. 2015 , publisher=

    2015

  35. [43]

    2003 , publisher=

    Topics in Optimal Transportation , author=. 2003 , publisher=

    2003

  36. [44]

    Operations Research , volume=

    Oracle-based robust optimization via online learning , author=. Operations Research , volume=. 2015 , publisher=

    2015

  37. [45]

    Theoretical Computer Science , volume=

    Online learning for min-max discrete problems , author=. Theoretical Computer Science , volume=. 2022 , publisher=

    2022

  38. [46]

    arXiv:1912.13213 , year=

    A Modern Introduction to Online Learning , author=. arXiv:1912.13213 , year=

    1912 arXiv

  39. [47]

    , title =

    Kantorovich, Leonid and Rubinstein, Gennady S. , title =. Vestnik Leningrad. Univ , pages =

  40. [48]

    Data-driven distributionally robust optimization using the

    Mohajerin Esfahani, Peyman and Kuhn, Daniel , journal=. Data-driven distributionally robust optimization using the. 2018 , publisher=

    2018

  41. [49]

    Nonconvex Optimization and its Applications , volume=

    On duality theory of conic linear problems , author=. Nonconvex Optimization and its Applications , volume=. 2001 , publisher=

    2001

  42. [50]

    Probability theory and related fields , volume=

    On the rate of convergence in Wasserstein distance of the empirical measure , author=. Probability theory and related fields , volume=. 2015 , publisher=

    2015

  43. [51]

    European Journal of Operational Research , volume=

    The nonlinear knapsack problem--algorithms and applications , author=. European Journal of Operational Research , volume=. 2002 , publisher=

    2002

  44. [52]

    Acta Numerica , volume=

    Distributionally robust optimization , author=. Acta Numerica , volume=. 2025 , publisher=

    2025

  45. [53]

    Operations Research , number=

    Nash Equilibria, Regularization, and Computation in Optimal Transport-Based Distributionally Robust Optimization , author=. Operations Research , number=

  46. [54]

    Generalization bounds for (

    An, Yang and Gao, Rui , journal=. Generalization bounds for (

  47. [55]

    arXiv:2504.08097 , year=

    Data Compression for Fast Online Stochastic Optimization , author=. arXiv:2504.08097 , year=

    arXiv

  48. [56]

    INFORMS Journal on Optimization , volume=

    Data-driven distributionally robust optimization over time , author=. INFORMS Journal on Optimization , volume=. 2023 , publisher=

    2023

  49. [57]

    Advances in Neural Information Processing Systems , pages=

    An online method for a class of distributionally robust optimization with non-convex objectives , author=. Advances in Neural Information Processing Systems , pages=

  50. [58]

    Ido Yerenburg , year=

  51. [59]

    Operations Research , volume=

    Online first-order framework for robust convex optimization , author=. Operations Research , volume=. 2018 , publisher=

    2018

  52. [60]

    Mathematical Programming , volume=

    Exploiting problem structure in optimization under uncertainty via online convex optimization , author=. Mathematical Programming , volume=. 2019 , publisher=

    2019

  53. [61]

    INFORMS Journal on Computing , volume=

    First-order algorithms for robust optimization problems via convex-concave saddle-point Lagrangian reformulation , author=. INFORMS Journal on Computing , volume=. 2025 , publisher=

    2025

  54. [62]

    Adversaries in online learning revisited:

    Pokutta, Sebastian and Xu, Huan , journal=. Adversaries in online learning revisited:

  55. [63]

    Foundations and Trends in Optimization , volume=

    Introduction to online convex optimization , author=. Foundations and Trends in Optimization , volume=. 2016 , publisher=

    2016

  56. [64]

    2022 , publisher=

    Introduction to Online Convex Optimization , author=. 2022 , publisher=

    2022

  57. [65]

    arXiv:2404.05377 , year=

    A Max-Min-Max Algorithm for Large-Scale Robust Optimization , author=. arXiv:2404.05377 , year=

    arXiv

  58. [66]

    Advances in Neural Information Processing Systems , pages=

    Stochastic gradient methods for distributionally robust optimization with f -divergences , author=. Advances in Neural Information Processing Systems , pages=

  59. [67]

    Advances in Neural Information Processing Systems , pages=

    Outlier-robust wasserstein dro , author=. Advances in Neural Information Processing Systems , pages=

  60. [68]

    The Thirty Seventh Annual Conference on Learning Theory , pages=

    Robust distribution learning with local and global adversarial corruptions , author=. The Thirty Seventh Annual Conference on Learning Theory , pages=

  61. [69]

    SIAM Journal on Optimization , volume=

    Robust stochastic approximation approach to stochastic programming , author=. SIAM Journal on Optimization , volume=. 2009 , publisher=

    2009

  62. [70]

    arXiv:2504.10796 , year=

    Wasserstein Distributionally Robust Regret Optimization , author=. arXiv:2504.10796 , year=

    arXiv

  63. [71]

    Available at SSRN 5493106 , year=

    Robustness and Regret Tolerance , author=. Available at SSRN 5493106 , year=

  64. [72]

    arXiv:2412.15406 , year=

    Distributionally Robust Regret Minimization , author=. arXiv:2412.15406 , year=

    arXiv

  65. [73]

    Production and Operations Management , volume=

    Regret in the newsvendor model with demand and yield randomness , author=. Production and Operations Management , volume=

  66. [74]

    Conference on Learning Theory , pages=

    Minimax regret optimization for robust machine learning under distribution shift , author=. Conference on Learning Theory , pages=

  67. [75]

    Mathematical Finance , volume=

    Coherent measures of risk , author=. Mathematical Finance , volume=

  68. [76]

    IEEE Transactions on Automatic Control , volume=

    Optimal stochastic linear systems with exponential performance criteria and their relation to deterministic differential games , author=. IEEE Transactions on Automatic Control , volume=

  69. [77]

    1990 , publisher=

    Risk-Sensitive Optimal Control , author=. 1990 , publisher=

    1990

  70. [78]

    Algorithms for

    Chow, Yinlam and Ghavamzadeh, Mohammad , booktitle=. Algorithms for

  71. [79]

    Advances in Neural Information Processing Systems , pages=

    On dynamic programming decompositions of static risk measures in markov decision processes , author=. Advances in Neural Information Processing Systems , pages=

  72. [80]

    Mathematics of Operations Research , volume=

    Q-learning for risk-sensitive control , author=. Mathematics of Operations Research , volume=

  73. [81]

    Advances in neural information processing systems , pages=

    Risk-aversion in multi-armed bandits , author=. Advances in neural information processing systems , pages=

  74. [82]

    IEEE Journal of Selected Topics in Signal Processing , volume=

    Risk-averse multi-armed bandit problems under mean-variance measure , author=. IEEE Journal of Selected Topics in Signal Processing , volume=. 2016 , publisher=

    2016

  75. [83]

    Exploration vs exploitation vs safety:

    Galichet, Nicolas and Sebag, Michele and Teytaud, Olivier , booktitle=. Exploration vs exploitation vs safety:

  76. [84]

    1998 , publisher=

    Variational Analysis , author=. 1998 , publisher=

    1998

  77. [85]

    Advances in Neural Information Processing Systems , pages=

    Distributionally robust logistic regression , author=. Advances in Neural Information Processing Systems , pages=

  78. [86]

    Distributionally robust inverse covariance estimation: The

    Nguyen, Viet Anh and Kuhn, Daniel and Mohajerin Esfahani, Peyman , journal=. Distributionally robust inverse covariance estimation: The

  79. [87]

    Bridging

    Nguyen, Viet Anh and Shafieezadeh-Abadeh, Soroosh and Kuhn, Daniel and Mohajerin Esfahani, Peyman , journal=. Bridging. 2023 , publisher=

    2023

  80. [88]

    Data-driven chance constrained programs over

    Chen, Zhi and Kuhn, Daniel and Wiesemann, Wolfram , journal=. Data-driven chance constrained programs over. 2024 , publisher=

    2024

  81. [89]

    Journal of Machine Learning Research , volume=

    Regularization via Mass Transportation , author=. Journal of Machine Learning Research , volume=

  82. [90]

    Modelling and Optimisation of Flows on Networks , pages=

    A User’s Guide to Optimal Transport , author=. Modelling and Optimisation of Flows on Networks , pages=. 2013 , editor=

    2013

  83. [91]

    1992 , publisher=

    Convergence of Probability Measures , author=. 1992 , publisher=

    1992

  84. [92]

    1974 , publisher=

    Conjugate Duality and Optimization , author=. 1974 , publisher=

    1974

  85. [93]

    1984 , publisher=

    Applied Nonlinear Analysis , author=. 1984 , publisher=

    1984

  86. [94]

    Mathematical Programming , volume=

    Data-driven inverse optimization with imperfect information , author=. Mathematical Programming , volume=. 2018 , publisher=

    2018

  87. [95]

    Wasserstein distributionally robust

    Shafieezadeh-Abadeh, Soroosh and Nguyen, Viet Anh and Kuhn, Daniel and Mohajerin Esfahani, Peyman , booktitle=. Wasserstein distributionally robust

  88. [96]

    Mathematical Programming , volume=

    Semi-discrete optimal transport: Hardness, regularization and numerical solution , author=. Mathematical Programming , volume=. 2023 , publisher=

    2023

  89. [97]

    Discrete optimal transport with independent marginals is \#

    Ta. Discrete optimal transport with independent marginals is \#. SIAM Journal on Optimization , volume=. 2023 , publisher=

    2023

  90. [98]

    Annals of Operations Research , volume=

    A framework for optimization under ambiguity , author=. Annals of Operations Research , volume=. 2012 , publisher=

    2012

  91. [99]

    arXiv:2010.05373 , year=

    Distributionally Robust Local Non-parametric Conditional Estimation , author=. arXiv:2010.05373 , year=

    2010 arXiv

  92. [100]

    Advances in Neural Information Processing Systems , pages=

    Optimistic distributionally robust optimization for nonparametric likelihood approximation , author=. Advances in Neural Information Processing Systems , pages=

  93. [101]

    Conic programming reformulations of two-stage distributionally robust linear programs over

    Hanasusanto, Grani A and Kuhn, Daniel , journal=. Conic programming reformulations of two-stage distributionally robust linear programs over. 2018 , publisher=

    2018

  94. [102]

    Operations Research , volume=

    A unified theory of robust and distributionally robust optimization via the primal-worst-equals-dual-best principle , author=. Operations Research , volume=. 2025 , publisher=

    2025

  95. [103]

    Mathematics of Operations Research , volume=

    Quantifying distributional model risk via optimal transport , author=. Mathematics of Operations Research , volume=. 2019 , publisher=

    2019

  96. [104]

    Blanchet, Jose and Kang, Yang and Murthy, Karthyek , journal=. Robust. 2019 , publisher=

    2019

  97. [105]

    Operations Research Letters , volume=

    Optimal uncertainty size in distributionally robust inverse covariance estimation , author=. Operations Research Letters , volume=. 2019 , publisher=

    2019

  98. [106]

    Confidence regions in

    Blanchet, Jose and Murthy, Karthyek and Si, Nian , journal=. Confidence regions in

  99. [107]

    Sample out-of-sample inference based on

    Blanchet, Jose and Kang, Yang , journal=. Sample out-of-sample inference based on

  100. [108]

    Asian Conference on Machine Learning , pages=

    Distributionally robust groupwise regularization estimator , author=. Asian Conference on Machine Learning , pages=

  101. [109]

    Applied Modeling Techniques and Data Analysis 1: Computational Data Analysis Methods and Tools , pages=

    Doubly Robust Data-driven Distributionally Robust Optimization , author=. Applied Modeling Techniques and Data Analysis 1: Computational Data Analysis Methods and Tools , pages=. 2021 , publisher=

    2021

  102. [110]

    IEEE Winter Simulation Conference , pages=

    Data-driven optimal transport cost selection for distributionally robust optimization , author=. IEEE Winter Simulation Conference , pages=

  103. [111]

    Mathematics of Operations Research , volume=

    Optimal transport-based distributionally robust optimization: Structural properties and iterative schemes , author=. Mathematics of Operations Research , volume=

  104. [112]

    Advances in Neural Information Processing Systems , pages=

    Multivariate distributionally robust convex regression under absolute error loss , author=. Advances in Neural Information Processing Systems , pages=

  105. [113]

    Data Analysis and Applications 3 , pages=

    Semi-supervised Learning Based on Distributionally Robust Optimization , author=. Data Analysis and Applications 3 , pages=. 2020 , publisher=

    2020

  106. [114]

    International Conference on Learning Representations , year=

    Certifying Some Distributional Robustness with Principled Adversarial Training , author=. International Conference on Learning Representations , year=

  107. [115]

    Advances in Neural Information Processing Systems , pages=

    Generalizing to unseen domains via adversarial data augmentation , author=. Advances in Neural Information Processing Systems , pages=

  108. [116]

    Distributionally robust stochastic optimization with

    Gao, Rui and Kleywegt, Anton , journal=. Distributionally robust stochastic optimization with. 2023 , publisher=

    2023

  109. [117]

    A Short and General Duality Proof for

    Zhang, Luhao and Yang, Jincheng and Gao, Rui , journal=. A Short and General Duality Proof for. 2024 , publisher=

    2024

  110. [118]

    Operations Research , volume=

    Wasserstein distributionally robust optimization and variation regularization , author=. Operations Research , volume=. 2024 , publisher=

    2024

  111. [119]

    arXiv:1701.04200 , year=

    Distributionally robust stochastic optimization with dependence structure , author=. arXiv:1701.04200 , year=

    arXiv

  112. [120]

    2023 , journal=

    Holistic Robust Data-Driven Decisions , author=. 2023 , journal=

    2023

  113. [121]

    Finite-sample guarantees for

    Gao, Rui , journal=. Finite-sample guarantees for. 2023 , publisher=

    2023

  114. [122]

    Robust hypothesis testing using

    Gao, Rui and Xie, Liyan and Xie, Yao and Xu, Huan , booktitle=. Robust hypothesis testing using

  115. [123]

    Data-driven risk-averse stochastic optimization with

    Zhao, Chaoyue and Guan, Yongpei , journal=. Data-driven risk-averse stochastic optimization with. 2018 , publisher=

    2018

  116. [124]

    Strong formulations for distributionally robust chance-constrained programs with left-hand side uncertainty under

    Ho-Nguyen, Nam and Kilin. Strong formulations for distributionally robust chance-constrained programs with left-hand side uncertainty under. INFORMS Journal on Optimization , volume=. 2023 , publisher=

    2023

  117. [125]

    Adversarial classification via distributional robustness with

    Ho-Nguyen, Nam and Wright, Stephen J , journal=. Adversarial classification via distributional robustness with. 2023 , publisher=

    2023

  118. [126]

    Distributionally Robust Chance-Constrained Programs with Right-Hand Side Uncertainty under

    Ho-Nguyen, Nam and K. Distributionally Robust Chance-Constrained Programs with Right-Hand Side Uncertainty under. Mathematical Programming , volume=

  119. [127]

    Chance-Constrained Set Covering with

    Shen, Haoming and Jiang, Ruiwei , journal=. Chance-Constrained Set Covering with

  120. [128]

    On distributionally robust chance constrained programs with

    Xie, Weijun , journal=. On distributionally robust chance constrained programs with

  121. [129]

    Advances in Neural Information Processing Systems , pages=

    Selecting optimal decisions via distributionally robust nearest-neighbor regression , author=. Advances in Neural Information Processing Systems , pages=

  122. [130]

    2006 , publisher=

    Elements of Information Theory , author=. 2006 , publisher=

    2006

  123. [131]

    Conference on Decision and Control , pages=

    A Distributionally Robust Optimization Approach for Outlier Detection , author=. Conference on Decision and Control , pages=

  124. [132]

    Journal of Machine Learning Research , volume=

    A robust learning approach for regression models based on distributionally robust optimization , author=. Journal of Machine Learning Research , volume=

  125. [133]

    Operations Research , volume=

    Two-stage sample robust optimization , author=. Operations Research , volume=. 2022 , publisher=

    2022

  126. [134]

    Tractable reformulations of distributionally robust two-stage stochastic programs with -

    Xie, Weijun , journal=. Tractable reformulations of distributionally robust two-stage stochastic programs with -

  127. [135]

    A First-Order Algorithmic Framework for

    Li, Jiajin and Huang, Sen and So, Anthony Man-Cho , booktitle=. A First-Order Algorithmic Framework for

  128. [136]

    Fast epigraphical projection-based incremental algorithms for

    Li, Jiajin and Chen, Caihua and So, Anthony Man-Cho , booktitle=. Fast epigraphical projection-based incremental algorithms for

  129. [137]

    IEEE Transactions on Automatic Control , volume=

    Cooperative data-driven distributionally robust optimization , author=. IEEE Transactions on Automatic Control , volume=. 2019 , publisher=

    2019

  130. [138]

    Conference on Decision and Control , pages=

    Regularized and distributionally robust data-enabled predictive control , author=. Conference on Decision and Control , pages=. 2019 , organization=

    2019

  131. [139]

    IEEE Transactions on Automatic Control , volume=

    Distributionally robust chance constrained data-enabled predictive control , author=. IEEE Transactions on Automatic Control , volume=. 2021 , publisher=

    2021

  132. [140]

    Minimax control of ambiguous linear stochastic systems using the

    Kim, Kihyun and Yang, Insoon , journal=. Minimax control of ambiguous linear stochastic systems using the

  133. [141]

    Learning-based distributionally robust motion control with

    Hakobyan, Astghik and Yang, Insoon , journal=. Learning-based distributionally robust motion control with

  134. [142]

    arXiv:2001.04727 , year=

    Wasserstein Distributionally Robust Motion Control for Collision Avoidance Using Conditional Value-at-Risk , author=. arXiv:2001.04727 , year=

    2001 arXiv

  135. [143]

    Wasserstein distributionally robust stochastic control:

    Yang, Insoon , journal=. Wasserstein distributionally robust stochastic control:

  136. [144]

    A convex optimization approach to distributionally robust

    Yang, Insoon , journal=. A convex optimization approach to distributionally robust. 2017 , publisher=

    2017

  137. [145]

    IEEE Transactions on Automatic Control , volume=

    Data-driven ambiguity sets with probabilistic guarantees for dynamic processes , author=. IEEE Transactions on Automatic Control , volume=

  138. [146]

    IEEE Transactions on Automatic Control , volume=

    Data assimilation and online optimization with performance guarantees , author=. IEEE Transactions on Automatic Control , volume=

  139. [147]

    European Control Conference , pages=

    Dynamic Evolution of Distributional Ambiguity Sets and Precision Tradeoffs in Data Assimilation , author=. European Control Conference , pages=. 2019 , organization=

    2019

  140. [148]

    2008 , publisher=

    Optimal Transport: Old and New , author=. 2008 , publisher=

    2008

  141. [149]

    1970 , publisher=

    Convex Analysis , author=. 1970 , publisher=

    1970

  142. [150]

    2011 , publisher=

    Convex Analysis and Monotone Operator Theory in Hilbert Spaces , author=. 2011 , publisher=

    2011

  143. [151]

    On sharpness of

    Isii, Keiiti , journal=. On sharpness of. 1962 , publisher=

    1962

  144. [152]

    Foundations and Trends in Optimization , volume=

    Proximal Algorithms , author=. Foundations and Trends in Optimization , volume=. 2014 , publisher=

    2014

  145. [153]

    Journal of Mathematical Analysis and Applications , volume=

    Duality in nonconvex optimization , author=. Journal of Mathematical Analysis and Applications , volume=. 1978 , publisher=

    1978

  146. [154]

    Archive for Rational Mechanics and Analysis , volume=

    A duality principle for non-convex optimisation and the calculus of variations , author=. Archive for Rational Mechanics and Analysis , volume=. 1979 , publisher=

    1979

  147. [155]

    Mathematical Programming , volume=

    A comment on ``computational complexity of stochastic programming problems'' , author=. Mathematical Programming , volume=. 2016 , publisher=

    2016

  148. [156]

    Mathematical Programming , volume=

    Computational complexity of stochastic programming problems , author=. Mathematical Programming , volume=. 2006 , publisher=

    2006

  149. [157]

    Operations Research , volume=

    Recovering best statistical guarantees via the empirical divergence-based distributionally robust optimization , author=. Operations Research , volume=. 2019 , publisher=

    2019

  150. [158]

    From data to decisions:

    Van Parys, Bart PG and Mohajerin Esfahani, Peyman and Kuhn, Daniel , journal=. From data to decisions:. 2021 , publisher=

    2021

  151. [159]

    Minimax statistical learning with

    Lee, Jaeho and Raginsky, Maxim , booktitle=. Minimax statistical learning with

  152. [160]

    Advances in Neural Information Processing Systems , pages=

    A minimax approach to supervised learning , author=. Advances in Neural Information Processing Systems , pages=

  153. [161]

    Theoretical analysis of adversarial learning:

    Tu, Zhuozhuo and Zhang, Jingwei and Tao, Dacheng , booktitle=. Theoretical analysis of adversarial learning:

  154. [162]

    Principled learning method for

    Kwon, Yongchan and Kim, Wonyoung and Won, Joong-Ho and Paik, Myunghee Cho , booktitle=. Principled learning method for

  155. [163]

    International Conference on Machine Learning , pages=

    On the Convergence and Robustness of Adversarial Training , author=. International Conference on Machine Learning , pages=

  156. [164]

    Extension of

    Hiriart-Urruty, J-B , journal=. Extension of. 1980 , publisher=

    1980

  157. [165]

    2009 , publisher=

    Variational Analysis , author=. 2009 , publisher=

    2009

  158. [166]

    Approximation and regularization of arbitrary functions in

    Attouch, H. Approximation and regularization of arbitrary functions in. Annales de l'Institut Henri Poincar\'e , volume=. 1993 , organization=

    1993

  159. [167]

    Annales de l'Institut Henri Poincar\'e , volume=

    Epigraphical analysis , author=. Annales de l'Institut Henri Poincar\'e , volume=. 1989 , organization=

    1989

  160. [168]

    Journal of Approximation Theory , volume=

    Towards minimal assumptions for the infimal convolution regularization , author=. Journal of Approximation Theory , volume=. 1991 , publisher=

    1991

  161. [169]

    Journal of Approximation Theory , volume=

    Proximal mappings , author=. Journal of Approximation Theory , volume=. 1998 , publisher=

    1998

  162. [170]

    2001 , publisher=

    Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications , author=. 2001 , publisher=

    2001

  163. [171]

    Mathematical Programming , volume=

    Distributionally robust joint chance constraints with second-order moment information , author=. Mathematical Programming , volume=. 2013 , publisher=

    2013

  164. [172]

    2014 , publisher=

    Lectures on Stochastic Programming: Modeling and Theory , author=. 2014 , publisher=

    2014

  165. [173]

    2017 , publisher=

    First-Order Methods in Optimization , author=. 2017 , publisher=

    2017

  166. [174]

    Journal of Convex Analysis , volume=

    Homogeneous functions and conjugacy , author=. Journal of Convex Analysis , volume=. 1998 , publisher=

    1998

  167. [175]

    Inf-convolution, sous-additivit

    Moreau, Jean Jacques , journal=. Inf-convolution, sous-additivit

  168. [176]

    Pacific Journal of Mathematics , volume=

    On general minimax theorems , author=. Pacific Journal of Mathematics , volume=

  169. [177]

    Gradient Flows:

    Ambrosio, Luigi and Gigli, Nicola and Savar. Gradient Flows:. 2008 , publisher=

    2008

  170. [178]

    Studia Mathematica , volume=

    Wasserstein metric and subordination , author=. Studia Mathematica , volume=

  171. [179]

    2009 , publisher=

    Convex Optimization Theory , author=. 2009 , publisher=

    2009

  172. [180]

    Zorzi, Mattia , journal=. Robust. 2016 , publisher=

    2016

  173. [181]

    On the robustness of the

    Zorzi, Mattia , journal=. On the robustness of the. 2017 , publisher=

    2017

  174. [182]

    IEEE Transactions on Information Theory , volume=

    Robust least-squares estimation with a relative entropy constraint , author=. IEEE Transactions on Information Theory , volume=. 2004 , publisher=

    2004

  175. [183]

    IEEE Transactions on Automatic Control , volume=

    Robust state space filtering under incremental model perturbations subject to a relative entropy tolerance , author=. IEEE Transactions on Automatic Control , volume=. 2012 , publisher=

    2012

  176. [184]

    Symmetric multilinear forms on

    Carando, Daniel and Rodr. Symmetric multilinear forms on. Linear Algebra and its Applications , volume=. 2019 , publisher=

    2019

  177. [185]

    Banach, Stefan , journal =

  178. [186]

    Mathematical Programming , volume=

    Implementable tensor methods in unconstrained convex optimization , author=. Mathematical Programming , volume=. 2021 , publisher=

    2021

  179. [187]

    Mathematical Programming , volume=

    Local convergence of tensor methods , author=. Mathematical Programming , volume=

  180. [188]

    Nagarajan, Vaishnavh and Kolter, J Zico , booktitle=

  181. [189]

    Improved training of

    Gulrajani, Ishaan and Ahmed, Faruk and Arjovsky, Martin and Dumoulin, Vincent and Courville, Aaron , booktitle=. Improved training of

  182. [190]

    Advances in Neural Information Processing Systems , pages=

    Stabilizing training of generative adversarial networks through regularization , author=. Advances in Neural Information Processing Systems , pages=

  183. [191]

    Advances in Neural Information Processing Systems , pages=

    Formal guarantees on the robustness of a classifier against adversarial manipulation , author=. Advances in Neural Information Processing Systems , pages=

  184. [192]

    Improving

    Jakubovitz, Daniel and Giryes, Raja , booktitle=. Improving

  185. [193]

    International Conference on Data Mining , pages=

    A unified gradient regularization family for adversarial examples , author=. International Conference on Data Mining , pages=

  186. [194]

    Neural Computation , volume=

    Unifying adversarial training algorithms with data gradient regularization , author=. Neural Computation , volume=. 2017 , publisher=

    2017

  187. [195]

    Schedae Informaticae , volume=

    Gradient Regularization Improves Accuracy of Discriminative Models , author=. Schedae Informaticae , volume=

  188. [196]

    Machine Learning with Applications , volume=

    Scaleable input gradient regularization for adversarial robustness , author=. Machine Learning with Applications , volume=. 2021 , pages=

    2021

  189. [197]

    Sensitivity analysis of

    Bartl, Daniel and Drapeau, Samuel and Ob \'oj, Jan and Wiesel, Johannes , journal =. Sensitivity analysis of

  190. [198]

    2013 , publisher=

    Lefkimmiatis, Stamatios and Ward, John Paul and Unser, Michael , journal=. 2013 , publisher=

    2013

  191. [199]

    Measuring complexity of learning schemes using

    Aziznejad, Shayan and Campos, Joaquim and Unser, Michael , journal=. Measuring complexity of learning schemes using

  192. [200]

    2013 , publisher=

    Lefkimmiatis, Stamatios and Unser, Michael , journal=. 2013 , publisher=

    2013

  193. [201]

    SIAM Journal on Imaging Sciences , volume=

    Total generalized variation , author=. SIAM Journal on Imaging Sciences , volume=

  194. [202]

    2014 , publisher=

    Hu, Yue and Ongie, Greg and Ramani, Sathish and Jacob, Mathews , journal=. 2014 , publisher=

    2014

  195. [203]

    2006 , publisher=

    Theory of Point Estimation , author=. 2006 , publisher=

    2006

  196. [204]

    1987 , publisher=

    Introduction to Optimization , author=. 1987 , publisher=

    1987

  197. [205]

    Multiplicative

    Pal, Soumik and Wong, Ting-Kam Leonard , journal=. Multiplicative. 2020 , publisher=

    2020

  198. [206]

    Annals of Probability , volume=

    Exponentially concave functions and a new information geometry , author=. Annals of Probability , volume=. 2018 , publisher=

    2018

  199. [207]

    Mathematics and Financial Economics , volume=

    The geometry of relative arbitrage , author=. Mathematics and Financial Economics , volume=. 2016 , publisher=

    2016

  200. [208]

    Logarithmic divergences from optimal transport and

    Wong, Ting-Kam Leonard , journal=. Logarithmic divergences from optimal transport and. 2018 , publisher=

    2018

  201. [209]

    Optimal versus naive diversification:

    DeMiguel, Victor and Garlappi, Lorenzo and Uppal, Raman , journal=. Optimal versus naive diversification:. 2009 , publisher=

    2009

  202. [210]

    Mathematics of Operations Research , volume=

    Robust sensitivity analysis for stochastic systems , author=. Mathematics of Operations Research , volume=. 2016 , publisher=

    2016

  203. [211]

    Sensitivity to serial dependency of input processes:

    Lam, Henry , journal=. Sensitivity to serial dependency of input processes:. 2018 , publisher=

    2018

  204. [212]

    Journal of Machine Learning Research , volume=

    Variance-based regularization with convex objectives , author=. Journal of Machine Learning Research , volume=

  205. [213]

    Statistics of robust optimization:

    Duchi, John C and Glynn, Peter W and Namkoong, Hongseok , journal=. Statistics of robust optimization:

  206. [214]

    Operations Research Letters , volume=

    Robust empirical optimization is almost the same as mean--variance optimization , author=. Operations Research Letters , volume=

  207. [215]

    Advances in Neural Information Processing Systems , pages=

    Distributionally Robust Optimization and Generalization in Kernel Methods , author=. Advances in Neural Information Processing Systems , pages=

  208. [216]

    2002 , publisher=

    A Primer of Real Analytic Functions , author=. 2002 , publisher=

    2002

  209. [217]

    2000 , publisher=

    Mathematical Methods and Models for Economists , author=. 2000 , publisher=

    2000

  210. [218]

    Operations Research , volume=

    Robust mean-covariance solutions for stochastic optimization , author=. Operations Research , volume=. 2007 , publisher=

    2007

  211. [219]

    1953 , publisher=

    Theory of Games and Economic Behavior , author=. 1953 , publisher=

    1953

  212. [220]

    Journal of Finance , volume=

    Portfolio Selection , author=. Journal of Finance , volume=

  213. [221]

    Management Science , volume=

    Robust growth-optimal portfolios , author=. Management Science , volume=

  214. [222]

    A descent lemma beyond

    Bauschke, Heinz H and Bolte, J. A descent lemma beyond. Mathematics of Operations Research , volume=

  215. [223]

    Mathematical Programming , volume=

    Golden ratio algorithms for variational inequalities , author=. Mathematical Programming , volume=. 2020 , publisher=

    2020

  216. [224]

    Journal of Multivariate Analysis , volume=

    A well-conditioned estimator for large-dimensional covariance matrices , author=. Journal of Multivariate Analysis , volume=. 2004 , publisher=

    2004

  217. [225]

    Proceedings of the IEEE , volume=

    Gradient-based learning applied to document recognition , author=. Proceedings of the IEEE , volume=. 1998 , publisher=

    1998

  218. [226]

    Mathematical Programming , volume=

    Uncertain convex programs: randomized solutions and confidence levels , author=. Mathematical Programming , volume=. 2005 , publisher=

    2005

  219. [227]

    Optimization Methods & Software , volume=

    Cutting-set methods for robust convex optimization with pessimizing oracles , author=. Optimization Methods & Software , volume=. 2009 , publisher=

    2009

  220. [228]

    Optimization Methods and Software , pages=

    On well-structured convex--concave saddle point problems and variational inequalities with monotone operators , author=. Optimization Methods and Software , pages=. 2021 , publisher=

    2021

  221. [229]

    2013 , publisher=

    Convergence of Probability Measures , author=. 2013 , publisher=

    2013

  222. [230]

    2000 , publisher=

    Topology , author=. 2000 , publisher=

    2000

  223. [231]

    Set-Valued and Variational Analysis , volume=

    Perspective functions: Properties, constructions, and examples , author=. Set-Valued and Variational Analysis , volume=. 2018 , publisher=

    2018

  224. [232]

    Journal of Optimization Theory and Applications , volume=

    Stochastic optimization problems with nondifferentiable cost functionals , author=. Journal of Optimization Theory and Applications , volume=. 1973 , publisher=

    1973

  225. [233]

    Econometrica , volume=

    Envelope theorems for arbitrary choice sets , author=. Econometrica , volume=

  226. [234]

    Wasserstein tube

    Aolaritei, Liviu and Fochesato, Marta and Lygeros, John and D. Wasserstein tube. Conference on Decision and Control , pages=

  227. [235]

    IEEE Transactions on Automatic Control , number =

    Distributional uncertainty propagation via optimal transport , author=. IEEE Transactions on Automatic Control , number =

  228. [236]

    Mathematical Programming (forthcoming) , pages=

    Wasserstein Distributionally Robust Estimation in High Dimensions: Performance Analysis and Optimal Hyperparameter Tuning , author=. Mathematical Programming (forthcoming) , pages=

  229. [237]

    International Conference on Learning Representations , year=

    Intriguing properties of neural networks , author=. International Conference on Learning Representations , year=

  230. [238]

    International Conference on Learning Representations , year=

    Explaining and Harnessing Adversarial Examples , author=. International Conference on Learning Representations , year=

  231. [239]

    International Conference on Learning Representations , year=

    Towards deep learning models resistant to adversarial attacks , author=. International Conference on Learning Representations , year=

  232. [240]

    Why do adversarial attacks transfer?

    Demontis, Ambra and Melis, Marco and Pintor, Maura and Jagielski, Matthew and Biggio, Battista and Oprea, Alina and Nita-Rotaru, Cristina and Roli, Fabio , booktitle=. Why do adversarial attacks transfer?

  233. [241]

    arXiv:1605.07277 , year=

    Transferability in machine learning: From phenomena to black-box attacks using adversarial samples , author=. arXiv:1605.07277 , year=

    arXiv

  234. [242]

    IEEE Symposium on Security and Privacy , pages=

    Distillation as a defense to adversarial perturbations against deep neural networks , author=. IEEE Symposium on Security and Privacy , pages=

  235. [243]

    Journal of the European Economic Association , volume=

    Pricing without priors , author=. Journal of the European Economic Association , volume=. 2008 , publisher=

    2008

  236. [244]

    Operations Research (forthcoming) , year=

    On the Robustness of Second-Price Auctions in Prior-Independent Mechanism Design , author=. Operations Research (forthcoming) , year=

  237. [245]

    Management Science , volume=

    Distributionally robust mechanism design , author=. Management Science , volume=. 2020 , publisher=

    2020

  238. [246]

    Higher order

    Salkeld, William , journal=. Higher order

  239. [247]

    Advances in Neural Information Processing Systems , pages=

    Wasserstein distributional robustness of neural networks , author=. Advances in Neural Information Processing Systems , pages=

  240. [248]

    arXiv:2209.10825 , year=

    Nonsmooth nonconvex-nonconcave minimax optimization: Primal-dual balancing and iteration complexity analysis , author=. arXiv:2209.10825 , year=

    arXiv

  241. [249]

    Adversarial classification:

    Garc. Adversarial classification:. Journal of Machine Learning Research , volume=

  242. [250]

    The many faces of adversarial risk:

    Pydi, Muni Sreenivas and Jog, Varun , journal=. The many faces of adversarial risk:

  243. [251]

    Journal of Machine Learning Research , volume=

    Existence and minimax theorems for adversarial surrogate risks in binary classification , author=. Journal of Machine Learning Research , volume=

  244. [252]

    Communications in Mathematical Sciences , volume=

    Extreme points of a ball about a measure with finite support , author=. Communications in Mathematical Sciences , volume=. 2017 , publisher=

    2017

  245. [253]

    Quantitative Finance , volume=

    Ambiguity in portfolio selection , author=. Quantitative Finance , volume=

  246. [254]

    2014 , publisher=

    Multistage Stochastic Optimization , author=. 2014 , publisher=

    2014

  247. [255]

    arXiv:2508.11858 , year=

    Optimality of Linear Policies in Distributionally Robust Linear Quadratic Control , author=. arXiv:2508.11858 , year=

  248. [256]

    Robust multidimensional pricing:

    Ko. Robust multidimensional pricing:. Mathematical Programming , volume=. 2022 , publisher=

    2022

  249. [257]

    arXiv:2603.29870 , year=

    Projection-Free Algorithms for Minimax Problems , author=. arXiv:2603.29870 , year=

  250. [258]

    Advances in Neural Information Processing Systems , pages=

    Projection-free methods for solving nonconvex-concave saddle point problems , author=. Advances in Neural Information Processing Systems , pages=

  251. [259]

    Mathematical Programming , volume=

    A unified single-loop alternating gradient projection algorithm for nonconvex-concave and convex-nonconcave minimax problems , author=. Mathematical Programming , volume=

  252. [260]

    Journal of Optimization Theory and Applications , volume=

    Subgradient methods for saddle-point problems , author=. Journal of Optimization Theory and Applications , volume=. 2009 , publisher=

    2009

  253. [261]

    Operations Research Letters , volume=

    Duality in robust optimization: primal worst equals dual best , author=. Operations Research Letters , volume=. 2009 , publisher=

    2009

  254. [262]

    Generalized

    Everett III, Hugh , journal=. Generalized

  255. [263]

    1994 , publisher=

    Interior-Point Polynomial Algorithms in Convex Programming , author=. 1994 , publisher=

    1994

  256. [264]

    Operations Research , volume=

    Sinkhorn distributionally robust optimization , author=. Operations Research , volume=

  257. [265]

    Regularization for

    Azizian, Wa. Regularization for. ESAIM: Control, Optimisation and Calculus of Variations , volume=. 2023 , publisher=

    2023

  258. [266]

    IEEE Journal on Selected Areas in Information Theory , volume=

    Flow-based distributionally robust optimization , author=. IEEE Journal on Selected Areas in Information Theory , volume=. 2024 , publisher=

    2024

  259. [267]

    Liu, Jiashuo and Wang, Tianyu and Lam, Henry and Namkoong, Hongseok and Blanchet, Jose , journal=

  260. [268]

    arXiv:2408.09672 , year=

    Regularization for adversarial robust learning , author=. arXiv:2408.09672 , year=

    arXiv

  261. [269]

    arXiv:2512.12550 , year=

    Iterative Sampling Methods for Sinkhorn Distributionally Robust Optimization , author=. arXiv:2512.12550 , year=

  262. [270]

    arXiv:2410.21231 , year=

    Vincent, Florian and Azizian, Wa. arXiv:2410.21231 , year=

  263. [271]

    International Conference on Machine Learning , pages=

    Convergence of Mean-Field Langevin Stochastic Descent-Ascent for Distributional Minimax Optimization , author=. International Conference on Machine Learning , pages=

  264. [272]

    Mathematical Programming , volume=

    Nonlinear distributionally robust optimization , author=. Mathematical Programming , volume=. 2025 , publisher=

    2025

  265. [273]

    Modified

    Kent, Carson and Li, Jiajin and Blanchet, Jose and Glynn, Peter W , booktitle=. Modified

  266. [274]

    Deterministic and Stochastic

    Yu, Di and Henderson, Shane G and Pasupathy, Raghu , journal=. Deterministic and Stochastic. 2025 , publisher=

    2025

  267. [275]

    SIAM Journal on Optimization , volume=

    Sparse inverse problems over measures: equivalence of the conditional gradient and exchange methods , author=. SIAM Journal on Optimization , volume=. 2019 , publisher=

    2019

  268. [276]

    Mathematical Programming , volume=

    Sparse optimization on measures with over-parameterized gradient descent , author=. Mathematical Programming , volume=. 2022 , publisher=

    2022

  269. [277]

    Advances in Neural Information Processing Systems , pages=

    On the global convergence of gradient descent for over-parameterized models using optimal transport , author=. Advances in Neural Information Processing Systems , pages=

  270. [278]

    Variational analysis in the

    Lanzetti, Nicolas and Terpin, Antonio and D. Variational analysis in the. arXiv:2406.10676 , year=

    arXiv

  271. [279]

    First-order conditions for optimization in the

    Lanzetti, Nicolas and Bolognani, Saverio and D. First-order conditions for optimization in the. SIAM Journal on Mathematics of Data Science , volume=. 2025 , publisher=

    2025

  272. [280]

    1997 , publisher=

    Introduction to Linear Optimization , author=. 1997 , publisher=

    1997

  273. [281]

    Advances in Neural Information Processing Systems , pages=

    Distributionally robust linear quadratic control , author=. Advances in Neural Information Processing Systems , pages=

  274. [282]

    Conference on Learning Theory , year=

    Optimal strategies and minimax lower bounds for online convex games , author=. Conference on Learning Theory , year=

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.