Distributionally Robust Optimization via Iterative Algorithms in Continuous Probability Spaces

Linglingzhi Zhu; Yao Xie; Yunqin Zhu

REVIEW 3 major objections 4 minor 1 cited by

Distributionally Robust Optimization via Iterative Algorithms in Continuous Probability Spaces

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Distributionally robust optimization over continuous worst-case distributions is solved by alternating subgradient steps in parameter space with a modified JKO scheme in Wasserstein space, giving $O(\epsilon^{-4})$ oracle complexity under…

desk verdict A coherent theoretical extension of flow-based DRO whose headline complexity claims rest on an oracle that Algorithm 2 does not realize; the promised numerical section is absent. read the letter →

arxiv 2412.20556 v2 pith:MHTEB3C4 submitted 2024-12-29 stat.ML cs.LGmath.OC

Linglingzhi Zhu , Yunqin Zhu , Yao Xie This is my paper · ORCID

classification stat.MLcs.LGmath.OC

keywords distributionallyrobustoptimizationWassersteinspaceBrenier'stheoremtransportmapmodifiedJKOschemeweakconvexityoraclecomplexitygeneralizedgeodesics

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 usually replaces the unknown worst-case distribution by a discrete measure supported on training points, which limits scalability and generalization. This paper tries to solve the robust minimax problem directly when the worst-case distribution is continuous, by representing it through Brenier's optimal transport map from a fixed continuous reference measure and optimizing over that map. The authors propose an iterative algorithm: subgradient descent on the decision parameter, with a modified Jordan-Kinderlehrer-Otto scheme to find the least favorable distribution, and they prove global convergence under weak convexity. The paper provides explicit guarantees: $O(\epsilon^{-4})$ subgradient calls and $O(\epsilon^{-4}\log(1/\epsilon))$ JKO steps for an $\epsilon$-stationary decision in the nonsmooth weakly convex setting, improved to $O(\epsilon^{-2})$ when the loss is smooth. This matters because robust classifiers trained this way are no longer confined to the training support and come with finite iteration bounds.

What carries the argument

The load-bearing machinery has three parts. Brenier's theorem gives, for a continuous reference $P$, a unique optimal transport map $T^Q_P$ with $W_2^2(Q,P) = \mathbb{E}_{\xi\sim P}\|\xi - T^Q_P(\xi)\|^2$, so choosing a distribution $Q$ is equivalent to choosing a map $T$. The modified JKO scheme (Algorithm 2) replaces the inner maximization by proximal steps with the term $W_\nu^2(Q, Q_i)$, where $W_\nu$ is the Wasserstein distance computed through optimal transport maps centered at a fixed continuous measure $\nu$; this generalized-geodesic proximal penalty is what makes the discrepancy strongly convex along the curves the analysis uses. The convergence argument then runs through the Moreau envelope $V_{1/(2\rho)}$ of the weakly convex max function $V(\varphi) := \max_Q H(\varphi,Q)$, whose gradient norm serves as the stationarity measure, and through a contraction estimate for the inner JKO iteration that yields an exponential rate whenever the discrepancy is $\lambda$-strongly convex along generalized geodesics centered at $\nu$ with $\lambda > \rho$.

What would settle it

Take a simple instance with a known solution, say a Gaussian reference $P$, a quadratic loss, and the $W_2^2$ discrepancy, and run Algorithm 2 with a neural-network transport map while monitoring the Wasserstein subgradient norm of the proximal objective. If for some accuracy $\epsilon'$ the required subgradient norm cannot be reached for any number of inner iterations, Assumption 4.2 fails and the claimed $O(\epsilon^{-4})$ and $O(\epsilon^{-2})$ bounds do not hold for that implementation; if it is reached, the observed inner iteration counts should match the exponential rate of Proposition 4.4.

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Extended reading notes

Core claim

On the paper's own terms, the central claim is Theorem 4.5: under weak convexity of the loss in the decision parameter, strong concavity of the inner objective along generalized geodesics, and an inexactness assumption on each modified JKO step, Algorithm 1 with step size $\eta = O(\epsilon^2)$ returns a $\varphi^*$ with $\|\nabla V_{1/(2\rho)}(\varphi^*)\| \le \epsilon$ using $O(\epsilon^{-4})$ subgradient oracle calls and $O(\epsilon^{-4}\log(1/\epsilon))$ modified JKO steps. Section 5 sharpens this: when the loss is smooth and the ambiguity set is compact, the same scheme reaches $\|\nabla V(\varphi^*)\| \le \epsilon$ with $O(\epsilon^{-2})$ gradient calls and $O(\epsilon^{-2}\log(1/\epsilon))$ JKO steps. The discovery is that the infinite-dimensional inner maximization over continuous distributions can be solved by proximal-point-like iterations in Wasserstein space rather than by finite linear programming over empirical measures, because Brenier's theorem turns distributions into transport maps and strong convexity along generalized geodesics gives exponential progress for the inner problem. The paper also shows that for the Wasserstein-2 discrepancy and for KL divergence to a log-concave reference, the needed strong convexity holds with regularization $\lambda$ large enough.

Load-bearing premise

The proof assumes that every modified JKO step actually returns a transport map whose Wasserstein subgradient of the proximal objective has norm at most $\epsilon'$; the paper does not give an algorithm that provably produces such a map, and the neural-network transport map maximization used in practice is nonconvex and unanalyzed, so the complexity guarantees apply only if this oracle can be realized.

Editorial extensions

If this is right

The least favorable distribution can be continuous, so the resulting robust decision function is defined on the whole input space rather than only on training points.
For weakly convex, possibly nonsmooth losses, the algorithm reaches an $\epsilon$-stationary point with $O(\epsilon^{-4})$ subgradient evaluations and $O(\epsilon^{-4}\log(1/\epsilon))$ modified JKO steps.
Under smoothness, the rates improve to $O(\epsilon^{-2})$ gradient evaluations and $O(\epsilon^{-2}\log(1/\epsilon))$ modified JKO steps.
The inner JKO iteration converges exponentially when the regularization parameter $\lambda$ exceeds the weak-concavity constant $\rho$ and the chosen discrepancy is strongly convex along generalized geodesics, covering $W_2^2$ and KL divergence with a log-concave reference.
The framework covers nonconvex models such as nonlinear regression, kernel SVM, and smooth neural networks, not just convex classifiers.

Reading between the lines

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

An implicit design choice the paper leaves open is the reference measure $\nu$ that centers the generalized geodesics; a poor choice could slow the inner JKO contraction or make the transport-map subproblem harder, so practical efficiency may hinge on $\nu$ rather than on the proven rates.
The complexity bounds count calls to an inexact JKO oracle that must itself be implemented, so the true end-to-end cost is the neural-network training cost inside Algorithm 2; proving convergence for that inner maximization is the natural next step.
The same Wasserstein-proximal template may apply to other infinite-dimensional robust problems, such as robust hypothesis testing or adversarial learning, whenever the discrepancy functional is strongly convex along fixed generalized geodesics.
The paper's final remark points toward finite-sample guarantees; a natural testable extension is to analyze the effect of replacing the true continuous reference $P$ by an empirical or kernel-density estimate on the convergence and robustness.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: the paper is a legitimate Wasserstein-space framework for continuous-LFD DRO, and the convergence analysis is mostly coherent. But its central complexity theorems are for an oracle that Algorithm 2 does not actually implement, and the abstract promises numerical results that are not in the submission.

What is new: fixing the base measure in generalized geodesic convexity is a real twist on the proximal-gradient/Wasserstein-proximal literature, and it lets the authors analyze a modified JKO scheme under weak concavity plus strong a.g.g. concavity. Theorem 3.3 is a clean weakly convex convergence argument, and Proposition 4.4’s exponential contraction under Assumption 4.2 checks out; I followed the algebra around (4.8) and the contraction step is sound. The smoothness section (Danskin, Lipschitz solution map, O(epsilon^{-2}) under smoothness) is a sensible improvement and the proofs are consistent. This is a genuine extension, not a repackaging.

The soft spot is exactly where the stress-test puts it. Assumption 4.2 asks that each inexact modified JKO step return a point with small Wasserstein subgradient norm. Algorithm 2’s Step 3 just says “solve” a parameterized transport-map maximization; no convergence, stationarity, or even feasibility guarantee is proved for that nonconvex theta-solver, and no argument shows the chosen transport-map class contains or certifies the exact JKO maximizer. So Theorem 4.5 and Theorem 5.3 are complexity bounds for an assumed oracle, not for a concrete algorithm. That does not invalidate the conditional theory, but it is a load-bearing gap between the stated results and any practical implementation. Relatedly, the abstract says numerical results demonstrate the method; the submitted version has no numerical section, and the conclusion explicitly defers finite-sample guarantees. Those two things should be reconciled.

On the citation pattern: the paper leans heavily on [XLCX24] and [CLTX24], both from overlapping groups, for several load-bearing lemmas. I do not see a circularity, and the imports look legitimate, but a referee should verify those lemmas carefully because a lot of weight sits on them.

Recommendation: send it to peer review. The theory is worth referee time, and the flaws are fixable: either construct or analyze a concrete inner solver, or clearly state that the complexity is for an idealized oracle; and add or explicitly withdraw the numerical claims. As submitted, it is a conditional contribution, not a fully supported one.

Referee Report

3 major / 4 minor

Summary. The paper studies distributionally robust optimization (DRO) when the worst-case distribution is continuous. It uses Brenier's theorem to rewrite the inner maximization over probability measures as an optimization over transport maps, and then proposes a two-loop algorithm: an outer subgradient update on the model parameters and an inner modified Jordan-Kinderlehrer-Otto (JKO) scheme to generate the least favorable distribution. The authors prove an O(ε^{-4}) oracle complexity for the weakly convex nonsmooth setting and an O(ε^{-2}) improvement under smoothness, both conditional on an inexact JKO oracle assumption. The abstract claims numerical results for neural transport-map implementations, but the submitted version contains no numerical section.

Significance. If fully established, the framework would provide a useful non-asymptotic convergence analysis for continuous DRO, complementing the discrete dual-reformulation literature and giving theoretical backing to neural-transport-map approaches. The formal analysis has clear strengths: the outer-loop weak-convexity argument in Theorem 3.3 is clean, the a.g.g. strong-concavity contraction in Proposition 4.4 is explicitly derived, and the complexity statements do not rely on fitted constants. However, the practical significance is currently limited because the central complexity theorem counts calls to an oracle whose existence is assumed but not realized by any concrete algorithm, and the advertised numerical validation is absent.

major comments (3)

[Section 4, Algorithm 2 and Theorem 4.5] The stated complexity bounds count oracle calls, not operations of a concrete algorithm. Assumption 4.2 postulates the existence of points Q_{i+1} whose Wasserstein subgradient norm is at most ε′, but Algorithm 2's Step 3 solves (or is assumed to solve) a nonconvex maximization over parameters θ of a transport map; the paper proves no convergence or stationarity guarantee for that θ-solver, no regularity such as strong concavity of the θ-objective, and no statement that the transport-map class {T_θ} contains or certifies the unconstrained JKO maximizer. Hence the product K·I in Theorem 4.5 is a count of oracle accesses under Assumption 4.2, not an operation count for Algorithm 2. The claim that the least favorable distribution is efficiently computable is therefore not established; please either supply a realizable inner solver with a convergence guarantee, or explicitly reframe the results as conditional on Assumption 4.2 and remove the algorithmic-efficiency claim.
[Abstract and Section 6] The abstract states that "Numerical results with neural network-based transport maps demonstrate that the proposed method enables both stable training of robust classifiers and effective worst-case inference for classification tasks," but the submitted manuscript contains no numerical experiments, figures, tables, or experimental setup, and the conclusion only lists finite-sample guarantees as future work. This claim should be supported by experiments or removed from the abstract and introduction.
[Section 5, Lemma 5.2 and Theorem 5.3] The smooth analysis requires λ>L, but this is not stated as an assumption. Lemma 5.2 begins with "from Lemma 4.3 we know that H(φ,·) is (λ−L)-strongly concave a.g.g."; Lemma 4.3 only gives (λ−ρ)-strong concavity under λ>ρ, and Assumption 5.1 does not relate λ to the gradient-Lipschitz constant L. Since all subsequent constants contain λ−L and can be undefined or negative when λ≤L, the theorem should either add λ>L explicitly or justify that ρ can be replaced by L while preserving λ>ρ.

minor comments (4)

[Section 4.2, Assumption 4.1 and Proposition 4.2] For the Wasserstein discrepancy D(Q,P)=1/2 W_2^2(Q,P), Proposition 4.2(i) establishes strong convexity only along generalized geodesics centered at P, not at an arbitrary ν∈P_2^r. Assumption 4.1(ii) and Proposition 4.4, however, use an arbitrary ν as the center of a.g.g. strong convexity and as the base of the proximal term W_ν^2. The paper should state that for the Wasserstein case one must take ν=P in Algorithm 2; otherwise the assumption is not satisfiable for the main Wasserstein example.
[Sections 3-4, imported lemmas] Several load-bearing results are imported without statements: the equivalence in (3.1) is attributed to [XCX24, Lemma A.1], and the proof of Proposition 4.4 uses [XLCX24, Lemmas 3.3 and 3.4] and [CLTX24, Lemma 4.1]. Since these lemmas carry the Wasserstein subdifferential calculus and the a.g.g. strong-convexity inequality, including their statements in an appendix would make the proof verifiable without consulting four external papers.
[Section 1, contributions list] The name "Jordan-Kinderleherer-Otto" is a typo; it should be "Jordan-Kinderlehrer-Otto".
[Section 3, Theorem 3.3] The stationarity measure V_{1/2ρ} uses the Moreau-envelope parameter r=2ρ, which requires ρ>0; the weakly convex case with ρ=0 is not covered by the theorem as stated and should be handled separately or excluded explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity by construction; central guarantees are conditional on an explicit oracle assumption, and imported lemmas are independent published results.

full rationale

The derivation chain is not circular. Theorem 3.3 follows from the epsilon-subproblem definition and weak convexity without reintroducing the conclusion as an input. Proposition 4.4 states a conditional contraction: under Assumption 4.2 (an epsilon-prime-accurate Wasserstein subgradient of each proximal subproblem) it bounds W_nu(Q_i,Q*) and the objective gap; this is a standard inexact proximal-point error bound, not a restatement of the theorem's target. Theorem 4.5 then combines these bounds and chooses epsilon-prime = O(epsilon) and eta = O(epsilon^2); the complexity counts are derived, not assumed. The load-bearing inequalities taken from the authors' prior work (transport-map equivalence in Section 3 after (3.1); Wasserstein subdifferential calculus and strong-concavity inequality in the proof of Proposition 4.4, citing [XLCX24, Lemma 3.3-3.4] and [CLTX24, Lemma 4.1]) are independent published statements rather than restatements of the present target, so they are legitimate support rather than circularity. The genuinely weak point is that Algorithm 2's parameterized transport-map maximization is not proved to realize Assumption 4.2, since the theta-problem is nonconvex and no convergence of its solver is shown; the theorem is therefore oracle-conditional for any concrete implementation. The conclusion also defers finite-sample guarantees, and the abstract's promised numerical results are absent. These are missing realization and support issues, not circularity by construction.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the regularized DRO formulation with fixed lambda, Brenier's theorem for continuous P, strong convexity of the discrepancy along generalized geodesics, and an inexact JKO oracle. No new physical or mathematical entities are introduced. The free parameters are algorithmic hyperparameters rather than fitted constants.

free parameters (3)

regularization parameter lambda
Chosen by bisection or by hand; the paper assumes lambda fixed and drops the constant lambda*delta, but does not analyze how lambda maps to the ambiguity radius delta.
outer step size eta = 1/sqrt(K) or (lambda-L)/(2(4lambda-3L)L)
Set in advance based on horizon K or on smoothness constants; not fitted to data but essential to the stated convergence rates.
JKO proximal step size gamma
Algorithm 2 parameter; the exponential convergence result in Proposition 4.4 depends on gamma, but no adaptive selection rule is provided.

assumptions (6)

domain assumption Reference distribution P has a continuous density, P in P^r_2.
Brenier's theorem and the transport-map equivalence in Section 3 require P to be absolutely continuous; the empirical distribution case is deferred to future work.
domain assumption Assumption 3.1: loss is rho-weakly convex and L-Lipschitz in the parameter phi for each xi.
This is the foundation for Lemma 3.1 and Theorem 3.3, and it excludes non-Lipschitz losses.
domain assumption Assumption 4.1(ii): the discrepancy D is lambda-strongly convex along generalized geodesics centered at nu, with lambda > rho.
Needed for strong concavity of H and for the exponential JKO rate. For Wasserstein-2 this only holds centered at P, not arbitrary nu.
ad hoc to paper Assumption 4.2: an inexact JKO oracle returns a point with Wasserstein subgradient norm bounded by epsilon-prime.
The oracle is central to Proposition 4.4 and Theorem 4.5, but no constructive algorithm is given to compute such a point.
domain assumption Assumption 5.1: loss is continuously differentiable with L-Lipschitz gradient and the support X is compact.
Required for the generalized Danskin theorem and the improved O(epsilon^-2) complexity in Section 5.
domain assumption For KL divergence, strong convexity a.g.g. requires P log-concave, p proportional to exp(-G) with G 1-strongly convex.
This restriction comes from Proposition 4.2(ii) and limits the KL-based variant to log-concave reference densities.

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Pith. "Pith review of Distributionally Robust Optimization via Iterative Algorithms in Continuous Probability Spaces." pith.science (2026). https://pith.science/paper/MHTEB3C4

@misc{pith2026241220556,
  author       = {Pith},
  title        = {Pith review of: Distributionally Robust Optimization via Iterative Algorithms in Continuous Probability Spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MHTEB3C4}},
  note         = {Machine review of arXiv:2412.20556}
}

read the original abstract

We study distributionally robust optimization (DRO) for robust inference when the worst-case distribution is continuous, leading to significant computational challenges due to the infinite-dimensional nature of the optimization problem. Unlike traditional discrete DRO approaches, which often suffer from scalability issues, limited generalization, and costly worst-case inference, our framework exploits Brenier's theorem to characterize the least favorable distribution as the pushforward of a transport map from a continuous reference measure. This characterization motivates our study of the minimax problem in Wasserstein space. We propose an iterative algorithmic framework with multiple variants and establish global convergence guarantees under mild assumptions, deriving complexity bounds in terms of subgradient evaluations and inexact Jordan-Kinderlehrer-Otto updates. Numerical results with neural network-based transport maps demonstrate that the proposed method enables both stable training of robust classifiers and effective worst-case inference for classification tasks.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Gradient Flow Sampler-based Distributionally Robust Optimization
math.OC 2025-10 conditional novelty 5.0 of 10

Entropy-regularized Wasserstein DRO can be solved by sampling from a Gibbs worst-case distribution with gradient-flow samplers, giving new WFR/SVGD algorithms and a principled recovery of WRM.

Reference graph

Works this paper leans on

32 extracted references · 27 canonical work pages · cited by 1 Pith paper

[1]

Convergence of descent methods for semi-algebraic and tame problems: Proximal algorithms, forward--backward splitting, and regularized G auss-- S eidel methods

Hedy Attouch, J \'e r \^o me Bolte, and Benar Fux Svaiter. Convergence of descent methods for semi-algebraic and tame problems: Proximal algorithms, forward--backward splitting, and regularized G auss-- S eidel methods. Mathematical Programming , 137(1):91--129, 2013

work page 2013
[2]

Gradient flows: In metric spaces and in the space of probability measures

Luigi Ambrosio, Nicola Gigli, and Giuseppe Savar \'e . Gradient flows: In metric spaces and in the space of probability measures . Springer Science & Business Media, 2008

work page 2008
[3]

Quantifying distributional model risk via optimal transport

Jose Blanchet and Karthyek Murthy. Quantifying distributional model risk via optimal transport. Mathematics of Operations Research , 44(2):565--600, 2019

work page 2019
[4]

Optimal transport-based distributionally robust optimization: Structural properties and iterative schemes

Jose Blanchet, Karthyek Murthy, and Fan Zhang. Optimal transport-based distributionally robust optimization: Structural properties and iterative schemes. Mathematics of Operations Research , 47(2):1500--1529, 2022

work page 2022
[5]

Polar factorization and monotone rearrangement of vector-valued functions

Yann Brenier. Polar factorization and monotone rearrangement of vector-valued functions. Communications on Pure and Applied Mathematics , 44(4):375--417, 1991

work page 1991
[6]

Moment problems and semidefinite optimization

Dimitris Bertsimas and Jay Sethuraman. Moment problems and semidefinite optimization. In Handbook of Semidefinite Programming: Theory, Algorithms, and Applications , pages 469--509. Springer, 2000

work page 2000
[7]

Robust solutions of optimization problems affected by uncertain probabilities

Aharon Ben-Tal, Dick Den Hertog, Anja De Waegenaere, Bertrand Melenberg, and Gijs Rennen. Robust solutions of optimization problems affected by uncertain probabilities. Management Science , 59(2):341--357, 2013

work page 2013
[8]

Mirror and preconditioned gradient descent in W asserstein space

Cl \'e ment Bonet, Th \'e o Uscidda, Adam David, Pierre-Cyril Aubin-Frankowski, and Anna Korba. Mirror and preconditioned gradient descent in W asserstein space. arXiv preprint arXiv:2406.08938 , 2024

arXiv 2024

Show all 32 references

[9]

Convergence of flow-based generative models via proximal gradient descent in W asserstein space

Xiuyuan Cheng, Jianfeng Lu, Yixin Tan, and Yao Xie. Convergence of flow-based generative models via proximal gradient descent in W asserstein space. IEEE Transactions on Information Theory , 2024

2024
[10]

A robust optimization perspective on stochastic programming

Xin Chen, Melvyn Sim, and Peng Sun. A robust optimization perspective on stochastic programming. Operations Research , 55(6):1058--1071, 2007

2007
[11]

Stochastic model-based minimization of weakly convex functions

Damek Davis and Dmitriy Drusvyatskiy. Stochastic model-based minimization of weakly convex functions. SIAM Journal on Optimization , 29(1):207--239, 2019

2019
[12]

Distributionally robust optimization under moment uncertainty with application to data-driven problems

Erick Delage and Yinyu Ye. Distributionally robust optimization under moment uncertainty with application to data-driven problems. Operations Research , 58(3):595--612, 2010

2010
[13]

Distributionally robust stochastic optimization with W asserstein distance

Rui Gao and Anton Kleywegt. Distributionally robust stochastic optimization with W asserstein distance. Mathematics of Operations Research , 48(2):603--655, 2023

2023
[14]

The variational formulation of the F okker-- P lanck equation

Richard Jordan, David Kinderlehrer, and Felix Otto. The variational formulation of the F okker-- P lanck equation. SIAM Journal on Mathematical Analysis , 29(1):1--17, 1998

1998
[15]

Modified F rank W olfe in probability space

Carson Kent, Jiajin Li, Jose Blanchet, and Peter W Glynn. Modified F rank W olfe in probability space. Advances in Neural Information Processing Systems , 34:14448--14462, 2021

2021
[16]

Distributionally robust optimization

Daniel Kuhn, Soroosh Shafieezadeh-Abadeh, and Wolfram Wiesemann. Distributionally robust optimization. arXiv preprint arXiv:2411.02549 , 2024

2024 arXiv
[17]

First-order conditions for optimization in the W asserstein space

Nicolas Lanzetti, Saverio Bolognani, and Florian D \"o rfler. First-order conditions for optimization in the W asserstein space. arXiv preprint arXiv:2209.12197 , 2022

2022 arXiv
[18]

A first-order algorithmic framework for distributionally robust logistic regression

Jiajin Li, Sen Huang, and Anthony Man-Cho So. A first-order algorithmic framework for distributionally robust logistic regression. Advances in Neural Information Processing Systems , 32, 2019

2019
[19]

Understanding notions of stationarity in nonsmooth optimization: A guided tour of various constructions of subdifferential for nonsmooth functions

Jiajin Li, Anthony Man-Cho So, and Wing-Kin Ma. Understanding notions of stationarity in nonsmooth optimization: A guided tour of various constructions of subdifferential for nonsmooth functions. IEEE Signal Processing Magazine , 37(5):18--31, 2020

2020
[20]

Data-driven distributionally robust optimization using the W asserstein metric: Performance guarantees and tractable reformulations

Peyman Mohajerin Esfahani and Daniel Kuhn. Data-driven distributionally robust optimization using the W asserstein metric: Performance guarantees and tractable reformulations. Mathematical Programming , 171(1):115--166, 2018

2018
[21]

Stochastic gradient methods for distributionally robust optimization with f -divergences

Hongseok Namkoong and John C Duchi. Stochastic gradient methods for distributionally robust optimization with f -divergences. Advances in Neural Information Processing Systems , 29, 2016

2016
[22]

Frameworks and results in distributionally robust optimization

Hamed Rahimian and Sanjay Mehrotra. Frameworks and results in distributionally robust optimization. Open Journal of Mathematical Optimization , 3:1--85, 2022

2022
[23]

Lectures on Stochastic Programming: Modeling and Theory

Alexander Shapiro, Darinka Dentcheva, and Andrzej Ruszczynski. Lectures on Stochastic Programming: Modeling and Theory . SIAM, 2021

2021
[24]

On duality theory of conic linear problems

Alexander Shapiro. On duality theory of conic linear problems. Nonconvex Optimization and its Applications , 57:135--155, 2001

2001
[25]

Distributionally robust stochastic programming

Alexander Shapiro. Distributionally robust stochastic programming. SIAM Journal on Optimization , 27(4):2258--2275, 2017

2017
[26]

The W asserstein proximal gradient algorithm

Adil Salim, Anna Korba, and Giulia Luise. The W asserstein proximal gradient algorithm. Advances in Neural Information Processing Systems , 33:12356--12366, 2020

2020
[27]

Certifying some distributional robustness with principled adversarial training

Aman Sinha, Hongseok Namkoong, Riccardo Volpi, and John Duchi. Certifying some distributional robustness with principled adversarial training. arXiv preprint arXiv:1710.10571 , 2017

2017 arXiv
[28]

Likelihood robust optimization for data-driven problems

Zizhuo Wang, Peter W Glynn, and Yinyu Ye. Likelihood robust optimization for data-driven problems. Computational Management Science , 13:241--261, 2016

2016
[29]

Information N ewton's flow: Second-order optimization method in probability space

Yifei Wang and Wuchen Li. Information N ewton's flow: Second-order optimization method in probability space. arXiv preprint arXiv:2001.04341 , 2020

2001 arXiv
[30]

Normalizing flow neural networks by JKO scheme

Chen Xu, Xiuyuan Cheng, and Yao Xie. Normalizing flow neural networks by JKO scheme. Advances in Neural Information Processing Systems , 36, 2024

2024
[31]

Flow-based distributionally robust optimization

Chen Xu, Jonghyeok Lee, Xiuyuan Cheng, and Yao Xie. Flow-based distributionally robust optimization. IEEE Journal on Selected Areas in Information Theory , 2024

2024
[32]

On linear optimization over W asserstein balls

Man-Chung Yue, Daniel Kuhn, and Wolfram Wiesemann. On linear optimization over W asserstein balls. Mathematical Programming , 195(1):1107--1122, 2022

2022

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Convergence of descent methods for semi-algebraic and tame problems: Proximal algorithms, forward--backward splitting, and regularized G auss-- S eidel methods

Hedy Attouch, J \'e r \^o me Bolte, and Benar Fux Svaiter. Convergence of descent methods for semi-algebraic and tame problems: Proximal algorithms, forward--backward splitting, and regularized G auss-- S eidel methods. Mathematical Programming , 137(1):91--129, 2013

work page 2013

[2] [2]

Gradient flows: In metric spaces and in the space of probability measures

Luigi Ambrosio, Nicola Gigli, and Giuseppe Savar \'e . Gradient flows: In metric spaces and in the space of probability measures . Springer Science & Business Media, 2008

work page 2008

[3] [3]

Quantifying distributional model risk via optimal transport

Jose Blanchet and Karthyek Murthy. Quantifying distributional model risk via optimal transport. Mathematics of Operations Research , 44(2):565--600, 2019

work page 2019

[4] [4]

Optimal transport-based distributionally robust optimization: Structural properties and iterative schemes

Jose Blanchet, Karthyek Murthy, and Fan Zhang. Optimal transport-based distributionally robust optimization: Structural properties and iterative schemes. Mathematics of Operations Research , 47(2):1500--1529, 2022

work page 2022

[5] [5]

Polar factorization and monotone rearrangement of vector-valued functions

Yann Brenier. Polar factorization and monotone rearrangement of vector-valued functions. Communications on Pure and Applied Mathematics , 44(4):375--417, 1991

work page 1991

[6] [6]

Moment problems and semidefinite optimization

Dimitris Bertsimas and Jay Sethuraman. Moment problems and semidefinite optimization. In Handbook of Semidefinite Programming: Theory, Algorithms, and Applications , pages 469--509. Springer, 2000

work page 2000

[7] [7]

Robust solutions of optimization problems affected by uncertain probabilities

Aharon Ben-Tal, Dick Den Hertog, Anja De Waegenaere, Bertrand Melenberg, and Gijs Rennen. Robust solutions of optimization problems affected by uncertain probabilities. Management Science , 59(2):341--357, 2013

work page 2013

[8] [8]

Mirror and preconditioned gradient descent in W asserstein space

Cl \'e ment Bonet, Th \'e o Uscidda, Adam David, Pierre-Cyril Aubin-Frankowski, and Anna Korba. Mirror and preconditioned gradient descent in W asserstein space. arXiv preprint arXiv:2406.08938 , 2024

arXiv 2024