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REVIEW 4 major objections 4 minor 36 references

Representation-Aware Distributionally Robust Optimization: A Knowledge Transfer Framework

T0 review · 4 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read The paper shows that making Wasserstein DRO representation-aware—via a transport cost that penalizes perturbations along known representation directions—is exactly equivalent to adding a seminorm shrinkage penalty toward the representation'

desk verdict Genuine extension of KG-WDRO with a clean duality result, but Theorem 2 has an inconsistent Σ definition that undermines the inference framework; needs major correction. read the letter →

arxiv 2509.09371 v2 pith:2Z2IDNBD submitted 2025-09-11 stat.ME cs.LG

Zitao Wang , Nian Si , Molei Liu This is my paper
classification stat.MEcs.LG MSC 62J0562F1290C25
keywords Wassersteindistributionallyrobustoptimizationrepresentationlearningtransferseminormregularizationconfidenceregionshyperparameterselectionsquare-rootlassoinvariantsubspace
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

The paper proposes READ, a distributionally robust estimation framework that uses external knowledge about a low-dimensional predictive representation to reshape the geometry of robustness. Instead of treating all covariate perturbations equally, READ's transport cost penalizes perturbations that move the estimated representation coordinates, so the worst-case adversary is steered away from the stable signal. The paper proves that for squared loss this representation-aware DRO objective is exactly a regularized regression whose penalty is a 'ridge-complement' seminorm: it shrinks the coefficient vector toward the column space of the representation, with per-direction strength controlled by alignment parameters. It then derives the limiting distribution of the resulting estimator, giving an explicit bias term, and uses that theory to construct confidence regions and to select both the Wasserstein radius and the alignment parameters adaptively. A sympathetic reader would care because this turns vague prior knowledge about stable structure into a principled, automatically tuned shrinkage that still protects against shifts orthogonal to the representation.

What carries the argument

The load-bearing object is the representation-aware transport cost c_{q,Λ}(x,u)=||x-u||_q^2 + ||Θ^T(x-u)||_Λ^2, an extended norm that makes perturbations along Θ-directions expensive. Its convex dual is the ridge-complement seminorm φ_{p,Λ}(β)=min_κ {||β-Θκ||_p^2 + ||κ||_{Λ^{-1}}^2}^{1/2}, which decomposes the parameter into a representation-aligned part (penalized by Λ^{-1}) and an orthogonal residual (penalized by the p-norm). This dual pair is the mechanism that converts representation-aware robustness into a computationally tractable regularized regression, and it also drives the asymptotic analysis: the confidence regions and the bias term in Theorem 2 are expressed through ψ_{p,Λ}, the

What would settle it

Simulate data with a shift that is not orthogonal to Θ (i.e., the future parameter β' = Θκ + ε' with large component along span(Θ)), then compare the coverage of READ's confidence region for β' against standard DRO; if READ's coverage falls below the nominal level while standard DRO's does not, the orthogonality assumption is violated. Alternatively, check Theorem 2 directly by Monte Carlo: for a known β*, compute the empirical limiting bias of β̂_{η,Λ} at δ=η/N and compare it to √η Σ^{-1}∇V(β*); a mismatch indicates an error in the variance or bias derivation.

Watch

Extended reading notes

Core claim

READ's central claim is that the representation-aware cost c_{q,Λ}(x,u)=||x-u||_q^2 + ||Θ^T(x-u)||_Λ^2 makes the Wasserstein DRO objective for linear regression equal to inf_β {√MSE_N(β) + √δ φ_{p,Λ}(β)}^2, where φ_{p,Λ}(β) = min_κ {||β-Θκ||_p^2 + ||κ||_{Λ^{-1}}^2}^{1/2}. The regularizer φ_{p,Λ} is a seminorm that vanishes on span(Θ), so the estimator is shrunk toward the representation's subspace, and the alignment matrix Λ controls how strongly each representation direction is trusted. Under mild regularity, the READ estimator satisfies √N(β̂_{η,Λ}-β*) ⇝ Σ^{-1}H + √η Σ^{-1}∇V(β*), with H a centered normal vector and V a sensitivity function; this bias-variance decomposition holds for all Λ

Load-bearing premise

The entire framework depends on the premise that an external representation Θ approximates a true invariant subspace Θ*, and that distribution shifts between environments are approximately orthogonal to its column space; if shifts also move the representation coordinates or Θ is a poor proxy for the true structure, READ's shrinkage can inject bias without delivering the promised robustness gain.

Editorial extensions

If this is right

  • If the equivalence in Corollary 1 holds, then any implementation of READ can be solved with standard cone programming, because the objective is a p-th order cone program.
  • The asymptotic characterization implies that, for any fixed Λ, the READ estimator has a Gaussian limit with a bias term proportional to √η Σ^{-1}∇V(β*), so inference is valid even when the radius is tuned by the proposed profile method.
  • Because the δ-selection rule is based on a quantile of ψ*_{p,Λ}(H) and is shown to hold for all Λ, the resulting confidence regions remain valid even after the alignment parameters are selected adaptively.
  • When the true parameter lies in the representation's span (perfect alignment), Theorem 3 shows the constrained estimator achieves the efficiency of the oracle restricted to that subspace, with strictly smaller asymptotic variance than unconstrained estimation.
  • In deployment settings where future distribution shifts are approximately orthogonal to the representation, the paper argues that READ's ambiguity set is aligned with the invariance structure, giving better coverage of future model parameters than standard DRO.

Reading between the lines

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

  • A direct testable consequence the author leaves implicit: the optimal alignment parameter Λ* selected by Algorithm 2 provides a data-driven measure of how informative the external representation is; if Λ* is near zero, the representation is uninformative for the target, and READ defaults to standard WDRO.
  • The same extended-norm duality used here could be applied to nonlinear representations by replacing Θ^T x with a feature map, yielding a kernelized ridge-complement regularizer; the paper does not explore this but the mechanism suggests it.
  • The variational representation in Proposition 6 could be used to construct non-asymptotic confidence sets via self-normalized empirical processes, extending the asymptotic regions to small samples; this is an extension beyond the paper's current claims.
  • The explicit bias formula in Example 2 gives a practical way to compare two candidate representations Θ1 and Θ2 without refitting: compute the asymptotic bias norm for each and prefer the one with smaller value, subject to cross-validation.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper introduces READ (REpresentation-Aware Distributionally Robust estimation), a Wasserstein DRO framework in which the transport cost incorporates an external representation matrix Θ and alignment hyperparameters Λ. The authors derive a dual reformulation showing that the READ objective for squared loss is equivalent to an MSE-plus-seminorm-regularized problem, with a similar result for logistic/hinge loss. They further propose an asymptotic theory for the READ estimator, a robust Wasserstein profile inference method for selecting the radius δ, an optimization-based procedure for selecting Λ, and a discussion of generalization to future environments. The empirical section reports simulations comparing READ with several transfer-learning and DRO baselines.

Significance. If the duality result and the asymptotic theory were correct, the paper would make a useful conceptual contribution: it connects representation-aware regularization to Wasserstein DRO, gives a tractable cone-program formulation, and proposes principled hyperparameter selection. The unification of seminorm regularizers under extended-norm transport costs is elegant and extends known results for square-root lasso and related estimators. However, the central asymptotic inference claim (Theorem 2) is internally inconsistent as written, the appendix containing all proofs is absent, and the abstract promises a single-cell multi-omics application that does not appear in the manuscript. These issues prevent the paper from being accepted in its current form.

major comments (4)
  1. [§4.2–§4.3, Theorem 2, Example 2, Proposition 5] The statement of Theorem 2 is inconsistent with the definition of Σ. In §4.2 and Assumption 2, Σ := E_{P*}[∇β h(X,Y;β*)]; for squared loss this is −E[XX^T]. Theorem 2 instead defines Σ := Cov_{P*}(h(X,Y;β*)). For h=(y−x^Tβ)x, ∇βh=−xx^T while Cov(h)=σ²E[xx^T], so the two quantities differ by a factor σ² and a sign. With the theorem's Σ=Cov(h), the limiting covariance of the first term becomes σ^{-2}Σ_x^{-1} instead of the OLS covariance σ²Σ_x^{-1}. A first-order expansion from Proposition 5 yields √N(β̂−β*) ⇝ Σ_x^{-1}H + √η Σ_x^{-1}∇βV(β*) (with H the normalized empirical score), not the expression stated in Theorem 2. Since Algorithm 1 (radius selection), Algorithm 2 (Λ selection), and Proposition 8 (validity after tuning) all invoke Theorem 2, the asymptotic inference framework is not established as written.
  2. [Appendix; Corollary 1; Propositions 2–6; Theorems 1–3] The manuscript repeatedly states that proofs are given in the Appendix, but no appendix is present in the submitted arXiv version. Thus Lemma 1, Propositions 2–6, Theorems 1–3, and the closed-form claims in Example 2 cannot be verified. These proofs are load-bearing: the duality reformulation, the asymptotic distribution, and the validity of the confidence regions all depend on them. The missing appendix is a blocking issue for evaluation.
  3. [Abstract and §7 vs. §6] The abstract states that 'Simulations and a single-cell multi-omics application demonstrate the advantages of READ,' and the conclusion refers to 'extensive simulations and a real-world study.' Section 6 contains simulations only; no single-cell or multi-omics data analysis appears anywhere in the full text. This is an unsupported empirical claim and must be either implemented in the manuscript or removed from the abstract and conclusion.
  4. [§5.3] The abstract claims that the resulting regions achieve 'higher coverage of future model parameters,' and the introduction promises robust generalization over future environments sharing the representation. Section 5.3 is a heuristic discussion: the decomposition β'=Θ*κ+ε' with Θ*^T ε'≈0 is an assumption, and no theorem or quantitative statement about coverage of future parameters is derived. Figure 1 is illustrative, not a formal guarantee. The future-coverage claim advertised in the abstract is therefore unsupported.
minor comments (4)
  1. [Throughout] The notation is badly corrupted: 'Λ91', 'p91 + q91', 'δ91_N', and 'λ91_m,N' presumably mean Λ^{-1}, 1/p+1/q, δ_N^{-1}, and λ_{m,N}^{-1}, respectively. The manuscript must be typeset cleanly before it can be reviewed reliably.
  2. [§3.1] 'Inspired by Lemma 1 and Theorem 2 & 3' references results that appear later; this forward reference should be corrected after renumbering.
  3. [§4.3, after Theorem 2] The displayed expression for the bias, 'ηα(Λ)/∥β*∥²_ΨΛ ∥ΨΛ(β*)∥²_2', is dimensionally unclear and uses notation not defined in the text; please clarify the precise form and derivation.
  4. [§5.2, Algorithm 2] Step 5 of Algorithm 2 says 'Perform a (cross) validation on a'; the validation criterion, the split, and the way a† is selected should be specified precisely.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the core READ duality is a self-contained convex-conjugate computation; the cited prior work and the representation-invariance generalization claim are not load-bearing reductions.

full rationale

The central formal claim (Corollary 1) is derived by applying Proposition 2 to the extended norm c_{q,Λ}; Proposition 4 computes the dual seminorm φ_{p,Λ} by a standard infimal-convolution/Fenchel conjugate calculation. The regularizer is therefore constructed from the cost by duality, not assumed to equal the DRO objective; no fitted parameter or target quantity is inserted into the equivalence. The asymptotic analysis (Theorem 1 and Theorem 2) is based on a first-order expansion of the dual objective in Proposition 5 and standard empirical-process convergence under Assumptions 1–6; it does not invoke a uniqueness theorem from the authors' prior work, nor does it define β* in terms of READ. Algorithms 1 and 2 select δ and Λ using the same asymptotic formulas, and Proposition 8 reuses those formulas; this is internal consistency rather than circularity—the chosen Λ is a fitted input, but the paper does not rename that fit as an independent prediction. The generalization discussion in Section 5.3 is explicitly conditional on the model β*=Θ*κ+ε with ε⊥spanΘ* and future perturbations nearly orthogonal to Θ*, and the simulation in Section 6.2.4 generates test environments from exactly that model; this weakens the empirical demonstration but is not an equation-level circular reduction. The paper contains self-citations (Wang et al. 2025; Blanchet et al. 2022 with co-author N. Si), but these are used for priority/comparison and for published RWPI machinery, not as unverified premises. Separately, Theorem 2 appears internally inconsistent because Σ is defined as E[∇βh] in Assumption 2/Example 2 but restated as Cov(h) in Theorem 2; for squared loss h=(Y−X^Tβ)X these differ by sign and a σ^2 factor. That is a genuine correctness concern for the inference claims, but it is not a circular reduction, so it does not affect the circularity score.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central derivation rests on standard convex duality (imported) and on the domain assumption that a stable low-dimensional representation Θ exists and shifts are representation-invariant. The hyperparameters Λ and δ are fitted to the data via Algorithms 1-2, so they count as free parameters; the elastic-net and grid-search constants in these algorithms are also chosen by hand. No new physical entities are introduced.

free parameters (4)
  • Alignment hyperparameters Λ = Selected by Algorithm 2 grid search over a,b; ablation example λ*=10
    Controls the cost penalty along each representation direction; tuned data-adaptively to minimize estimated bias.
  • Wasserstein radius δ = δα(Λ)=η_α(Λ)/N, with η_α estimated by Monte Carlo in Algorithm 1
    Defines the ambiguity set size; selected to give 1-α coverage of β*.
  • Elastic-net tuning ν, μ = ν=0 or 0.3, μ=1 in simulations
    Used in Algorithm 2 Step 3 to compute κ_init; chosen by hand.
  • Grid parameters a,b = Grid search over a,b, cross-validation on a
    Parameterize Λ(a,b)=a diag(|κ_init|^b); grid ranges not fully specified.
assumptions (5)
  • standard math Strong duality of Wasserstein DRO with l.s.c. costs (Proposition 1)
    Imported from Esfahani and Kuhn 2018, Blanchet and Murthy 2019, Gao and Kleywegt 2023; used in Section 2.2.
  • standard math Toland's duality for nonconvex programs
    Used to derive Propositions 2 and 3 in Section 2.3.
  • domain assumption Low-rank representation model X=Θ*U+ε_X, Y=α^T U+ε_Y with Θ*ᵀΘ*=I (LowRank)
    Introduced in Section 3.1; motivates the representation-aware cost and underlies all simulations.
  • domain assumption Representation invariance: future shifts β'=Θ*κ+ε' with ε' nearly orthogonal to Θ*
    Stated in Section 5.3; basis for the generalization guarantee claim.
  • domain assumption Regularity Assumptions 1-7 (boundedness, positive definiteness, subspace estimation rate d_op(Θ_N,Θ)=O_p(N^{-γ}), γ>1/2)
    Needed for Theorems 1-3 (Sections 4.2-4.4); restricts the data-generating distributions.

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Cite this review

Pith. "Pith review of Representation-Aware Distributionally Robust Optimization: A Knowledge Transfer Framework." pith.science (2026). https://pith.science/paper/2Z2IDNBD

@misc{pith2026250909371,
  author       = {Pith},
  title        = {Pith review of: Representation-Aware Distributionally Robust Optimization: A Knowledge Transfer Framework},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2Z2IDNBD}},
  note         = {Machine review of arXiv:2509.09371}
}
read the original abstract

Distributionally robust optimization (DRO) protects statistical learning against distributional shifts by optimizing the worst-case performance over a set of perturbed distributions. However, standard DRO formulations often treat all feature perturbations equally. This can be unnecessarily conservative when external knowledge suggests that the predictive signal is embedded in a low-dimensional representation of covariates. We propose REpresentation-Aware Distributionally robust estimation (READ), a Wasserstein DRO framework that uses external representations to guide the geometry of robustness. Rather than uniformly perturbing all covariate directions, READ increases the transport cost of perturbations that change representation coordinates, thereby reshaping the dual regularization toward the representation subspace. Meanwhile, it preserves protection against variations orthogonal to the representation. We study READ in two regimes. First, for inference on the current target, we characterize our estimator asymptotically and develop a Wasserstein profile inference approach to construct representation-aligned confidence regions while enabling automatic hyperparameter tuning. Second, for deployment to future populations that differ from the current target but are generated from the same representation-invariant random-coefficient model, we show that the resulting regions achieve higher coverage of future model parameters than standard methods. Simulations and a single-cell multi-omics application demonstrate the advantages of READ in multi-source and multitask transfer learning settings.

Figures

Figures reproduced from arXiv: 2509.09371 by the authors.

Figure 1
Figure 1. Simulation based on the data-generating process in Example [PITH_FULL_IMAGE:figures/full_fig_p025_1.png] view at source ↗
Figure 2
Figure 2. Performance comparison in Setting I across varying [PITH_FULL_IMAGE:figures/full_fig_p028_2.png] view at source ↗
Figure 3
Figure 3. Performance comparison in Setting II across varying [PITH_FULL_IMAGE:figures/full_fig_p029_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Bias norm of βDRO(λ) as a function of λ, illustrating the effect of adaptive tuning. We clearly observe the importance of adaptive tuning of Λ provided by Algorithm 2 for READ, in contrast to KG-DRO, which fixes λ = ∞ in all directions of Θ. In Experiment I, without sp…
Figure 5
Figure 5. Figure 5: Performance comparison in Setting IV across varying [PITH_FULL_IMAGE:figures/full_fig_p031_5.png]

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Works this paper leans on

36 extracted references · 9 linked inside Pith

  1. [1]

    , " * write output.state after.block = add.period write newline

    ENTRY address author booktitle chapter edition editor howpublished institution journal key month note number organization pages publisher school series title type url volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sentence := ...

  2. [2]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

  3. [3]

    Bastani, H. (2020). Predicting with proxies: Transfer learning in high dimension. Management Science , 67(5):2964--2984

    2020

  4. [4]

    Belloni, A., Chernozhukov, V., and Wang, L. (2011). Square-root lasso: Pivotal recovery of sparse signals via conic programming. Biometrika , 98(4):791--806

    2011

  5. [5]

    Blanchet, J., Kang, Y., and Murthy, K. (2019). Robust wasserstein profile inference and applications to machine learning. Journal of Applied Probability , 56(3):830--857

    2019

  6. [6]

    and Murthy, K

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

    2019

  7. [7]

    Blanchet, J., Murthy, K., and Nguyen, V. A. (2021). Statistical analysis of wasserstein distributionally robust estimators. In Tutorials in Operations Research: Emerging optimization methods and modeling techniques with applications , pages 227--254. INFORMS

    2021

  8. [8]

    Blanchet, J., Murthy, K., and Si, N. (2022). Confidence regions in wasserstein distributionally robust estimation. Biometrika , 109(2):295--315

    2022

Show all 36 references
  1. [9]

    Cai, T., Li, M., and Liu, M. (2024). Semi-supervised triply robust inductive transfer learning. Journal of the American Statistical Association , pages 1--14

    2024

  2. [10]

    T., Toh, K.-C., and Zhang, Y

    Chu, H. T., Toh, K.-C., and Zhang, Y. (2022). On regularized square-root regression problems: Distributionally robust interpretation and fast computations. Journal of Machine Learning Research , 23(308):1--39

    2022

  3. [11]

    Gao, R. (2023). Finite-sample guarantees for wasserstein distributionally robust optimization: Breaking the curse of dimensionality. Operations Research , 71(6):2291--2306

    2023

  4. [12]

    Gao, R., Chen, X., and Kleywegt, A. J. (2022). Wasserstein distributionally robust optimization and variation regularization. Operations Research , 72(3):1177--1191

    2022

  5. [13]

    and Kleywegt, A

    Gao, R. and Kleywegt, A. (2023). Distributionally robust stochastic optimization with wasserstein distance. Mathematics of Operations Research , 48(2):603--655

    2023

  6. [14]

    Gu, T., Han, Y., and Duan, R. (2024). Robust angle-based transfer learning in high dimensions. Journal of the Royal Statistical Society Series B: Statistical Methodology

    2024

  7. [15]

    F., and Ma, C

    Gui, Y., Barber, R. F., and Ma, C. (2024). Distributionally robust risk evaluation with an isotonic constraint. arXiv preprint arXiv:2407.06867

    2024 arXiv

  8. [16]

    He, Z., Sun, Y., and Li, R. (2024). Transfusion: Covariate-shift robust transfer learning for high-dimensional regression. In International Conference on Artificial Intelligence and Statistics , pages 703--711. PMLR

    2024

  9. [17]

    and Zhang, X

    Hu, X. and Zhang, X. (2023). Optimal parameter-transfer learning by semiparametric model averaging. Journal of Machine Learning Research , 24(358):1--53

    2023

  10. [18]

    T., and Li, H

    Li, S., Cai, T. T., and Li, H. (2021). Transfer learning for high-dimensional linear regression: Prediction, estimation and minimax optimality. Journal of the Royal Statistical Society Series B: Statistical Methodology , 84(1):149--173

    2021

  11. [19]

    Lin, Z., Zhao, J., Wang, F., and Wang, H. (2024). Profiled transfer learning for high dimensional linear model. arXiv preprint arXiv:2406.00701

    2024 arXiv

  12. [20]

    and Kuhn, D

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

    2018

  13. [21]

    Rockafellar, R. T. and Wets, R. J.-B. (1998). Variational Analysis , volume 317 of Grundlehren der mathematischen Wissenschaften . Springer, Berlin, Heidelberg

    1998

  14. [22]

    M., and Kuhn, D

    Rychener, Y., Esteban-P \'e rez, A., Morales, J. M., and Kuhn, D. (2024). Wasserstein distributionally robust optimization with heterogeneous data sources. arXiv preprint arXiv:2407.13582

    2024 arXiv

  15. [23]

    M., and Kuhn, D

    Shafieezadeh-Abadeh, S., Esfahani, P. M., and Kuhn, D. (2015). Distributionally robust logistic regression. In Advances in Neural Information Processing Systems , volume 28

    2015

  16. [24]

    Shafieezadeh-Abadeh, S., Kuhn, D., and Esfahani, P. M. (2019). Regularization via mass transportation. Journal of Machine Learning Research , 20(103):1--68

    2019

  17. [25]

    Shen, X., B\" u hlmann, P., and Taeb, A. (2025). Causality-oriented robustness: exploiting general noise interventions in linear structural causal models. Journal of the American Statistical Association , 0(ja):1--20

    2025

  18. [26]

    and Feng, Y

    Tian, Y. and Feng, Y. (2023). Transfer learning under high-dimensional generalized linear models. Journal of the American Statistical Association , 118(544):2684--2697

    2023

  19. [27]

    Tian, Y., Gu, Y., and Feng, Y. (2025). Learning from similar linear representations: Adaptivity, minimaxity, and robustness. arXiv prepreint arXiv: 2303.17765

    2025 arXiv

  20. [28]

    Toland, J. F. (1978). Duality in nonconvex optimization. Journal of Mathematical Analysis and Applications , 66(2):399--415

    1978

  21. [29]

    Toland, J. F. (1979). A duality principle for non-convex optimisation and the calculus of variations. Archive for Rational Mechanics and Analysis , 71:41--61

    1979

  22. [30]

    S ola, M., B\" u hlmann, P., and Shen, X. (2025). Causality-inspired robustness for nonlinear models via representation learning. arXiv preprint arXiv:2505.12868

    2025 arXiv

  23. [31]

    Wang, Z., Wang, Z., Liu, M., and Si, N. (2025). Knowledge-guided wasserstein distributionally robust optimization. In Forty-second International Conference on Machine Learning

    2025

  24. [32]

    Wellner, J. et al. (2013). Weak convergence and empirical processes: with applications to statistics . Springer Science & Business Media

    2013

  25. [33]

    Xiong, X., Guo, Z., and Cai, T. (2023). Distributionally robust transfer learning. arXiv preprint arXiv:2309.06534

    2023 arXiv

  26. [34]

    Zhan, K., Xiong, X., Guo, Z., Cai, T., and Liu, M. (2024). Domain adaptation optimized for robustness in mixture populations. arXiv preprint arXiv:2407.20073

    2024 arXiv

  27. [35]

    Zhang, X., Liu, H., and Zhang, X. (2025). Sufficiency-principled transfer learning via model averaging. arXiv preprint arXiv: 2507.15416

    2025 arXiv

  28. [36]

    Zhou, D., Li, M., Cai, T., and Liu, M. (2024). Model-assisted and knowledge-guided transfer regression for the underrepresented population. arXiv preprint arXiv: 2410.06484

    2024 arXiv

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Reviewed August 4, 2026 · model on record in the stance chip above.