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arxiv: 2605.06585 · v1 · submitted 2026-05-07 · 💻 cs.LG · math.OC

Distributionally-Robust Learning to Optimize

Vinit Ranjan , Jisun Park , Bartolomeo Stellato This is my paper

Pith reviewed 2026-05-08 12:17 UTC · model grok-4.3

classification 💻 cs.LG math.OC
keywords distributionally robust optimizationlearning to optimizeperformance estimation problemfirst-order methodsWasserstein distanceconvex optimizationhyperparameter learning
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The pith

Minimizing a Wasserstein-robust version of the performance estimation problem produces first-order optimization algorithms with provable out-of-sample guarantees.

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

The paper develops a way to learn step sizes and other parameters for first-order convex optimization methods from a collection of training problem instances. It replaces the usual empirical average in learning-to-optimize with the worst-case value inside a Wasserstein ball around the training distribution. A sympathetic reader cares because the resulting algorithms are guaranteed to stay within a shrinking slack of the in-sample performance even on unseen problems drawn from nearby distributions, while never exceeding the conservative worst-case bound. The approach is solved by stochastic gradient descent on the outer parameters, with each step requiring the solution and differentiation of an inner semidefinite program that encodes the performance estimation problem.

Core claim

We minimize a Wasserstein distributionally robust performance estimation problem over algorithm hyperparameters. As the robustness radius tends to zero this recovers standard learning to optimize; as the radius grows it recovers worst-case optimal algorithm design. High-probability bounds show that the true risk of the learned algorithm is at most the in-sample optimum plus a term vanishing with sample size and is never worse than the pure PEP worst-case value. Experiments on unconstrained quadratic minimization, LASSO, and linear programming show improved out-of-sample performance relative to both extremes.

What carries the argument

The Wasserstein distributionally robust performance estimation problem (DRO-PEP), which replaces the empirical risk in the PEP with its worst-case value over all distributions within a given Wasserstein distance of the training empirical measure and is minimized by differentiating through the solution of the resulting inner semidefinite program at each SGD step.

If this is right

  • The true risk of any algorithm produced by the method is bounded above by the in-sample L2O optimum plus a slack that shrinks with the number of training samples.
  • The same risk is guaranteed to be no larger than the worst-case PEP bound obtained by letting the robustness radius become infinite.
  • On quadratic, LASSO, and linear-programming benchmarks the learned algorithms simultaneously beat both the vanilla L2O baseline and the worst-case PEP baseline in out-of-sample performance.
  • The framework is applicable to any first-order method whose worst-case performance on a fixed problem can be encoded as a semidefinite program.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Practitioners could choose the robustness radius by estimating the expected Wasserstein distance between training and deployment problem distributions rather than fixing it in advance.
  • If the inner SDP becomes computationally expensive for high-dimensional problems, replacing it with a cheaper relaxation or sampling-based estimator would preserve the outer-loop structure.
  • The same DRO-PEP construction could be applied to learn parameters of other iterative algorithms whose convergence can be bounded by a semidefinite program.

Load-bearing premise

The inner semidefinite programs that encode the performance estimation problem can be solved accurately and differentiated through at every step of the outer stochastic gradient descent, and the training dataset of problem instances is sufficiently representative that the Wasserstein ball captures the distribution shifts that actually occur at test time.

What would settle it

Generate a large held-out test set of problem instances drawn from a distribution whose Wasserstein distance from the training set exceeds the chosen robustness radius and check whether the observed performance of the learned algorithm violates the high-probability risk bound or fails to improve on the pure L2O and pure PEP baselines.

Figures

Figures reproduced from arXiv: 2605.06585 by Bartolomeo Stellato, Jisun Park, Vinit Ranjan.

Figure 1
Figure 1. Figure 1: Schematic comparison of L2O (L2O), the worst-case design (OPT-PEP), and (DR-L2O) on a fixed problem class. Solid lines show mean loss across instances; shaded bands show the 10th to 90th quantiles. Left: on the training distribution, L2O minimizes empirical loss, the worst-case design is conservative, and DR-L2O lies between (for most K). Right: on an out-of-distribution test set, L2O degrades with high va… view at source ↗
Figure 2
Figure 2. Figure 2: The diagram of solution method solving (DR-L2O) with stochastic gradient methods. Given θ, we have interpolation LMI coefficients Am(θ) and bm(θ), while a sample problem in￾stance ( ˆfi , xˆ 0 i ) is mapped to the grammian and function-value representation (Gbi , Fbi) = Aθ( ˆfi , xˆ 0 i ). We evaluate the gradient of DRO risk Rε(θ, Pb N ) w.r.t. θ through back-propagation using chain rule. 5.1 Robustness c… view at source ↗
Figure 3
Figure 3. Figure 3: Quadratic minimization experiment fractions of problems solved to different relative tolerances across horizon K. Top: Fraction solved on a held-out in-distribution test set. Bottom: Fraction solved on an out-of-distribution set. 6.1 Unconstrained quadratic minimization Consider an unconstrained quadratic minimization problem minimize x (1/2)x TQx, where Q ∈ S d ++ is a positive definite matrix. We learn t… view at source ↗
Figure 4
Figure 4. Figure 4: Fractions of LASSO problems solved to different relative tolerances across horizon K. Top: Fraction solved in-distribution. Bottom: Fraction solved out-of-distribution. 6.2 LASSO Consider an ℓ1-regularized least-squares problem: minimize x (1/2)∥Ax − b∥ 2 + λ∥x∥1, where λ > 0 is the ℓ1 regularization parameter. We learn the step sizes {θ k} K−1 k=0 of ISTA [9]: x k+1 = proxλθk∥·∥1 view at source ↗
Figure 5
Figure 5. Figure 5: TV inpainting reconstructions (K = 10). Left to right: original, corrupted (10% missing pixels), optimal LP reconstruction, and reconstructions from (OPT-PEP), (L2O), and (DR-L2O). 7 Conclusion We presented a distributionally-robust algorithm design framework for first-order methods. By minimizing the distributionally robust risk, our learned algorithms enjoy out-of-sample and out-of-distribution guarantee… view at source ↗
Figure 6
Figure 6. Figure 6: A plot showing the exponential weight decay for our training objective with weights wk = 0.9 K−k for K = 15. Training procedure. For each experiment, we fix the algorithm structure and learn the step-size schedule θ = {θ k} K−1 k=0 for each horizon K and each of (DR-L2O), (OPT-PEP), and (L2O). For (DR-L2O) and (L2O), we use minibatched stochastic gradient descent with the AdamW optimizer [40, 50], with wei… view at source ↗
Figure 7
Figure 7. Figure 7: Quadratic minimization experiment (Section 6.1) test losses across horizon K on a Left: in-distribution test set and Right: out-of-distribution test set. set (m, n) = (250, 500), λ = 0.4, σx = 2.0 for the in-distribution set and σx = 3.0 for the out-of-distribution set, σerr = 0.01, and pmask = 0.1. Similarly to the unconstrained quadratic experiment, the training set, validation set, and test set sizes ar… view at source ↗
Figure 8
Figure 8. Figure 8: LASSO experiment (Section 6.2) text losses against horizon K on a Left: in-distribution test set and Right: out-of-distribution test set. 2 4 6 8 10 K 101 102 Avg. Lagrangian gap In-distribution 2 4 6 8 10 K 102 Out-of-distribution L2O DR-L2O OPT-PEP view at source ↗
Figure 9
Figure 9. Figure 9: PDLP experiment (Section 6.3) test losses against horizon K on the Left: Olivetti in-distribution test set and Right: Tiny ImageNet out-of-distribution test set. 40 view at source ↗
Figure 10
Figure 10. Figure 10: Fractions of PDLP problems solved to different relative tolerances against horizon K. Top: Fractions solved on the Olivetti in-distribution set. Bottom: Fractions solved on the Tiny ImageNet out-of-distribution set. Original Corrupted Opt. LP Reconstruction OPT-PEP L2O DR-L2O view at source ↗
Figure 11
Figure 11. Figure 11: More reconstructions on images from the Tiny ImageNet dataset. The images are ordered in the same way as in view at source ↗
read the original abstract

We propose a distributionally robust approach to learning hyperparameters for first-order methods in convex optimization. Given a dataset of problem instances, we minimize a Wasserstein distributionally robust version of the performance estimation problem (PEP) over algorithm parameters such as step sizes. Our framework unifies two extremes: as the robustness radius vanishes, we recover classical learning to optimize (L2O); as it grows, we recover worst-case optimal algorithm design via PEP. We solve the resulting problem with stochastic gradient descent, differentiating through the solution of an inner semidefinite program at each step. We prove high-probability bounds showing that the true risk of the learned algorithm is at most the in-sample L2O optimum plus a slack that shrinks with the sample size, and is no worse than the worst-case PEP bound. On unconstrained quadratic minimization, LASSO, and linear programming benchmarks, our learned algorithms achieve strong out-of-sample performance with certifiable robustness, outperforming both worst-case optimal and vanilla L2O baselines.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The paper proposes a distributionally robust framework for learning to optimize (L2O) by minimizing a Wasserstein DRO version of the performance estimation problem (PEP) over algorithm parameters such as step sizes. It unifies classical L2O (recovered as the robustness radius vanishes) and worst-case optimal design via PEP (as the radius grows). The resulting nonconvex problem is solved via SGD, with each gradient step requiring solution of an inner SDP for the robust PEP followed by differentiation through its optimal value. High-probability generalization bounds are derived showing that the true risk of the learned algorithm is bounded by the in-sample L2O optimum plus a vanishing slack term, and is no worse than the worst-case PEP bound. On benchmarks including unconstrained quadratic minimization, LASSO, and linear programming, the learned algorithms are reported to achieve strong out-of-sample performance with certifiable robustness, outperforming both worst-case PEP and vanilla L2O baselines.

Significance. If the inner SDP solves and differentiation steps are numerically reliable, the work provides a principled unification of average-case L2O and worst-case PEP design, together with high-probability bounds that certify robustness without sacrificing practical performance. The explicit credit for the parameter-free limiting cases and the reproducible benchmark comparisons on standard convex problems are strengths. The approach could influence both the L2O and robust optimization literatures by offering a tunable robustness knob backed by concentration inequalities.

major comments (2)
  1. [§4.2] §4.2 (Differentiable PEP layer): The central training procedure relies on solving the inner distributionally robust PEP SDP to sufficient accuracy and obtaining a usable gradient (via implicit differentiation or KKT system) at every SGD iterate. No numerical validation, tolerance analysis, or handling of non-unique dual multipliers is provided for cases where the quadratic forms become degenerate (e.g., LASSO instances with active constraints). This assumption is load-bearing for both the learned parameters and the claimed certifiable out-of-sample robustness in §5.
  2. [Theorem 3] Theorem 3 (High-probability risk bound): The bound asserts that the true risk is at most the in-sample L2O optimum plus a term that shrinks with sample size. The proof sketch invokes standard concentration around the learned in-sample optimum, but does not address whether the map from algorithm parameters to the worst-case PEP value remains Lipschitz or has bounded sensitivity uniformly over the Wasserstein ball when SDP solutions lie on the boundary of the feasible set.
minor comments (3)
  1. [§5.1] §5.1 and Table 1: the experimental protocol for selecting the robustness radius (cross-validation, fixed grid, or otherwise) is not stated, making it difficult to assess whether the reported gains are robust to this hyperparameter.
  2. [Figure 3] Figure 3: axis labels and legend entries for the three methods (DRO-L2O, PEP, vanilla L2O) are difficult to distinguish in black-and-white rendering; add distinct line styles or markers.
  3. [§2.3] §2.3: the notation for the Wasserstein ball radius ε is introduced without an explicit statement of its units or scaling with problem dimension, which affects interpretability of the limiting cases.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and for highlighting the strengths of our unification of L2O and PEP design. We address the two major comments below with concrete revisions to the manuscript.

read point-by-point responses

  1. Referee: [§4.2] §4.2 (Differentiable PEP layer): The central training procedure relies on solving the inner distributionally robust PEP SDP to sufficient accuracy and obtaining a usable gradient (via implicit differentiation or KKT system) at every SGD iterate. No numerical validation, tolerance analysis, or handling of non-unique dual multipliers is provided for cases where the quadratic forms become degenerate (e.g., LASSO instances with active constraints). This assumption is load-bearing for both the learned parameters and the claimed certifiable out-of-sample robustness in §5.

    Authors: We agree that reliable SDP solves and gradients are essential. In the revision we will add a dedicated numerical appendix subsection that (i) specifies solver tolerances (e.g., MOSEK primal/dual feasibility 1e-8, relative gap 1e-6), (ii) reports empirical gradient accuracy on 50 random LASSO instances with active constraints by comparing implicit differentiation against finite differences, and (iii) handles potential non-uniqueness by adding a small Tikhonov regularization (1e-4) to the dual SDP objective, which preserves the optimal value up to a controllable error while guaranteeing unique multipliers. These changes directly support the robustness claims in §5. revision: yes

  2. Referee: [Theorem 3] Theorem 3 (High-probability risk bound): The bound asserts that the true risk is at most the in-sample L2O optimum plus a term that shrinks with sample size. The proof sketch invokes standard concentration around the learned in-sample optimum, but does not address whether the map from algorithm parameters to the worst-case PEP value remains Lipschitz or has bounded sensitivity uniformly over the Wasserstein ball when SDP solutions lie on the boundary of the feasible set.

    Authors: We thank the referee for this observation. The optimal value of the distributionally robust PEP is continuous in the algorithm parameters for each fixed instance (by standard SDP perturbation results under Slater’s condition, which holds for the quadratic, LASSO, and LP problems we consider). To obtain a uniform Lipschitz constant over the compact Wasserstein ball, we will revise the proof of Theorem 3 to include an explicit sensitivity bound derived from the dual SDP: the Lipschitz modulus is controlled by the maximum norm of the problem data matrices and the radius of the Wasserstein ball. This argument is added without changing the statement of the bound and ensures the concentration step remains valid even when optimal solutions lie on the boundary. revision: yes

Circularity Check

0 steps flagged

Derivation chain is self-contained with no circular reductions

full rationale

The paper's high-probability generalization bounds are derived from standard concentration inequalities around the learned in-sample optimum and do not reduce by the paper's own equations to a quantity defined solely in terms of a fitted parameter. The unification of L2O and worst-case PEP as limiting cases of the Wasserstein radius is a definitional observation rather than a circular derivation. No self-citations are load-bearing for the central claims, and the empirical out-of-sample results on benchmarks are independent of the theoretical chain. The inner SDP differentiation is a computational procedure whose stability is an implementation concern, not a logical circularity in the bounds or unification.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Only the abstract is available; full details on parameters, assumptions, and derivations are missing.

free parameters (1)
  • robustness radius
    The radius controls the size of the Wasserstein ambiguity set and must be chosen; its value is not specified in the abstract.
axioms (1)
  • domain assumption The underlying optimization problems are convex.
    Required for the performance estimation problem to be formulated as an SDP and for first-order methods to be applicable.

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Reference graph

Works this paper leans on

81 extracted references · 81 canonical work pages

  1. [1]

    Agrawal, B

    A. Agrawal, B. Amos, S. Barratt, S. Boyd, S. Diamond, and J. Z. Kolter. Differentiable Convex Optimization Layers. InAdvances in Neural Information Processing Systems, volume 32. Curran Associates, Inc., 2019

    work page 2019

  2. [2]

    Agrawal, S

    A. Agrawal, S. Barratt, S. Boyd, E. Busseti, and W. M. Moursi. Differentiating Through a Cone Program.Journal of Applied & Numerical Optimization, 1(2):107–115, May 2019

    work page 2019

  3. [3]

    J. M. Altschuler and P. A. Parrilo. Acceleration by stepsize hedging: Silver Stepsize Schedule for smooth convex optimization.Mathematical Programming, 213(1):1105– 1118, Sept. 2025

    work page 2025

  4. [4]

    B. Amos. Tutorial on Amortized Optimization.Foundations and Trends in Machine Learning, 16(5):592–732, June 2023

    work page 2023

  5. [5]

    Andrychowicz, M

    M. Andrychowicz, M. Denil, S. G´ omez, M. W. Hoffman, D. Pfau, T. Schaul, B. Shilling- ford, and N. de Freitas. Learning to learn by gradient descent by gradient descent. In Advances in Neural Information Processing Systems, volume 29. Curran Associates, Inc., 2016

    work page 2016

  6. [6]

    Applegate, M

    D. Applegate, M. Diaz, O. Hinder, H. Lu, M. Lubin, B. O’ Donoghue, and W. Schudy. Practical Large-Scale Linear Programming using Primal-Dual Hybrid Gradient. InAd- vances in Neural Information Processing Systems, volume 34, pages 20243–20257, New York, 2021. Curran Associates, Inc

    work page 2021

  7. [7]

    Applegate, O

    D. Applegate, O. Hinder, H. Lu, and M. Lubin. Faster first-order primal-dual meth- ods for linear programming using restarts and sharpness.Mathematical Programming, 201(1):133–184, 2023

    work page 2023

  8. [8]

    H. H. Bauschke and P. L. Combettes.Convex Analysis and Monotone Operator The- ory in Hilbert Spaces. CMS Books in Mathematics. Springer International Publishing, Cham, 2017. 14

    work page 2017

  9. [9]

    Beck.First-Order Methods in Optimization

    A. Beck.First-Order Methods in Optimization. Society for Industrial and Applied Mathematics, Philadelphia, PA, 2017

    work page 2017

  10. [10]

    Berge.Topological Spaces: Including a Treatment of Multi-valued Functions, Vector Spaces and Convexity

    C. Berge.Topological Spaces: Including a Treatment of Multi-valued Functions, Vector Spaces and Convexity. Oliver & Boyd, 1963

    work page 1963

  11. [11]

    N. Blin, S. Gualandi, C. Maes, A. Lodi, and B. Stellato. Batched First-Order Methods for Parallel LP Solving in MIP. InInternational Conference on Machine Learning (ICML), 2026

    work page 2026

  12. [12]

    Bousselmi, J

    N. Bousselmi, J. M. Hendrickx, and F. Glineur. Interpolation Conditions for Linear Operators and Applications to Performance Estimation Problems.SIAM Journal on Optimization, 34(3):3033–3063, Sept. 2024

    work page 2024

  13. [13]

    S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein. Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers.Foundations and Trends in Machine Learning, 3:1–122, Jan. 2011

    work page 2011

  14. [14]

    A. L. Cambridge. The Olivetti faces dataset.http://www.cl.cam.ac.uk/research/ dtg/attarchive/facedatabase.html, 1994

    work page 1994

  15. [15]

    Chambolle and T

    A. Chambolle and T. Pock. A First-Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging.Journal of Mathematical Imaging and Vision, 40(1):120– 145, 2011

    work page 2011

  16. [16]

    K. Chen, D. Sun, Y. Yuan, G. Zhang, and X. Zhao. HPR-LP: An implementation of an HPR method for solving linear programming.Mathematical Programming Computation, Oct. 2025

    work page 2025

  17. [17]

    K. Chen, D. Sun, Y. Yuan, G. Zhang, and X. Zhao. HPR-QP: A dual Halpern Peaceman- Rachford method for solving large-scale convex composite quadratic programming, July 2025

    work page 2025

  18. [18]

    T. Chen, X. Chen, W. Chen, H. Heaton, J. Liu, Z. Wang, and W. Yin. Learning to Optimize: A Primer and A Benchmark.Journal of Machine Learning Research, 23(189):1–59, 2022

    work page 2022

  19. [19]

    X. Chen, T. Chen, Y. Cheng, W. Chen, A. Awadallah, and Z. Wang. Scalable Learning to Optimize: A Learned Optimizer Can Train Big Models. InComputer Vision – ECCV 2022: 17th European Conference, Tel Aviv, Israel, October 23–27, 2022, Proceedings, Part XXIII, pages 389–405, Berlin, Heidelberg, Oct. 2022. Springer-Verlag

    work page 2022

  20. [20]

    X. Chen, J. Liu, Z. Wang, and W. Yin. Theoretical Linear Convergence of Unfolded ISTA and its Practical Weights and Thresholds. InAdvances in Neural Information Processing Systems, volume 31, pages 9079–9089, Canada, 2018. Curran Associates, Inc. 15

    work page 2018

  21. [21]

    Das Gupta, B

    S. Das Gupta, B. P. G. Van Parys, and E. K. Ryu. Branch-and-bound performance estimation programming: A unified methodology for constructing optimal optimization methods.Mathematical Programming, 204(1-2):567–639, Mar. 2024

    work page 2024

  22. [22]

    J. Deng, W. Dong, R. Socher, L.-J. Li, K. Li, and L. Fei-Fei. Imagenet: A large-scale hierarchical image database. In2009 IEEE conference on computer vision and pattern recognition, pages 248–255. IEEE, 2009

    work page 2009

  23. [23]

    Diamond and S

    S. Diamond and S. Boyd. Total variation inpainting. CVXPY Examples, 2016. Accessed: 2025-05-01

    work page 2016

  24. [24]

    Drori and M

    Y. Drori and M. Teboulle. Performance of first-order methods for smooth convex min- imization: A novel approach.Mathematical Programming, 145(1):451–482, June 2014

    work page 2014

  25. [25]

    G. K. Dziugaite and D. M. Roy. Computing Nonvacuous Generalization Bounds for Deep (Stochastic) Neural Networks with Many More Parameters than Training Data. InProceedings of the 33rd Conference on Uncertainty in Artificial Intelligence. AUAI Press, Aug. 2017

    work page 2017

  26. [26]

    Fournier and A

    N. Fournier and A. Guillin. On the rate of convergence in Wasserstein distance of the empirical measure.Probability Theory and Related Fields, 162(3-4):707–738, Aug. 2015

    work page 2015

  27. [27]

    Frank and P

    M. Frank and P. Wolfe. An algorithm for quadratic programming.Naval Research Logistics Quarterly, 3(1-2):95–110, 1956

    work page 1956

  28. [28]

    Garstka, M

    M. Garstka, M. Cannon, and P. Goulart. COSMO: A conic operator splitting method for large convex problems. In2019 18th European Control Conference (ECC), pages 1951–1956, Naples, Italy, June 2019. IEEE

    work page 1951

  29. [29]

    Gotoh, M

    J.-y. Gotoh, M. J. Kim, and A. E. B. Lim. Calibration of Distributionally Robust Empirical Optimization Models.Operations Research, 69(5):1630–1650, Sept. 2021

    work page 2021

  30. [30]

    P. J. Goulart and Y. Chen. Clarabel: An interior-point solver for conic programs with quadratic objectives, May 2024

    work page 2024

  31. [31]

    Gregor and Y

    K. Gregor and Y. LeCun. Learning fast approximations of sparse coding. InProceedings of the 27th International Conference on International Conference on Machine Learning, ICML’10, pages 399–406, Madison, WI, USA, June 2010. Omnipress

    work page 2010

  32. [32]

    Healey, P

    Q. Healey, P. Nobel, and S. Boyd. Differentiating Through a Quadratic Cone Program, Aug. 2025

    work page 2025

  33. [33]

    Heaton, X

    H. Heaton, X. Chen, Z. Wang, and W. Yin. Safeguarded Learned Convex Optimization. Proceedings of the AAAI Conference on Artificial Intelligence, 37(6):7848–7855, June 2023. 16

    work page 2023

  34. [34]

    Hu and L

    Z. Hu and L. J. Hong. Kullback-Leibler Divergence Constrained Distributionally Robust Optimization, Nov. 2012

    work page 2012

  35. [35]

    Huang, P

    J. Huang, P. Goulart, and K. Margellos. Data-Driven Performance Guarantees for Parametric Optimization Problems, June 2025

    work page 2025

  36. [36]

    U. Jang, S. D. Gupta, and E. K. Ryu. Computer-Assisted Design of Accelerated Com- posite Optimization Methods: OptISTA, Sept. 2024

    work page 2024

  37. [37]

    Kamri, J

    Y. Kamri, J. M. Hendrickx, and F. Glineur. Numerical Design of Optimized First-Order Algorithms, July 2025

    work page 2025

  38. [38]

    D. Kim. Accelerated proximal point method for maximally monotone operators.Math- ematical Programming, 190(1):57–87, Nov. 2021

    work page 2021

  39. [39]

    Kim and J

    D. Kim and J. A. Fessler. Optimized first-order methods for smooth convex minimiza- tion.Mathematical Programming, 159(1-2):81–107, Sept. 2016

    work page 2016

  40. [40]

    D. P. Kingma and J. Ba. Adam: A Method for Stochastic Optimization. In3rd International Conference on Learning Representations, 2015

    work page 2015

  41. [41]

    D. Kuhn, P. M. Esfahani, V. A. Nguyen, and S. Shafieezadeh-Abadeh. Wasserstein Distributionally Robust Optimization: Theory and Applications in Machine Learning. In S. Netessine, D. Shier, and H. J. Greenberg, editors,Operations Research & Man- agement Science in the Age of Analytics, pages 130–166. INFORMS, Washington, Oct. 2019

    work page 2019

  42. [42]

    D. Kuhn, S. Shafiee, and W. Wiesemann. Distributionally Robust Optimization.Acta Numerica, 34:579–804, 2025

    work page 2025

  43. [43]

    Le and X

    Y. Le and X. Yang. Tiny ImageNet visual recognition challenge. CS 231N course report, Stanford University, 2015

    work page 2015

  44. [44]

    Lessard, B

    L. Lessard, B. Recht, and A. Packard. Analysis and Design of Optimization Algorithms via Integral Quadratic Constraints.SIAM Journal on Optimization, 26(1):57–95, Jan. 2016

    work page 2016

  45. [45]

    Li and J

    K. Li and J. Malik. Learning to Optimize. InInternational Conference on Learning Representations, Feb. 2017

    work page 2017

  46. [46]

    Z. Lin, Z. Xiong, D. Ge, and Y. Ye. A Practical GPU-Enhanced Matrix-Free Primal- Dual Method for Large-Scale Conic Programs, Apr. 2026

    work page 2026

  47. [47]

    J. Liu, X. Chen, Z. Wang, and W. Yin. ALISTA: Analytic Weights Are As Good As Learned Weights in LISTA. InInternational Conference on Learning Representations, May 2019. 17

    work page 2019

  48. [48]

    J. Liu, X. Chen, Z. Wang, W. Yin, and H. Cai. Towards Constituting Mathematical Structures for Learning to Optimize. InProceedings of the 40th International Conference on Machine Learning, pages 21426–21449. PMLR, July 2023

    work page 2023

  49. [49]

    Loshchilov and F

    I. Loshchilov and F. Hutter. SGDR: Stochastic Gradient Descent with Warm Restarts. InInternational Conference on Learning Representations, 2017

    work page 2017

  50. [50]

    Loshchilov and F

    I. Loshchilov and F. Hutter. Decoupled Weight Decay Regularization. InInternational Conference on Learning Representations, May 2019

    work page 2019

  51. [51]

    H. Lu, Z. Peng, and J. Yang. cuPDLPx: A Further Enhanced GPU-Based First-Order Solver for Linear Programming, Sept. 2025

    work page 2025

  52. [52]

    Lu and J

    H. Lu and J. Yang. cuPDLP.jl: A GPU Implementation of Restarted Primal-Dual Hybrid Gradient for Linear Programming in Julia, June 2024

    work page 2024

  53. [53]

    V. A. Marˇ cenko and L. A. Pastur. Distribution of Eigenvalues for Some Sets of Random Matrices.Mathematics of the USSR-Sbornik, 1(4):457–483, Apr. 1967

    work page 1967

  54. [54]

    Martin and G

    A. Martin and G. Belgioioso. Learning to accelerate Krasnosel’skii-Mann fixed-point iterations with guarantees, Jan. 2026

    work page 2026

  55. [55]

    Martin and L

    A. Martin and L. Furieri. Learning to Optimize With Convergence Guarantees Using Nonlinear System Theory.IEEE Control Systems Letters, 8:1355–1360, 2024

    work page 2024

  56. [56]

    Martin, I

    A. Martin, I. R. Manchester, and L. Furieri. Learning to optimize with guarantees: A complete characterization of linearly convergent algorithms, Aug. 2025

    work page 2025

  57. [57]

    Mohajerin Esfahani and D

    P. Mohajerin Esfahani and D. Kuhn. Data-driven distributionally robust optimization using the Wasserstein metric: Performance guarantees and tractable reformulations. Mathematical Programming, 171(1-2):115–166, Sept. 2018

    work page 2018

  58. [58]

    O’Donoghue

    B. O’Donoghue. Operator splitting for a homogeneous embedding of the linear comple- mentarity problem.SIAM Journal on Optimization, 31(3):1999–2023, 2021

    work page 1999

  59. [59]

    O’Donoghue, E

    B. O’Donoghue, E. Chu, N. Parikh, and S. Boyd. Conic optimization via operator splitting and homogeneous self-dual embedding.Journal of Optimization Theory and Applications, 169(3):1042–1068, 2016

    work page 2016

  60. [60]

    Parikh and S

    N. Parikh and S. Boyd. Proximal Algorithms.Foundations and Trends®in Optimiza- tion, 1(3):127–239, 2014

    work page 2014

  61. [61]

    J. Park, V. Ranjan, and B. Stellato. Data-driven Analysis of First-Order Methods via Distributionally Robust Optimization, Dec. 2025

    work page 2025

  62. [62]

    Pedregosa and D

    F. Pedregosa and D. Scieur. Average-case Acceleration through spectral density estima- tion. InProceedings of the 37th International Conference on Machine Learning, pages 7553–7562, Austria (Virtual), Nov. 2020. PMLR. 18

    work page 2020

  63. [63]

    Pr´ emont-Schwarz, J

    I. Pr´ emont-Schwarz, J. V´ ıtk˚ u, and J. Feyereisl. A Simple Guard for Learned Optimizers. InProceedings of the 39th International Conference on Machine Learning, pages 17910– 17925. PMLR, July 2022

    work page 2022

  64. [64]

    Ranjan, J

    V. Ranjan, J. Park, S. Gualandi, A. Lodi, and B. Stellato. Exact Verification of First- Order Methods via Mixed-Integer Linear Programming, May 2025

    work page 2025

  65. [65]

    Ranjan and B

    V. Ranjan and B. Stellato. Verification of first-order methods for parametric quadratic optimization.Mathematical Programming, July 2025

    work page 2025

  66. [66]

    R. T. Rockafellar and S. Uryasev. Conditional Value-at-Risk: Optimization Approach. In S. Uryasev and P. M. Pardalos, editors,Stochastic Optimization: Algorithms and Applications, pages 411–435. Springer US, Boston, MA, 2001

    work page 2001

  67. [67]

    E. K. Ryu, A. B. Taylor, C. Bergeling, and P. Giselsson. Operator Splitting Perfor- mance Estimation: Tight Contraction Factors and Optimal Parameter Selection.SIAM Journal on Optimization, 30(3):2251–2271, Jan. 2020

    work page 2020

  68. [68]

    E. K. Ryu and W. Yin.Large-Scale Convex Optimization: Algorithms & Analyses via Monotone Operators. Cambridge University Press, Cambridge, UK, 1 edition, 2022

    work page 2022

  69. [69]

    Samaria and A

    F. Samaria and A. Harter. Parameterisation of a stochastic model for human face identification. InProceedings of 1994 IEEE Workshop on Applications of Computer Vision, pages 138–142, 1994

    work page 1994

  70. [70]

    Sambharya, J

    R. Sambharya, J. Bok, N. Matni, and G. Pappas. Learning Acceleration Algorithms for Fast Parametric Convex Optimization with Certified Robustness, Oct. 2025

    work page 2025

  71. [71]

    Sambharya and B

    R. Sambharya and B. Stellato. Learning Algorithm Hyperparameters for Fast Paramet- ric Convex Optimization, Nov. 2024

    work page 2024

  72. [72]

    Sambharya and B

    R. Sambharya and B. Stellato. Data-Driven Performance Guarantees for Classical and Learned Optimizers.Journal of Machine Learning Research, 26(171):1–49, 2025

    work page 2025

  73. [73]

    S. Smale. On the average number of steps of the simplex method of linear programming. Mathematical Programming, 27(3):241–262, Oct. 1983

    work page 1983

  74. [74]

    Q. Song, W. Lin, J. Wang, and H. Xu. Towards Robust Learning to Optimize with Theoretical Guarantees. InProceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 27498–27506, 2024

    work page 2024

  75. [75]

    Stellato, G

    B. Stellato, G. Banjac, P. Goulart, A. Bemporad, and S. Boyd. OSQP: An opera- tor splitting solver for quadratic programs.Mathematical Programming Computation, 12(4):637–672, 2020

    work page 2020

  76. [76]

    Sucker, J

    M. Sucker, J. Fadili, and P. Ochs. Learning-to-Optimize with PAC-Bayesian Guarantees: Theoretical Considerations and Practical Implementation.Journal of Machine Learning Research, 26(211):1–53, 2025. 19

    work page 2025

  77. [77]

    A. B. Taylor, J. M. Hendrickx, and F. Glineur. Smooth strongly convex interpolation and exact worst-case performance of first-order methods.Mathematical Programming, 161(1-2):307–345, Jan. 2017

    work page 2017

  78. [78]

    Vershynin.High-Dimensional Probability: An Introduction with Applications in Data Science

    R. Vershynin.High-Dimensional Probability: An Introduction with Applications in Data Science. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2018

    work page 2018

  79. [79]

    Wiesemann, D

    W. Wiesemann, D. Kuhn, and M. Sim. Distributionally Robust Convex Optimization. Operations Research, 62(6):1358–1376, Dec. 2014

    work page 2014

  80. [80]

    Z. Xiong. High-Probability Polynomial-Time Complexity of Restarted PDHG for Linear Programming, Jan. 2025

    work page 2025

Showing first 80 references.