## What we are interested in

We research and develop algorithms for continuous optimization, with a particular emphasis on solving high-dimensional optimization problems with first-order methods. We prove convergence rates and guarantees for a variety of settings, such as:
*Conditional gradient* aka *Frank-Wolfe* methods, which perform constrained optimization by accessing the feasible set via a linear optimization oracle.
*Proximal* methods, which yield efficient algorithms for nonsmooth optimization and practically reshape smooth ones.
*Online learning* algorithms which consist of playing a sequential game in which the algorithms predict and receive a loss in an online way and try to minimize overall regret; they have numerous applications to optimization via reductions.
*Accelerated* methods which combine online learning tools with other optimization techniques to exploit and improve convergence of other more simple algorithms, often to optimality.
And *block-iterative* algorithms which leverage parallelism and distributed computation for faster algorithmic performance.

## đź§‘â€ŤđźŽ“ Members

## đź”¬ Projects

Motivated by nonsmooth problems in machine learning, we solve the problem of minimizing an abs-smooth function subject to closed convex constraints. New theory and algorithms are developed using linear minimization oracles to enforce constraints and abs-linearization methods to handle nonsmoothness.

We will investigate mixed-integer optimization with convex objectives using error-adaptive convex solvers in branch-and-bound. Focusing on improving lower bounds and balancing computational costs, we aim to develop a faster branch-and-bound methodology by leveraging modern MILP techniques and error-adaptive methods. Key aspects include warm-starting and controlled inexactness in early termination.

We develop theory and algorithms for 0-1 decision making in optimization problems constrained by partial differential equations. By exploring extended formulations, we achieve new stationarity concepts through sequential exact and approximative relaxation of adjoint-based primal-dual optimality conditions.

In this project, we study algorithms that promote sparsity. We develop PageRank optimization algorithms that scale with solution sparsity and investigate Riemannian optimization using manifold geometry. Additionally, we develop algorithms for efficient fair resource allocation based on established fairness axioms.

Worst-case complexity bounds are increasingly insufficient to explain the (often superior) real-world performance of optimization and learning algorithms. We consider data-dependent rates, approximation guarantees, and complexity bounds to provide guarantees much more in line with actual performance.

## đź’¬ Talks and posters

### Conference and workshop talks

- Nov 2023
*Bounding Geometric Penalties in First-order Riemannian Optimization*by Christophe Roux

Seminar "Modern Methods in Applied Stochastics and Nonparametric Statistics", Berlin

### Research seminar talks

- Apr 2024
*Bounding Geometric Penalties in Riemannian Optimization*by Christophe Roux

Research Seminar, SaarbrĂĽcken

## đź“ť Publications and preprints

- Carderera, A., BesanĂ§on, M., and Pokutta, S. (2024). Scalable Frank-Wolfe on Generalized Self-concordant Functions Via Simple Steps.
*SIAM Journal on Optimization*,*34*(3), 2231â€“2258. DOI: 10.1137/23M1616789 [summary] [slides] [poster] [code]## [BibTeX]

- Braun, G., GuzmĂˇn, C., and Pokutta, S. (2024). Corrections to â€śLower Bounds on the Oracle Complexity of Nonsmooth Convex Optimization Via Information Theory.â€ť
*IEEE Transactions on Information Theory*,*70*, 5408â€“5409. DOI: 10.1109/TIT.2024.3357200## [BibTeX]

- Braun, G., Pokutta, S., and Woodstock, Z. (2024).
*Flexible Block-iterative Analysis for the Frank-Wolfe Algorithm*. [arXiv]## [BibTeX]

- Deza, A., Onn, S., Pokutta, S., and Pournin, L. (2024). Kissing Polytopes.
*SIAM Journal on Discrete Mathematics*. [arXiv]## [BibTeX]

- MartĂnez-Rubio, D., Roux, C., and Pokutta, S. (2024). Convergence and Trade-offs in Riemannian Gradient Descent and Riemannian Proximal Point.
*Proceedings of International Conference on Machine Learning*. [arXiv]## [BibTeX]

- BesanĂ§on, M., Garcia, J. D., Legat, B., and Sharma, A. (2023). Flexible Differentiable Optimization Via Model Transformations.
*INFORMS Journal on Computing*. DOI: 10.1287/ijoc.2022.0283 [URL] [arXiv]## [BibTeX]

- Wirth, E., Pena, J., and Pokutta, S. (2023).
*Accelerated Affine-invariant Convergence Rates of the Frank-Wolfe Algorithm with Open-loop Step-sizes*. [arXiv]## [BibTeX]

- Kreimeier, T., Pokutta, S., Walther, A., and Woodstock, Z. (2023). On a Frank-Wolfe Approach for Abs-smooth Functions.
*Optimization Methods and Software*. [arXiv]## [BibTeX]

- Criscitiello, C., MartĂnez-Rubio, D., and Boumal, N. (2023). Open Problem: Polynomial Linearly-convergent Method for G-convex Optimization?
*Proceedings of Annual Conference on Learning Theory*. [arXiv]## [BibTeX]

- MartĂnez-Rubio, D., and Pokutta, S. (2023). Accelerated Riemannian Optimization: Handling Constraints with a Prox to Bound Geometric Penalties.
*Proceedings of Annual Conference on Learning Theory*. [URL] [arXiv] [poster]## [BibTeX]

- MartĂnez-Rubio, D., Roux, C., Criscitiello, C., and Pokutta, S. (2023).
*Accelerated Riemannian Min-Max Optimization Ensuring Bounded Geometric Penalties*. [arXiv]## [BibTeX]

- MartĂnez-Rubio, D., Wirth, E., and Pokutta, S. (2023). Accelerated and Sparse Algorithms for Approximate Personalized PageRank and Beyond.
*Proceedings of Annual Conference on Learning Theory*. [arXiv]## [BibTeX]

- Scieur, D., Kerdreux, T., MartĂnez-Rubio, D., dâ€™Aspremont, A., and Pokutta, S. (2023).
*Strong Convexity of Sets in Riemannian Manifolds*. [arXiv]## [BibTeX]

- Woodstock, Z., and Pokutta, S. (2023).
*Splitting the Conditional Gradient Algorithm*. [arXiv]## [BibTeX]

- Wirth, E., Kerdreux, T., and Pokutta, S. (2023). Acceleration of Frank-Wolfe Algorithms with Open Loop Step-sizes.
*Proceedings of International Conference on Artificial Intelligence and Statistics*. [arXiv]## [BibTeX]

- Wirth, E., Kera, H., and Pokutta, S. (2023). Approximate Vanishing Ideal Computations at Scale.
*Proceedings of International Conference on Learning Representations*. [arXiv] [slides]## [BibTeX]

- Wilken, S. E., BesanĂ§on, M., KratochvĂl, M., Kuate, C. A. F., Trefois, C., Gu, W., and EbenhĂ¶h, O. (2022). Interrogating the Effect of Enzyme Kinetics on Metabolism Using Differentiable Constraint-based Models.
*Metabolic Engineering*.## [BibTeX]

- BĂşi, M. N., Combettes, P. L., and Woodstock, Z. (2022). Block-activated Algorithms for Multicomponent Fully Nonsmooth Minimization.
*Proceedings of ICASSP 2022 - 2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)*, 5428â€“5432. DOI: 10.1109/ICASSP43922.2022.9747479 [URL]## [BibTeX]

- BesanĂ§on, M., Carderera, A., and Pokutta, S. (2022). FrankWolfe.jl: A High-Performance and Flexible Toolbox for Frank-Wolfe Algorithms and Conditional Gradients.
*INFORMS Journal on Computing*. [arXiv] [summary] [slides] [code]## [BibTeX]

- Braun, G., Pokutta, S., and Weismantel, R. (2022).
*Alternating Linear Minimization: Revisiting von Neumannâ€™s Alternating Projections*. [arXiv] [slides] [video]## [BibTeX]

- Criado, F., MartĂnez-Rubio, D., and Pokutta, S. (2022). Fast Algorithms for Packing Proportional Fairness and Its Dual.
*Proceedings of Conference on Neural Information Processing Systems*. [arXiv] [poster]## [BibTeX]

- Braun, G., Carderera, A., Combettes, C., Hassani, H., Karbasi, A., Mokhtari, A., and Pokutta, S. (2022).
*Conditional Gradient Methods*. [arXiv]## [BibTeX]

- HunkenschrĂ¶der, C., Pokutta, S., and Weismantel, R. (2022). Optimizing a Low-dimensional Convex Function Over a High-dimensional Cube.
*SIAM Journal on Optimization*. [arXiv]## [BibTeX]

- Macdonald, J., BesanĂ§on, M., and Pokutta, S. (2022). Interpretable Neural Networks with Frank-Wolfe: Sparse Relevance Maps and Relevance Orderings.
*Proceedings of International Conference on Machine Learning*. [arXiv] [poster] [video]## [BibTeX]

- Wirth, E., and Pokutta, S. (2022). Conditional Gradients for the Approximately Vanishing Ideal.
*Proceedings of International Conference on Artificial Intelligence and Statistics*. [arXiv] [summary] [poster] [code]## [BibTeX]

- Carderera, A., BesanĂ§on, M., and Pokutta, S. (2021). Simple Steps Are All You Need: Frank-Wolfe and Generalized Self-concordant Functions.
*Proceedings of Conference on Neural Information Processing Systems*,*34*, 5390â€“5401. [URL] [summary] [slides] [poster] [code]## [BibTeX]

- Kerdreux, T., Roux, C., dâ€™Aspremont, A., and Pokutta, S. (2021). Linear Bandits on Uniformly Convex Sets.
*Journal of Machine Learning Research*,*22*(284), 1â€“23. [URL] [arXiv] [summary]## [BibTeX]

- Braun, G., and Pokutta, S. (2021).
*Dual Prices for Frankâ€“Wolfe Algorithms*. [arXiv]## [BibTeX]

- Roux, C., Wirth, E., Pokutta, S., and Kerdreux, T. (2021).
*Efficient Online-bandit Strategies for Minimax Learning Problems*. [arXiv]## [BibTeX]

- Braun, G., Pokutta, S., and Zink, D. (2019). Affine Reductions for LPs and SDPs.
*Mathematical Programming*,*173*, 281â€“312. DOI: 10.1007/s10107-017-1221-9 [arXiv]## [BibTeX]

- Braun, G., Pokutta, S., and Zink, D. (2019). Lazifying Conditional Gradient Algorithms.
*The Journal of Machine Learning Research*,*20*(71), 1â€“42. [URL] [arXiv]## [BibTeX]

- Braun, G., Pokutta, S., Tu, D., and Wright, S. (2019). Blended Conditional Gradients: the Unconditioning of Conditional Gradients.
*Proceedings of International Conference on Machine Learning*,*97*, 735â€“743. [URL] [arXiv] [summary] [slides] [poster] [code]## [BibTeX]

- Braun, G., Pokutta, S., and Roy, A. (2018). Strong Reductions for Extended Formulations.
*Mathematical Programming*,*172*, 591â€“620. DOI: 10.1007/s10107-018-1316-y [arXiv]## [BibTeX]

- Braun, G., Brown-Cohen, J., Huq, A., Pokutta, S., Raghavendra, P., Roy, A., Weitz, B., and Zink, D. (2017). The Matching Problem Has No Small Symmetric SDP.
*Mathematical Programming*,*165*, 643â€“662. DOI: 10.1007/s10107-016-1098-z [arXiv]## [BibTeX]

- Braun, G., GuzmĂˇn, C., and Pokutta, S. (2017). Unifying Lower Bounds on the Oracle Complexity of Nonsmooth Convex Optimization.
*IEEE Transactions on Information Theory*,*63*(7), 4709â€“4724. DOI: 10.1109/TIT.2017.2701343 [arXiv]## [BibTeX]

- Braun, G., Jain, R., Lee, T., and Pokutta, S. (2017). Information-theoretic Approximations of the Nonnegative Rank.
*Computational Complexity*,*26*, 147â€“197. DOI: 10.1007/s00037-016-0125-z [URL]## [BibTeX]

- Braun, G., Pokutta, S., and Zink, D. (2017). Lazifying Conditional Gradient Algorithms.
*Proceedings of International Conference on Machine Learning*,*70*, 566â€“575. [URL] [arXiv] [slides] [poster]## [BibTeX]

- Braun, G., Pokutta, S., and Roy, A. (2016). Strong Reductions for Extended Formulations.
*Proceedings of Conference on Integer Programming and Combinatorial Optimization*,*9682*, 350â€“361. DOI: 10.1007/978-3-319-33461-5_29 [arXiv]## [BibTeX]

- Braun, G., Firorini, S., and Pokutta, S. (2016). Average Case Polyhedral Complexity of the Maximum Stable Set Problem.
*Mathematical Programming*,*160*(1), 407â€“431. DOI: 10.1007/s10107-016-0989-3 [arXiv]## [BibTeX]

- Braun, G., and Pokutta, S. (2016). Common Information and Unique Disjointness.
*Algorithmica*,*76*(3), 597â€“629. DOI: 10.1007/s00453-016-0132-0 [URL]## [BibTeX]

- Braun, G., and Pokutta, S. (2016).
*An Efficient High-probability Algorithm for Linear Bandits*. [arXiv]## [BibTeX]

- Braun, G., Brown-Cohen, J., Huq, A., Pokutta, S., Raghavendra, P., Roy, A., Weitz, B., and Zink, D. (2016). The Matching Problem Has No Small Symmetric SDP.
*Proceedings of Symposium on Discrete Algorithms*, 1067â€“1078. DOI: 10.1137/1.9781611974331.ch75 [arXiv]## [BibTeX]

- Braun, G., Firorini, S., Pokutta, S., and Steurer, D. (2015). Approximation Limits of Linear Programs (beyond Hierarchies).
*Mathematics of Operations Research*,*40*(3), 756â€“772. DOI: 10.1287/moor.2014.0694 [arXiv]## [BibTeX]

- Braun, G., Pokutta, S., and Xie, Y. (2015). Info-greedy Sequential Adaptive Compressed Sensing.
*IEEE Journal of Selected Topics in Signal Processing*,*9*(4), 601â€“611. DOI: 10.1109/JSTSP.2015.2400428 [arXiv]## [BibTeX]

- Braun, G., Pokutta, S., and Zink, D. (2015). Inapproximability of Combinatorial Problems Via Small LPs and SDPs.
*Proceedings of Annual Symposium on Theory of Computing*, 107â€“116. DOI: 10.1145/2746539.2746550 [arXiv] [video]## [BibTeX]

- Braun, G., and Pokutta, S. (2015). The Matching Polytope Does Not Admit Fully-polynomial Size Relaxation Schemes.
*Proceedings of Symposium on Discrete Algorithms*, 837â€“846. DOI: 10.1137/1.9781611973730.57 [arXiv]## [BibTeX]

- Braun, G., Firorini, S., and Pokutta, S. (2014). Average Case Polyhedral Complexity of the Maximum Stable Set Problem.
*Proceedings of Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques*,*28*, 515â€“530. DOI: 10.4230/LIPIcs.APPROX-RANDOM.2014.515 [URL] [arXiv]## [BibTeX]

- Braun, G., Pokutta, S., and Xie, Y. (2014). Info-greedy Sequential Adaptive Compressed Sensing.
*Proceedings of Allerton Conference on Communication, Control, and Computing (Allerton)*, 858â€“865. DOI: 10.1109/ALLERTON.2014.7028544 [arXiv]## [BibTeX]

- Braun, G., and Pokutta, S. (2013). Common Information and Unique Disjointness.
*Proceedings of Annual Symposium on Foundations of Computer Science*, 688â€“697. [URL]## [BibTeX]

- Braun, G., and Pokutta, S. (2012). An Algebraic Approach to Symmetric Extended Formulations.
*Proceedings of International Symposium on Combinatorial Optimization*,*7422*, 141â€“152. DOI: 10.1007/978-3-642-32147-4_14 [arXiv]## [BibTeX]

- Braun, G., Firorini, S., Pokutta, S., and Steurer, D. (2012). Approximation Limits of Linear Programs (beyond Hierarchies).
*Proceedings of Annual Symposium on Foundations of Computer Science*, 480â€“489. DOI: 10.1109/FOCS.2012.10 [arXiv]## [BibTeX]