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Decentralized and Equitable Optimal Transport (2403.04259v2)
Published 7 Mar 2024 in math.OC and cs.LG
Abstract: This paper considers the decentralized (discrete) optimal transport (D-OT) problem. In this setting, a network of agents seeks to design a transportation plan jointly, where the cost function is the sum of privately held costs for each agent. We reformulate the D-OT problem as a constraint-coupled optimization problem and propose a single-loop decentralized algorithm with an iteration complexity of O(1/{\epsilon}) that matches existing centralized first-order approaches. Moreover, we propose the decentralized equitable optimal transport (DE-OT) problem. In DE-OT, in addition to cooperatively designing a transportation plan that minimizes transportation costs, agents seek to ensure equity in their individual costs. The iteration complexity of the proposed method to solve DE-OT is also O(1/{\epsilon}). This rate improves existing centralized algorithms, where the best iteration complexity obtained is O(1/{\epsilon}2).
- G. Monge, “Memory on the theory of cut and fill,” Mem. Math. Phys. Acad. Royal Sci., pp. 666–704, 1781.
- L. V. Kantorovich, “On the translocation of masses,” in Dokl. Akad. Nauk. USSR (NS), vol. 37, 1942, pp. 199–201.
- E. F. Montesuma, F. N. Mboula, and A. Souloumiac, “Recent advances in optimal transport for machine learning,” arXiv preprint arXiv:2306.16156, 2023.
- M. Scetbon, L. Meunier, J. Atif, and M. Cuturi, “Equitable and optimal transport with multiple agents,” in AISTATS, 2021.
- M. Huang, S. Ma, and L. Lai, “On the convergence of projected alternating maximization for equitable and optimal transport,” arXiv preprint arXiv:2109.15030, 2021.
- J. Altschuler, J. Niles-Weed, and P. Rigollet, “Near-linear time approximation algorithms for optimal transport via sinkhorn iteration,” Advances in neural information processing systems, vol. 30, 2017.
- G. Peyré, M. Cuturi, et al., “Computational optimal transport: With applications to data science,” Foundations and Trends® in Machine Learning, vol. 11, no. 5-6, pp. 355–607, 2019.
- M. Cuturi, “Sinkhorn distances: Lightspeed computation of optimal transport,” Advances in neural information processing systems, vol. 26, 2013.
- T. Lin, N. Ho, and M. I. Jordan, “On the efficiency of entropic regularized algorithms for optimal transport,” Journal of Machine Learning Research, vol. 23, no. 137, pp. 1–42, 2022.
- P. Dvurechensky, A. Gasnikov, and A. Kroshnin, “Computational optimal transport: Complexity by accelerated gradient descent is better than by sinkhorn’s algorithm,” in International conference on machine learning. PMLR, 2018, pp. 1367–1376.
- A. Chambolle and J. P. Contreras, “Accelerated bregman primal-dual methods applied to optimal transport and wasserstein barycenter problems,” SIAM Journal on Mathematics of Data Science, vol. 4, no. 4, pp. 1369–1395, 2022.
- A. Jambulapati, A. Sidford, and K. Tian, “A direct tilde {{\{{O}}\}}(1/epsilon) iteration parallel algorithm for optimal transport,” Advances in Neural Information Processing Systems, vol. 32, 2019.
- G. Li, Y. Chen, Y. Chi, H. V. Poor, and Y. Chen, “Fast computation of optimal transport via entropy-regularized extragradient methods,” arXiv preprint arXiv:2301.13006, 2023.
- D. A. Lorenz, P. Manns, and C. Meyer, “Quadratically regularized optimal transport,” Applied Mathematics & Optimization, vol. 83, no. 3, pp. 1919–1949, 2021.
- D. A. Pasechnyuk, M. Persiianov, P. Dvurechensky, and A. Gasnikov, “Algorithms for euclidean regularised optimal transport,” arXiv preprint arXiv:2307.00321, 2023.
- R. Zhang and Q. Zhu, “Consensus-based distributed discrete optimal transport for decentralized resource matching,” IEEE Transactions on Signal and Information Processing over Networks, vol. 5, no. 3, pp. 511–524, 2019.
- J. Hughes and J. Chen, “Fair and distributed dynamic optimal transport for resource allocation over networks,” in 2021 55th Annual Conference on Information Sciences and Systems (CISS). IEEE, 2021, pp. 1–6.
- X. Wang, H. Xu, and M. Yang, “Decentralized entropic optimal transport for privacy-preserving distributed distribution comparison,” arXiv preprint arXiv:2301.12065, 2023.
- A. Falsone, I. Notarnicola, G. Notarstefano, and M. Prandini, “Tracking-ADMM for distributed constraint-coupled optimization,” Automatica, vol. 117, p. 108962, 2020.
- T.-H. Chang, M. Hong, and X. Wang, “Multi-agent distributed optimization via inexact consensus ADMM,” IEEE Transactions on Signal Processing, vol. 63, no. 2, pp. 482–497, 2014.
- T.-H. Chang, “A proximal dual consensus ADMM method for multi-agent constrained optimization,” IEEE Transactions on Signal Processing, vol. 64, no. 14, pp. 3719–3734, 2016.
- Y. Su, Q. Wang, and C. Sun, “Distributed primal-dual method for convex optimization with coupled constraints,” IEEE Transactions on Signal Processing, vol. 70, pp. 523–535, 2021.
- S. A. Alghunaim, K. Yuan, and A. H. Sayed, “A proximal diffusion strategy for multiagent optimization with sparse affine constraints,” IEEE Transactions on Automatic Control, vol. 65, no. 11, pp. 4554–4567, 2019.
- J. Li and H. Su, “Implicit tracking-based distributed constraint-coupled optimization,” IEEE Transactions on Control of Network Systems, 2022.
- J. Li, Q. An, and H. Su, “Proximal nested primal-dual gradient algorithms for distributed constraint-coupled composite optimization,” Applied Mathematics and Computation, vol. 444, p. 127801, 2023.
- F. Iutzeler and L. Condat, “Distributed projection on the simplex and ℓ1subscriptℓ1\ell_{1}roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ball via ADMM and gossip,” IEEE Signal Processing Letters, vol. 25, no. 11, pp. 1650–1654, 2018.