Genetic column generation: Fast computation of high-dimensional multi-marginal optimal transport problems

• URL: http://arxiv.org/abs/2103.12624v1
• Date: Tue, 23 Mar 2021 15:24:50 GMT
• Title: Genetic column generation: Fast computation of high-dimensional multi-marginal optimal transport problems
• Authors: Gero Friesecke, Andreas S. Schulz, and Daniela V\"ogler
• Abstract summary: We use a genetic learning method tailormade for MMOT in which the dual state within CG plays the role of an "adversary" On a sequence of benchmark problems with up to 120 gridpoints and up to 30 marginals, our method always found the exacts.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-sa/4.0/
• Abstract: We introduce a simple, accurate, and extremely efficient method for numerically solving the multi-marginal optimal transport (MMOT) problems arising in density functional theory. The method relies on (i) the sparsity of optimal plans [for $N$ marginals discretized by $\ell$ gridpoints each, general Kantorovich plans require $\ell^N$ gridpoints but the support of optimizers is of size $O(\ell\cdot N)$ [FV18]], (ii) the method of column generation (CG) from discrete optimization which to our knowledge has not hitherto been used in MMOT, and (iii) ideas from machine learning. The well-known bottleneck in CG consists in generating new candidate columns efficiently; we prove that in our context, finding the best new column is an NP-complete problem. To overcome this bottleneck we use a genetic learning method tailormade for MMOT in which the dual state within CG plays the role of an "adversary", in loose similarity to Wasserstein GANs. On a sequence of benchmark problems with up to 120 gridpoints and up to 30 marginals, our method always found the exact optimizers. Moreover, empirically the number of computational steps needed to find them appears to scale only polynomially when both $N$ and $\ell$ are simultaneously increased (while keeping their ratio fixed to mimic a thermodynamic limit of the particle system).

Related papers

• Scalable 3D Registration via Truncated Entry-wise Absolute Residuals [65.04922801371363]
  A $3$D registration approach can process more than ten million ($107$) point pairs with over $99%$ random outliers. We call our method TEAR, as it involves minimizing an outlier-robust loss that computes Truncated Entry-wise Absolute Residuals.
  arXiv  Detail & Related papers  (2024-04-01T04:43:39Z)
• Towards Efficient Pareto-optimal Utility-Fairness between Groups in Repeated Rankings [7.6275971668447005]
  We tackle the problem of computing a sequence of rankings with the guarantee of the Pareto-optimal balance between consumers and producers of items. We introduce a novel approach to the above problem by using the Expohedron - a permutahedron whose points represent all exposures of items. We further propose an efficient method by relaxing our optimization problem on the Expohedron's circumscribed $n$-sphere, which significantly improve the running time.
  arXiv  Detail & Related papers  (2024-02-22T05:48:54Z)
• Orthogonal Directions Constrained Gradient Method: from non-linear equality constraints to Stiefel manifold [16.099883128428054]
  We propose a novel algorithm, the Orthogonal Directions Constrained Method (ODCGM) ODCGM only requires computing a projection onto a vector space. We show that ODCGM exhibits the near-optimal oracle complexities.
  arXiv  Detail & Related papers  (2023-03-16T12:25:53Z)
• Pseudonorm Approachability and Applications to Regret Minimization [73.54127663296906]
  We convert high-dimensional $ell_infty$-approachability problems to low-dimensional pseudonorm approachability problems. We develop an algorithmic theory of pseudonorm approachability, analogous to previous work on approachability for $ell$ and other norms.
  arXiv  Detail & Related papers  (2023-02-03T03:19:14Z)
• Fast Computation of Optimal Transport via Entropy-Regularized Extragradient Methods [75.34939761152587]
  Efficient computation of the optimal transport distance between two distributions serves as an algorithm that empowers various applications. This paper develops a scalable first-order optimization-based method that computes optimal transport to within $varepsilon$ additive accuracy.
  arXiv  Detail & Related papers  (2023-01-30T15:46:39Z)
• Explicit Second-Order Min-Max Optimization Methods with Optimal Convergence Guarantee [86.05440220344755]
  We propose and analyze inexact regularized Newton-type methods for finding a global saddle point of emphcon unconstrained min-max optimization problems. We show that the proposed methods generate iterates that remain within a bounded set and that the iterations converge to an $epsilon$-saddle point within $O(epsilon-2/3)$ in terms of a restricted function.
  arXiv  Detail & Related papers  (2022-10-23T21:24:37Z)
• Functional Constrained Optimization for Risk Aversion and Sparsity Control [7.561780884831967]
  Risk and sparsity requirements need to be enforced simultaneously in many applications, e.g., in portfolio optimization, assortment planning, and radiation planning. We propose a Level Conditional Gradient (LCG) method, which generates a convex or sparse trajectory for these challenges. We show that the method achieves a level single-set projection of the optimal value an inner conditional approximation (CGO) for solving mini-max sub gradient.
  arXiv  Detail & Related papers  (2022-10-11T02:51:51Z)
• Enhancing Column Generation by a Machine-Learning-Based Pricing Heuristic for Graph Coloring [5.278929511653199]
  Column Generation (CG) is an effective method for solving large-scale optimization problems. We propose a Machine-Learning Pricing Heuristic that can generate many high-quality columns efficiently.
  arXiv  Detail & Related papers  (2021-12-08T03:58:25Z)
• On the complexity of the optimal transport problem with graph-structured cost [9.24979291231758]
  Multi-marginal optimal transport (MOT) is a generalization of optimal transport to multiple marginals. The usage of MOT has been largely impeded by its computational complexity which scales exponentially in the number of marginals.
  arXiv  Detail & Related papers  (2021-10-01T19:29:59Z)
• Minimax Optimization with Smooth Algorithmic Adversaries [59.47122537182611]
  We propose a new algorithm for the min-player against smooth algorithms deployed by an adversary. Our algorithm is guaranteed to make monotonic progress having no limit cycles, and to find an appropriate number of gradient ascents.
  arXiv  Detail & Related papers  (2021-06-02T22:03:36Z)
• FANOK: Knockoffs in Linear Time [73.5154025911318]
  We describe a series of algorithms that efficiently implement Gaussian model-X knockoffs to control the false discovery rate on large scale feature selection problems. We test our methods on problems with $p$ as large as $500,000$.
  arXiv  Detail & Related papers  (2020-06-15T21:55:34Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.