Multimarginal Optimal Transport (MOT) has recently attracted significant interest due to its many applications. However, in most applications, the success of MOT is severely hindered by a lack of sub-exponential time algorithms. This paper develops a general theory about "structural properties" that make MOT tractable. We identify two such properties: decomposability of the cost into either (i) local interactions and simple global interactions; or (ii) low-rank interactions and sparse interactions. We also provide strong evidence that (iii) repulsive costs make MOT intractable by showing that several such problems of interest are NP-hard to solve-even approximately. These three structures are quite general, and collectively they encompass many (if not most) current MOT applications. We demonstrate our results on a variety of applications in machine learning, statistics, physics, and computational geometry.

This paper studies the value of switching actions in the Prediction From Experts problem (PFE) and Adversarial Multiarmed Bandits problem (MAB). First, we revisit the well-studied and practically motivated setting of PFE with switching costs. Many algorithms achieve the minimax optimal order for both regret and switches in expectation; however, high probability guarantees are an open problem. We present the first algorithms that achieve this optimal order for both quantities with high probability. This also implies the first high probability guarantees for several other problems, and, in particular, is efficiently adaptable to online combinatorial optimization with limited switching. Next, to investigate the value of switching actions more granularly, we introduce the switching budget setting, which limits algorithms to a fixed number of (costless) switches. Using this result and several reductions, we unify previous work and completely characterize the complexity of this switching budget setting up to small polylogarithmic factors: for both PFE and MAB, for all switching budgets, and for both expectation and high probability guarantees. Interestingly, as the switching budget decreases, the minimax regret rate admits a phase transition for PFE but not for MAB. These results recover and generalize the known minimax rates for the (arbitrary) switching cost setting.

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The Sinkhorn "distance," a variant of the Wasserstein distance with entropic regularization, is an increasingly popular tool in machine learning and statistical inference. However, the time and memory requirements of standard algorithms for computing this distance grow quadratically with the size of the data, making them prohibitively expensive on massive data sets. In this work, we show that this challenge is surprisingly easy to circumvent: combining two simple techniques-the Nyström method and Sinkhorn scaling-provably yields an accurate approximation of the Sinkhorn distance with significantly lower time and memory requirements than other approaches. We prove our results via new, explicit analyses of the Nyström method and of the stability properties of Sinkhorn scaling. We validate our claims experimentally by showing that our approach easily computes Sinkhorn distances on data sets hundreds of times larger than can be handled by other techniques.