Thibault Séjourné

Thibault Séjourné, Samitha Samaranayake, Siddhartha Banerjee

Mobility-on-Demand platforms are a fast growing component of the urban transit ecosystem. Though a growing literature addresses the question of how to make individual MoD platforms more efficient, much less is known about the cost of market fragmentation, i.e., the impact on welfare due to splitting the demand between multiple independent platforms. Our work aims to quantify how much platform fragmentation degrades the efficiency of the system. In particular, we focus on a setting where demand is exogenously split between multiple platforms, and study the increase in the supply rebalancing cost incurred by each platform to meet this demand, vis-a-vis the cost incurred by a centralized platform serving the aggregate demand. We show under a large-market scaling, this Price-of-Fragmentation undergoes a phase transition, wherein, depending on the nature of the exogenous demand, the additional cost due to fragmentation either vanishes or grows unbounded. We provide conditions that characterize which regime applies to any given system, and discuss implications of this on how such platforms should be regulated.

Clément Bonet, Kimia Nadjahi, Thibault Séjourné et al.

Optimal transport (OT) is a powerful framework to compare probability measures, a fundamental task in many statistical and machine learning problems. Substantial advances have been made in designing OT variants which are either computationally and statistically more efficient or robust. Among them, sliced OT distances have been extensively used to mitigate optimal transport's cubic algorithmic complexity and curse of dimensionality. In parallel, unbalanced OT was designed to allow comparisons of more general positive measures, while being more robust to outliers. In this paper, we bridge the gap between those two concepts and develop a general framework for efficiently comparing positive measures. We notably formulate two different versions of sliced unbalanced OT, and study the associated topology and statistical properties. We then develop a GPU-friendly Frank-Wolfe like algorithm to compute the corresponding loss functions, and show that the resulting methodology is modular as it encompasses and extends prior related work. We finally conduct an empirical analysis of our loss functions and methodology on both synthetic and real datasets, to illustrate their computational efficiency, relevance and applicability to real-world scenarios including geophysical data.

Thibault Séjourné, Gabriel Peyré, François-Xavier Vialard

Optimal Transport (OT) has recently emerged as a central tool in data sciences to compare in a geometrically faithful way point clouds and more generally probability distributions. The wide adoption of OT into existing data analysis and machine learning pipelines is however plagued by several shortcomings. This includes its lack of robustness to outliers, its high computational costs, the need for a large number of samples in high dimension and the difficulty to handle data in distinct spaces. In this review, we detail several recently proposed approaches to mitigate these issues. We insist in particular on unbalanced OT, which compares arbitrary positive measures, not restricted to probability distributions (i.e. their total mass can vary). This generalization of OT makes it robust to outliers and missing data. The second workhorse of modern computational OT is entropic regularization, which leads to scalable algorithms while lowering the sample complexity in high dimension. The last point presented in this review is the Gromov-Wasserstein (GW) distance, which extends OT to cope with distributions belonging to different metric spaces. The main motivation for this review is to explain how unbalanced OT, entropic regularization and GW can work hand-in-hand to turn OT into efficient geometric loss functions for data sciences.

Thibault Séjourné, François-Xavier Vialard, Gabriel Peyré

Unbalanced optimal transport (UOT) extends optimal transport (OT) to take into account mass variations to compare distributions. This is crucial to make OT successful in ML applications, making it robust to data normalization and outliers. The baseline algorithm is Sinkhorn, but its convergence speed might be significantly slower for UOT than for OT. In this work, we identify the cause for this deficiency, namely the lack of a global normalization of the iterates, which equivalently corresponds to a translation of the dual OT potentials. Our first contribution leverages this idea to develop a provably accelerated Sinkhorn algorithm (coined 'translation invariant Sinkhorn') for UOT, bridging the computational gap with OT. Our second contribution focusses on 1-D UOT and proposes a Frank-Wolfe solver applied to this translation invariant formulation. The linear oracle of each steps amounts to solving a 1-D OT problems, resulting in a linear time complexity per iteration. Our last contribution extends this method to the computation of UOT barycenter of 1-D measures. Numerical simulations showcase the convergence speed improvement brought by these three approaches.

Kilian Fatras, Thibault Séjourné, Nicolas Courty et al.

Optimal transport distances have found many applications in machine learning for their capacity to compare non-parametric probability distributions. Yet their algorithmic complexity generally prevents their direct use on large scale datasets. Among the possible strategies to alleviate this issue, practitioners can rely on computing estimates of these distances over subsets of data, {\em i.e.} minibatches. While computationally appealing, we highlight in this paper some limits of this strategy, arguing it can lead to undesirable smoothing effects. As an alternative, we suggest that the same minibatch strategy coupled with unbalanced optimal transport can yield more robust behavior. We discuss the associated theoretical properties, such as unbiased estimators, existence of gradients and concentration bounds. Our experimental study shows that in challenging problems associated to domain adaptation, the use of unbalanced optimal transport leads to significantly better results, competing with or surpassing recent baselines.

Thibault Séjourné, François-Xavier Vialard, Gabriel Peyré

Comparing metric measure spaces (i.e. a metric space endowed with aprobability distribution) is at the heart of many machine learning problems. The most popular distance between such metric measure spaces is theGromov-Wasserstein (GW) distance, which is the solution of a quadratic assignment problem. The GW distance is however limited to the comparison of metric measure spaces endowed with a probability distribution. To alleviate this issue, we introduce two Unbalanced Gromov-Wasserstein formulations: a distance and a more tractable upper-bounding relaxation.They both allow the comparison of metric spaces equipped with arbitrary positive measures up to isometries. The first formulation is a positive and definite divergence based on a relaxation of the mass conservation constraint using a novel type of quadratically-homogeneous divergence. This divergence works hand in hand with the entropic regularization approach which is popular to solve large scale optimal transport problems. We show that the underlying non-convex optimization problem can be efficiently tackled using a highly parallelizable and GPU-friendly iterative scheme. The second formulation is a distance between mm-spaces up to isometries based on a conic lifting. Lastly, we provide numerical experiments onsynthetic examples and domain adaptation data with a Positive-Unlabeled learning task to highlight the salient features of the unbalanced divergence and its potential applications in ML.

Thibault Séjourné, Jean Feydy, François-Xavier Vialard et al.

Optimal transport induces the Earth Mover's (Wasserstein) distance between probability distributions, a geometric divergence that is relevant to a wide range of problems. Over the last decade, two relaxations of optimal transport have been studied in depth: unbalanced transport, which is robust to the presence of outliers and can be used when distributions don't have the same total mass; entropy-regularized transport, which is robust to sampling noise and lends itself to fast computations using the Sinkhorn algorithm. This paper combines both lines of work to put robust optimal transport on solid ground. Our main contribution is a generalization of the Sinkhorn algorithm to unbalanced transport: our method alternates between the standard Sinkhorn updates and the pointwise application of a contractive function. This implies that entropic transport solvers on grid images, point clouds and sampled distributions can all be modified easily to support unbalanced transport, with a proof of linear convergence that holds in all settings. We then show how to use this method to define pseudo-distances on the full space of positive measures that satisfy key geometric axioms: (unbalanced) Sinkhorn divergences are differentiable, positive, definite, convex, statistically robust and avoid any "entropic bias" towards a shrinkage of the measures' supports.