Learning Generative Models with Sinkhorn Divergences

The ability to compare two degenerate probability distributions (i.e. two probability distributions supported on two distinct low-dimensional manifolds living in a much higher-dimensional space) is a crucial problem arising in the estimation of generative models for high-dimensional observations such as those arising in computer vision or natural language. It is known that optimal transport metrics can represent a cure for this problem, since they were specifically designed as an alternative to information divergences to handle such problematic scenarios. Unfortunately, training generative machines using OT raises formidable computational and statistical challenges, because of (i) the computational burden of evaluating OT losses, (ii) the instability and lack of smoothness of these losses, (iii) the difficulty to estimate robustly these losses and their gradients in high dimension. This paper presents the first tractable computational method to train large scale generative models using an optimal transport loss, and tackles these three issues by relying on two key ideas: (a) entropic smoothing, which turns the original OT loss into one that can be computed using Sinkhorn fixed point iterations; (b) algorithmic (automatic) differentiation of these iterations. These two approximations result in a robust and differentiable approximation of the OT loss with streamlined GPU execution. Entropic smoothing generates a family of losses interpolating between Wasserstein (OT) and Maximum Mean Discrepancy (MMD), thus allowing to find a sweet spot leveraging the geometry of OT and the favorable high-dimensional sample complexity of MMD which comes with unbiased gradient estimates. The resulting computational architecture complements nicely standard deep network generative models by a stack of extra layers implementing the loss function

Reinforcement learning for personalized dialogue management

Language systems have been of great interest to the research community and have recently reached the mass market through various assistant platforms on the web. Reinforcement Learning methods that optimize dialogue policies have seen successes in past years and have recently been extended into methods that personalize the dialogue, e.g. take the personal context of users into account. These works, however, are limited to personalization to a single user with whom they require multiple interactions and do not generalize the usage of context across users. This work introduces a problem where a generalized usage of context is relevant and proposes two Reinforcement Learning (RL)-based approaches to this problem. The first approach uses a single learner and extends the traditional POMDP formulation of dialogue state with features that describe the user context. The second approach segments users by context and then employs a learner per context. We compare these approaches in a benchmark of existing non-RL and RL-based methods in three established and one novel application domain of financial product recommendation. We compare the influence of context and training experiences on performance and find that learning approaches generally outperform a handcrafted gold standard

Régularisation Entropique du Transport Optimal pour le Machine Learning

This thesis proposes theoretical and numerical contributions to use Entropy-regularized Optimal Transport (EOT) for machine learning. We introduce Sinkhorn Divergences (SD), a class of discrepancies between probability measures based on EOT which interpolates between two other well-known discrepancies: Optimal Transport (OT) and Maximum Mean Discrepancies (MMD). We develop an efficient numerical method to use SD for density fitting tasks, showing that a suitable choice of regularization can improve performance over existing methods. We derive a sample complexity theorem for SD which proves that choosing a large enough regularization parameter allows to break the curse of dimensionality from OT, and recover asymptotic rates similar to MMD. We propose and analyze stochastic optimization solvers for EOT, which yield online methods that can cope with arbitrary measures and are well suited to large scale problems, contrarily to existing discrete batch solvers.Le Transport Optimal régularisé par l’Entropie (TOE) permet de définir les Divergences de Sinkhorn (DS), une nouvelle classe de distance entre mesures de probabilités basées sur le TOE. Celles-ci permettent d’interpoler entre deux autres dis- tances connues: le Transport Optimal (TO) et l’Ecart Moyen Maxi- mal (EMM). Les DS peuvent être utilisées pour apprendre des modèles probabilistes avec de meilleures performances que les algorithmes existants pour une régularisation adéquate. Ceci est justifié par un théorème sur l’approximation des SD par des échantillons, prouvant qu’une régularisation suffisante per- met de se débarrasser de la malédiction de la dimension du TO, et l’on retrouve à l’infini le taux de convergence des EMM. Enfin, nous présentons de nouveaux algorithmes de résolution pour le TOE basés sur l’optimisation stochastique ‘en-ligne’ qui, contrairement à l’état de l’art, ne se restreignent pas aux mesures discrètes et s’adaptent bien aux problèmes de grande dimension

Transport Optimal pour l'Apprentissage Automatique

This thesis proposes theoretical and numerical contributions to use Entropy-regularized Optimal Transport (EOT) for machine learning. We introduce Sinkhorn Divergences (SD), a class of discrepancies between probability measures based on EOT which interpolates between two other well-known discrepancies: Optimal Transport (OT) and Maximum Mean Discrepancies (MMD). We develop an ecient numerical method to use SD for density fitting tasks, showing that a suitable choice of regularization can improve performance over existing methods. We derive a sample complexity theorem for SD which proves that choosing a large enough regularization parameter allows to break the curse of dimensionality from OT, and recover asymptotic rates similar to MMD.We propose and analyze stochastic optimization solvers for EOT, which yield online methods that can cope with arbitrary measures and are well suited to large scale problems, contrarily to existing discrete batch solvers.Le Transport Optimal régularisé par l’Entropie (TOE) permet de définir les Divergences de Sinkhorn (DS), une nouvelle classe de distance entre mesures de probabilités basées sur le TOE. Celles-ci permettent d’interpoler entre deux autres distances connues : le Transport Optimal (TO) et l’Ecart Moyen Maximal (EMM). Les DS peuvent être utilisées pour apprendre des modèles probabilistes avec de meilleures performances que les algorithmes existants pour une régularisation adéquate. Ceci est justifié par un théorème sur l’approximation des SD par des échantillons, prouvant qu’une régularisation sus ante permet de se débarrasser de la malédiction de la dimension du TO, et l’on retrouve à l’infini le taux de convergence des EMM. Enfin, nous présentons de nouveaux algorithmes de résolution pour le TOE basés sur l’optimisation stochastique « en-ligne » qui, contrairement à l’état de l’art, ne se restreignent pas aux mesures discrètes et s’adaptent bien aux problèmes de grande dimension