In real-world settings involving consequential decision-making, the deployment of machine learning systems generally requires both reliable uncertainty quantification and protection of individuals' privacy. We present a framework that treats these two desiderata jointly. Our framework is based on conformal prediction, a methodology that augments predictive models to return prediction sets that provide uncertainty quantification -- they provably cover the true response with a user-specified probability, such as 90%. One might hope that when used with privately-trained models, conformal prediction would yield privacy guarantees for the resulting prediction sets; unfortunately this is not the case. To remedy this key problem, we develop a method that takes any pre-trained predictive model and outputs differentially private prediction sets. Our method follows the general approach of split conformal prediction; we use holdout data to calibrate the size of the prediction sets but preserve privacy by using a privatized quantile subroutine. This subroutine compensates for the noise introduced to preserve privacy in order to guarantee correct coverage. We evaluate the method with experiments on the CIFAR-10, ImageNet, and CoronaHack datasets.

With the increasing demand for large-scale training of machine learning models, consensus-based distributed optimization methods have recently been advocated as alternatives to the popular parameter server framework. In this paradigm, each worker maintains a local estimate of the optimal parameter vector, and iteratively updates it by waiting and averaging all estimates obtained from its neighbors, and then corrects it on the basis of its local dataset. However, the synchronization phase can be time consuming due to the need to wait for \textit{stragglers}, i.e., slower workers. An efficient way to mitigate this effect is to let each worker wait only for updates from the fastest neighbors before updating its local parameter. The remaining neighbors are called \textit{backup workers.} To minimize the globally training time over the network, we propose a fully distributed algorithm to dynamically determine the number of backup workers for each worker. We show that our algorithm achieves a linear speedup for convergence (i.e., convergence performance increases linearly with respect to the number of workers). We conduct extensive experiments on MNIST and CIFAR-10 to verify our theoretical results.

Optimal Transport (OT) defines geometrically meaningful "Wasserstein" distances, used in machine learning applications to compare probability distributions. However, a key bottleneck is the design of a "ground" cost which should be adapted to the task under study. In most cases, supervised metric learning is not accessible, and one usually resorts to some ad-hoc approach. Unsupervised metric learning is thus a fundamental problem to enable data-driven applications of Optimal Transport. In this paper, we propose for the first time a canonical answer by computing the ground cost as a positive eigenvector of the function mapping a cost to the pairwise OT distances between the inputs. This map is homogeneous and monotone, thus framing unsupervised metric learning as a non-linear Perron-Frobenius problem. We provide criteria to ensure the existence and uniqueness of this eigenvector. In addition, we introduce a scalable computational method using entropic regularization, which - in the large regularization limit - operates a principal component analysis dimensionality reduction. We showcase this method on synthetic examples and datasets. Finally, we apply it in the context of biology to the analysis of a high-throughput single-cell RNA sequencing (scRNAseq) dataset, to improve cell clustering and infer the relationships between genes in an unsupervised way.

Many reinforcement learning algorithms can be seen as versions of approximate policy iteration (API). While standard API often performs poorly, it has been shown that learning can be stabilized by regularizing each policy update by the KL-divergence to the previous policy. Popular practical algorithms such as TRPO, MPO, and VMPO replace regularization by a constraint on KL-divergence of consecutive policies, arguing that this is easier to implement and tune. In this work, we study this implementation choice in more detail. We compare the use of KL divergence as a constraint vs. as a regularizer, and point out several optimization issues with the widely-used constrained approach. We show that the constrained algorithm is not guaranteed to converge even on simple problem instances where the constrained problem can be solved exactly, and in fact incurs linear expected regret. With approximate implementation using softmax policies, we show that regularization can improve the optimization landscape of the original objective. We demonstrate these issues empirically on several bandit and RL environments.

We propose a novel approach to analyze generalization error for discretizations of Langevin diffusion, such as the stochastic gradient Langevin dynamics (SGLD). For an $\epsilon$ tolerance of expected generalization error, it is known that a first order discretization can reach this target if we run $\Omega(\epsilon^{-1} \log (\epsilon^{-1}) )$ iterations with $\Omega(\epsilon^{-1})$ samples. In this article, we show that with additional smoothness assumptions, even first order methods can achieve arbitrarily runtime complexity. More precisely, for each $N>0$, we provide a sufficient smoothness condition on the loss function such that a first order discretization can reach $\epsilon$ expected generalization error given $\Omega( \epsilon^{-1/N} \log (\epsilon^{-1}) )$ iterations with $\Omega(\epsilon^{-1})$ samples.

Causal inference for extremes has only be considered during the past few years. That observations of climate extremes such as floods, hurricanes, and droughts, but also man-made catastrophes like industry fire, terrorist attacks, or crashes of financial markets have been in the focus of research is convincingly documented in the journal Extremes. On the other hand, it is a fundamental problem to assess causality of risks. Often rare events are interconnected; for example, floods disseminate through a river network, and credit markets might fail due to some endogenous systemic risk propagation. Hence, it is necessary to not only understand dependencies between rare events, but also their causal structure.

The knowledge that data lies close to a particular submanifold of the ambient Euclidean space may be useful in a number of ways. For instance, one may want to automatically mark any point far away from the submanifold as an outlier, or to use its geodesic distance to measure similarity between points. Classical problems for manifold learning are often posed in a very high dimension, e.g. for spaces of images or spaces of representations of words. Today, with deep representation learning on the rise in areas such as computer vision and natural language processing, many problems of this kind may be transformed into problems of moderately high dimension, typically of the order of hundreds. Motivated by this, we propose a manifold learning technique suitable for moderately high dimension and large datasets. The manifold is learned from the training data in the form of an intersection of quadric hypersurfaces -- simple but expressive objects. At test time, this manifold can be used to introduce an outlier score for arbitrary new points and to improve a given similarity metric by incorporating learned geometric structure into it.

Addressing fairness concerns about machine learning models is a crucial step towards their long-term adoption in real-world automated systems. While many approaches have been developed for training fair models from data, little is known about the effects of data corruption on these methods. In this work we consider fairness-aware learning under arbitrary data manipulations. We show that an adversary can force any learner to return a biased classifier, with or without degrading accuracy, and that the strength of this bias increases for learning problems with underrepresented protected groups in the data. We also provide upper bounds that match these hardness results up to constant factors, by proving that two natural learning algorithms achieve order-optimal guarantees in terms of both accuracy and fairness under adversarial data manipulations.

Given the abundance of applications of ranking in recent years, addressing fairness concerns around automated ranking systems becomes necessary for increasing the trust among end-users. Previous work on fair ranking has mostly focused on application-specific fairness notions, often tailored to online advertising, and it rarely considers learning as part of the process. In this work, we show how to transfer numerous fairness notions from binary classification to a learning to rank context. Our formalism allows us to design a method for incorporating fairness objectives with provable generalization guarantees. An extensive experimental evaluation shows that our method can improve ranking fairness substantially with no or only little loss of model quality.

Mini-batch optimal transport (m-OT) has been successfully used in practical applications that involve probability measures with intractable density, or probability measures with a very high number of supports. The m-OT solves several sparser optimal transport problems and then returns the average of their costs and transportation plans. Despite its scalability advantage, m-OT is not a proper metric between probability measures since it does not satisfy the identity property. To address this problem, we propose a novel mini-batching scheme for optimal transport, named Batch of Mini-batches Optimal Transport (BoMb-OT), that can be formulated as a well-defined distance on the space of probability measures. Furthermore, we show that the m-OT is a limit of the entropic regularized version of the proposed BoMb-OT when the regularized parameter goes to infinity. We carry out extensive experiments to show that the new mini-batching scheme can estimate a better transportation plan between two original measures than m-OT. It leads to a favorable performance of BoMb-OT in the matching and color transfer tasks. Furthermore, we observe that BoMb-OT also provides a better objective loss than m-OT for doing approximate Bayesian computation, estimating parameters of interest in parametric generative models, and learning non-parametric generative models with gradient flow.