There is intense interest in applying machine learning to problems of causal inference in fields such as healthcare, economics and education. In particular, individual-level causal inference has important applications such as precision medicine. We give a new theoretical analysis and family of algorithms for predicting individual treatment effect (ITE) from observational data, under the assumption known as strong ignorability. The algorithms learn a "balanced" representation such that the induced treated and control distributions look similar. We give a novel, simple and intuitive generalization-error bound showing that the expected ITE estimation error of a representation is bounded by a sum of the standard generalization-error of that representation and the distance between the treated and control distributions induced by the representation. We use Integral Probability Metrics to measure distances between distributions, deriving explicit bounds for the Wasserstein and Maximum Mean Discrepancy (MMD) distances. Experiments on real and simulated data show the new algorithms match or outperform the state-of-the-art.

Estimating individual treatment effect: generalization bounds and algorithms

https://par.nsf.gov/servlets/purl/10182988

A geometric view of optimal transportation and generative model

This paper presents a novel two-step approach for the fundamental problem of learning an optimal map from one distribution to another. First, we learn an optimal transport (OT) plan, which can be thought as a one-to-many map between the two distributions. To that end, we propose a stochastic dual approach of regularized OT, and show empirically that it scales better than a recent related approach when the amount of samples is very large. Second, we estimate a \textit{Monge map} as a deep neural network learned by approximating the barycentric projection of the previously-obtained OT plan. This parameterization allows generalization of the mapping outside the support of the input measure. We prove two theoretical stability results of regularized OT which show that our estimations converge to the OT plan and Monge map between the underlying continuous measures. We showcase our proposed approach on two applications: domain adaptation and generative modeling.

Large-Scale Optimal Transport and Mapping Estimation

In this work, we provide faster algorithms for approximating the optimal transport distance, e.g. earth mover's distance, between two discrete probability distributions $\mu, \nu \in \Delta^n$. Given a cost function $C : [n] \times [n] \to \mathbb{R}_{\geq 0}$ where $C(i,j) \leq 1$ quantifies the penalty of transporting a unit of mass from $i$ to $j$, we show how to compute a coupling $X$ between $r$ and $c$ in time $\widetilde{O}\left(n^2 /\epsilon \right)$ whose expected transportation cost is within an additive $\epsilon$ of optimal. This improves upon the previously best known running time for this problem of $\widetilde{O}\left(\text{min}\left\{ n^{9/4}/\epsilon, n^2/\epsilon^2 \right\}\right)$. 
We achieve our results by providing reductions from optimal transport to canonical optimization problems for which recent algorithmic efforts have provided nearly-linear time algorithms. Leveraging nearly linear time algorithms for solving packing linear programs and for solving the matrix balancing problem, we obtain two separate proofs of our stated running time. Further, one of our algorithms is easily parallelized and can be implemented with depth $\widetilde{O}(1/\epsilon)$. 
Moreover, we show that further algorithmic improvements to our result would be surprising in the sense that any improvement would yield an $o(n^{2.5})$ algorithm for \textit{maximum cardinality bipartite matching}, for which currently the only known algorithms for achieving such a result are based on fast-matrix multiplication.

Towards Optimal Running Times for Optimal Transport

This short article revisits some of the ideas introduced in arXiv:1701.07875 and arXiv:1705.07642 in a simple setup. This sheds some lights on the connexions between Variational Autoencoders (VAE), Generative Adversarial Networks (GAN) and Minimum Kantorovitch Estimators (MKE).

GAN and VAE from an Optimal Transport Point of View

Optimal transport (OT) defines a powerful framework to compare probability distributions in a geometrically faithful way. However, the practical impact of OT is still limited because of its computational burden. We propose a new class of stochastic optimization algorithms to cope with large-scale problems routinely encountered in machine learning applications. These methods are able to manipulate arbitrary distributions (either discrete or continuous) by simply requiring to be able to draw samples from them, which is the typical setup in high-dimensional learning problems. This alleviates the need to discretize these densities, while giving access to provably convergent methods that output the correct distance without discretization error. These algorithms rely on two main ideas: (a) the dual OT problem can be re-cast as the maximization of an expectation ; (b) entropic regularization of the primal OT problem results in a smooth dual optimization optimization which can be addressed with algorithms that have a provably faster convergence. We instantiate these ideas in three different setups: (i) when comparing a discrete distribution to another, we show that incremental stochastic optimization schemes can beat Sinkhorn's algorithm, the current state-of-the-art finite dimensional OT solver; (ii) when comparing a discrete distribution to a continuous density, a semi-discrete reformulation of the dual program is amenable to averaged stochastic gradient descent, leading to better performance than approximately solving the problem by discretization ; (iii) when dealing with two continuous densities, we propose a stochastic gradient descent over a reproducing kernel Hilbert space (RKHS). This is currently the only known method to solve this problem, apart from computing OT on finite samples. We backup these claims on a set of discrete, semi-discrete and continuous benchmark problems.

Genevay Aude

Papers

Stochastic Optimization for Large-scale Optimal Transport