1,273 research outputs found
Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration
Computing optimal transport distances such as the earth mover's distance is a
fundamental problem in machine learning, statistics, and computer vision.
Despite the recent introduction of several algorithms with good empirical
performance, it is unknown whether general optimal transport distances can be
approximated in near-linear time. This paper demonstrates that this ambitious
goal is in fact achieved by Cuturi's Sinkhorn Distances. This result relies on
a new analysis of Sinkhorn iteration, which also directly suggests a new greedy
coordinate descent algorithm, Greenkhorn, with the same theoretical guarantees.
Numerical simulations illustrate that Greenkhorn significantly outperforms the
classical Sinkhorn algorithm in practice
A Numerical Method to solve Optimal Transport Problems with Coulomb Cost
In this paper, we present a numerical method, based on iterative Bregman
projections, to solve the optimal transport problem with Coulomb cost. This is
related to the strong interaction limit of Density Functional Theory. The first
idea is to introduce an entropic regularization of the Kantorovich formulation
of the Optimal Transport problem. The regularized problem then corresponds to
the projection of a vector on the intersection of the constraints with respect
to the Kullback-Leibler distance. Iterative Bregman projections on each
marginal constraint are explicit which enables us to approximate the optimal
transport plan. We validate the numerical method against analytical test cases
Robust Optimal Risk Sharing and Risk Premia in Expanding Pools
We consider the problem of optimal risk sharing in a pool of cooperative
agents. We analyze the asymptotic behavior of the certainty equivalents and
risk premia associated with the Pareto optimal risk sharing contract as the
pool expands. We first study this problem under expected utility preferences
with an objectively or subjectively given probabilistic model. Next, we develop
a robust approach by explicitly taking uncertainty about the probabilistic
model (ambiguity) into account. The resulting robust certainty equivalents and
risk premia compound risk and ambiguity aversion. We provide explicit results
on their limits and rates of convergence, induced by Pareto optimal risk
sharing in expanding pools
On the Effectiveness of Richardson Extrapolation in Machine Learning
Richardson extrapolation is a classical technique from numerical analysis that can improve the approximation error of an estimation method by combining linearly several estimates obtained from different values of one of its hyperparameters, without the need to know in details the inner structure of the original estimation method. The main goal of this paper is to study when Richardson extrapolation can be used within machine learning, beyond the existing applications to step-size adaptations in stochastic gradient descent. We identify two situations where Richardson interpolation can be useful: (1) when the hyperparameter is the number of iterations of an existing iterative optimization algorithm, with applications to averaged gradient descent and Frank-Wolfe algorithms (where we obtain asymptotically rates of on polytopes, where is the number of iterations), and (2) when it is a regularization parameter, with applications to Nesterov smoothing techniques for minimizing non-smooth functions (where we obtain asymptotically rates close to for non-smooth functions), and ridge regression. In all these cases, we show that extrapolation techniques come with no significant loss in performance, but with sometimes strong gains, and we provide theoretical justifications based on asymptotic developments for such gains, as well as empirical illustrations on classical problems from machine learning
- …