94 research outputs found
Sharp asymptotic and finite-sample rates of convergence of empirical measures in Wasserstein distance
The Wasserstein distance between two probability measures on a metric space
is a measure of closeness with applications in statistics, probability, and
machine learning. In this work, we consider the fundamental question of how
quickly the empirical measure obtained from independent samples from
approaches in the Wasserstein distance of any order. We prove sharp
asymptotic and finite-sample results for this rate of convergence for general
measures on general compact metric spaces. Our finite-sample results show the
existence of multi-scale behavior, where measures can exhibit radically
different rates of convergence as grows
Uncoupled isotonic regression via minimum Wasserstein deconvolution
Isotonic regression is a standard problem in shape-constrained estimation
where the goal is to estimate an unknown nondecreasing regression function
from independent pairs where . While this problem is well understood both statistically and
computationally, much less is known about its uncoupled counterpart where one
is given only the unordered sets and . In this work, we leverage tools from optimal transport theory to derive
minimax rates under weak moments conditions on and to give an efficient
algorithm achieving optimal rates. Both upper and lower bounds employ
moment-matching arguments that are also pertinent to learning mixtures of
distributions and deconvolution.Comment: To appear in Information and Inference: a Journal of the IM
Entropic optimal transport is maximum-likelihood deconvolution
We give a statistical interpretation of entropic optimal transport by showing
that performing maximum-likelihood estimation for Gaussian deconvolution
corresponds to calculating a projection with respect to the entropic optimal
transport distance. This structural result gives theoretical support for the
wide adoption of these tools in the machine learning community
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
Online learning in repeated auctions
Motivated by online advertising auctions, we consider repeated Vickrey
auctions where goods of unknown value are sold sequentially and bidders only
learn (potentially noisy) information about a good's value once it is
purchased. We adopt an online learning approach with bandit feedback to model
this problem and derive bidding strategies for two models: stochastic and
adversarial. In the stochastic model, the observed values of the goods are
random variables centered around the true value of the good. In this case,
logarithmic regret is achievable when competing against well behaved
adversaries. In the adversarial model, the goods need not be identical and we
simply compare our performance against that of the best fixed bid in hindsight.
We show that sublinear regret is also achievable in this case and prove
matching minimax lower bounds. To our knowledge, this is the first complete set
of strategies for bidders participating in auctions of this type
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