The Gaussian-smoothed optimal transport (GOT) framework, recently proposed by Goldfeld et al., scales to high dimensions in estimation and provides an alternative to entropy regularization. This paper provides convergence guarantees for estimating the GOT distance under more general settings. For the Gaussian-smoothed $p$-Wasserstein distance in $d$ dimensions, our results require only the existence of a moment greater than $d + 2p$. For the special case of sub-gamma distributions, we quantify the dependence on the dimension $d$ and establish a phase transition with respect to the scale parameter. We also prove convergence for dependent samples, only requiring a condition on the pairwise dependence of the samples measured by the covariance of the feature map of a kernel space. A key step in our analysis is to show that the GOT distance is dominated by a family of kernel maximum mean discrepancy (MMD) distances with a kernel that depends on the cost function as well as the amount of Gaussian smoothing. This insight provides further interpretability for the GOT framework and also introduces a class of kernel MMD distances with desirable properties. The theoretical results are supported by numerical experiments.The Gaussian-smoothed optimal transport (GOT) framework, recently proposed by Goldfeld et al., scales to high dimensions in estimation and provides an alternative to entropy regularization. This paper provides convergence guarantees for estimating the GOT distance under more general settings. For the Gaussian-smoothed $p$-Wasserstein distance in $d$ dimensions, our results require only the existence of a moment greater than $d + 2p$. For the special case of sub-gamma distributions, we quantify the dependence on the dimension $d$ and establish a phase transition with respect to the scale parameter. We also prove convergence for dependent samples, only requiring a condition on the pairwise dependence of the samples measured by the covariance of the feature map of a kernel space. A key step in our analysis is to show that the GOT distance is dominated by a family of kernel maximum mean discrepancy (MMD) distances with a kernel that depends on the cost function as well as the amount of Gaussian smoothing. This insight provides further interpretability for the GOT framework and also introduces a class of kernel MMD distances with desirable properties. The theoretical results are supported by numerical experiments.
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Smoothed Analysis of Online and Differentially Private Learning
Practical and pervasive needs for robustness and privacy in algorithms have inspired the design of
online adversarial and differentially private learning algorithms. The primary quantity that characterizes
learnability in these settings is the Littlestone dimension of the class of hypotheses [Alon et al., 2019,
Ben-David et al., 2009]. This characterization is often interpreted as an impossibility result because
classes such as linear thresholds and neural networks have infinite Littlestone dimension. In this paper,
we apply the framework of smoothed analysis [Spielman and Teng, 2004], in which adversarially chosen
inputs are perturbed slightly by nature. We show that fundamentally stronger regret and error guarantees are possible with smoothed adversaries than with worst-case adversaries. In particular, we obtain
regret and privacy error bounds that depend only on the VC dimension and the bracketing number of a
hypothesis class, and on the magnitudes of the perturbations.
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- Award ID(s):
- 2006737
- NSF-PAR ID:
- 10310908
- Date Published:
- Journal Name:
- Advances in neural information processing systems
- Volume:
- 33
- ISSN:
- 1049-5258
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- The DOI is not currently available.
